On generalized localization of Fourier inversion for distributions

In this paper we study the behavior of spherical Fourier integrals and pointwise convergence and summability of Fourier inversion. We consider generalized localization principle which in classical <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:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">L_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spaces was investigated by P.Sjölin, A.Carbery, F.Soria, and Sh.Alimov. Proceeding with these studies in this paper we establish sharp conditions for generalized localization in the class of finitely supported distributions.

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