Riesz sequence

Paley-Wiener criterion

Let {e_{n}} be an orthonormal basis for a Hilbert space H and let {x_{n}} be "close" to {e_{n}} in the sense that

: \left| \sum a_{i} (e_{i} - x_{i})\right| \leq \lambda \sqrt{\sum |a_{i}|^{2}}

for some constant \lambda , 0 \leq \lambda , and arbitrary scalars a_{1},\dotsc, a_{n} (n = 1,2,3,\dotsc) . Then {x_{n}} 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 L2(R), let

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

and let \hat{\varphi} denote the Fourier transform of {\varphi}. 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_{n}\left|\hat{\varphi}(\omega + 2\pi n)\right|^2\leq C

The first of the above conditions is the definition for ({\varphi_n}) 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]. It is a foundational result in the theory of non-harmonic Fourier series.

Let \Lambda = {\lambda_n}{n \in \mathbb{Z}} be a sequence of real numbers such that : \sup{n \in \mathbb{Z}} |\lambda_n - n| Then the sequence of complex exponentials {e^{i \lambda_n t}}_{n \in \mathbb{Z}} forms a Riesz basis for L^2[-\pi, \pi].

This theorem demonstrates the stability of the standard orthonormal basis {e^{int}}_{n \in \mathbb{Z}} (up to normalization) under perturbations of the frequencies n.

The constant 1/4 is sharp; if \sup_{n \in \mathbb{Z}} |\lambda_n - n| = 1/4, the sequence may fail to be a Riesz basis, such as:\lambda_n= \begin{cases}n-\frac{1}{4}, & n0 \ 0, & n=0 \ n+\frac{1}{4}, & nWhen \Lambda = {\lambda_n}{n \in \mathbb{Z}} are allowed to be complex, the theorem holds under the condition \sup{n \in \mathbb{Z}} |\lambda_n - n| . Whether the constant is sharp is an open question.

Notes

References