Journal of Approximation Theory最新文献_第9页

Random sections of ℓp-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators 的随机部分ℓp-椭球、最优恢复和对角算子的Gelfand数

IF 0.9 3区数学

Journal of Approximation Theory Pub Date : 2023-09-01 DOI: 10.1016/j.jat.2023.105919

Aicke Hinrichs , Joscha Prochno , Mathias Sonnleitner

{"title":"Random sections of ℓp-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators","authors":"Aicke Hinrichs , Joscha Prochno , Mathias Sonnleitner","doi":"10.1016/j.jat.2023.105919","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105919","url":null,"abstract":"<div>We study the circumradius of a random section of an <math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-ellipsoid, <math><mrow><mn>0</mn><mo><</mo><mi>p</mi><mo>≤</mo><mi>∞</mi></mrow></math>, and compare it with the minimal circumradius over all sections with subspaces of the same codimension. Our main result is an upper bound for random sections, which we prove using techniques from asymptotic geometric analysis if <math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>∞</mi></mrow></math> and compressed sensing if <math><mrow><mn>0</mn><mo><</mo><mi>p</mi><mo>≤</mo><mn>1</mn></mrow></math>. This can be interpreted as a bound on the quality of random (Gaussian) information for the recovery of vectors from an <math><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub></math>-ellipsoid for which the radius of optimal information is given by the Gelfand numbers of a diagonal operator. In the case where the semiaxes decay polynomially and <math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>∞</mi></mrow></math>, we conjecture that, as the amount of information increases, the radius of random information either decays like the radius of optimal information or is bounded from below by a constant, depending on whether the exponent of decay is larger than the critical value <math><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac></mrow></math> or not. If <math><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mn>2</mn></mrow></math>, we prove this conjecture by providing a matching lower bound. This extends the recent work of Hinrichs et al. [Random sections of ellipsoids and the power of random information, Trans. Amer. Math. Soc., 2021] for the case <math><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></math>.</div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50191929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Telescoping continued fractions for the error term in Stirling’s formula 斯特林公式中误差项的伸缩连分式

IF 0.9 3区数学

Journal of Approximation Theory Pub Date : 2023-09-01 DOI: 10.1016/j.jat.2023.105943

Gaurav Bhatnagar , Krishnan Rajkumar

引用次数: 0

Sampling discretization error of integral norms for function classes with small smoothness 小光滑度函数类积分范数的采样离散误差

IF 0.9 3区数学

Journal of Approximation Theory Pub Date : 2023-09-01 DOI: 10.1016/j.jat.2023.105913

V.N. Temlyakov

引用次数: 0

Sharp estimates for Jacobi heat kernels in conic domains 二次域Jacobi热核的尖锐估计

IF 0.9 3区数学

Journal of Approximation Theory Pub Date : 2023-09-01 DOI: 10.1016/j.jat.2023.105921

Dawid Hanrahan , Dariusz Kosz

引用次数: 0

Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials 一类与广义Charlier多项式相关的Sobolev正交多项式的渐近分析

IF 0.9 3区数学

Journal of Approximation Theory Pub Date : 2023-09-01 DOI: 10.1016/j.jat.2023.105918

Diego Dominici , Juan José Moreno-Balcázar

引用次数: 2

Strong uniqueness and alternation theorems for relative Chebyshev centers 相对切比雪夫中心的强唯一性和交变定理

IF 0.9 3区数学

Journal of Approximation Theory Pub Date : 2023-09-01 DOI: 10.1016/j.jat.2023.105917

F.E. Levis , C.V. Ridolfi , L. Zabala

引用次数: 0

On a conjecture concerning interpolation by bivariate Bernstein polynomials 关于二元Bernstein多项式插值的一个猜想

IF 0.9 3区数学

Journal of Approximation Theory Pub Date : 2023-09-01 DOI: 10.1016/j.jat.2023.105920

Michael S. Floater

引用次数: 0

Stable decomposition of homogeneous Mixed-norm Triebel–Lizorkin spaces 齐次混合范数triiebel - lizorkin空间的稳定分解

IF 0.9 3区数学

Journal of Approximation Theory Pub Date : 2023-08-22 DOI: 10.1016/j.jat.2023.105958

Morten Nielsen

引用次数: 0

On semi-classical weight functions on the unit circle 单位圆上的半经典权函数

IF 0.9 3区数学

Journal of Approximation Theory Pub Date : 2023-08-16 DOI: 10.1016/j.jat.2023.105957

Cleonice F. Bracciali , Karina S. Rampazzi , Luana L. Silva Ribeiro

引用次数: 0

Hermite–Padé approximation and integrability Hermite–Padé近似与可积性

IF 0.9 3区数学

Journal of Approximation Theory Pub Date : 2023-08-01 DOI: 10.1016/j.jat.2023.105910

Adam Doliwa, Artur Siemaszko

{"title":"Hermite–Padé approximation and integrability","authors":"Adam Doliwa, Artur Siemaszko","doi":"10.1016/j.jat.2023.105910","DOIUrl":"https://doi.org/10.1016/j.jat.2023.105910","url":null,"abstract":"<div>We show that solution to the Hermite–Padé type I approximation problem leads in a natural way to a subclass of solutions of the Hirota (discrete Kadomtsev–Petviashvili) system and of its adjoint linear problem. Our result explains the appearance of various ingredients of the integrable systems theory in application to multiple orthogonal polynomials, numerical algorithms, random matrices, and in other branches of mathematical physics and applied mathematics where the Hermite–Padé approximation problem is relevant. We present also the geometric algorithm, based on the notion of Desargues maps, of construction of solutions of the problem in the projective space over the field of rational functions. As a byproduct we obtain the corresponding generalization of the Wynn recurrence. We isolate the boundary data of the Hirota system which provide solutions to Hermite–Padé problem showing that the corresponding reduction lowers dimensionality of the system. In particular, we obtain certain equations which, in addition to the known ones given by Paszkowski, can be considered as direct analogs of the Frobenius identities. We study the place of the reduced system within the integrability theory, which results in finding multidimensional (in the sense of number of variables) extension of the discrete-time Toda chain equations.</div>","PeriodicalId":54878,"journal":{"name":"Journal of Approximation Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50186872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3