Constructive Approximation最新文献

{"title":"Bounds on Orthonormal Polynomials for Restricted Measures","authors":"D. S. Lubinsky","doi":"10.1007/s00365-023-09671-z","DOIUrl":"https://doi.org/10.1007/s00365-023-09671-z","url":null,"abstract":"<p>Suppose that <span>(nu )</span> is a given positive measure on <span>(left[ -1,1right] )</span>, and that <span>(mu )</span> is another measure on the real line, whose restriction to <span>( left( -1,1right) )</span> is <span>(nu )</span>. We show that one can bound the orthonormal polynomials <span>(p_{n}left( mu ,yright) )</span> for <span>(mu )</span> and <span>(yin mathbb {R})</span>, by the supremum of <span>(left| S_{J}left( yright) p_{n-J}left( S_{J}^{2}nu ,yright) right| )</span>, where the sup is taken over all <span>(0le Jle n)</span> and all monic polynomials <span>(S_{J})</span> of degree <i>J</i> with zeros in an appropriate set.</p>","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138743301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Monotone Discretization of Anisotropic Differential Operators Using Voronoi’s First Reduction","authors":"Frédéric Bonnans, Guillaume Bonnet, Jean-Marie Mirebeau","doi":"10.1007/s00365-023-09672-y","DOIUrl":"https://doi.org/10.1007/s00365-023-09672-y","url":null,"abstract":"<p>We consider monotone discretization schemes, using adaptive finite differences on Cartesian grids, of partial differential operators depending on a strongly anisotropic symmetric positive definite matrix. For concreteness, we focus on a linear anisotropic elliptic equation, but our approach extends to divergence form or non-divergence form diffusion, and to a variety of first and second order Hamilton–Jacobi–Bellman PDEs. The design of our discretization stencils relies on a matrix decomposition technique coming from the field of lattice geometry, and related to Voronoi’s reduction of positive quadratic forms. We show that it is efficiently computable numerically, in dimension up to four, and yields sparse and compact stencils. However, some of the properties of this decomposition, related with the regularity and the local connectivity of the numerical scheme stencils, are far from optimal. We thus present fixes and variants of the decomposition that address these defects, leading to stability and convergence results for the numerical schemes.</p>","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138530721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Applications of the Lipschitz Summation Formula and a Generalization of Raabe’s Cosine Transform","authors":"Atul Dixit, Rahul Kumar","doi":"10.1007/s00365-023-09668-8","DOIUrl":"https://doi.org/10.1007/s00365-023-09668-8","url":null,"abstract":"","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135219382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Polynomial Approximation on $$C^2$$-Domains","authors":"Feng Dai, Andriy Prymak","doi":"10.1007/s00365-023-09669-7","DOIUrl":"https://doi.org/10.1007/s00365-023-09669-7","url":null,"abstract":"","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135511519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Shrinking Schauder Frames and Their Associated Bases","authors":"Kevin Beanland, Daniel Freeman","doi":"10.1007/s00365-023-09667-9","DOIUrl":"https://doi.org/10.1007/s00365-023-09667-9","url":null,"abstract":"","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135994142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stable Gabor Phase Retrieval in Gaussian Shift-Invariant Spaces via Biorthogonality","authors":"Philipp Grohs, Lukas Liehr","doi":"10.1007/s00365-023-09629-1","DOIUrl":"https://doi.org/10.1007/s00365-023-09629-1","url":null,"abstract":"Abstract We study the phase reconstruction of signals f belonging to complex Gaussian shift-invariant spaces $$V^infty (varphi )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>V</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>φ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> from spectrogram measurements $$|{mathcal {G}} f(X)|$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>G</mml:mi> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> where $${mathcal {G}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>G</mml:mi> </mml:math> is the Gabor transform and $$X subseteq {{mathbb {R}}}^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊆</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on | f | result in stability estimates in the situation where one aims to reconstruct f on compacts intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements (Grohs and Liehr in Injectivity of Gabor phase retrieval from lattice measurements. Appl. Comput. Harmon. Anal. 62, 173–193 (2023)) we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond $$V^infty (varphi )$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:msup> <mml:mi>V</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>φ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , such as Paley–Wiener spaces.","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The WCGA in $$L^p(log L)^{alpha }$$ Spaces","authors":"G. Garrigós","doi":"10.1007/s00365-023-09664-y","DOIUrl":"https://doi.org/10.1007/s00365-023-09664-y","url":null,"abstract":"","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48554265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modulated Bi-Orthogonal Polynomials on the Unit Circle: The 2j-kdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$2j-k$$","authors":"R. Gharakhloo, Nicholas S. Witte","doi":"10.1007/s00365-022-09604-2","DOIUrl":"https://doi.org/10.1007/s00365-022-09604-2","url":null,"abstract":"","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2023-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49163230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Rodrigues’ Descendants of a Polynomial and Boutroux Curves","authors":"Rikard Bøgvad, Christian Hägg, Boris Shapiro","doi":"10.1007/s00365-023-09657-x","DOIUrl":"https://doi.org/10.1007/s00365-023-09657-x","url":null,"abstract":"Abstract Motivated by the classical Rodrigues’ formula, we study below the root asymptotic of the polynomial sequence $$begin{aligned} {mathcal {R}}_{[alpha n],n,P}(z)=frac{mathop {}!textrm{d}^{[alpha n]}P^n(z)}{mathop {}!textrm{d}z^{[alpha n]}}, n= 0,1,dots end{aligned}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> <mml:mo>]</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mrow /> <mml:mspace /> <mml:msup> <mml:mtext>d</mml:mtext> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>P</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mrow /> <mml:mspace /> <mml:mtext>d</mml:mtext> <mml:msup> <mml:mi>z</mml:mi> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where P ( z ) is a fixed univariate polynomial, $$alpha $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>α</mml:mi> </mml:math> is a fixed positive number smaller than $$deg P$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>deg</mml:mo> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> , and $$[alpha n]$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> stands for the integer part of $$alpha n$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> . Our description of this asymptotic is expressed in terms of an explicit harmonic function uniquely determined by the plane rational curve emerging from the application of the saddle point method to the integral representation of the latter polynomials using Cauchy’s formula for higher derivatives. As a consequence of our method, we conclude that this curve is birationally equivalent to the zero locus of the bivariate algebraic equation satisfied by the Cauchy transform of the asymptotic root-counting measure for the latter polynomial sequence. We show that this harmonic function is also associated with an abelian differential having only purely imaginary periods and the latter plane curve belongs to the class of Boutroux curves initially introduced in Bertola (Anal Math Phys 1: 167–211","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135478842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectral Properties of Sierpinski Measures on $$mathbb {R}^n$$","authors":"X. Dai, Xiaoye Fu, Zhigang Yan","doi":"10.1007/s00365-023-09654-0","DOIUrl":"https://doi.org/10.1007/s00365-023-09654-0","url":null,"abstract":"","PeriodicalId":50621,"journal":{"name":"Constructive Approximation","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2023-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45021574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}