Riesz Sequence

In mathematics, a sequence of vectors (''x''_''n'') in a Hilbert space $(H,\langle\cdot,\cdot\rangle)$ is called a Riesz sequence if there exist constants 0 such that

:$c\left( \sum_n , a_n, ^2 \right) \leq \left\Vert \sum_n a_n x_n \right\Vert^2 \leq C \left( \sum_n , a_n, ^2 \right)$

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

:$\overline = H$.

Alternatively, one can define the Riesz basis as a family of the form $\left\_^ = \left\_^$, where $\left\_^$ is an orthonormal basis for $H$ and $U : H \rightarrow H$ is a bounded bijective operator. 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 $H$ and let $\$ be "close" to $\$ in the sense that

:$\left\, \sum a_ (e_ - x_)\right\, \leq \lambda \sqrt$

for some constant $\lambda$, $0 \leq \lambda < 1$, and arbitrary scalars $a_,\dotsc, a_$ $(n = 1,2,3,\dotsc)$ . Then $\$ is a Riesz basis for $H$.

Theorems

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

Let $\varphi$ be in the ''L''^''p'' space ''L''²(R), let

:$\varphi_n(x) = \varphi(x-n)$

and let $\hat$ denote the Fourier transform of $$. Define constants ''c'' and ''C'' with 0. Then the following are equivalent:

:$1. \quad \forall (a_n) \in \ell^2,\ \ c\left( \sum_n , a_n, ^2 \right) \leq \left\Vert \sum_n a_n \varphi_n \right\Vert^2 \leq C \left( \sum_n , a_n, ^2 \right)$

:$2. \quad c\leq\sum_\left, \hat(\omega + 2\pi n)\^2\leq C$

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 L^2 \pi, \pi/math>. It is a foundational result in the theory of non-harmonic Fourier series.

Let $\Lambda = \_$ be a sequence of real numbers such that
:$\sup_ , \lambda_n - n, < \frac$
Then the sequence of complex exponentials $\_$ forms a Riesz basis for L^2 \pi, \pi/math>.

This theorem demonstrates the stability of the standard orthonormal basis $\_$ (up to normalization) under perturbations of the frequencies $n$.

The constant 1/4 is sharp; if $\sup_ , \lambda_n - n, = 1/4$, the sequence may fail to be a Riesz basis, such as:$\lambda_n= \beginn-\frac, & n>0 \\ 0, & n=0 \\ n+\frac, & n<0\end$When $\Lambda = \_$ are allowed to be complex, the theorem holds under the condition $\sup_ , \lambda_n - n, < \frac$. Whether the constant is sharp is an open question.

See also

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

Notes

References

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{{PlanetMath attribution, id=7152, title=Riesz basis

Functional analysis