Riesz sequence#Paley-Wiener criterion
In mathematics, a sequence of vectors (xn) in a Hilbert space is called a Riesz sequence if there exist constants
:
for all sequences of scalars (an) in the ℓp space ℓ2. A Riesz sequence is called a Riesz basis if
:.
Alternatively, one can define the Riesz basis as a family of the form , where is an orthonormal basis for and is a bounded bijective operator. Hence, Riesz bases need not be orthonormal, i.e., they are a generalization of orthonormal bases.{{sfn | Antoine | Balazs | 2012}}
Paley-Wiener criterion
{{distinguish|Paley-Wiener theorem}}
Let be an orthonormal basis for a Hilbert space and let be "close" to in the sense that
:
for some constant , , and arbitrary scalars . Then is a Riesz basis for .{{sfn | Young | 2001 | p=35}}{{sfn | Paley | Wiener | 1934 | p=100}}
Theorems
If H is a finite-dimensional space, then every basis of H is a Riesz basis.
Let be in the Lp space L2(R), let
:
and let denote the Fourier transform of . Define constants c and C with
:
:
The first of the above conditions is the definition for () to form a Riesz basis for the space it spans.
Kadec 1/4 Theorem
{{see also|Frame (linear algebra)#Non-harmonic Fourier series}}
The Kadec 1/4 theorem, sometimes called the Kadets 1/4 theorem, provides a specific condition under which a sequence of complex exponentials forms a Riesz basis for the Lp space . It is a foundational result in the theory of non-harmonic Fourier series.
Let be a sequence of real numbers such that
:
Then the sequence of complex exponentials forms a Riesz basis for .{{sfn | Young | 2001 | p=36}}
This theorem demonstrates the stability of the standard orthonormal basis (up to normalization) under perturbations of the frequencies .
The constant 1/4 is sharp; if , the sequence may fail to be a Riesz basis, such as:{{sfn|Young|2001|p=37}}When are allowed to be complex, the theorem holds under the condition . Whether the constant is sharp is an open question.{{sfn|Young|2001|p=37}}
See also
Notes
{{Reflist}}
References
- {{cite journal | last=Antoine | first=J.-P. | last2=Balazs | first2=P. | title=Frames, Semi-Frames, and Hilbert Scales | journal=Numerical Functional Analysis and Optimization | volume=33 | issue=7-9 | date=2012 | issn=0163-0563 | doi=10.1080/01630563.2012.682128| arxiv=1203.0506 }}
- {{Citation | last1=Christensen | first1=Ole | title=Frames, Riesz bases, and Discrete Gabor/Wavelet expansions | url=https://www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00903-X/S0273-0979-01-00903-X.pdf | year=2001 | journal=Bulletin of the American Mathematical Society |series=New Series | volume=38 | issue=3 | pages=273–291 | doi=10.1090/S0273-0979-01-00903-X | doi-access=free }}
- {{Citation | last=Mallat | first=Stéphane | title=A Wavelet Tour of Signal Processing: The Sparse Way | url=https://www.di.ens.fr/~mallat/papiers/WaveletTourChap1-2-3.pdf | year=2008 | isbn=9780123743701 | edition = 3rd | pages=46–47 }}
- {{cite book |authorlink1=Raymond Paley | authorlink2=Norbert Wiener| last=Paley | first=Raymond E. A. C. | last2=Wiener | first2=Norbert | title=Fourier Transforms in the Complex Domain | publisher=American Mathematical Soc. | publication-place=Providence, RI | date=1934 | isbn=978-0-8218-1019-4}}
- {{cite book | last=Young | first=Robert M. | title=An Introduction to Non-Harmonic Fourier Series, Revised Edition, 93 | publisher=Academic Press | date=2001 | isbn=978-0-12-772955-8}}
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