Ralph S. Phillips

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Ralph Saul Phillips (March 9, 1913 – November 23, 1998) was an American mathematician known for his influential work in functional analysis, operator theory, and scattering theory. He earned his Ph.D. from the University of Michigan in 1939 and held academic positions at institutions including the University of Southern California and the University of Chicago before joining Stanford University in 1960.

Phillips made significant contributions to the study of partial differential equations, harmonic analysis, and wave propagation. Much of his research centered on the mathematical foundations of scattering phenomena, an area closely related to physics and engineering. One of his notable collaborations was with Peter D. Lax, with whom he co-authored key works on scattering theory.

His impact is felt both through his numerous publications and through the work of those he mentored. Over the course of his career, Phillips published influential papers and monographs that helped shape modern approaches to PDEs and scattering theory, leaving a lasting legacy in the mathematical sciences.

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Ramanujan graphs

1988-09-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Extremals of determinants of Laplacians

1988-09-01

Brad Osgood , Ralph S. Phillips , Peter Sarnak

Dissipative operators in a Banach space

1961-06-01

G. Lumer , Ralph S. Phillips

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Local boundary conditions for dissipative symmetric linear differential operators

1960-08-01

Peter D. Lax , Ralph S. Phillips

Scattering theory for automorphic functions

1980-01-01

Peter D. Lax , Ralph S. Phillips

This paper is an expository account of our 1976 monograph [6] on Scattering theory for automorphic functions.Several improvements have been incorporated: a more direct proof of the meromorphic character of … This paper is an expository account of our 1976 monograph [6] on Scattering theory for automorphic functions.Several improvements have been incorporated: a more direct proof of the meromorphic character of the Eisenstein series, an explicit formula for the translation representations and a simpler derivation of the spectral representations.Our hyperbolic approach to the Selberg trace formula is also included.

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Dissipative operators and hyperbolic systems of partial differential equations

1959-01-01

Ralph S. Phillips

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The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces

1982-01-01

Peter D. Lax , Ralph S. Phillips

On linear transformations

1940-01-01

Ralph S. Phillips

The purpose of this paper is to give a characterization of linear and completely continuous transformations both on the common Banach spaces to an arbitrary Banach space and vice versa. … The purpose of this paper is to give a characterization of linear and completely continuous transformations both on the common Banach spaces to an arbitrary Banach space and vice versa. There is an abundant literature on this subject. Among the earliest papers, the now famous paper of Radon [241 should be mentioned. Here linear transformations on LP to Lq (1 <p, q< oc) are characterized in a manner suggestive of the methods used in the present paper. The works of Gelfand [12], Dunford [6], Kantorovitch and Vulich [17], and Dunford and Pettis [9] contain much material on this subject supplementary to that treated here. In the interest of completeness we have restated a few of the results obtained by Gelfand [12 ], and Gowurin [13 ]. The principal tools used in our characterizations are certain abstractly valued function spaces. One such space is the class of all additive set functions x(r) on all Lebesgue measurable subsets r of (0, 1) to a Banach space X where for all linear functionals x on X and for all subdivisions 7r=` (Irl, T2, , *r ) of (0, 1) into disjoint sets,

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Compact isospectral sets of surfaces

1988-09-01

Brad Osgood , Ralph S. Phillips , Peter Sarnak

The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces

1982-05-01

Peter D. Lax , Ralph S. Phillips

Hecke operators and distributing points on the sphere I

1986-01-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Perturbation theory for semi-groups of linear operators

1953-01-01

Ralph S. Phillips

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Explicit expanders and the Ramanujan conjectures

1986-01-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Article Free Access Share on Explicit expanders and the Ramanujan conjectures Authors: A Lubotzky Institute of Mathematics and Computer Science, Hebrew University, Jerusalem, Israel Institute of Mathematics and Computer Science, … Article Free Access Share on Explicit expanders and the Ramanujan conjectures Authors: A Lubotzky Institute of Mathematics and Computer Science, Hebrew University, Jerusalem, Israel Institute of Mathematics and Computer Science, Hebrew University, Jerusalem, IsraelView Profile , R Phillips Department of Mathematics, Stanford University, Stanford, California Department of Mathematics, Stanford University, Stanford, CaliforniaView Profile , P Sarnak Department of Mathematics, Stanford University, Stanford, California Department of Mathematics, Stanford University, Stanford, CaliforniaView Profile Authors Info & Claims STOC '86: Proceedings of the eighteenth annual ACM symposium on Theory of computingNovember 1986 Pages 240–246https://doi.org/10.1145/12130.12154Online:01 November 1986Publication History 77citation1,237DownloadsMetricsTotal Citations77Total Downloads1,237Last 12 Months64Last 6 weeks9 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

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Exponential decay of solutions of the wave equation in the exterior of a star‐shaped obstacle

1963-11-01

Peter D. Lax , Cathleen S. Morawetz , Ralph S. Phillips

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On cusp forms for co-finite subgroups ofPSL(2,?)

1985-06-01

Ralph S. Phillips , Peter Sarnak

Geodesics in homology classes

1987-06-01

Ralph S. Phillips , Peter Sarnak

On Weakly Compact Subsets of a Banach Space

1943-01-01

Ralph S. Phillips

In the last few years use of the Banach space weak topology has proven exceedingly fruitful in the hands of Bourbaki (4), Smulian (24), Kakutani (14), Ghantmacher and Smulian (9), … In the last few years use of the Banach space weak topology has proven exceedingly fruitful in the hands of Bourbaki (4), Smulian (24), Kakutani (14), Ghantmacher and Smulian (9), Sirvint (23), Dunford and Pettis (8), and others. These investigations have been concerned either with sequentially compact subsets or with bicompact subsets. In this paper an attempt has been made to bridge the gap between these two extremes by considering Ma-compact subsets where K, is any cardinal number. One could define a subset to be t.-compact if every covering of power K, of the set's weak closure contained a finite subcovering. Bourgin has done this, treating the problem from a topological point of view (5). We have preferred an analysis approach, which, we believe, permits a much more direct and essentially simpler development.

An Inversion Formula for Laplace Transforms and Semi-Groups of Linear Operators

1954-03-01

Ralph S. Phillips

This paper is primarily concerned with the problem of determining necessary and sufficient conditions that a closed linear operator be the infinitesimal generator of a semi-group of linear bounded transformations. … This paper is primarily concerned with the problem of determining necessary and sufficient conditions that a closed linear operator be the infinitesimal generator of a semi-group of linear bounded transformations. The first results in this direction were published independently by E. Hille [3] and K. Yosida [8] in 1948. They found necessary and sufficient conditions on the resolvent R(X; U) of an operator U in order that U generate a semi-group T(s) such that 1f T(s) 11 f exp (ws) for some real w. Hille has recently given another proof of this theorem [4] based entirely on the Post-Widder inversion formula for Laplace transforms. Yosida, in his proof, introduced the approximating semi-groups

On matrices whose real linear combinations are non-singular

1965-04-01

J. F. Adams , Peter D. Lax , Ralph S. Phillips

Let A be either the real field R, or the complex field C, or the skew field Q of quaternions. Let A1, A2, , Ak be nXn matrices with entries … Let A be either the real field R, or the complex field C, or the skew field Q of quaternions. Let A1, A2, , Ak be nXn matrices with entries from A. Consider a typical linear combination J= X1A with real coefficients Xi; we shall say that the set { A } the property if such a linear combination is nonsingular (invertible) except when all the coefficients Xi are zero. We shall write A(n) for the maximum number of such matrices which form a set with the property P. We shall write AH(n) for the maximum number of matrices which form a set with the property P. (Here, if A = R, the word Hermitian merely means symmetric; if A= Q it is defined using the usual conjugation in Q.) Our aim is to determine the numbers A(n), AH(n). Of course, it is possible to word the problem more invariantly. Let W be a set of matrices which is a vector space of dimension k over R; we will say that W the property if every nonzero w in W is nonsingular (invertible). We now ask for the maximum possible dimension of such a space. In [1], the first named author has proved that R(n) equals the socalled Radon-Hurwitz function, defined below. In this note we determine RH(n), C(n), CH(n), Q(n) and QH(n) by deriving inequalities between them and R(n). The elementary constructions needed to prove these inequalities can also be used to give a simplified description of the Radon-Hurwitz matrices. The study of sets of real symmetric matrices {Aj} with the property P may be motivated as follows. For such a set, the system of partial differential equations

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Hecke operators and distributing points on S2. II

1987-07-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Absolutely Convergent Fourier Expansions for Non-Commutative Normed Rings

1942-07-01

S. Bochner , Ralph S. Phillips

Scattering theory for dissipative hyperbolic systems

1973-10-01

Peter D. Lax , Ralph S. Phillips

Decaying modes for the wave equation in the exterior of an obstacle

1969-11-01

Peter D. Lax , Ralph S. Phillips

Abstract In this paper we study the dependence of the set of ‘exterior’ eigenvalues {λ k } of Δ on the geometry of the obstacle 𝒪. In particular we show … Abstract In this paper we study the dependence of the set of ‘exterior’ eigenvalues {λ k } of Δ on the geometry of the obstacle 𝒪. In particular we show that the real eigenvalues, corresponding to purely decaying modes, depend monotonically on the obstacle 𝒪, both for the Dirichlet and Neumann boundary conditions . From this we deduce, by comparison with spheres—for which the eigenvalues {λ k } can be determined as roots of special functions—upper and lower bounds for the density of the real {λ k }, and upper and lower bounds for λ 1 , the rate of decay of the fundamental real decaying mode. We also consider the wave equation with a positive potential and establish an analogous monotonicity theorem for such problems. We obtain a second proof for the above Dirichlet problem in the limit as the potential becomes infinite on 𝒪. Finally we derive an integral equation for the decaying modes; this equation bears strong resemblance to one appearing in the transport theory of mono‐energetic neutrons in homogeneous media, and can be used to demonstrate the existence of infinitely many modes.

The adjoint semi-group

1955-06-01

Ralph S. Phillips

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Hodge theory on hyperbolic manifolds

1990-04-01

Rafe Mazzeo , Ralph S. Phillips

Scattering Theory for the Acoustic Equation in an Even Number of Space Dimensions

1972-01-01

Peter D. Lax , Ralph S. Phillips

Abstract : The paper extends the general theory of scattering developed by the authors for hyperbolic systems in an odd number of spacial dimensions to the even dimensional case. As … Abstract : The paper extends the general theory of scattering developed by the authors for hyperbolic systems in an odd number of spacial dimensions to the even dimensional case. As before a key role is played by the incoming and outgoing subspaces and the corresponding translation representations - in this case the Radon transform. For the even dimensional case these two subspaces are no longer orthogonal. This eliminates from the previous theory the associated semigroup of operators which was so useful in characterizing the poles of the scattering matrix. (Modified author abstract)

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The Circle Problem in the Hyperbolic Plane

1994-04-01

Ralph S. Phillips , Zeév Rudnick

Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. I

1984-05-01

Peter D. Lax , Ralph S. Phillips

Abstract This paper deals with the spectral theory of the Laplace‐Beltrami operator Δ acting on automorphic functions in n ‐dimensional hyperbolic space H n . We study discrete subgroups Γ … Abstract This paper deals with the spectral theory of the Laplace‐Beltrami operator Δ acting on automorphic functions in n ‐dimensional hyperbolic space H n . We study discrete subgroups Γ which have a fundamental polyhedron F with a finite number of sides and infinite volume. Concerning these we have shown previously that the spectrum of Δ contains at most a finite number of point eigenvalues in [‐(1/2( n ‐ 1)) 2 , 0], and none less than (1/2( n ‐1)) 2 . Here we prove that the spectrum of Δ is absolutely continuous and of infinite multiplicity in (‐∞, ‐(1/2( n ‐ 1)) 2 ). Our approach uses the non‐Euclidean wave equation introduced by Faddeev and Pavlov, Energy E F is defined as ( u t , u t )‐(u, Lu ), where the bracket is the L 2 scalar product over a fundamental polyhedron with respect to the invariant volume of the hyperbolic metric. Energy is conserved under the group of operator U(t) relating initial data to data at time t. We construct two isometric representations of the space of automorphic data by L 2 (R, N) which transmute the action of U(t) into translation. These representations are given explicitly in terms of integrals of the data over horospheres. In Part II we shall show the completeness of these representations. u tt ‐Lu = 0, L = Δ + (1/2( n ‐ 1)) 2 .

Integration in a convex linear topological space

1940-01-01

Ralph S. Phillips

This paper is concerned with a theory of integration for functions with values in a convex linear topological space.We consider an integral which is essentially an extension to this general … This paper is concerned with a theory of integration for functions with values in a convex linear topological space.We consider an integral which is essentially an extension to this general space of the integral studied by Garrett Birkhoff [l] in a Banach space.By imposing different convex neighborhood topologies on a Banach space, we obtain as instances of our integral those defined by Birkhoff [l], Dunford [2], Gelfand [3], and Pettis [4].Let/(s) be a function on an abstract set 5 to the real numbers, and let a(o) be a nonnegative measure function on an additive family of subsets S of S. A necessary and sufficient condition for the Lebesgue integral to exist is that for each e>0 there exist a partition Ae of 5 into a denumerable set of sets (<r,-) such that for any two orderings of these sets, (o-,-1) and {a?), lim sup X) oc(o-t1)-supSitai1f(si) -lim inf £a(ff,*)-inf,i8li«/(sf) < e.

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Scattering Theory for Automorphic Functions. (AM-87)

1977-12-31

Peter D. Lax , Ralph S. Phillips

The application by Fadeev and Pavlov of the Lax-Phillips scattering theory to the automorphic wave equation led Professors Lax and Phillips to reexamine this development within the framework of their … The application by Fadeev and Pavlov of the Lax-Phillips scattering theory to the automorphic wave equation led Professors Lax and Phillips to reexamine this development within the framework of their theory. This volume sets forth the results of that work in the form of new or more straightforward treatments of the spectral theory of the Laplace-Beltrami operator over fundamental domains of finite area; the meromorphic character over the whole complex plane of the Eisenstein series; and the Selberg trace formula. CONTENTS: 1. Introduction. 2. An abstract scattering theory. 3. A modified theory for second order equations with an indefinite energy form. 4. The Laplace-Beltrami operator for the modular group. 5. The automorphic wave equation. 6. Incoming and outgoing subspaces for the automorphic wave equations. 7. The scattering matrix for the automorphic wave equation. 8. The general case. 9. The Selberg trace formula.

The laplacian for domains in hyperbolic space and limit sets of Kleinian groups

1985-01-01

Ralph S. Phillips , Peter Sarnak

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Dissipative hyperbolic systems

1957-01-01

Ralph S. Phillips

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Perturbation Theory for Semi-Groups of Linear Operators

1953-03-01

Ralph S. Phillips

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Semi-groups of positive contraction operators

1962-01-01

Ralph S. Phillips

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Perturbation theory for the Laplacian on automorphic functions

1992-01-01

Ralph S. Phillips , Peter Sarnak

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma subset-of PSL left-parenthesis 2 comma double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo>⊂</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>PSL(2, </mml:mtext> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma subset-of PSL left-parenthesis 2 comma double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mo>⊂</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>PSL(2, </mml:mtext> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>)</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma \subset {\text {PSL(2, }}\mathbb {R}{\text {)}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a discrete subgroup with quotient <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma minus upper H"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:mi class="MJX-variant" mathvariant="normal">∖</mml:mi> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Gamma \backslash H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of finite volume but not compact. The spectrum of the Laplacian on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> automorphic functions is unstable under perturbations; however, it becomes much more manageable when the scattering frequencies are adjoined (with multiplicity equal to the order of the pole of the determinant of the scattering matrix at these points). This augmented set shows up in a natural way in a one-sided version of the Selberg trace formula and is the actual spectrum of the generator of a cut-off wave equation. Applying standard perturbation theory to this operator, it is proved that the augmented spectrum is real analytic in Teichmüller space. The same operator is used to derive Fermi’s Golden Rule in this setting. It turns out that the proper multiplicity to be attached to the Laplacian eigenvalue at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="one fourth"> <mml:semantics> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> <mml:annotation encoding="application/x-tex">\frac {1}{4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is twice the dimension of cusp forms plus <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu equals trace left-bracket normal upper Phi plus upper I right-bracket slash 2"> <mml:semantics> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext> tr</mml:mtext> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext> + </mml:mtext> </mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>/2</mml:mtext> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu = {\text { tr}}[\Phi {\text { + }}I]{\text {/2}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; here <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal">Φ</mml:mi> <mml:annotation encoding="application/x-tex">\Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the scattering matrix at this point. It is shown that the generic value of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu"> <mml:semantics> <mml:mi>μ</mml:mi> <mml:annotation encoding="application/x-tex">\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Teichmüller space of the once punctured torus and the six-times punctured sphere is zero. This is also true of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi>χ</mml:mi> <mml:annotation encoding="application/x-tex">\chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-twisted spectral problem, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi"> <mml:semantics> <mml:mi>χ</mml:mi> <mml:annotation encoding="application/x-tex">\chi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a character for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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An example of huygens' principle

1978-07-01

Peter D. Lax , Ralph S. Phillips

The weyl theorem and the deformation of discrete groups

1985-11-01

Ralph S. Phillips , Peter Sarnak

The scattering of sound waves by an obstacle

1977-03-01

Peter D. Lax , Ralph S. Phillips

Abstract In this paper we study the scattering of acoustic waves by an obstacle 𝒪. We establish the following relation between the scattering kernel S ( s , θ, ω) … Abstract In this paper we study the scattering of acoustic waves by an obstacle 𝒪. We establish the following relation between the scattering kernel S ( s , θ, ω) and the support function h 𝒪 of the obstacle: The right endpoint of the support of S ( s , θ, ω) as function of s is h 𝒪 (θ‐ω); h 𝒪 is defined by For Dirichlet boundary condition the result is proved in full generality, for Neumann condition only for backscattering, i.e., for θ = ‐ω. Since the convex hull of 𝒪 can be recovered from knowledge of h 𝒪 , the above result may be useful in reconstructing 𝒪 from scattering data.

Scattering theory

1964-01-01

Peter D. Lax , Ralph S. Phillips

Let H be a Hubert space, U(t) a group of unitary operators.A closed subspace D + of H will be called outgoing if it has the following properties:A prototype of … Let H be a Hubert space, U(t) a group of unitary operators.A closed subspace D + of H will be called outgoing if it has the following properties:A prototype of the above situation is when H is L 2 ( -°°, °° î N), i.e., the space of square integrable functions on the whole real axis whose values lie in some accessory Hubert space N t U(t) is translation by t, and D+ is L%(0 f oo ; N).THEOREM l. 8 If D + is outgoing for the group U(t), then H can be represented isometrically as L%(-~ oo, oo ; N) so that U(t) is translation and D + is the space of functions with support on the positive reals.This representation is unique up to isomorphisms of N.We shall call this representation an outgoing translation representation of the group.Taking the Fourier transform we obtain an outgoing spectral representation of the group U(t), where elements of D + are represented as functions in A+(N) f that is the Fourier transform of L(0, <& ; N).According to the Paley-Wiener theorem A+(N) consists of boundary values of functions with values in N, analytic in the upper half-plane whose square integrals along lines Imz = const are uniformly bounded.An incoming subspace P_ is defined similarly and an analogous representation theorem holds, D~ being represented by functions with support on the negative axis, that is, by L 2 ( -<*>, 0; NJ).N-and N are unitarily equivalent and will henceforth be identified.In the application to the wave equation there is a natural identification of N and N-.Let D + and £>_ be outgoing and incoming subspaces respectively for the same unitary group, and suppose that D+ and D-are orthogonal.To each function f (EH there are associated two functions k~ and k+, the respective incoming and outgoing translation representa-

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The spectrum of fermat curves

1991-03-01

Ralph S. Phillips , Peter Sarnak

Spectral theory for semi-groups of linear operators

1951-01-01

Ralph S. Phillips

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The paley‐wiener theorem for the radon transform

1970-05-01

Peter D. Lax , Ralph S. Phillips

Abstract We prove that the support of a complex‐valued function f in ℝ k is contained in a convex set K if and only if the support of its Radon … Abstract We prove that the support of a complex‐valued function f in ℝ k is contained in a convex set K if and only if the support of its Radon transform k ( s , ω) is, for each ω, contained in s ≦ S K (ω); here S K is the support function of the set K . This theorem is used to determine the propagation speeds of hyperbolic differential equations with constant coefficients, to prove the nonexistence of point spectrum for a certain class of partial differential operators, and to give a simple reduction of Lions' convolution theorem to the one‐dimensional convolution theorem of Titchmarsh.

Translation representations for the solution of the non‐euclidean wave equation

1979-09-01

Peter D. Lax , Ralph S. Phillips

On Matrices Whose Real Linear Combinations are Nonsingular

1965-04-01

J. F. Adams , Peter D. Lax , Ralph S. Phillips

Maass cusp forms

1985-06-01

Jean-Marc Deshouillérs , Henryk Iwaniec , Ralph S. Phillips +1 more

It is shown that, under certain standard assumptions, such as extended Riemann hypotheses, the scattering matrix ϕ( s ) for generic Γ ≤ SL(2, R) is unexpectedly of order 2. … It is shown that, under certain standard assumptions, such as extended Riemann hypotheses, the scattering matrix ϕ( s ) for generic Γ ≤ SL(2, R) is unexpectedly of order 2. This leads to the conjecture that the generic cofinite Γ has very few Maass cusp forms.

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The wave equation in exterior domains

1962-01-01

Peter D. Lax , Ralph S. Phillips

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Cusp forms for character varieties

1994-01-01

Ralph S. Phillips , Peter Sarnak

Bessel Function Approximations

1950-04-01

Ralph S. Phillips , Henry Malin

On a theorem due to Sz.-Nagy

1959-03-01

Ralph S. Phillips

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On the extension problem for dissipative operators

1968-05-01

Michael G. Crandall , Ralph S. Phillips

Semi-Groups of Contraction Operators

2011-01-01

Ralph S. Phillips

Scattering theory for twisted automorphic functions

1998-01-01

Ralph S. Phillips

The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group $\Gamma$ with … The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group $\Gamma$ with an irreducible unitary representation $\rho$ and satisfying $u(\gamma z)=\rho (\gamma )u(z)$. The Lax-Phillips approach is used with the wave equation playing a central role. Incoming and outgoing subspaces are employed to obtain corresponding unitary translation representations, $R_-$ and $R_+$, for the solution operator. The scattering operator, which maps $R_-f$ into $R_+f$, is unitary and commutes with translation. The spectral representation of the scattering operator is a multiplicative operator, which can be expressed in terms of the constant term of the Eisenstein series. When the dimension of $\rho$ is one, the elements of the scattering operator cannot vanish. However when $\dim (\rho )>1$ this is no longer the case.

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Dissolving a cusp form in the presence of multiplicities

1997-06-01

Chris Judge , Ralph S. Phillips

Perturbation theory for twisted automorphic functions

1997-03-01

Ralph S. Phillips

Conjugacy Classes of Γ(2) and Spectral Rigidity

1995-07-01

Ralph S. Phillips

The free group Γ(2) is generated by A = (1 2, 0 1) and B = (10, -21), and setting X(ξ,η)(A) = exp(2πiξ), X(ξ,η)(B) = exp(2πiη) defines a unitary character … The free group Γ(2) is generated by A = (1 2, 0 1) and B = (10, -21), and setting X(ξ,η)(A) = exp(2πiξ), X(ξ,η)(B) = exp(2πiη) defines a unitary character on Γ(2) for 0 ≤ ξ,η < 1. A program is devised to compute μ(tr) = Σχ(ξ,η)(conj. class), summed over all primitive conjugacy classes of Γ(2) of trace tr. Combined with a Luo-Sarnak theorem, this yields lower bounds for the spectral variance for a large sampling of characters in the range 0 < ξ,η < 1. The results indicate that the Berry conjecture for spectral rigidity does not hold for this set of classically chaotic systems. The program is also used to compute θ(X) = Σln(N({γ}), summed over all primitive conjugacy classes of Γ(2) of norm N({γ}) ≤ x. The function 0(x) is asymptotic to x, and the remainder can be written as θ(x) - x = x β . The values of β(x) are computed for all traces between 3202 and 4802 (here x = tr 2 -2). The β's cluster around 0.6, attaining a maximum of 2/3. Finally, it is proved that the remainder 0(x) - x has a negative bias by showing that the mean normalized remainder converges to a negative limit.

Conjugacy classes of Γ(2) and spectral rigidity

1995-01-01

Ralph S. Phillips

The free group<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma left-parenthesis 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\Gamma (2)</mml:annotation></mml:semantics></mml:math></inline-formula>is generated by<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals left-parenthesis 1 2 comma 0 … The free group<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma left-parenthesis 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\Gamma (2)</mml:annotation></mml:semantics></mml:math></inline-formula>is generated by<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals left-parenthesis 1 2 comma 0 1 right-parenthesis"><mml:semantics><mml:mrow><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mspace width="thickmathspace" /><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mspace width="thickmathspace" /><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">A = (1\;2,0\;1)</mml:annotation></mml:semantics></mml:math></inline-formula>and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B equals left-parenthesis 1 0 comma minus 2 1 right-parenthesis"><mml:semantics><mml:mrow><mml:mi>B</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mspace width="thickmathspace" /><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mspace width="thickmathspace" /><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">B = (1\;0, - 2\;1)</mml:annotation></mml:semantics></mml:math></inline-formula>, and setting<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi Subscript left-parenthesis xi comma eta right-parenthesis Baseline left-parenthesis upper A right-parenthesis equals exp left-parenthesis 2 pi i xi right-parenthesis comma chi Subscript left-parenthesis xi comma eta right-parenthesis Baseline left-parenthesis upper B right-parenthesis equals exp left-parenthesis 2 pi i eta right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>χ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>exp</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>i</mml:mi><mml:mi>ξ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>χ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>B</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>exp</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>i</mml:mi><mml:mi>η</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">{\chi _{(\xi ,\eta )}}(A) = \exp (2\pi i\xi ),{\chi _{(\xi ,\eta )}}(B) = \exp (2\pi i\eta )</mml:annotation></mml:semantics></mml:math></inline-formula>defines a unitary character on<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma left-parenthesis 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\Gamma (2)</mml:annotation></mml:semantics></mml:math></inline-formula>for<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to xi comma eta greater-than 1"><mml:semantics><mml:mrow><mml:mn>0</mml:mn><mml:mo>≤</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">0 \leq \xi ,\eta > 1</mml:annotation></mml:semantics></mml:math></inline-formula>. A program is devised to compute<disp-formula content-type="math/mathml">\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu left-parenthesis trace right-parenthesis equals sigma-summation chi Subscript left-parenthesis xi comma eta right-parenthesis Baseline left-parenthesis conj period class right-parenthesis comma"><mml:semantics><mml:mrow><mml:mi>μ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext>tr</mml:mtext></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>χ</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">(</mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext>conj</mml:mtext></mml:mrow><mml:mo>.</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext>class</mml:mtext></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">\mu ({\text {tr}}) = \sum {{\chi _{(\xi ,\eta )}}({\text {conj}}. {\text {class}}),}</mml:annotation></mml:semantics></mml:math>\]</disp-formula>summed over all primitive conjugacy classes of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma left-parenthesis 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\Gamma (2)</mml:annotation></mml:semantics></mml:math></inline-formula>of trace tr. Combined with a Luo-Sarnak theorem, this yields lower bounds for the spectral variance for a large sampling of characters in the range<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 greater-than xi comma eta greater-than 1"><mml:semantics><mml:mrow><mml:mn>0</mml:mn><mml:mo>></mml:mo><mml:mi>ξ</mml:mi><mml:mo>,</mml:mo><mml:mi>η</mml:mi><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">0 > \xi ,\eta > 1</mml:annotation></mml:semantics></mml:math></inline-formula>. The results indicate that the Berry conjecture for spectral rigidity does not hold for this set of classically chaotic systems. The program is also used to compute<disp-formula content-type="math/mathml">\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta left-parenthesis x right-parenthesis equals sigma-summation ln left-parenthesis upper N left-parenthesis StartSet gamma EndSet right-parenthesis right-parenthesis comma"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>∑</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>ln</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mi>γ</mml:mi><mml:mo fence="false" stretchy="false">}</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">\theta (x) = \sum {\ln (N(\{ \gamma \} )),}</mml:annotation></mml:semantics></mml:math>\]</disp-formula>summed over all primitive conjugacy classes of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma left-parenthesis 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">Γ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\Gamma (2)</mml:annotation></mml:semantics></mml:math></inline-formula>of norm<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N left-parenthesis StartSet gamma EndSet right-parenthesis less-than-or-equal-to x"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo fence="false" stretchy="false">{</mml:mo><mml:mi>γ</mml:mi><mml:mo fence="false" stretchy="false">}</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>≤</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">N(\{ \gamma \} ) \leq x</mml:annotation></mml:semantics></mml:math></inline-formula>. The function<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta left-parenthesis x right-parenthesis"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\theta (x)</mml:annotation></mml:semantics></mml:math></inline-formula>is asymptotic to<italic>x</italic>, and the remainder can be written as<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue theta left-parenthesis x right-parenthesis minus x EndAbsoluteValue equals x Superscript beta"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>θ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mi>x</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mi>x</mml:mi><mml:mi>β</mml:mi></mml:msup></mml:mrow></mml:mrow><mml:annotation encoding="application/x-tex">|\theta (x) - x| = {x^\beta }</mml:annotation></mml:semantics></mml:math></inline-formula>. The values of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta left-parenthesis x right-parenthesis"><mml:semantics><mml:mrow><mml:mi>β</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\beta (x)</mml:annotation></mml:semantics></mml:math></inline-formula>are computed for all traces between 3202 and 4802 (here<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x equals trace squared minus 2"><mml:semantics><mml:mrow><mml:mi>x</mml:mi><mml:mo>=</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mtext>tr</mml:mtext><mml:mn>2</mml:mn></mml:msup></mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">x = {\text {tr}^2} - 2</mml:annotation></mml:semantics></mml:math></inline-formula>). The<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta"><mml:semantics><mml:mi>β</mml:mi><mml:annotation encoding="application/x-tex">\beta</mml:annotation></mml:semantics></mml:math></inline-formula>’s cluster around 0.6, attaining a maximum of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 slash 3"><mml:semantics><mml:mrow><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">2/3</mml:annotation></mml:semantics></mml:math></inline-formula>. Finally, it is proved that the remainder<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta left-parenthesis x right-parenthesis minus x"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\theta (x) - x</mml:annotation></mml:semantics></mml:math></inline-formula>has a negative bias by showing that the mean normalized remainder converges to a negative limit.

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The Circle Problem in the Hyperbolic Plane

1994-04-01

Ralph S. Phillips , Zeév Rudnick

Cusp forms for character varieties

1994-01-01

Ralph S. Phillips , Peter Sarnak

Scattering theory for symmetric spaces of noncompact type

1993-10-01

Ralph S. Phillips , Mehrdad Shahshahani

Automorphic spectrum and Fermi’s Golden Rule

1992-12-01

Ralph S. Phillips , Peter Sarnak

Translation representation for automorphic solutions of the wave equation in non‐euclidean spaces, IV

1992-02-01

Peter D. Lax , Ralph S. Phillips

Abstract In Part I of this series of papers we have defined the incoming and outgoing translation representations for automorphic solutions of the hyperbolic wave equations; in Part II we … Abstract In Part I of this series of papers we have defined the incoming and outgoing translation representations for automorphic solutions of the hyperbolic wave equations; in Part II we have proved the completeness of these representations when the fundamental polyhedron F has a finite number of sides with a finite or infinite volume, but is not compact. In Part IV we present a proof of completeness which is simpler than our original proof contained in Section 7 of Part II for the case when F has cusps of less than maximal rank; and we supply a proof for the case, not covered in Section 7, when the parabolic subgroup associated with such cusps contains twists.

Perturbation theory for the Laplacian on automorphic functions

1992-01-01

Ralph S. Phillips , Peter Sarnak

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The spectrum of fermat curves

1991-03-01

Ralph S. Phillips , Peter Sarnak

Hodge theory on hyperbolic manifolds

1990-04-01

Rafe Mazzeo , Ralph S. Phillips

Extremals of determinants of Laplacians

1988-09-01

Brad Osgood , Ralph S. Phillips , Peter Sarnak

Compact isospectral sets of surfaces

1988-09-01

Brad Osgood , Ralph S. Phillips , Peter Sarnak

Ramanujan graphs

1988-09-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Scattering theory for the wave equation on a hyperbolic manifold

1987-10-01

Ralph S. Phillips , Bettina Wiskott , Alex Woo

Hecke operators and distributing points on S2. II

1987-07-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Geodesics in homology classes

1987-06-01

Ralph S. Phillips , Peter Sarnak

Scattering theory for the wave equation on a hyperbolic manifold

1987-01-01

Ralph S. Phillips , Bettina Wiskott , Alex Woo

Scattering Theory for the Wave Equation on a Hyperbolic Manifold

1987-01-01

Ralph S. Phillips

Hecke operators and distributing points on the sphere I

1986-01-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

Explicit expanders and the Ramanujan conjectures

1986-01-01

Alexander Lubotzky , Ralph S. Phillips , Peter Sarnak

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The weyl theorem and the deformation of discrete groups

1985-11-01

Ralph S. Phillips , Peter Sarnak

On cusp forms for co-finite subgroups ofPSL(2,?)

1985-06-01

Ralph S. Phillips , Peter Sarnak

Maass cusp forms

1985-06-01

Jean-Marc Deshouillérs , Henryk Iwaniec , Ralph S. Phillips +1 more

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Translation Representations for Automorphic Solutions of The Wave Equation in Non-Euclidean Spaces; The Case of Finite Volume

1985-06-01

Peter D. Lax , Ralph S. Phillips

Let $\Gamma$ be a discrete subgroup of automorphisms of ${{\mathbf {H}}^n}$, with fundamental polyhedron of finite volume, finite number of sides, and $N$ cusps. Denote by ${\Delta _\Gamma }$ the … Let $\Gamma$ be a discrete subgroup of automorphisms of ${{\mathbf {H}}^n}$, with fundamental polyhedron of finite volume, finite number of sides, and $N$ cusps. Denote by ${\Delta _\Gamma }$ the Laplace-Beltrami operator acting on functions automorphic with respect to $\Gamma$. We give a new short proof of the fact that ${\Delta _\Gamma }$ has absolutely continuous spectrum of uniform multiplicity $N$ on $( - \infty ,{((n - 1)/2)^2})$, plus a standard discrete spectrum. We show that this property of the spectrum is unchanged under arbitrary perturbation of the metric on a compact set. Our method avoids Eisenstein series entirely and proceeds instead by constructing explicitly a translation representation for the associated wave equation.

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On the spectrum of the Hecke groups

1985-03-01

Ralph S. Phillips , Peter Sarnak

Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. III

1985-03-01

Peter D. Lax , Ralph S. Phillips

Abstract In Part III we show that the translation representations defined in Part I, and shown to be complete in Part II, are incoming and outgoing in the sense of … Abstract In Part III we show that the translation representations defined in Part I, and shown to be complete in Part II, are incoming and outgoing in the sense of propagation of signals along rays. We give a new proof of the absolute continuity of the spectrum below ‐¼ n 2 , and point out its implications for local energy decay. We give a new proof of completeness in dimensions 2 and 3 when energy is positive. Finally, we define and prove completeness of the translation representations when the metric is perturbed on a compact set.

Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume

1985-01-01

Peter D. Lax , Ralph S. Phillips

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a discrete subgroup of automorphisms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a discrete subgroup of automorphisms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper H Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">H</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {H}}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with fundamental polyhedron of finite volume, finite number of sides, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> cusps. Denote by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript normal upper Gamma"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Delta _\Gamma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the Laplace-Beltrami operator acting on functions automorphic with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ</mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We give a new short proof of the fact that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript normal upper Gamma"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi mathvariant="normal">Γ</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{\Delta _\Gamma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has absolutely continuous spectrum of uniform multiplicity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis negative normal infinity comma left-parenthesis left-parenthesis n minus 1 right-parenthesis slash 2 right-parenthesis squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">( - \infty ,{((n - 1)/2)^2})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, plus a standard discrete spectrum. We show that this property of the spectrum is unchanged under arbitrary perturbation of the metric on a compact set. Our method avoids Eisenstein series entirely and proceeds instead by constructing explicitly a translation representation for the associated wave equation.

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The laplacian for domains in hyperbolic space and limit sets of Kleinian groups

1985-01-01

Ralph S. Phillips , Peter Sarnak

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Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. II

1984-11-01

Peter D. Lax , Ralph S. Phillips

Abstract In this part we prove that the translation representation defined in Part I for the non‐Euclidean wave equation is complete. We show how one can derive from this translation … Abstract In this part we prove that the translation representation defined in Part I for the non‐Euclidean wave equation is complete. We show how one can derive from this translation representation a spectral representation for the Laplace‐Beltrami operator over geometrically finite fundamental polyhedra with infinite volume, without the use of Eisenstein series.

Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. I

1984-05-01

Peter D. Lax , Ralph S. Phillips

The Spectrum of the Laplacian for Domains in Hyperbolic Space and Limit Sets of Kleinian Groups

1984-01-01

Ralph S. Phillips

A local paley‐wiener theorem for the radon transform of L2 functions in a non‐euclidean setting

1982-07-01

Peter D. Lax , Ralph S. Phillips

The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces

1982-05-01

Peter D. Lax , Ralph S. Phillips

The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces

1982-01-01

Peter D. Lax , Ralph S. Phillips

Translation representations for the solution of the non‐euclidean wave equation II

1981-05-01

Peter D. Lax , Ralph S. Phillips

The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces

1981-01-01

Peter D. Lax , Ralph S. Phillips

The counting numbers for discrete subgroups of motions in Euclidean and non-Euclidean spaces are obtained using the wave equation as the principal tool. In dimensions 2 and 3 the error … The counting numbers for discrete subgroups of motions in Euclidean and non-Euclidean spaces are obtained using the wave equation as the principal tool. In dimensions 2 and 3 the error estimates are close to the best known.

Scattering theory for automorphic functions

1980-01-01

Peter D. Lax , Ralph S. Phillips

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Translation representations for the solution of the non‐euclidean wave equation

1979-09-01

Peter D. Lax , Ralph S. Phillips

An example of huygens' principle

1978-07-01

Peter D. Lax , Ralph S. Phillips

Scattering theory for domains with non-smooth boundaries

1978-06-01

Peter D. Lax , Ralph S. Phillips

§6. INCOMING AND OUTGOING SUBSPACES FOR THE AUTOMORPHIC WAVE EQUATION

1977-12-31

Peter D. Lax , Ralph S. Phillips

§4. THE LAPLACE-BELTRAMI OPERATOR FOR THE MODULAR GROUP

1977-12-31

Peter D. Lax , Ralph S. Phillips

§3. A MODIFIED THEORY FOR SECOND ORDER EQUATIONS WITH AN INDEFINITE ENERGY FORM

1977-12-31

Peter D. Lax , Ralph S. Phillips

§5. THE AUTOMORPHIC WAVE EQUATIONS

1977-12-31

Peter D. Lax , Ralph S. Phillips

§7. THE SCATTERING MATRIX FOR THE AUTOMORPHIC WAVE EQUATION

1977-12-31

Peter D. Lax , Ralph S. Phillips

Scattering Theory for Automorphic Functions. (AM-87)

1977-12-31

Peter D. Lax , Ralph S. Phillips

Coauthor	Papers Together
Peter D. Lax	37
Peter Sarnak	16
Alexander Lubotzky	4
Bettina Wiskott	2
Alex Woo	2
Brad Osgood	2
Cathleen S. Morawetz	2
Mehrdad Shahshahani	1
J. F. Adams	1
Michael G. Crandall	1
Zeév Rudnick	1
Rafe Mazzeo	1
Henry Malin	1
Jean‐Marc Deshouillers	1
S. Bochner	1
J. F. Adams	1
Henryk Iwaniec	1
Heather A. Dye	1
Chris Judge	1
G. Lumer	1
Leonard Sarason	1

Functional Analysis and Semi-groups

1996-02-06

Einar Hille , Robert A. Phillips

Part One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: … Part One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: Subadditive functions Semi-modules Addition theorem in a Banach algebra Semi-groups in the strong topology Generator and resolvent Generation of semi-groups Part Three. Advanced Analytical Theory of Semi-Groups: Perturbation theory Adjoint theory Operational calculus Spectral theory Holomorphic semi-groups Applications to ergodic theory Part Four. Special Semi-groups and Applications: Translations and powers Trigonometric semi-groups Semi-groups in $L_p(-\infty,\infty)$ Semi-groups in Hilbert space Miscellaneous applications Part Five. Extensions of the theory: Notes on Banach algebras Lie semi-groups Functions on vectors to vectors Bibliography Index.

Scattering theory for automorphic functions

1980-01-01

Peter D. Lax , Ralph S. Phillips

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The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces

1982-05-01

Peter D. Lax , Ralph S. Phillips

The Asymptotic Distribution of Lattice Points in Euclidean and Non-Euclidean Spaces

1982-01-01

Peter D. Lax , Ralph S. Phillips

Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. I

1984-05-01

Peter D. Lax , Ralph S. Phillips

The limit set of a Fuchsian group

1976-01-01

S. J. Patterson

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Translation representations for the solution of the non‐euclidean wave equation

1979-09-01

Peter D. Lax , Ralph S. Phillips

Pseudo-laplaciens. I

1982-01-01

Yves Colin de Verdìère

We construct, on a 2 or 3-dimensional Riemannian manifold, the self-adjoint extensions Δ α,x 0 (α∈R/πZ) of the Laplace operator restricted to the functions vanishing in some neigbhourhood of some … We construct, on a 2 or 3-dimensional Riemannian manifold, the self-adjoint extensions Δ α,x 0 (α∈R/πZ) of the Laplace operator restricted to the functions vanishing in some neigbhourhood of some point x 0 of X. We compute explicitely the eigenvalues of Δ α,x 0 .

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Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Applications to Dirichlet Series

1956-09-01

Atle Selberg

In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these … In The following lectures we shall give a brief sketch of some representative parts of certain investigations that have been undertaken during the last five years. The center of these investigations is a general relation which can be considered as a generalization of the so-called Poisson summation formula (in one or more dimensions). This relation we here refer to as the trace-formula.

On cusp forms for co-finite subgroups ofPSL(2,?)

1985-06-01

Ralph S. Phillips , Peter Sarnak

�ber eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichlet scher Reihen durch Funktionalgleichungen

1949-12-01

Hans Maaß

Spectral theory of automorphic functions

1982-01-01

Alexei B. Venkov

Functional Analysis and Semi-Groups

1949-04-01

Richard G. Cooke

Symmetric positive linear differential equations

1958-08-01

Kurt Friedrichs

Dissipative operators and hyperbolic systems of partial differential equations

1959-01-01

Ralph S. Phillips

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The spectrum of fermat curves

1991-03-01

Ralph S. Phillips , Peter Sarnak

Disappearance of Cusp Forms in Special Families

1994-03-01

Scott A. Wolpert

In his 1954 G6ttingen lectures, Atle Selberg [Sb] posed the question of the asymptotic density of the poles of the Eisenstein series. For r a cofinite non-cocompact subgroup of PSL(2; … In his 1954 G6ttingen lectures, Atle Selberg [Sb] posed the question of the asymptotic density of the poles of the Eisenstein series. For r a cofinite non-cocompact subgroup of PSL(2; R) the question is known to be equivalent to the asymptotic count of the number of eigenvalues for the LaplaceBeltrami operator D, acting in L2(H/r), where H is the upper halfplane with hyperbolic metric. In general the spectral decomposition of D consists of the continuous band [1/4, oo), with multiplicity the number of cusps of H/r, and a discrete component, possibly in [1/4, 0o), spanned by eigenfunctions W1i, WP2* .. (see [Hj1], [Kb], [LxP]). For the eigenvalues Aj, Dpj + Ajpj = O, consider the spectral counting function N(R) = # Ij v 'N 0 (see [DIPS]; [PS2], Cor. 2.3). Luo [Lu] recently established the unconditional result; his argument does not involve the Lindel6f hypothesis. Deshouillers et

Maass cusp forms

1985-06-01

Jean-Marc Deshouillérs , Henryk Iwaniec , Ralph S. Phillips +1 more

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A local paley‐wiener theorem for the radon transform of L2 functions in a non‐euclidean setting

1982-07-01

Peter D. Lax , Ralph S. Phillips

Translation representations for the solution of the non‐euclidean wave equation II

1981-05-01

Peter D. Lax , Ralph S. Phillips

Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. Teil I

1973-12-01

J�rgen Elstrodt

On one-parameter semi-groups of linear transformations

1951-04-01

Ralph S. Phillips

for 0 O. We shall show that the first hypothesis for this theorem implies the second. We shall also show by means of an example that weak measurability is not … for 0 O. We shall show that the first hypothesis for this theorem implies the second. We shall also show by means of an example that weak measurability is not sufficient to imply the boundedness of IIT(t)II in any interval [5, 1/a]. We being by proving the following lemma.

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The Selberg Trace Formula for PSL(2,ℝ)

1983-01-01

Dennis A. Hejhal

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Cusp forms for character varieties

1994-01-01

Ralph S. Phillips , Peter Sarnak

Local boundary conditions for dissipative symmetric linear differential operators

1960-08-01

Peter D. Lax , Ralph S. Phillips

Lectures on Quasiconformal Mappings

2006-07-14

Lars V. Ahlfors

The Ahlfors Lectures: Acknowledgments Differentiable quasiconformal mappings The general definition Extremal geometric properties Boundary correspondence The mapping theorem Teichmuller spaces Editors' notes The Additional Chapters: A supplement to Ahlfors's lectures … The Ahlfors Lectures: Acknowledgments Differentiable quasiconformal mappings The general definition Extremal geometric properties Boundary correspondence The mapping theorem Teichmuller spaces Editors' notes The Additional Chapters: A supplement to Ahlfors's lectures Complex dynamics and quasiconformal mappings Hyperbolic structures on three-manifolds that fiber over the circle.

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Scattering Theory for Automorphic Functions. (AM-87)

1977-12-31

Peter D. Lax , Ralph S. Phillips

THE PARABOLIC DIFFERENTIAL EQUATIONS AND THE ASSOCIATED SEMI-GROUPS OF TRANSFORMATIONS

2015-01-01

William Feller

HARMONIC ANALYSIS ON RIEMANNIAN SYMMETRIC SPACES OF NEGATIVE CURVATURE AND SCATTERING THEORY

1976-06-30

M A Semenov-Tjan-Šanskiĭ

Harmonic analysis on a Riemannian symmetric space can be connected with the study of a nonstationary system of equations that has been constructed with respect to the ring of Laplace … Harmonic analysis on a Riemannian symmetric space can be connected with the study of a nonstationary system of equations that has been constructed with respect to the ring of Laplace operators. The scattering theory for this system generalizes the scattering theory for hyperbolic equations constructed by Lax and Phillips. The paper contains a series of new spectral theorems generalizing the Harish-Chandra theorem and a formulation of a causality principle for scattering operators.Bibliography: 23 items.

Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene. I.

1956-12-01

Heinz Huber

The laplacian operator on a Riemann surface, III

1976-01-01

S. J. Patterson

A General Theory of Limits

1922-04-01

Eliakim Hastings Moore , Hal L. Smith

I. As to Notions.-In this paper we investigate a simple general limit of which, as will appear in ? 5, the various classical limits of analysis are actually instances.* The … I. As to Notions.-In this paper we investigate a simple general limit of which, as will appear in ? 5, the various classical limits of analysis are actually instances.* The general limit in question is an obvious generalization of the following two limits: 1. An infinite sequence {an} of real or (ordinary) complex numbers an (n = 1, 2, *..) converges to a number a as a limit, in notation: Lnoo an = a,-as clearly defined about a century ago,-in case for every positive number e there exists a positive integer ne of such a nature that for every integer n i ne it is true that in absolute value ana is. at most e. Here the numerical sequence {an } may be considered as a numerically valued function a (an In) of the positive integer n (or on the range [n] of positive integers n), viz., a(n) an for every n. 2. Relative to a general (i.e., any particular) class e[q] of general elements q and the class ( = [s] of all finite classes s of elements q, a numerically valued function a (a (s) Is) on the range S converges to a number a as limit, in notation: L,a(s) = a, in case for every positive number e there exists a class Se of such a nature that for every class s including Se it is true that in absolute value a(s) a is at most e. The limit (2) belongs to General Analysis, i.e., to that doctrine of analysis in which a general class, here {D, plays a fundamental role. The limit (2), introducedt in 1915 by the senior author, plays a central role in his second theory$ of Linear Integral Equations in General Analysis. This general theory has as notable instances: (a) Hilbert's theory of limited quadratic forms in a denumerable infinitude of variables; here the Hilbert space [a] of infinite sequences a (a (n) I n) of real numbers with convergent Ina (n) a (n) plays a central role; (b) an analogous theory in which the corresponding role

Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups

1984-01-01

Dennis Sullivan

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Die Resolvente zum Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. Teil II

1973-06-01

J�rgen Elstrodt

Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II

1961-08-01

Heinz Huber

Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. II

1967-12-01

Walter Roelcke

Scattering Theory for the Acoustic Equation in an Even Number of Space Dimensions

1972-01-01

Peter D. Lax , Ralph S. Phillips

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The Surjectivity of Invariant Differential Operators on Symmetric Spaces I

1973-11-01

Sigurđur Helgason

Schottky Groups and Mumford Curves

1980-01-01

Lothar Gerritzen , Marius van der Put

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A lattice‐point problem in hyperbolic space

1975-06-01

S. J. Patterson

Meromorphic families of compact operators

1968-12-01

Stanly Steinberg

Dissipative hyperbolic systems

1957-01-01

Ralph S. Phillips

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Asymptotic completeness for quantum mechanical potential scattering

1978-08-01

Volker Enß

Perturbation Theory for Linear Operators

1995-01-01

Tosio Kato

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The decay of solutions of the exterior initial‐boundary value problem for the wave equation

1961-08-01

Cathleen S. Morawetz

On integration in vector spaces

1938-01-01

B. J. Pettis

Introduction.Several authors ([3]-[10], inclusive; [I5])f have already given generalized Lebesgue integrals for functions x(s) whose values lie in a Banach space (P-space) 3c.fIn the following pages another definition, § based … Introduction.Several authors ([3]-[10], inclusive; [I5])f have already given generalized Lebesgue integrals for functions x(s) whose values lie in a Banach space (P-space) 3c.fIn the following pages another definition, § based on the linear functionals over 3c and on the ordinary Lebesgue integral, || will be given and its properties and relationships to other integrals discussed.We consider functions x(s) defined to a P-space X = [x] from an abstract space S= [s] possessing both a c-field 2 of "measurable" sets having S as an element and a non-negative bounded c.a. (completely additive) "measure" function a(E) defined over 2. Notational conventions are as follows: 3c denotes the P-space conjugate to 3c ([l], p. 188),/=/(x), g = g(x), • ■ ■ denote elements of 3c, and F = F(f), G = G(f), ■ ■ ■ elements of 3c, the conjugate space of 3c; and for real-valued functions Greek letters will be used.When the abstract functions x(s), y(s), • • ■ , or the real functions 4>(s), \p(s), ■ ■ ■ are considered as elements of a functional space, they will sometimes be written x(:), y(:) or <f>(:), \f/(:).The "zero" element of 3c will be denoted by d.The contents of the paper may be outlined as follows.The first section compares measurability of functions under the strong and weak topologies of 3c.The second defines the (3c) integral and utilizes an essential lemma, first stated by Orlicz, to prove the complete additivity and absolute continuity of the integral.The linear operations from Lp to 3c and from Ï to Lp defined by integrable functions are investigated in §3, and these results are expressed in terms of real-valued kernels in §7.Approximation and convergence theorems occupy §4 and lead to §5 and the relationships between the (3c) integral and the integrals given by other definitions.A few remarks on completely continuous transformations and on differentiation account for § §6 and 8, respectively, and four examples form §9. In conclusion a few open questions are cited.

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