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On the strong divergence of Hilbert transform approximations and a problem of Ul’yanov
- Source :
- Journal of Approximation Theory. 204:34-60
- Publication Year :
- 2016
- Publisher :
- Elsevier BV, 2016.
-
Abstract
- This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.
- Subjects :
- Numerical Analysis
Sequence
Conjecture
Applied Mathematics
General Mathematics
010102 general mathematics
Mathematical analysis
020206 networking & telecommunications
02 engineering and technology
01 natural sciences
Combinatorics
symbols.namesake
Uniform norm
Subsequence
0202 electrical engineering, electronic engineering, information engineering
symbols
Hilbert transform
0101 mathematics
Divergence (statistics)
Finite set
Fourier series
Analysis
Mathematics
Subjects
Details
- ISSN :
- 00219045
- Volume :
- 204
- Database :
- OpenAIRE
- Journal :
- Journal of Approximation Theory
- Accession number :
- edsair.doi...........f22f3c32620557b60b7d8fa7599e6241