Our principal results fall into three main classes. First, a large number of formulae from the classical theory of special functions are given appropriate generalizations. Some of these turn out …
Our principal results fall into three main classes. First, a large number of formulae from the classical theory of special functions are given appropriate generalizations. Some of these turn out to have applications to lattice-point problems and to the theory of non-central Wishart distributions in statistics. Secondly, the L2-theory of the Hankel transform is established with the generalized Bessel functions furnishing the kernel, i.e. the transformation g(A) = JM>oA y(AM)f(M) (det M)7 dM is a self-reciprocal unitary correspondence of the Hilbert space of functions for which f A>O I f(A) 12 (det A)7 dA < c* onto itself. Here Py is a real number greater than -1 and A and M are positive definite matrices. In this connection there are two results we wish to emphasize. (1) A complete set of eigenfunctions for the Hankel transform is given in the form' etr(- A)L(') (2A), t running over a certain index class, with the L() (A) as polynomials in the entries of the matrix A. These polynomials enjoy generalized versions of nearly all the properties of the Laguerre polynomials to which they reduce in the scalar case. (2) The ordinary multi-dimensional Fourier transform of a function of mk variables satisfying a certain generalized radiality condition reduces to a Hankel transform. More precisely, arrange the mk variables in a k X m matrix T; then if the function depends only on R = T'T, T' being the transposed matrix, the Fourier transform may be computed in terms of the Hankel transform of order -y = 2 (k - m - 1) defined for functions of positive semidefinite m X m matrices R. The third class of results concerns the properties of harmonic polynomials in several variables having a certain matrix homogeneity. We call a polynomial, P(T), in the entries of the k X m matrix T, an H-polynomial if (1) P(T) is a harmonic function of mk variables and (2) P(TZ) = (det Z)VP(T) for some integer v and all m X m symmetric matrices Z. These H-polynomials behave like Stieffel-manifold (in contrast to spherical) harmonics. They are related in a natural way to generalized Gegenbauer polynomials which are in turn defined as hypergeometric functions.
Let G be a locally compact group and H a closed subgroup. Then H is always a set of local spectral synthesis with respect to the algebra A p (G), …
Let G be a locally compact group and H a closed subgroup. Then H is always a set of local spectral synthesis with respect to the algebra A p (G), where A 2 (G) is the Fourier algebra in the sense of Eymard. Global synthesis holds if and only if a certain condition (C) is satisfied; it is whenever the subgroup H is amenable or normal. Global synthesis implies that each convolution operator on L p (G) with support in H which is the ultraweak limit of measures carried by H. The problem of passing from local to global synthesis is examined in an abstract context.
In the martingale context, the dual Banach space to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is …
In the martingale context, the dual Banach space to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is BMO in analogy with the result of Charles Fefferman [4] for the classical case. This theorem is an easy consequence of decomposition theorems for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-martingales which involve the notion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regulated <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-martingales where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p less-than-or-equal-to normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 > p \leq \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The strongest decomposition theorem is for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p = \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and this provides full information about BMO. The weaker <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> decomposition is fundamental in the theory of martingale transforms.
The space PF p (G) of p-pseudofunctions on a locally compact group G is the completion of L 1 (G) for the norm of convolvers of L p (G). In …
The space PF p (G) of p-pseudofunctions on a locally compact group G is the completion of L 1 (G) for the norm of convolvers of L p (G). In case the group G is amenable, the dual Banach space of PF p (G) may be identified with a certain algebra B p (G) of continuous functions on G. The algebra B p (G) is already known, but here it is shown that B p is a functor of locally compact groups. When p=2 we have that PF 2 (G) is the C * -algebra of G whose dual is FS(G), the algebra of Fourier-Stieltjes transforms. Thus for an amenable group, element of B n (G) generalizes the notion of Fourier-Stieltjes transform with which it coincides in case p=2.
The present article is intended as a survey of the title subject.The material is closely intertwined with all branches of harmonic analysis.Historically, the rigorous foundations of the theory arise in …
The present article is intended as a survey of the title subject.The material is closely intertwined with all branches of harmonic analysis.Historically, the rigorous foundations of the theory arise in Riemann's treatment of trigonometric series, and spectral theory is essentially equivalent to the study of formal multiplication.The original motivation for the modern treatment, due to Wiener, Carleman, and Beurling, came from the study of integral equations with convolution kernels.There are applications to linear partial differential equations with constant coefficients, and there is a very close connection with problems about entire functions bounded on a line.My own concern with the topic commenced with questions about the theory of approximation for multiple Fourier transforms.When I first looked at the subject the basic material did not appear very well organized, and certain elementary facts were not recognized as immediately obvious to experts in the field.Thus, in 1954, I set about to record what was known with certain useful additions.Shortly after the original version of this work was finished in 1956, the paper of Domar appeared.(An author's name in small capitals indicates a reference to the bibliography.)There was a considerable overlap, for, although the two papers were quite independent, both were heavily influenced by unpublished notes of Beurling.I have revised the paper in an attempt to suppress details which may be found elsewhere.Also, I have taken advantage of more recent work by others and myself to improve the material of the last half of the paper.After a preliminary section introducing much of the notation and basic definitions, the contents of this paper are divided into six parts.The headings are:1.The spectrum.2. The point spectrum.3. Potential theory and spectral analysis.4. The spectral synthesis problem. 5. Representations.6. Examples of spectral synthesis.
Let G be a finite-dimensional real vector space.A proper negativedefinite function defined on G is a complex-valued function \p with the property that for each t > 0 exp{-#(*)} = …
Let G be a finite-dimensional real vector space.A proper negativedefinite function defined on G is a complex-valued function \p with the property that for each t > 0 exp{-#(*)} = f exp(if*)P(J, do where P(t, •) is a (Radon) probability measure on the dual space, G'.We shall be concerned with real, homogeneous, negative-definite functions, i.e., those negative-definite functions for which there exists a positive constant a such that for each scalar X and each vector xEG we haveThe associated probability measures here correspond to the symmetric stable laws of Paul Levy [ô].It is very easy to see that one must have a ^2.We shall restrict our attention to the range l^a^2 and treat \plla instead of \\i.Let us make a definition and alter the notation slightly.Definition.Suppose 1 ^p á 2. A continuous non-negative function ip defined on G is an ¿"-norm if (i) ^(Xx) = |X|^(x) for each real X and each xEG, and (ii) \¡/p is a proper negative-definite function.The concern of this paper is what are the Lp-norms on G.In one sense the question has been answered by Levy; see Theorem 1 below.It does not appear to me, however, that the connection between normed vector spaces and symmetric stable laws is obvious from Levy's presentation, and I think the connection is an illuminating one.The central idea of the present article is that the terminology uLp-norm" is apt.It is well known (cf.[2]) that if 0 <ß^ 1 then \p? is a proper negative-definite function whenever \p is.A fortiori, an L^-norm is an Lr-norm for lúr^p.The simplest example of an L2-norm is a function \p of the form ^(x) = | £x| where t;EG'.Since sums and positive multiples of negative-definite functions are negative-definite, the Lvnorms on G form a cone which contains, in particular, functions of the type
The present article is intended as a survey of the title subject.The material is closely intertwined with all branches of harmonic analysis.Historically, the rigorous foundations of the theory arise in …
The present article is intended as a survey of the title subject.The material is closely intertwined with all branches of harmonic analysis.Historically, the rigorous foundations of the theory arise in Riemann's treatment of trigonometric series, and spectral theory is essentially equivalent to the study of formal multiplication.The original motivation for the modern treatment, due to Wiener, Carleman, and Beurling, came from the study of integral equations with convolution kernels.There are applications to linear partial differential equations with constant coefficients, and there is a very close connection with problems about entire functions bounded on a line.My own concern with the topic commenced with questions about the theory of approximation for multiple Fourier transforms.When I first looked at the subject the basic material did not appear very well organized, and certain elementary facts were not recognized as immediately obvious to experts in the field.Thus, in 1954, I set about to record what was known with certain useful additions.Shortly after the original version of this work was finished in 1956, the paper of Domar appeared.(An author's name in small capitals indicates a reference to the bibliography.)There was a considerable overlap, for, although the two papers were quite independent, both were heavily influenced by unpublished notes of Beurling.I have revised the paper in an attempt to suppress details which may be found elsewhere.Also, I have taken advantage of more recent work by others and myself to improve the material of the last half of the paper.After a preliminary section introducing much of the notation and basic definitions, the contents of this paper are divided into six parts.The headings are:1.The spectrum.2. The point spectrum.3. Potential theory and spectral analysis.4. The spectral synthesis problem. 5. Representations.6. Examples of spectral synthesis.
AccessScience is an authoritative and dynamic online resource that contains incisively written, high-quality educational material covering all major scientific disciplines. An acclaimed gateway to scientific knowledge, AccessScience is continually expanding …
AccessScience is an authoritative and dynamic online resource that contains incisively written, high-quality educational material covering all major scientific disciplines. An acclaimed gateway to scientific knowledge, AccessScience is continually expanding the ways it can demonstrate and explain core, trustworthy scientific information that inspires and guides users to deeper knowledge.
Suppose fELlfnLP. f is said to have the Wiener closure property,2 (C), if the translates of f span LP. Since fEL1, the Fourier transform 7 is well defined. Let Z(f) …
Suppose fELlfnLP. f is said to have the Wiener closure property,2 (C), if the translates of f span LP. Since fEL1, the Fourier transform 7 is well defined. Let Z(f) be the set of zeros of J. One would like to reformulate (C) in terms of structural properties of the closed set Z(f). The problem seems quite difficult; in this note we show that (C) is nearly equivalent to a uniqueness property of Z(f).3 It is assumed that the notion of the spectrum' of a bounded continuous function is familiar. DEFINITION. A closed set is of type Us if the only bounded continuous function in B with spectrum contained in the set is the null function.' We shall say that f has property (U) if Z(f) is of type U2 where 1/p+1/q= 1. Pollard, [4], has observed, what is true for any locally compact Abelian group, that
In the martingale context, the dual Banach space to H. is BMO in analogy with the result of Charles Fefferman [4] for the classical case.This theorem is an easy consequence …
In the martingale context, the dual Banach space to H. is BMO in analogy with the result of Charles Fefferman [4] for the classical case.This theorem is an easy consequence of decomposition theorems for H.-martingales which involve the notion ofL -regulated ¿.-martingales where 1 < p < oo.The strongest decomposition theorem is for p = oo, and this provides full information about BMO.The weaker p = 2 decomposition is fundamental in the theory of martingale transforms.Introduction.Shortly after Charles Fefferman [A] proved that the dual Banach space to the Hardy space H. (in this context we view Hl as the space of functions on the circle which together with their conjugate function belong to Lj) was equivalent to the space BMO of functions of bounded mean oscillation treated by John and Nirenberg [9], a martingale analogue was proved by Fefferman and Stein, A. Garsia, and me; see [5] and [6].In this Richard Gundy played a role which, at least for me, was of the highest value.There are related matters in [7] and [ll].The martingale spaces BMO are defined for 1 < p < oo by the John-Nirenberg conditions JN plus a supplementary condition which is needed to handle the case of general martingales; see §2 below for the definitions.In the classical case, that of martingales on a dyadic stochastic base, John and Nirenberg proved that, while the defining conditions for membership in BMO appear to be more stringent with increasing p, they are all equivalent.The John-Nirenberg results remain valid for general martingales; the statement is Theorem JN in §2.Although this theorem is stated right after a section of preliminaries and the basic definitions, it is one of the deepest results in the paper and much of the subsequently developed machinery is used in the proof.The duality of H. with BMO in both the classical and martingale cases is proved in two parts.The easy step is to show that every bounded linear functional
Let X denote the unit circle and LP, 1 <p < oo, the usual Lebesgue space. Given fCLP there is a harmonic function u in the unit disc with LP …
Let X denote the unit circle and LP, 1 <p < oo, the usual Lebesgue space. Given fCLP there is a harmonic function u in the unit disc with LP boundary value f. Set f*(x) =supr<, I u(r, x) j. The HardyLittlewood Maximal Theorem2 asserts that 'there exists a constant BP such that jff*jfp<Bpjjfjjp. A similar theorem is given in higher dimensions by H. E. Rauch [2] and K. T. Smith [3] where X is now the unit sphere in n-space. These results are obtained by first proving a maximal ergodic theorem and then passing over to the maximal theorem. The purpose of this note is to remark that the maximal theorem is a trivial deduction from a maximal ergodic theorem which is itself completely standard, so that, in effect, there is very little to prove. Before presenting the general procedure, I give an example which illustrates everything. Let X be the real line and take fELP. The harmonic function in the upper half plane with boundary values f is
It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G 2 as groups of automorphisms of the Cayley algebras. There is, however, a …
It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G 2 as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let W be a complex 7-dimensional vector space. Among the alternating 3-forms on W there is a connected dense open subset Ψ( W ) of “maximal” forms. If ψ ∈ Ψ( W ) then the subgroup of AUT C ( W ) consisting of the invertible complex-linear transformations S such that ψ ( S•, S•, S• ) = ψ (•, •, •) is denoted G ( ψ ), and, in Proposition 3.6. we prove where G 1 ( ψ ) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra of type G 2 is defined in terms of the alternating 3-form ψ alone without the need to specify an invariant quadratic form. In the real case the result is more striking.
We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods.If/£71, the Fourier transform is J(t)=fexp ( -itx)f(x)dx and …
We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods.If/£71, the Fourier transform is J(t)=fexp ( -itx)f(x)dx and Z(f) denotes the set of zeros of/.Given (/>£T,°° we may form the convolution focp(x) =ff(y)<f>(x -y)dy.The spectrum of <p is defined by h.(<p) =C\Z(f) where the intersection is taken
We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold.
We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold.
We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold.
We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold.
We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods.If/£71, the Fourier transform is J(t)=fexp ( -itx)f(x)dx and …
We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods.If/£71, the Fourier transform is J(t)=fexp ( -itx)f(x)dx and Z(f) denotes the set of zeros of/.Given (/>£T,°° we may form the convolution focp(x) =ff(y)<f>(x -y)dy.The spectrum of <p is defined by h.(<p) =C\Z(f) where the intersection is taken
Abstract The action of Lie groups as transitive groups of restricted contact transformations of compact manifolds are classified.
Abstract The action of Lie groups as transitive groups of restricted contact transformations of compact manifolds are classified.
Abstract If a Lie group acts faithfully as a transitive group of contact transformations of a compact manifold it is either compact with centre of dimension at most 1 or …
Abstract If a Lie group acts faithfully as a transitive group of contact transformations of a compact manifold it is either compact with centre of dimension at most 1 or non-compact simple. The latter case is described
Let Pt be the fundamental solution of the heat equation on a symmetric space of noncompact type with pole at o. We investigate the estimate for Pt(o) as t → …
Let Pt be the fundamental solution of the heat equation on a symmetric space of noncompact type with pole at o. We investigate the estimate for Pt(o) as t → ∞, which results from a formula of Flensted-Jensen.
Let Pt be the fundamental solution of the heat equation on a symmetric space of noncompact type with pole at o. We investigate the estimate for Pt(o) as t → …
Let Pt be the fundamental solution of the heat equation on a symmetric space of noncompact type with pole at o. We investigate the estimate for Pt(o) as t → ∞, which results from a formula of Flensted-Jensen.
AccessScience is an authoritative and dynamic online resource that contains incisively written, high-quality educational material covering all major scientific disciplines. An acclaimed gateway to scientific knowledge, AccessScience is continually expanding …
AccessScience is an authoritative and dynamic online resource that contains incisively written, high-quality educational material covering all major scientific disciplines. An acclaimed gateway to scientific knowledge, AccessScience is continually expanding the ways it can demonstrate and explain core, trustworthy scientific information that inspires and guides users to deeper knowledge.
Abstract If a Lie group acts faithfully as a transitive group of contact transformations of a compact manifold it is either compact with centre of dimension at most 1 or …
Abstract If a Lie group acts faithfully as a transitive group of contact transformations of a compact manifold it is either compact with centre of dimension at most 1 or non-compact simple. The latter case is described
We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold.
We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold.
We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold.
We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold.
Abstract The action of Lie groups as transitive groups of restricted contact transformations of compact manifolds are classified.
Abstract The action of Lie groups as transitive groups of restricted contact transformations of compact manifolds are classified.
It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G 2 as groups of automorphisms of the Cayley algebras. There is, however, a …
It is now customary to give concrete descriptions of the exceptional simple Lie groups of type G 2 as groups of automorphisms of the Cayley algebras. There is, however, a more elementary description. Let W be a complex 7-dimensional vector space. Among the alternating 3-forms on W there is a connected dense open subset Ψ( W ) of “maximal” forms. If ψ ∈ Ψ( W ) then the subgroup of AUT C ( W ) consisting of the invertible complex-linear transformations S such that ψ ( S•, S•, S• ) = ψ (•, •, •) is denoted G ( ψ ), and, in Proposition 3.6. we prove where G 1 ( ψ ) is identified with the exceptional simple complex Lie group of dimension 14. Thus the complex Lie algebra of type G 2 is defined in terms of the alternating 3-form ψ alone without the need to specify an invariant quadratic form. In the real case the result is more striking.
In the martingale context, the dual Banach space to H. is BMO in analogy with the result of Charles Fefferman [4] for the classical case.This theorem is an easy consequence …
In the martingale context, the dual Banach space to H. is BMO in analogy with the result of Charles Fefferman [4] for the classical case.This theorem is an easy consequence of decomposition theorems for H.-martingales which involve the notion ofL -regulated ¿.-martingales where 1 < p < oo.The strongest decomposition theorem is for p = oo, and this provides full information about BMO.The weaker p = 2 decomposition is fundamental in the theory of martingale transforms.Introduction.Shortly after Charles Fefferman [A] proved that the dual Banach space to the Hardy space H. (in this context we view Hl as the space of functions on the circle which together with their conjugate function belong to Lj) was equivalent to the space BMO of functions of bounded mean oscillation treated by John and Nirenberg [9], a martingale analogue was proved by Fefferman and Stein, A. Garsia, and me; see [5] and [6].In this Richard Gundy played a role which, at least for me, was of the highest value.There are related matters in [7] and [ll].The martingale spaces BMO are defined for 1 < p < oo by the John-Nirenberg conditions JN plus a supplementary condition which is needed to handle the case of general martingales; see §2 below for the definitions.In the classical case, that of martingales on a dyadic stochastic base, John and Nirenberg proved that, while the defining conditions for membership in BMO appear to be more stringent with increasing p, they are all equivalent.The John-Nirenberg results remain valid for general martingales; the statement is Theorem JN in §2.Although this theorem is stated right after a section of preliminaries and the basic definitions, it is one of the deepest results in the paper and much of the subsequently developed machinery is used in the proof.The duality of H. with BMO in both the classical and martingale cases is proved in two parts.The easy step is to show that every bounded linear functional
The space PF p (G) of p-pseudofunctions on a locally compact group G is the completion of L 1 (G) for the norm of convolvers of L p (G). In …
The space PF p (G) of p-pseudofunctions on a locally compact group G is the completion of L 1 (G) for the norm of convolvers of L p (G). In case the group G is amenable, the dual Banach space of PF p (G) may be identified with a certain algebra B p (G) of continuous functions on G. The algebra B p (G) is already known, but here it is shown that B p is a functor of locally compact groups. When p=2 we have that PF 2 (G) is the C * -algebra of G whose dual is FS(G), the algebra of Fourier-Stieltjes transforms. Thus for an amenable group, element of B n (G) generalizes the notion of Fourier-Stieltjes transform with which it coincides in case p=2.
In the martingale context, the dual Banach space to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is …
In the martingale context, the dual Banach space to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is BMO in analogy with the result of Charles Fefferman [4] for the classical case. This theorem is an easy consequence of decomposition theorems for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-martingales which involve the notion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regulated <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-martingales where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p less-than-or-equal-to normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 > p \leq \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The strongest decomposition theorem is for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p = \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and this provides full information about BMO. The weaker <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> decomposition is fundamental in the theory of martingale transforms.
Let G be a locally compact group and H a closed subgroup. Then H is always a set of local spectral synthesis with respect to the algebra A p (G), …
Let G be a locally compact group and H a closed subgroup. Then H is always a set of local spectral synthesis with respect to the algebra A p (G), where A 2 (G) is the Fourier algebra in the sense of Eymard. Global synthesis holds if and only if a certain condition (C) is satisfied; it is whenever the subgroup H is amenable or normal. Global synthesis implies that each convolution operator on L p (G) with support in H which is the ultraweak limit of measures carried by H. The problem of passing from local to global synthesis is examined in an abstract context.
Let G be a finite-dimensional real vector space.A proper negativedefinite function defined on G is a complex-valued function \p with the property that for each t > 0 exp{-#(*)} = …
Let G be a finite-dimensional real vector space.A proper negativedefinite function defined on G is a complex-valued function \p with the property that for each t > 0 exp{-#(*)} = f exp(if*)P(J, do where P(t, •) is a (Radon) probability measure on the dual space, G'.We shall be concerned with real, homogeneous, negative-definite functions, i.e., those negative-definite functions for which there exists a positive constant a such that for each scalar X and each vector xEG we haveThe associated probability measures here correspond to the symmetric stable laws of Paul Levy [ô].It is very easy to see that one must have a ^2.We shall restrict our attention to the range l^a^2 and treat \plla instead of \\i.Let us make a definition and alter the notation slightly.Definition.Suppose 1 ^p á 2. A continuous non-negative function ip defined on G is an ¿"-norm if (i) ^(Xx) = |X|^(x) for each real X and each xEG, and (ii) \¡/p is a proper negative-definite function.The concern of this paper is what are the Lp-norms on G.In one sense the question has been answered by Levy; see Theorem 1 below.It does not appear to me, however, that the connection between normed vector spaces and symmetric stable laws is obvious from Levy's presentation, and I think the connection is an illuminating one.The central idea of the present article is that the terminology uLp-norm" is apt.It is well known (cf.[2]) that if 0 <ß^ 1 then \p? is a proper negative-definite function whenever \p is.A fortiori, an L^-norm is an Lr-norm for lúr^p.The simplest example of an L2-norm is a function \p of the form ^(x) = | £x| where t;EG'.Since sums and positive multiples of negative-definite functions are negative-definite, the Lvnorms on G form a cone which contains, in particular, functions of the type
Let X denote the unit circle and LP, 1 <p < oo, the usual Lebesgue space. Given fCLP there is a harmonic function u in the unit disc with LP …
Let X denote the unit circle and LP, 1 <p < oo, the usual Lebesgue space. Given fCLP there is a harmonic function u in the unit disc with LP boundary value f. Set f*(x) =supr<, I u(r, x) j. The HardyLittlewood Maximal Theorem2 asserts that 'there exists a constant BP such that jff*jfp<Bpjjfjjp. A similar theorem is given in higher dimensions by H. E. Rauch [2] and K. T. Smith [3] where X is now the unit sphere in n-space. These results are obtained by first proving a maximal ergodic theorem and then passing over to the maximal theorem. The purpose of this note is to remark that the maximal theorem is a trivial deduction from a maximal ergodic theorem which is itself completely standard, so that, in effect, there is very little to prove. Before presenting the general procedure, I give an example which illustrates everything. Let X be the real line and take fELP. The harmonic function in the upper half plane with boundary values f is
The present article is intended as a survey of the title subject.The material is closely intertwined with all branches of harmonic analysis.Historically, the rigorous foundations of the theory arise in …
The present article is intended as a survey of the title subject.The material is closely intertwined with all branches of harmonic analysis.Historically, the rigorous foundations of the theory arise in Riemann's treatment of trigonometric series, and spectral theory is essentially equivalent to the study of formal multiplication.The original motivation for the modern treatment, due to Wiener, Carleman, and Beurling, came from the study of integral equations with convolution kernels.There are applications to linear partial differential equations with constant coefficients, and there is a very close connection with problems about entire functions bounded on a line.My own concern with the topic commenced with questions about the theory of approximation for multiple Fourier transforms.When I first looked at the subject the basic material did not appear very well organized, and certain elementary facts were not recognized as immediately obvious to experts in the field.Thus, in 1954, I set about to record what was known with certain useful additions.Shortly after the original version of this work was finished in 1956, the paper of Domar appeared.(An author's name in small capitals indicates a reference to the bibliography.)There was a considerable overlap, for, although the two papers were quite independent, both were heavily influenced by unpublished notes of Beurling.I have revised the paper in an attempt to suppress details which may be found elsewhere.Also, I have taken advantage of more recent work by others and myself to improve the material of the last half of the paper.After a preliminary section introducing much of the notation and basic definitions, the contents of this paper are divided into six parts.The headings are:1.The spectrum.2. The point spectrum.3. Potential theory and spectral analysis.4. The spectral synthesis problem. 5. Representations.6. Examples of spectral synthesis.
The present article is intended as a survey of the title subject.The material is closely intertwined with all branches of harmonic analysis.Historically, the rigorous foundations of the theory arise in …
The present article is intended as a survey of the title subject.The material is closely intertwined with all branches of harmonic analysis.Historically, the rigorous foundations of the theory arise in Riemann's treatment of trigonometric series, and spectral theory is essentially equivalent to the study of formal multiplication.The original motivation for the modern treatment, due to Wiener, Carleman, and Beurling, came from the study of integral equations with convolution kernels.There are applications to linear partial differential equations with constant coefficients, and there is a very close connection with problems about entire functions bounded on a line.My own concern with the topic commenced with questions about the theory of approximation for multiple Fourier transforms.When I first looked at the subject the basic material did not appear very well organized, and certain elementary facts were not recognized as immediately obvious to experts in the field.Thus, in 1954, I set about to record what was known with certain useful additions.Shortly after the original version of this work was finished in 1956, the paper of Domar appeared.(An author's name in small capitals indicates a reference to the bibliography.)There was a considerable overlap, for, although the two papers were quite independent, both were heavily influenced by unpublished notes of Beurling.I have revised the paper in an attempt to suppress details which may be found elsewhere.Also, I have taken advantage of more recent work by others and myself to improve the material of the last half of the paper.After a preliminary section introducing much of the notation and basic definitions, the contents of this paper are divided into six parts.The headings are:1.The spectrum.2. The point spectrum.3. Potential theory and spectral analysis.4. The spectral synthesis problem. 5. Representations.6. Examples of spectral synthesis.
We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods.If/£71, the Fourier transform is J(t)=fexp ( -itx)f(x)dx and …
We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods.If/£71, the Fourier transform is J(t)=fexp ( -itx)f(x)dx and Z(f) denotes the set of zeros of/.Given (/>£T,°° we may form the convolution focp(x) =ff(y)<f>(x -y)dy.The spectrum of <p is defined by h.(<p) =C\Z(f) where the intersection is taken
Suppose fELlfnLP. f is said to have the Wiener closure property,2 (C), if the translates of f span LP. Since fEL1, the Fourier transform 7 is well defined. Let Z(f) …
Suppose fELlfnLP. f is said to have the Wiener closure property,2 (C), if the translates of f span LP. Since fEL1, the Fourier transform 7 is well defined. Let Z(f) be the set of zeros of J. One would like to reformulate (C) in terms of structural properties of the closed set Z(f). The problem seems quite difficult; in this note we show that (C) is nearly equivalent to a uniqueness property of Z(f).3 It is assumed that the notion of the spectrum' of a bounded continuous function is familiar. DEFINITION. A closed set is of type Us if the only bounded continuous function in B with spectrum contained in the set is the null function.' We shall say that f has property (U) if Z(f) is of type U2 where 1/p+1/q= 1. Pollard, [4], has observed, what is true for any locally compact Abelian group, that
We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods.If/£71, the Fourier transform is J(t)=fexp ( -itx)f(x)dx and …
We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods.If/£71, the Fourier transform is J(t)=fexp ( -itx)f(x)dx and Z(f) denotes the set of zeros of/.Given (/>£T,°° we may form the convolution focp(x) =ff(y)<f>(x -y)dy.The spectrum of <p is defined by h.(<p) =C\Z(f) where the intersection is taken
Our principal results fall into three main classes. First, a large number of formulae from the classical theory of special functions are given appropriate generalizations. Some of these turn out …
Our principal results fall into three main classes. First, a large number of formulae from the classical theory of special functions are given appropriate generalizations. Some of these turn out to have applications to lattice-point problems and to the theory of non-central Wishart distributions in statistics. Secondly, the L2-theory of the Hankel transform is established with the generalized Bessel functions furnishing the kernel, i.e. the transformation g(A) = JM>oA y(AM)f(M) (det M)7 dM is a self-reciprocal unitary correspondence of the Hilbert space of functions for which f A>O I f(A) 12 (det A)7 dA < c* onto itself. Here Py is a real number greater than -1 and A and M are positive definite matrices. In this connection there are two results we wish to emphasize. (1) A complete set of eigenfunctions for the Hankel transform is given in the form' etr(- A)L(') (2A), t running over a certain index class, with the L() (A) as polynomials in the entries of the matrix A. These polynomials enjoy generalized versions of nearly all the properties of the Laguerre polynomials to which they reduce in the scalar case. (2) The ordinary multi-dimensional Fourier transform of a function of mk variables satisfying a certain generalized radiality condition reduces to a Hankel transform. More precisely, arrange the mk variables in a k X m matrix T; then if the function depends only on R = T'T, T' being the transposed matrix, the Fourier transform may be computed in terms of the Hankel transform of order -y = 2 (k - m - 1) defined for functions of positive semidefinite m X m matrices R. The third class of results concerns the properties of harmonic polynomials in several variables having a certain matrix homogeneity. We call a polynomial, P(T), in the entries of the k X m matrix T, an H-polynomial if (1) P(T) is a harmonic function of mk variables and (2) P(TZ) = (det Z)VP(T) for some integer v and all m X m symmetric matrices Z. These H-polynomials behave like Stieffel-manifold (in contrast to spherical) harmonics. They are related in a natural way to generalized Gegenbauer polynomials which are in turn defined as hypergeometric functions.
Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact …
Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.
Here (P is any nondecreasing absolutely continuous function satisfying a growth condition; the choice of c and C depend only on the rate of growth of (P.Finally, the assumptions of …
Here (P is any nondecreasing absolutely continuous function satisfying a growth condition; the choice of c and C depend only on the rate of growth of (P.Finally, the assumptions of most of our theorems cannot be substantially weakened.
is pertinent.The "group ring" considered in [30] consists of the present group algebra modified, in case it does not contain an identity, by the adjunction of an identity.
is pertinent.The "group ring" considered in [30] consists of the present group algebra modified, in case it does not contain an identity, by the adjunction of an identity.
Let $u$ be harmonic in the upper half-plane and $0 < p < \infty$. Then $u = \text {Re} F$ for some analytic function $F$ of the Hardy class ${H^p}$ …
Let $u$ be harmonic in the upper half-plane and $0 < p < \infty$. Then $u = \text {Re} F$ for some analytic function $F$ of the Hardy class ${H^p}$ if and only if the nontangential maximal function of $u$ is in ${L^p}$. A general integral inequality between the nontangential maximal function of $u$ and that of its conjugate function is established.
We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods.If/£71, the Fourier transform is J(t)=fexp ( -itx)f(x)dx and …
We are concerned with analyzing the spectrum of a bounded measurable function on the real line by means of certain summability methods.If/£71, the Fourier transform is J(t)=fexp ( -itx)f(x)dx and Z(f) denotes the set of zeros of/.Given (/>£T,°° we may form the convolution focp(x) =ff(y)<f>(x -y)dy.The spectrum of <p is defined by h.(<p) =C\Z(f) where the intersection is taken
is pertinent.The "group ring" considered in [30] consists of the present group algebra modified, in case it does not contain an identity, by the adjunction of an identity.
is pertinent.The "group ring" considered in [30] consists of the present group algebra modified, in case it does not contain an identity, by the adjunction of an identity.
This paper grew out of discussions with S. Bochner.It would be hard now to disengage his contributions from mine.We shall characterize those families (pt)o<t<«, of finite positive measures on a …
This paper grew out of discussions with S. Bochner.It would be hard now to disengage his contributions from mine.We shall characterize those families (pt)o<t<«, of finite positive measures on a Lie group Q which are weakly continuous and form a semi-group under convolution.No generality is lost in assuming that the pt are probability measures.Paul Levy [3] obtained a characterization when Q is 'R., the additive group of reals: The characteristic function <pt(^)=fcii"pt(d<r) has the form (1) <pt(t) = exp iitai, -lb? + I f (e* -1-*-^JG(do-)l where b is non-negative and G is a positive measure on <r\ such that G(0) =0 and f<r2G(da) is finite.Let us translate this result.The definition Stf-(t) = ff(r+a)pt(da) yields a semi-group (St) of transformations which are defined at least for bounded continuous functions.It is easy to prove that the limit Mf-(r) = lim-[<>,/• (r)-/(r)] t\o t(2) = af (r) + bf"(r) + J [f(r + *)-f(r) -^-jG(d<r) exists if/ is the Fourier transform of a function vanishing rapidly at infinity.Moreover, M determines the family (pt).This second form extends to a semi-group (pt) on a Lie group Q.Let us first assume(3) lim pt(E) = 1, E a neighborhood of t,(here e is the neutral element).Then the pt define a semi-group (St) of transformations on a Banach space J of functions with the property that S(f->f as / decreases to 0. Thus (St) has an infinitesimal generator; one would expect it to have an expression like (2).
This paper is concerned with the theory of ideals in the algebra L1 of integrable functions on a locally compact abelian group.After some preliminaries an analytical proof is given of …
This paper is concerned with the theory of ideals in the algebra L1 of integrable functions on a locally compact abelian group.After some preliminaries an analytical proof is given of the known theorem that an analytic function of a Fourier transform represents again a Fourier transform (p.406).Then, in part I, the continuous homomorphisms of closed ideals I of L1 upon C, the field of complex numbers, are studied.Any such homomorphism is given by a Fourier transform and, if I o is its kernel, the quotient-algebra I/Io, normed in the usual way, is not only algebraically isomorphic, but also isometric with C (Theorem 1.2).Another result states that homomorphic groups have homomorphic L'-algebras and that a corresponding property of isometry holds (Theorem 1.3).In part II, which may be read independently of part I, a theorem of S. Mandelbrojt and S. Agmon, which generalizes Wiener's theorem on the translates of a function in Ll, is extended to groups (Theorem 2.2).Several generalizations of Wiener's classical theorem have been published in the past few years; references to the literature are given on p. 422.The rest of part II is devoted to some applications (pp.422-425).In conclusion it should be said that the work is carried out in abstract generality, with the methods, and in the spirit, of analysis, which is then applied to algebra.
Introduction Quasi-analytic functions Szasz's theorem Certain integral expansions A class of singular integral equations Entire functions of the exponential type The closure of sets of complex exponential functions Non-harmonic Fourier …
Introduction Quasi-analytic functions Szasz's theorem Certain integral expansions A class of singular integral equations Entire functions of the exponential type The closure of sets of complex exponential functions Non-harmonic Fourier series and a gap theorem Generalized harmonic analysis in the complex domain The harmonic analysis of random functions Bibliography Index.
Let G be a locally compact group and H a closed subgroup. Then H is always a set of local spectral synthesis with respect to the algebra A p (G), …
Let G be a locally compact group and H a closed subgroup. Then H is always a set of local spectral synthesis with respect to the algebra A p (G), where A 2 (G) is the Fourier algebra in the sense of Eymard. Global synthesis holds if and only if a certain condition (C) is satisfied; it is whenever the subgroup H is amenable or normal. Global synthesis implies that each convolution operator on L p (G) with support in H which is the ultraweak limit of measures carried by H. The problem of passing from local to global synthesis is examined in an abstract context.
for some bounded function h on the boundary of the disc. The function h(z) determines a function h(g) on G by setting h(g) = h(g(O)). If h(z) is harmonic, it …
for some bounded function h on the boundary of the disc. The function h(z) determines a function h(g) on G by setting h(g) = h(g(O)). If h(z) is harmonic, it may be shown that h(g) is annihilated by a certain class of differential operators on G. The Poisson formula (1) may be used to express h(g), and we find that here it takes on a particularly simple form. Namely, if we denote by m the normalized Lebesgue measure on {j z I = 1}, and by gm, the transform of this measure by the group element g E G, then it can be seen that (1) becomes
In the martingale context, the dual Banach space to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is …
In the martingale context, the dual Banach space to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is BMO in analogy with the result of Charles Fefferman [4] for the classical case. This theorem is an easy consequence of decomposition theorems for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{H_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-martingales which involve the notion of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regulated <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-martingales where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 greater-than p less-than-or-equal-to normal infinity"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 > p \leq \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The strongest decomposition theorem is for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p = \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and this provides full information about BMO. The weaker <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> decomposition is fundamental in the theory of martingale transforms.
Abstract A revised and expanded second edition of Reiter's classic text Classical Harmonic Analysis and Locally Compact Groups (Clarendon Press 1968). It deals with various developments in analysis centring around …
Abstract A revised and expanded second edition of Reiter's classic text Classical Harmonic Analysis and Locally Compact Groups (Clarendon Press 1968). It deals with various developments in analysis centring around the fundamental work of Wiener, Carleman, and especially A. Weil. It starts with the classical theory of Fourier transforms in euclidean space, continues with a study of certain general function algebras, and then discusses functions defined on locally compact groups. The aim is, firstly, to bring out clearly the relations between classical analysis and group theory , and secondly, to study basic properties of functions on abelian and non-abelian groups. The book gives a systematic introduction to these topics and endeavours to provide tools for further research. In the new edition relevant material is added that was not yet available at the time of the first edition.
This paper grew out of discussions with S. Bochner.It would be hard now to disengage his contributions from mine.We shall characterize those families (pt)o<t<«, of finite positive measures on a …
This paper grew out of discussions with S. Bochner.It would be hard now to disengage his contributions from mine.We shall characterize those families (pt)o<t<«, of finite positive measures on a Lie group Q which are weakly continuous and form a semi-group under convolution.No generality is lost in assuming that the pt are probability measures.Paul Levy [3] obtained a characterization when Q is 'R., the additive group of reals: The characteristic function <pt(^)=fcii"pt(d<r) has the form (1) <pt(t) = exp iitai, -lb? + I f (e* -1-*-^JG(do-)l where b is non-negative and G is a positive measure on <r\ such that G(0) =0 and f<r2G(da) is finite.Let us translate this result.The definition Stf-(t) = ff(r+a)pt(da) yields a semi-group (St) of transformations which are defined at least for bounded continuous functions.It is easy to prove that the limit Mf-(r) = lim-[<>,/• (r)-/(r)] t\o t(2) = af (r) + bf"(r) + J [f(r + *)-f(r) -^-jG(d<r) exists if/ is the Fourier transform of a function vanishing rapidly at infinity.Moreover, M determines the family (pt).This second form extends to a semi-group (pt) on a Lie group Q.Let us first assume(3) lim pt(E) = 1, E a neighborhood of t,(here e is the neutral element).Then the pt define a semi-group (St) of transformations on a Banach space J of functions with the property that S(f->f as / decreases to 0. Thus (St) has an infinitesimal generator; one would expect it to have an expression like (2).
Le thdor~me que nous allons 4tablir dans cette Note fair pattie d'une thdorie spectrale encore inpubli4e, et sa d4monstration primordiale reposait sur l'usage de la thdorie gdn6rale des int~grales de …
Le thdor~me que nous allons 4tablir dans cette Note fair pattie d'une thdorie spectrale encore inpubli4e, et sa d4monstration primordiale reposait sur l'usage de la thdorie gdn6rale des int~grales de Fourier.M6me si le thdor~me en question appartient ~ cette branche de l'Analyse, on peut l'6tablir par des mdthodes dldmentaires et c'est pour cette raison-lg que nous le publions ici sdpardment avec quelques applications immddiates. Convergence non uniforme des fonetions born~es et continuessur un ensemble ouvert.Soit 0 un ensemble ouvert de points, et ~, ainsi que Tj; T2, ... des fonctions borndes et continues sur O. En ddsignant d'une maniSre gdn~rale par I1~11 la borne sup4rieure du module ]~] sur O, lu convergence uniforme de la suite T, vers ~p se traduit par la condition lim II~v-9onll = o.?1=Qo Nous allons dans cette recherche employer une notion particuli~re de convergence, appellde dans la suite convergence ~troite et ddfinie ainsi: la suite ~,~ sera dire 6troitement convergente sur O, si 1 ~ elie converge uniformdment sur tout ensemble fermd indus dans O, 2 ~ lim II {P,, II = II ~P II < c~ o~ ~ ddsigne la fonetion limite de la suite.n~c~
The present paper, an edited excerpt from my dissertation, arose from the suggestion of S. Bochner that I try to extend the maximal theorem of Hardy and Littlewood (2) to …
The present paper, an edited excerpt from my dissertation, arose from the suggestion of S. Bochner that I try to extend the maximal theorem of Hardy and Littlewood (2) to functions analytic in the solid unit hypersphere
1. Introduction. A well-known inequality of Hardy-Littlewood reads as follows (4): if p > 1 and f > 0, then , where is defined as the supremum of the numbers …
1. Introduction. A well-known inequality of Hardy-Littlewood reads as follows (4): if p > 1 and f > 0, then , where is defined as the supremum of the numbers the constant depends on p only. The statement obtained by putting p = 1 is false; its substitute reads: the constants depend on p but not on f.
Introduction. In this paper we shall be concerned with two problems. The first is a problem of Littlewood [1] in classical Fourier analysis concerning a lower bound for the LI …
Introduction. In this paper we shall be concerned with two problems. The first is a problem of Littlewood [1] in classical Fourier analysis concerning a lower bound for the LI norm of certain exponential sums. The second is the problem of determining all the idempotent measures on a locally compact abelian group. This second problem we shall solve entirely and thus complete a line of investigation begun by ilelson [3] and Rudin [5]. The problem of idempotent measures is related to the question of describing all homomorphisms of the algebra l' (G) into the algebra L' (H) where G and H are two locally compact abelian groups. We shall treat this problem in a subsequent paper. As will be explained, the problem of Littlewood is closely connected to the problem of idempotent measures, and though the first is stated on the circle group the method of proof will be indispensable for the analysis on the more general class of abelian groups. We now state the problem of Littlewood. Consider an exponential sum
Abstract The action of Lie groups as transitive groups of restricted contact transformations of compact manifolds are classified.
Abstract The action of Lie groups as transitive groups of restricted contact transformations of compact manifolds are classified.