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Register a new userBasis properties of the p,q-sine functions
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We improve the currently known thresholds for basisness of the family of periodically dilated p,q-sine functions. Our findings rely on a Beurling decomposition of the corresponding change of coordinates in terms of shift operators of infinite multiplicity. We also determine refined bounds on the Riesz constant associated to this family. These results seal mathematical gaps in the existing literature on the subject.
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Let $D^alpha, alpha>0$, be the Vladimirov-Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity $D^alpha D^{-alpha}f=f$ was known only for the case where $f$ has a compact support. Following a result by Samko about the fractional Laplacian of real analysis, we extend the above identity in terms of $L^p$-convergence of truncated integrals. Differences between real and non-Archimedean cases are discussed.
This paper deals with the inequalities devoted to the comparison between the norm of a function on a compact hypergroup and the norm of its Fourier coefficients. We prove the classical Paley inequality in the setting of compact hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley (Pitt) inequalities in the noncommutative context. We establish Hormanders $L^p$-$L^q$ Fourier multiplier theorem on compact hypergroups for $1<p leq 2 leq q<infty$ as an application of Hausdorff-Young-Paley inequality. We examine our results for the hypergroups constructed from the conjugacy classes of compact Lie groups and for a class of countable compact hypergroups.
We prove the exponential law $mathcal A(E times F, G) cong mathcal A(E,mathcal A(F,G))$ (bornological isomorphism) for the following classes $mathcal A$ of test functions: $mathcal B$ (globally bounded derivatives), $W^{infty,p}$ (globally $p$-integrable derivatives), $mathcal S$ (Schwartz space), $mathcal D$ (compact sport, $mathcal B^{[M]}$ (globally Denjoy_Carleman), $W^{[M],p}$ (Sobolev_Denjoy_Carleman), $mathcal S_{[L]}^{[M]}$ (Gelfand_Shilov), and $mathcal D^{[M]}$. Here $E, F, G$ are convenient vector spaces (finite dimensional in the cases of $W^{infty,p}$, $mathcal D$, $W^{[M],p}$, and $mathcal D^{[M]})$, and $M=(M_k)$ is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms $operatorname{Diff} mathcal B$, $operatorname{Diff} W^{infty,p}$, $operatorname{Diff} mathcal S$, and $operatorname{Diff}mathcal D$ are $C^infty$ Lie groups, and $operatorname{Diff} mathcal B^{{M}}$, $operatorname{Diff}W^{{M},p}$, $operatorname{Diff} mathcal S_{{L}}^{{M}}$, and $operatorname{Diff}mathcal D^{[M]}$, for non-quasianalytic $M$, are $C^{{M}}$ Lie groups, where $operatorname{Diff}mathcal A = {operatorname{Id} +f : f in mathcal A(mathbb R^n,mathbb R^n), inf_{x in mathbb R^n} det(mathbb I_n+ df(x))>0}$. We also discuss stability under composition.
A Borel probability measure $mu$ on a locally compact group is called a spectral measure if there exists a subset of continuous group characters which forms an orthogonal basis of the Hilbert space $L^2(mu)$. In this paper, we characterize all spectral measures in the field $mathbb{Q}_p$ of $p$-adic numbers.
We prove the converse to a result of Karlin [Trans. AMS 1964], and also strengthen his result and two results of Schoenberg [Ann. of Math. 1955]. One of the latter results concerns zeros of Laplace transforms of multiply positive functions. The other results study which powers $alpha$ of two specific kernels are totally non-negative of order $pgeq 2$ (denoted TN$_p$); both authors showed this happens for $alphageq p-2$, and Schoenberg proved that it does not for $alpha<p-2$. We show more strongly that for every $p times p$ submatrix of either kernel, up to a shift, its $alpha$th power is totally positive of order $p$ (TP$_p$) for every $alpha > p-2$, and is not TN$_p$ for every non-integer $alphain(0,p-2)$. In particular, these results reveal critical exponent phenomena in total positivity. We also prove the converse to a 1968 result of Karlin, revealing yet another critical exponent phenomenon - for Laplace transforms of all Polya Frequency (PF) functions. We further classify the powers preserving all TN$_p$ Hankel kernels on intervals, and isolate individual kernels encoding these powers. We then transfer results on preservers by Polya-Szego (1925), Loewner/Horn (1969), and Khare-Tao (in press), from positive matrices to Hankel TN$_p$ kernels. Another application constructs individual matrices encoding the Loewner convex powers. This complements Jains results (2020) for Loewner positivity, which we strengthen to total positivity. Remarkably, these (strengthened) results of Jain, those of Schoenberg and Karlin, the latters converse, and the above Hankel kernels all arise from a single symmetric rank-two kernel and its powers: $max(1+xy,0)$. We also provide a novel characterization of PF functions and sequences of order $pgeq 3$, following Schoenbergs 1951 result for $p=2$. We correct a small gap in his paper, in the classification of discontinuous PF functions.
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