Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2020-10-24 , DOI: 10.1007/s00009-020-01626-z Marcell Gaál , Zsuzsanna Nagy-Csiha
This paper is concerned with a Delsarte-type extremal problem. Denote by \({\mathcal {P}}(G)\) the set of positive definite continuous functions on a locally compact abelian group G. We consider the function class, which was originally introduced by Gorbachev,
$$\begin{aligned}&{\mathcal {G}}(W, Q)_G = \left\{ f \in {\mathcal {P}}(G) \cap L^1(G)~:\right. \\&\qquad \qquad \qquad \qquad \qquad \left. f(0) = 1, ~ {\text {supp}}{f_+} \subseteq W,~ {\text {supp}}{\widehat{f}} \subseteq Q \right\} \end{aligned}$$where \(W\subseteq G\) is closed and of finite Haar measure and \(Q\subseteq {\widehat{G}}\) is compact. We also consider the related Delsarte-type problem of finding the extremal quantity
$$\begin{aligned} {\mathcal {D}}(W,Q)_G = \sup \left\{ \int _{G} f(g) \mathrm{d}\lambda _G(g) ~ : ~ f \in {\mathcal {G}}(W,Q)_G\right\} . \end{aligned}$$The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem \({\mathcal {D}}(W,Q)_G\). The existence of the extremal function has recently been established by Berdysheva and Révész in the most immediate case where \(G={\mathbb {R}}^d\). So, the novelty here is that we consider the problem in the general setting of locally compact abelian groups. In this way, our result provides a far reaching generalization of the former work of Berdysheva and Révész.
中文翻译：
关于Delsarte极值问题的极值函数的存在
本文涉及Delsarte型极值问题。用\（{\ mathcal {P}}（G）\）表示局部紧阿贝尔群G上的正定连续函数集。我们考虑最初由戈尔巴乔夫（Gorbachev）引入的函数类，
$$ \ begin {aligned}＆{\ mathcal {G}}（W，Q）_G = \ left \ {f \ in {\ mathcal {P}}（G）\ cap L ^ 1（G）〜：\\对。\\＆\ qquad \ qquad \ qquad \ qquad \ qquad \ left。f（0）= 1，〜{\ text {supp}} {f_ +} \ subseteq W，〜{\ text {supp}} {\ widehat {f}} \ subseteq Q \ right \} \ end {aligned} $$其中\（W \ subseteq G \）是封闭的，并且具有有限的Haar度量，而\（Q \ subseteq {\ widehat {G}} \）是紧凑的。我们还考虑了寻找极值量的相关Delsarte型问题
$$ \ begin {aligned} {\ mathcal {D}}（W，Q）_G = \ sup \ left \ {\ int _ {G} f（g）\ mathrm {d} \ lambda _G（g）〜： 〜f \ in {\ mathcal {G}}（W，Q）_G \ right \}中。\ end {aligned} $$本文的主要目的是证明Delsarte型极值问题\（{\ mathcal {D}}（W，Q）_G \）的极值函数的存在。最近，在最直接的\（G = {\ mathbb {R}} ^ d \）情况下，Berdysheva和Révész确定了极值函数的存在。因此，这里的新颖之处在于我们在局部紧凑的阿贝尔群的一般设置中考虑了这个问题。这样，我们的结果对Berdysheva和Révész的前作进行了深远的概括。
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