On the existence of an extremal function for the Delsarte extremal problem

Abstract In the general setting of a locally compact Abelian group G , the Delsarte extremal problem asks for the supremum of integrals over the collection of continuous positive definite functions $$f \colon G \to \mathbb{R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>G</mml:mi> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> satisfying $$f(0) = 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and having $$supp f_{+} \subset \Omega$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mi>u</mml:mi> <mml:mi>p</mml:mi> <mml:mi>p</mml:mi> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> for some measurable subset $$\Omega$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> of finite measure. In this paper, we consider the question of the existence of an extremal function for the Delsarte extremal problem. In particular, we show that there exists an extremal function for the Delsarte problem when $$\Omega$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> is closed, extending previously known existence results to a larger class of functions.