Riesz sequence

In mathematics, a sequence of vectors (xn) in a Hilbert space $$(H,\langle\cdot,\cdot\rangle)$$ is called a Riesz sequence if there exist constants $$0<c\le C<+\infty$$ 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 (an) in the ℓp space ℓ2. A Riesz sequence is called a Riesz basis if


 * $$\overline{\mathop{\rm span} (x_n)} = H$$.

Alternatively, one can define the Riesz basis as a family of the form $$ \left\{x_{n} \right\}_{n=1}^{\infty} = \left\{ Ue_{n} \right\}_{n=1}^{\infty} $$, where $$ \left\{e_{n} \right\}_{n=1}^{\infty} $$ 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 $$ \{e_{n}\} $$ 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 $$ \lambda $$, $$ 0 \leq \lambda < 1 $$, 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 Lp 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 $$0<c\le C<+\infty$$. 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.