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.