On the representation of smooth functions on the sphere using finitely many bits
type: Item
instance of: scholarly article
title: On the representation of smooth functions on the sphere using finitely many bits
zbMATH Open document ID: 1082.41020
published in: Applied and Computational Harmonic Analysis
publication date: 2005-06-13T00:00:00Z
review text: The paper deals with approximation of smooth functions defined on the \(q\)-dimensional Euclidean sphere \(S^q\) by polynomial operators. Let \({\mathcal P}_l(t)\) be the Legendre polynomial of degree \(l\) and let \(h(t)\) be some \([0, \infty) \to {\mathbb R}\) function vanishing for sufficiently large \(t\). For \(t \in {\mathbb R}\) and \(n=0,1,\ldots\), let \[ \Phi_n(t)=\sum_{l=0}^\infty h(l/2^n) {d_l^q \over \omega_q}{\mathcal P}_l(t), \] where \(d_l^q\) is the linear dimension of the set of all spherical harmonics of degree \(l\) and \(\omega_q\) is the volume of \(S^q\). Given a sequence \(\{\mu_n\}\) of signed Borel measures on \(S^q\), the summability operator is defined by \[ \sigma_n(f, {\mathbf x})=\int_{S_q} \Phi_n ({\mathbf x}\cdot \xi)\,d\mu_n(\xi). \] The difference \(\tau_n(f)=\sigma_n(f)-\sigma_{n-1}(f)\) is the polynomial frame operator. The most interesting cases are when \(\mu_n\) are the volume measures on \(S^q\) for all \(n\) and when \(\mu_n\) are the so-called M-Z quadrature measures. In the latter case, the author proves that under certain uniformity conditions \(f=\sum_{n=0}^\infty \tau_n(f)\), with the series convergent in the sense of \(L_p(S^q)\). Various properties of this expansion are proved. In particular, by uniform quantization of the values of polynomial operators one can obtain a bit coding of functions \(f\) from the Besov classes, with the number of bits that is within only a logarithmic factor of their metric entropy. Moreover, the coding has a localization property: from the total set of bits one can identify for each spherical cap a certain subset of bits that represents the function on this cap. The cardinality of this subset depends on the local smoothness of the function and the volume of the cap.
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 41A35
zbMATH DE Number: 2175611
zbMATH Keywords: approximation on the sphere
zbMATH Keywords: polynomial frames
zbMATH Keywords: metric entropy
author: H. N. Mhaskar
reviewed by: Yuly Makovoz
MaRDI profile type: Publication
full work available at URL: j.acha.2004.11.004
OpenAlex ID: W2002337104
cites work: On the importance of combining wavelet-based nonlinear approximation with coding strategies
cites work: On the mathematical foundations of learning
cites work: On simple oversampled A/D conversion in L/sup 2/(R)
cites work: Approximating a bandlimited function using very coarsely quantized data: a family of stable sigma-delta modulators of arbitrary order
cites work: Q4273944
cites work: Unconditional bases and bit-level compression
cites work: Quantum approximation. II: Sobolev embeddings
cites work: Entropy, universal coding, approximation, and bases properties
cites work: Q4871618
cites work: Q4320142
cites work: On the representation of band limited functions using finitely many bits
cites work: Polynomial operators and local smoothness classes on the unit interval
cites work: Approximation properties of zonal function networks using scattered data on the sphere
cites work: Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature
cites work: On the representation of band-dominant functions on the sphere using finitely many bits
cites work: Polynomial frames on the sphere
cites work: Spherical harmonics
cites work: Tractability of approximation for weighted Korobov spaces on classical and quantum computers
cites work: Über die Approximationsordnung bei Kugelfunktionen und algebraischen Polynomen
cites work: Q3810338
cites work: Q5640160
cites work: Weak convergence and empirical processes. With applications to statistics
DOI: 10.1016/J.ACHA.2004.11.004
description: scientific article; zbMATH DE number 2175611