On the Existence of an Extremal Function in the Delsarte Extremal Problem
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On the Existence of an Extremal Function in the Delsarte Extremal Problem Marcell Ga´al and Zsuzsanna Nagy-Csiha Abstract. This paper is concerned with a Delsarte-type extremal problem. Denote by 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, G(W, Q)G = f ∈ P(G) ∩ L1 (G) : f (0) = 1, supp f+ ⊆ W, supp f ⊆ Q is where W ⊆ G is closed and of finite Haar measure and Q ⊆ G compact. We also consider the related Delsarte-type problem of finding the extremal quantity f (g)dλG (g) : f ∈ G(W, Q)G . D(W, Q)G = sup G
The main objective of the current paper is to prove the existence of an extremal function for the Delsarte-type extremal problem D(W, Q)G . The existence of the extremal function has recently been established by Berdysheva and R´ev´esz in the most immediate case where G = Rd . 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 R´ev´esz. Mathematics Subject Classification. Primary 43A35, 43A40; Secondary 43A25, 43A70. Keywords. LCA groups, fourier transform, positive definite functions, Delsarte’s extremal problem.
1. Introduction The Fourier analytic formulation of the so-called Delsarte extremal problem on Rd incorporates the calculation of the numerical quantity 1 sup f (0) = sup f (x)dx, d (2π) 2 Rd 0123456789().: V,-vol
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provided that (i) (ii) (iii) (iv)
f ∈ L1 (Rd ), f is continuous and bounded on Rd , f (0) = 1, f (x) ≤ 0 for x ≥ 2 and f(y) ≥ 0. The last property (iv) can be interpreted as f being positive definite; see the precise definition of positive definiteness in the forthcoming section.
The Delsarte extremal problem has generated broad interest because of its intimate connections to different problems from various branches of mathematics. First of all, the linear programming bound of Delsarte is useful in coding and design theory as well. Second, let us mention that relying on Delsarte’s problem, upper bounds can be derived for the sphere packing density of Rd [2,8,17,23,24]. Moreover, Gorbachev and Tikhonov [10] worked out a further concrete application of the Delsarte problem for the so-called Wiener problem. A few of years ago, Viazovska [22] solved the sphere packing problem in dimension 8, combining the Delsarte extremal problem with modular form techniques. Subsequently, in the paper [4], Cohn et al. resolved the problem also in dimension 24. Besides solving the Delsarte problem, further challenging and closely related questions come into picture. As for recent investigations in this direction, we refer to the seminal paper of Berdysheva and R´ev´esz [1]. They have pointed out the independence of the extremal constant from the underlying function class. Furthermore, they showed the existence of an extremal function in band-limited cases. The main ob
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