Stability condition and Riesz bounds for exponential splines
type: Item
instance of: scholarly article
title: Stability condition and Riesz bounds for exponential splines
zbMATH Open document ID: 1543.41007
author: Evgenia Vasiljevna Mishchenko
publication date: 2024-08-14T00:00:00Z
review text: Let \(p\in\mathbb{R}\setminus\{0\}\) and let \(m\in\mathbb{N}\). The author introduces a family of exponential splines \(U_{m,p}\) recursively as follows:\N\begin{align*}\NU_{1,p} (x) &:= \begin{cases} \frac{p e^{p x}}{e^p-1}, & x\in [0,1);\\\N0, & \text{else.} \end{cases}\\\NU_{m+1, p} (x)&:= (U_{m,p} * \chi_{[0,1)}) (x), \quad m\in \mathbb{N},\N\end{align*}\Nwhere \(*\) denotes convolution and \(\chi_{[0,1)}\) the indicator function of \([0,1)\). The last equation can be rewritten in the form\N\[\NU_{m+1, p} = Q_{m} * \varphi_p,\N\]\Nwhere \(Q_m\) denotes the cardinal B-splines of order \(m\) and \(\varphi_p := U_{1,p}\).\N\NThe stability of the integer translates of \(U_{m,p}\) for arbitrary \(m\) and \(p\) as defined above is established and the Riesz bounds computed. Furthermore, the method employed to obtain this result allows the computation of the Riesz bounds for the convolution of a cardinal B-splines \(Q_{m}\) of arbitrary order and a function with an appropriate Fourier transform.
reviewed by: Peter Massopust
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 40A30
zbMATH DE Number: 7896836
zbMATH Keywords: E-spline
zbMATH Keywords: Riesz basis
zbMATH Keywords: Riesz bounds
zbMATH Keywords: functional series
MaRDI profile type: Publication
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published in: Sibirskie Elektronnye Matematicheskie Izvestiya
DOI: 10.33048/SEMI.2023.20.088
description: scientific article; zbMATH DE number 7896836