A new characterization of the bounded operators commuting with Hankel translation

Abstract

In this note we prove that a bounded operator in the Zemanian space h m commutes with the Hankel translation if, and only if, it commutes with the Bessel operator S m x ÀmÀ1a2 Dx 2m1 Dx ÀmÀ1a2 .A. H. Zemanian [12] introduced for every m P R the same space h m constituted by all those complex valued and smooth functions 0 0xY x P 0Y I such thatis finite for every mY k P N. The space h m is endowed with the topology generated by the family fg m mYk g mYkPN of seminorms. Thus h m is a Fre  chet space. In 12] it was established that the Hankel transformation h m defined by h m 0y I 0 xy 1a2 J m xy0xdxY 0 P h m and y P 0Y I Y 1 where J m denotes the Bessel function of the first kind and order m, is an automorphism of h m , provided that m^À 1 2 . D. T. Haimo [7] and I. I. Hirschman [8] investigated the Hankel convolution on L p -spaces for a variant of the Hankel transformation closely connected to (1). After straightforward manipulations, the results in [7] and [8] allow to define, for every m b À 1 2 , a convolution for the Hankel transformation h m as follows. We denote by L m the space of measurable functions 0 on 0Y I such that I 0 j0xjx m1a2 dx`I . Let 0 and y be in L m . The Hankel convolution 05y of 0 and y is defined by 05yx I 0 0yt x yydyY x P 0Y I Y

The Role of Nonpolynomiality in Uniform Approximation by RBF Networks of Hankel Translates

“…Equation (62) is a consequence of the fact that commutes with Hankel translations on H (cf. [33]). Finally, (63) derives from (23) and the identity…”