Sums of translates generalize logarithms of weighted algebraic polynomials. The paper presents the solution to the minimax and maximin problems on the real axis for sums of translates. We prove …
Sums of translates generalize logarithms of weighted algebraic polynomials. The paper presents the solution to the minimax and maximin problems on the real axis for sums of translates. We prove that there is a unique function that is extremal in both problems. The key in our proof is a reduction to the problem on a segment. For this, we work out an analogue of the Mhaskar-Rakhmanov-Saff theorem, too.
Following P. Fenton, we investigate sum of translates functions $F(\mathbf{x},t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to {\underline{\mathbb{R}}}:=\mathbb{R}\cup\{-\infty\}$ is a "sufficiently non-degenerate" and upper-bounded "field function", and $K:[-1,1]\to {\underline{\mathbb{R}}}$ is a fixed "kernel …
Following P. Fenton, we investigate sum of translates functions $F(\mathbf{x},t):=J(t)+\sum_{j=1}^n \nu_j K(t-x_j)$, where $J:[0,1]\to {\underline{\mathbb{R}}}:=\mathbb{R}\cup\{-\infty\}$ is a "sufficiently non-degenerate" and upper-bounded "field function", and $K:[-1,1]\to {\underline{\mathbb{R}}}$ is a fixed "kernel function", concave both on $(-1,0)$ and $(0,1)$, $\mathbf{x}:=(x_1,\ldots,x_n)$ with $0\le x_1\le\dots\le x_n\le 1$, and $\nu_1,\dots,\nu_n>0$ are fixed. We analyze the behavior of the local maxima vector $\mathbf{m}:=(m_0,m_1,\ldots,m_n)$, where $m_j:=m_j(\mathbf{x}):=\sup_{x_j\le t\le x_{j+1}} F(\mathbf{x},t)$, with $x_0:=0$, $x_{n+1}:=1$; and study the optimization (minimax and maximin) problems $\inf_{\mathbf{x}}\max_j m_j(\mathbf{x})$ and $\sup_{\mathbf{x}}\min_j m_j(\mathbf{x})$. The main result is the equality of these quantities, and provided $J$ is upper semicontinuous, the existence of extremal configurations and their description as equioscillation points $\mathbf{w}$. In our previous papers we obtained results for the case of singular kernels, i.e., when $K(0)=-\infty$ and the field $J$ was assumed to be upper semicontinuous. In this work we get rid of these assumptions and prove common generalizations of Fenton's and our previous results, arriving at the greatest generality in the setting of concave kernel functions.
For n 2 , we obtain the extremal values of the minimax problem for exponential sums μ(n) := min |x|=1 max ⎧⎨ ⎩ ∣∣∣∣∣ n−1 ∑ k=0 xk ∣∣∣∣∣ , …
For n 2 , we obtain the extremal values of the minimax problem for exponential sums μ(n) := min |x|=1 max ⎧⎨ ⎩ ∣∣∣∣∣ n−1 ∑ k=0 xk ∣∣∣∣∣ , ∣∣∣∣∣ n−1 ∑ k=0 xkn ∣∣∣∣∣ ⎫⎬ ⎭ . Moreover, we show that the polynomial with coefficients 0 and 1 derived from μ(n) does not have zeros on the unit circle. Mathematics subject classification (2000): 52C15; 05B40.
The purpose of this document is just to provide additional information about the concepts of maxima and minima in MM. This paper must simply be considered as a clarification of …
The purpose of this document is just to provide additional information about the concepts of maxima and minima in MM. This paper must simply be considered as a clarification of these notions. The notion of dynamics is also addressed. Mamba implementations of the operators are also provided.
Este trabalho apresenta uma pesquisa sobre problemas de maximos e minimos da Geometria Euclidiana. Inicialmente apresentamos alguns resultados preliminares seguidos de suas demonstracoes que em sua essencia usam conceitos basicos …
Este trabalho apresenta uma pesquisa sobre problemas de maximos e minimos da Geometria Euclidiana. Inicialmente apresentamos alguns resultados preliminares seguidos de suas demonstracoes que em sua essencia usam conceitos basicos de geometria. Em seguida apresentamos alguns problemas de maximizacao de area e de minimizacao de perimetro em triângulos e poligonos convexos, culminando com uma prova da desigualdade isoperimetrica para poligonos e comentario do caso geral. Resolvemos alguns problemas classicos de geometria que estao relacionados com valores extremos e apresentamos outros como problemas propostos.
This paper contains a number of results on the logarithmic asymptotics and the asymptotic distribution of zeros of polynomials that are orthonormal on the real axis or semiaxis with respect …
This paper contains a number of results on the logarithmic asymptotics and the asymptotic distribution of zeros of polynomials that are orthonormal on the real axis or semiaxis with respect to weight functions of the type . Bibliography: 10 titles.
It is shown that if f(z) is entire and satisfies lim log Af(/-,/)/(log r)2 = o < oo then for a sequence of r -» oo This proves a long-standing …
It is shown that if f(z) is entire and satisfies lim log Af(/-,/)/(log r)2 = o < oo then for a sequence of r -» oo This proves a long-standing conjecture of P. D. Barry.1. Introduction.Suppose that w(z) is subharmonic in the plane and let B(r, u) = max|2|_r w(z), A(r, u) = inf|z|_r u(z).It was proved by P. D. Barry [2, Theorem 8 and comments in §7.4then, on a sequence of r -» oo, A(r, u) > B(r, u) -m2a + o(l).The constant m2a is known to be best possible over the class of subharmonic functions but Barry has shown [2, pp.484-485] that a better constant is possible over the class of functions u(z) = log|/(z)|, where / is entire.He conjectured (see [2, p. 485]; also [1, p. 130]) that the right constant is log C rather than m2a, where fr il + exp(-(2/c-l)/4a)l2.C-/i1ll-exp{-(2^-l)/4a}j' (U)an example [2, p. 484] shows that this would be best possible.The aim here is to prove Barry's conjecture.Theorem.Suppose that /(z) is an entire function satisfying -log M(r,f) = a<oo (1 3) '•-»«' (log r)Then given e > 0 there are certain arbitrarily large values of r for which where C is the number (1.2).Here M(r,f) = max,z,_r|/(z)| and m{r,f) = minw_r|/(z)|.
We extend some equilibrium-type results first conjectured by Ambrus, Ball and Erdélyi, and then proved recenly by Hardin, Kendall and Saff. We work on the torus T ≃ [ 0 …
We extend some equilibrium-type results first conjectured by Ambrus, Ball and Erdélyi, and then proved recenly by Hardin, Kendall and Saff. We work on the torus T ≃ [ 0 , 2 π ) , but the motivation comes from an analogous setup on the unit interval, investigated earlier by Fenton. The problem is to minimize — with respect to the arbitrary translates y 0 = 0 , y j ∈ T , j = 1 , ⋯ , n — the maximum of the sum function F : = K 0 + ∑ j = 1 n K j ( · − y j ) , where the functions K j are certain fixed 'kernel functions'. In our setting, the function F has singularities at functions y j , while in between these nodes it still behaves regularly. So one can consider the maxima m i on each subinterval between the nodes y j , and minimize max F = max i m i . Also the dual question of maximization of min i m i arises. Hardin, Kendall and Saff considered one even kernel, K j = K for j = 0 , ⋯ , n , and Fenton considered the case of the interval [ − 1 , 1 ] with two fixed kernels K 0 = J and K j = K for j = 1 , ⋯ , n . Here we build up a systematic treatment when all the kernel functions can be different without assuming them to be even. As an application we generalize a result of Bojanov about Chebyshev-type polynomials with prescribed zero order.
It is proven that for any system of n points z_1, ..., z_n on the (complex) unit circle, there exists another point z of norm 1, such that $$\sum 1/|z-z_k|^2 …
It is proven that for any system of n points z_1, ..., z_n on the (complex) unit circle, there exists another point z of norm 1, such that $$\sum 1/|z-z_k|^2 \leq n^2/4.$$ Equality holds iff the point system is a rotated copy of the nth unit roots. Two proofs are presented: one uses a characterisation of equioscillating rational functions, while the other is based on Bernstein's inequality.
Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and …
Minimax and maximin problems are investigated for a special class of functions on the interval $[0,1]$. These functions are sums of translates of positive multiples of one kernel function and a very general external field function. Due to our very general setting the minimax, equioscillation and characterization results obtained extend those of Bojanov, Fenton, Hardin, Kendall, Saff, Ambrus, Ball and Erdélyi. Moreover, we discover a surprising intertwining phenomenon of interval maxima, which provides new information even in the most classical extremal problem of Chebyshev. Bibliography: 25 titles.