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.

Paley-Wiener criterion

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 . Hence, Riesz bases need not be orthonormal.[1]

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 . 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

References

  1. Young, Robert M. (2001). An Introduction to Non-Harmonic Fourier Series, Revised Edition. Academic Press. p. 35. ISBN 978-0-12-772955-8.

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