Discrete Fourier-Laplace transforms of Lipschitz functions in the spaces \(S^{(p,q)}(\sigma^{m-1})\)
Let \(\sigma^{m-1}\subset \mathbb R^m\), \(m\ge3\), be the unit sphere and let \((L_p(\sigma^{m-1}),\|\cdot\|_p)\) be the \(L_p\) space with respect to the normalized surface measure on \(\sigma^{m-1}\). Let \(S^{(p,q)}(\sigma^{m-1})\), \(0<p,q<\infty\), be the space of \(L_1\) functions \(f\) such that the quantity \(\|f\|_{S^{(p,q)}}^q:=\sum_0^\infty \|Y^\lambda_n f\|_p^q\) is finite, where \(Y^\lambda_n\), \(\lambda=(m-2)/2\), denotes the projection operator onto the subspace of \(L_2(\sigma^{m-1})\) of spherical harmonics of degree \(n\). In the paper the main results of \textit{S. El Ouadih} and \textit{R. Daher} [Integral Transforms Spec. Funct. 31, No.~12, 1010--1019 (2020; Zbl 1476.43003)], Theorems 3.1 and 4.1 proved in the \(L_2\) setting, are generalized to Theorems 3.1 and 4.1 in the \(S^{(p,q)}\) setting, with \(p,q\) fixed. In this generalization \(L_2(\sigma^{m-1})\) and \(\|\cdot\|_2\) are replaced by \(S^{(p,q)}(\sigma^{m-1})\) and \(\|\cdot\|_{S^{(p,q)}}\), \(\mathrm{Lip}(\delta,2)\) and \(\mathrm{Lip}(\delta,\gamma,2)\) by \(\mathrm{Lip}(\delta,p,q)\) and \(\mathrm{Lip}(\delta,\gamma,p,q)\), \(L_2\) norms of projections \(Y^\lambda_nf\) by their \(L_p\) (quasi) norms, the exponent 2 by the exponent \(q\), respectively. See the review of [loc. cit.] for more comments.
scientific article; zbMATH DE number 7355471
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