Riesz sequence

In mathematics, a sequence of vectors in a Hilbert space is called a Riesz sequence if there exist constants such that
for every finite scalar sequence and hence, for all.
A Riesz sequence is called a Riesz basis if
Equivalently, a Riesz basis for is a family of the form, where is an orthonormal basis for and is a bounded bijective operator. Subsequently, there exist constants such that
Hence, Riesz bases need not be orthonormal, i.e., they are a generalization of orthonormal bases.

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.

Theorems

If H is a finite-dimensional space, then every basis of H is a Riesz basis.
Let be in the L^p space L², 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.

Kadec 1/4 Theorem

The Kadec 1/4 theorem, sometimes called the Kadets 1/4 theorem, provides a specific condition under which a sequence of complex exponentials forms a Riesz basis for the Lp space. It is a foundational result in the theory of non-harmonic Fourier series.
Let be a sequence of real numbers such that
Then the sequence of complex exponentials forms a Riesz basis for.
This theorem demonstrates the stability of the standard orthonormal basis under perturbations of the frequencies.
The constant 1/4 is sharp; if, the sequence may fail to be a Riesz basis, such as:When are allowed to be complex, the theorem holds under the condition. Whether the constant is sharp is an open question.