In mathematics, a sequence of vectors (xn) in a Hilbert space ( H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (H,\langle \cdot ,\cdot \rangle )} is called a Riesz sequence if there exist constants 0 < c ≤ C < ∞ {\displaystyle 0<c\leq C<\infty } such that
c ∑ n = 1 ∞ | a n | 2 ≤ ‖ ∑ n = 1 ∞ a n x n ‖ 2 ≤ C ∑ n = 1 ∞ | a n | 2 , {\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 { a n } {\displaystyle \{a_{n}\}} and hence, for all { a n } n = 1 ∞ ∈ ℓ 2 {\displaystyle \{a_{n}\}_{n=1}^{\infty }\in \ell ^{2}} . A Riesz sequence is called a Riesz basis if
s p a n ( x n ) ¯ = H . {\displaystyle {\overline {\mathop {\rm {span}} (x_{n})}}=H.}
Equivalently, a Riesz basis for H {\displaystyle H} is a family of the form { x n } n = 1 ∞ = { U e n } n = 1 ∞ {\displaystyle \left\{x_{n}\right\}_{n=1}^{\infty }=\left\{Ue_{n}\right\}_{n=1}^{\infty }} , where { e n } n = 1 ∞ {\displaystyle \left\{e_{n}\right\}_{n=1}^{\infty }} is an orthonormal basis for H {\displaystyle H} and U : H → H {\displaystyle U:H\rightarrow H} is a bounded bijective operator. Subsequently, there exist constants 0 < c ≤ C < ∞ {\displaystyle 0<c\leq C<\infty } such that
c ‖ f ‖ 2 ≤ ∑ n = 1 ∞ | ⟨ f , x n ⟩ | 2 ≤ C ‖ f ‖ 2 , ∀ f ∈ H . {\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.
Paley-Wiener criterion
Let { e n } {\displaystyle \{e_{n}\}} be an orthonormal basis for a Hilbert space H {\displaystyle H} and let { x n } {\displaystyle \{x_{n}\}} be "close" to { e n } {\displaystyle \{e_{n}\}} in the sense that
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