Riesz sequence

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

for all sequences of scalars (a_n) in the ℓ^p space ℓ². A Riesz sequence is called a Riesz basis if

.

Theorems

If H is a finite-dimensional space, then every basis of H is a Riesz basis.

Let φ be in the L^p space L²(R), let

φ_n(x) = φ(x − n)

and let denote the Fourier transform of φ. Define constants c and C with . Then the following are equivalent:

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

See also

• Orthonormal basis
• Hilbert space
• Frame of a vector space

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.