Pub Date : 2023-12-18DOI: 10.1007/s00365-023-09671-z
D. S. Lubinsky
Suppose that (nu ) is a given positive measure on (left[ -1,1right] ), and that (mu ) is another measure on the real line, whose restriction to ( left( -1,1right) ) is (nu ). We show that one can bound the orthonormal polynomials (p_{n}left( mu ,yright) ) for (mu ) and (yin mathbb {R}), by the supremum of (left| S_{J}left( yright) p_{n-J}left( S_{J}^{2}nu ,yright) right| ), where the sup is taken over all (0le Jle n) and all monic polynomials (S_{J}) of degree J with zeros in an appropriate set.
{"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}
Pub Date : 2023-12-01DOI: 10.1007/s00365-023-09672-y
Frédéric Bonnans, Guillaume Bonnet, Jean-Marie Mirebeau
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.
{"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}
Pub Date : 2023-10-24DOI: 10.1007/s00365-023-09668-8
Atul Dixit, Rahul Kumar
{"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}
Pub Date : 2023-10-17DOI: 10.1007/s00365-023-09667-9
Kevin Beanland, Daniel Freeman
{"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}
Pub Date : 2023-10-04DOI: 10.1007/s00365-023-09629-1
Philipp Grohs, Lukas Liehr
Abstract We study the phase reconstruction of signals f belonging to complex Gaussian shift-invariant spaces $$V^infty (varphi )$$ V∞(φ) from spectrogram measurements $$|{mathcal {G}} f(X)|$$ |Gf(X)| where $${mathcal {G}}$$ G is the Gabor transform and $$X subseteq {{mathbb {R}}}^2$$ X⊆R2 . 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 $$V^infty (varphi )$$ V∞(φ) , such as Paley–Wiener spaces.
{"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}
Pub Date : 2023-06-21DOI: 10.1007/s00365-022-09604-2
R. Gharakhloo, Nicholas S. Witte
{"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}
Pub Date : 2023-05-30DOI: 10.1007/s00365-023-09657-x
Rikard Bøgvad, Christian Hägg, Boris Shapiro
Abstract Motivated by the classical Rodrigues’ formula, we study below the root asymptotic of the polynomial sequence $$begin{aligned} {mathcal {R}}_{[alpha n],n,P}(z)=frac{mathop {}!textrm{d}^{[alpha n]}P^n(z)}{mathop {}!textrm{d}z^{[alpha n]}}, n= 0,1,dots end{aligned}$$ R[αn],n,P(z)=d[αn]Pn(z)dz[αn],n=0,1,⋯ where P ( z ) is a fixed univariate polynomial, $$alpha $$ α is a fixed positive number smaller than $$deg P$$ degP , and $$[alpha n]$$ [αn] stands for the integer part of $$alpha n$$ αn . 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
受经典Rodrigues公式的启发，我们研究了多项式序列$$begin{aligned} {mathcal {R}}_{[alpha n],n,P}(z)=frac{mathop {}!textrm{d}^{[alpha n]}P^n(z)}{mathop {}!textrm{d}z^{[alpha n]}}, n= 0,1,dots end{aligned}$$ R [α n]， n, P (z) = d [α n] P n (z) d z [α n]， n = 0,1，⋯其中P (z)是一个固定的单变量多项式，$$alpha $$ α是一个小于$$deg P$$ deg P的固定正数，$$[alpha n]$$ [α n]表示$$alpha n$$ α n的整数部分。我们对这个渐近的描述是用一个显式调和函数来表示的，这个显式调和函数是由一个平面有理曲线唯一确定的，这个曲线是利用柯西高阶导数公式将鞍点法应用于后一阶多项式的积分表示而产生的。由于我们的方法，我们得出结论，这条曲线与后一个多项式序列的渐近根计数测度的柯西变换所满足的二元代数方程的零轨迹是双等效的。我们证明了这个调和函数也与只有纯虚周期的阿贝尔微分有关，后者的平面曲线属于Bertola (Anal Math Phys 1: 167-211, 2011)， Bertola和Mo (Adv Math 220(1): 154-218, 2009)中最初引入的Boutroux曲线类。作为附加的相关信息，我们导出了一个线性常微分方程，由$${{mathcal {R}}_{[alpha n],n,P}(z)}$$ R [α n]， n, P (z){以及更一般函数的幂的高阶导数满足。}
{"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}
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