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Riesz sequence

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In mathematics, a sequence of vectors (xn) in a Hilbert space is called a Riesz sequence if there exist constants such that

for all sequences of scalars (an) in the p space2. 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.[1]

Paley-Wiener criterion

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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 .[2][3]

Theorems

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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 . Then the following are equivalent:

The first of the above conditions is the definition for () to form a Riesz basis for the space it spans.

See also

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Notes

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References

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  • Antoine, J.-P.; Balazs, P. (2012). "Frames, Semi-Frames, and Hilbert Scales". Numerical Functional Analysis and Optimization. 33 (7–9). arXiv:1203.0506. doi:10.1080/01630563.2012.682128. ISSN 0163-0563.
  • Christensen, Ole (2001), "Frames, Riesz bases, and Discrete Gabor/Wavelet expansions" (PDF), Bulletin of the American Mathematical Society, New Series, 38 (3): 273–291, doi:10.1090/S0273-0979-01-00903-X
  • Mallat, Stéphane (2008), A Wavelet Tour of Signal Processing: The Sparse Way (PDF) (3rd ed.), pp. 46–47, ISBN 9780123743701
  • Paley, Raymond E. A. C.; Wiener, Norbert (1934). Fourier Transforms in the Complex Domain. Providence, RI: American Mathematical Soc. ISBN 978-0-8218-1019-4.
  • Young, Robert M. (2001). An Introduction to Non-Harmonic Fourier Series, Revised Edition, 93. Academic Press. ISBN 978-0-12-772955-8.

This article incorporates material from Riesz sequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. This article incorporates material from Riesz basis on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.