Riesz sequence
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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 , where is an orthonormal basis for and is a bounded bijective operator. Hence, Riesz bases need not be orthonormal, i.e., they are a generalization of orthonormal bases.[1]
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 .[2][3]
Theorems
Summarize
Perspective
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.
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 .[4]
This theorem demonstrates the stability of the standard orthonormal basis (up to normalization) under perturbations of the frequencies .
The constant 1/4 is sharp; if , the sequence may fail to be a Riesz basis, such as:[5]When are allowed to be complex, the theorem holds under the condition . Whether the constant is sharp is an open question.[5]
See also
Notes
References
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