Author
A. C. Schaeffer
Bio: A. C. Schaeffer is an academic researcher from Stanford University. The author has contributed to research in topics: Exponential type & Borel summation. The author has an hindex of 10, co-authored 15 publications receiving 2682 citations.
Papers
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2,320 citations
TL;DR: A generalization of Bernstein's theorem has been applied to almost periodic functions as discussed by the authors, which has been shown to be applicable to a much wider class of functions than polynomials.
Abstract: Introduction. Weierstrass was the first to prove that an arbitrary continuous function which is defined over a closed finite interval may be uniformly approximated by a sequence of polynomials. The more difficult problem of best approximation by polynomials had earlier been initiated by Tchebycheff. A number of years later, in the early part of the present century, de la Vallée Poussin raised the following question of best approximation : Is it possible to approximate every polygonal line by polynomials of degree n with an error of o(l/n) as n becomes large? (He had proved that the approximation can be carried out with an error of 0(1 /n). This question was answered in the negative by Serge Bernstein in a prize-winning essay on problems of best approximation. In this paper Bernstein proved and made considerable use of an inequality concerning the derivatives of polynomials. This inequality and a related (and earlier) one by Andrew Markoff have been the starting point of a considerable literature. I t has been found for example that the underlying ideas of these two inequalities are applicable to a much wider class of functions than polynomials. These inequalities have supplied one approach to questions concerning the derivatives of quasi-analytic functions. A generalization of Bernstein's theorem has been applied to almost periodic functions. In discussing a mathematical theory we may emphasize either its applications or the salient points of the theory itself. The applications of Bernstein's inequality to problems of approximation (where it has probably had its greatest success) have been treated in the literature ; see for example Dunham Jackson's book in the Colloquium Publications of the American Mathematical Society. On the other hand I am unaware of any recent résumé of the literature which has been suggested by the theorems of Markoff and Bernstein, so I shall discuss some of the investigations which have centered about these theorems.
164 citations
Book•
31 Dec 1950
TL;DR: One day, you will discover a new adventure and knowledge by spending more money, but when will you think that you need to obtain those all requirements when having much money?
Abstract: One day, you will discover a new adventure and knowledge by spending more money. But when? Do you think that you need to obtain those all requirements when having much money? Why don't you try to get something simple at first? That's something that will lead you to know more about the world, adventure, some places, history, entertainment, and more? It is your own time to continue reading habit. One of the books you can enjoy now is coefficient regions for schlicht functions here.
130 citations
TL;DR: In this article, it was shown that the best possible dominant over the complex plane of the class of functions f{z] cannot have complex zeros, and that these results contain two theorems of S. Bernstein.
Abstract: and on the real axis ƒ(z) is real and bounded by 1. First it is shown that the function cos \\z—f(z) cannot have complex zeros. Moreover its real zeros are simple at the points where the strict inequality \\f(z)\\ <1 is satisfied. This theorem is then used to find a \"best possible\" dominant over the complex plane of the class of functions f{z). Finally it is shown that these results contain two theorems of S. Bernstein.
75 citations
71 citations
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Book•
01 Jan 1998
TL;DR: An introduction to a Transient World and an Approximation Tour of Wavelet Packet and Local Cosine Bases.
Abstract: Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.
17,693 citations
TL;DR: This work construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity, by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction.
Abstract: We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.
8,588 citations
TL;DR: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied and the notion of time-frequency localization is made precise, within this framework, by two localization theorems.
Abstract: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied. The first procedure is the short-time or windowed Fourier transform; the second is the wavelet transform, in which high-frequency components are studied with sharper time resolution than low-frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time-frequency density. Finally, the notion of time-frequency localization is made precise, within this framework, by two localization theorems. >
6,180 citations
TL;DR: It is proved that replacing the usual quadratic regularizing penalties by weighted 𝓁p‐penalized penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem.
Abstract: We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary preassigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted p-penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem. Use of such p-penalized problems with p < 2 is often advocated when one expects the underlying ideal noiseless solution to have a sparse expansion with respect to the basis under consideration. To compute the corresponding regularized solutions, we analyze an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. © 2004 Wiley Periodicals, Inc.
4,339 citations
Posted Content•
Abstract: We consider linear inverse problems where the solution is assumed to have a sparse expansion on an arbitrary pre-assigned orthonormal basis. We prove that replacing the usual quadratic regularizing penalties by weighted l^p-penalties on the coefficients of such expansions, with 1 < or = p < or =2, still regularizes the problem. If p < 2, regularized solutions of such l^p-penalized problems will have sparser expansions, with respect to the basis under consideration. To compute the corresponding regularized solutions we propose an iterative algorithm that amounts to a Landweber iteration with thresholding (or nonlinear shrinkage) applied at each iteration step. We prove that this algorithm converges in norm. We also review some potential applications of this method.
3,640 citations