Riesz sequence

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 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

is an orthonormal basis for

is a bounded bijective operator.

Hence, Riesz bases need not be orthonormal, i.e., they are a generalization of orthonormal bases.

be an orthonormal basis for a Hilbert space

in the sense that for some constant

, and arbitrary scalars

is a Riesz basis for

[2][3] If H is a finite-dimensional space, then every basis of H is a Riesz basis.

φ

be in the Lp space L2(R), let and let

φ ^

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.

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.