Riesz sequence

In mathematics, a sequence of vectors (x_n) in a Hilbert space ${\displaystyle (H,\langle \cdot ,\cdot \rangle )}$ is called a Riesz sequence if there exist constants ${\displaystyle 0<c\leq C<\infty }$ such that $${\displaystyle c\sum _{n=1}^{\infty }|a_{n}|^{2}\leq \left\Vert \sum _{n=1}^{\infty }a_{n}x_{n}\right\Vert ^{2}\leq C\sum _{n=1}^{\infty }|a_{n}|^{2},}$$ for every finite scalar sequence ${\displaystyle \{a_{n}\}}$ and hence, for all ${\displaystyle \{a_{n}\}_{n=1}^{\infty }\in \ell ^{2}}$.¹²

A Riesz sequence is called a Riesz basis if $${\displaystyle {\overline {\mathop {\rm {span}} (x_{n})}}=H.}$$ Equivalently, a Riesz basis for ${\displaystyle H}$ is a family of the form ${\displaystyle \left\{x_{n}\right\}_{n=1}^{\infty }=\left\{Ue_{n}\right\}_{n=1}^{\infty }}$, where ${\displaystyle \left\{e_{n}\right\}_{n=1}^{\infty }}$ is an orthonormal basis for ${\displaystyle H}$ and ${\displaystyle U:H\rightarrow H}$ is a bounded bijective operator. Subsequently, there exist constants ${\displaystyle 0<c\leq C<\infty }$ such that³ $${\displaystyle c\|f\|^{2}\leq \sum _{n=1}^{\infty }|\langle f,x_{n}\rangle |^{2}\leq C\|f\|^{2},\quad \forall f\in H.}$$ Hence, Riesz bases need not be orthonormal, i.e., they are a generalization of orthonormal bases.⁴

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

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

for some constant ${\displaystyle \lambda }$, ${\displaystyle 0\leq \lambda <1}$, and arbitrary scalars ${\displaystyle a_{1},\dotsc ,a_{n}}$ ${\displaystyle (n=1,2,3,\dotsc )}$ . Then ${\displaystyle \{x_{n}\}}$ is a Riesz basis for ${\displaystyle H}$.⁵⁶

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

Let ${\displaystyle \varphi }$ be in the L^p space L²(R), let

${\displaystyle \varphi _{n}(x)=\varphi (x-n)}$

and let ${\displaystyle {\hat {\varphi }}}$ denote the Fourier transform of ${\displaystyle {\varphi }}$. Define constants c and C with ${\displaystyle 0<c\leq C<+\infty }$. Then the following are equivalent:⁷

${\displaystyle 1.\quad \forall (a_{n})\in \ell ^{2},\ \ c\left(\sum _{n}|a_{n}|^{2}\right)\leq \left\Vert \sum _{n}a_{n}\varphi _{n}\right\Vert ^{2}\leq C\left(\sum _{n}|a_{n}|^{2}\right)}$

${\displaystyle 2.\quad c\leq \sum _{n}\left|{\hat {\varphi }}(\omega +2\pi n)\right|^{2}\leq C}$

The first of the above conditions is the definition for (${\displaystyle {\varphi _{n}}}$) to form a Riesz basis for the space it spans.

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 ${\displaystyle L^{2}[-\pi ,\pi ]}$. It is a foundational result in the theory of non-harmonic Fourier series.

Let ${\displaystyle \Lambda =\{\lambda _{n}\}_{n\in \mathbb {Z} }}$ be a sequence of real numbers such that

${\displaystyle \sup _{n\in \mathbb {Z} }|\lambda _{n}-n|<{\frac {1}{4}}}$

Then the sequence of complex exponentials ${\displaystyle \{e^{i\lambda _{n}t}\}_{n\in \mathbb {Z} }}$ forms a Riesz basis for ${\displaystyle L^{2}[-\pi ,\pi ]}$.⁸

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

The constant 1/4 is sharp; if ${\displaystyle \sup _{n\in \mathbb {Z} }|\lambda _{n}-n|=1/4}$, the sequence may fail to be a Riesz basis, such as:⁹$${\displaystyle \lambda _{n}={\begin{cases}n-{\frac {1}{4}},&n>0\\0,&n=0\\n+{\frac {1}{4}},&n<0\end{cases}}}$$When ${\displaystyle \Lambda =\{\lambda _{n}\}_{n\in \mathbb {Z} }}$ are allowed to be complex, the theorem holds under the condition ${\displaystyle \sup _{n\in \mathbb {Z} }|\lambda _{n}-n|<{\frac {\log 2}{\pi }}}$. Whether the constant is sharp is an open question.⁹