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 \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\} and hence, for all \{a_n\}_{n=1}^{\infty}\in \ell^{2} . A Riesz sequence is called a Riesz basis if \overline{\mathop{\rm span} (x_n)} = H. Equivalently, a Riesz basis for H is 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. Subsequently, there exist constants 0 such that 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.

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 \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\} and hence, for all \{a_n\}_{n=1}^{\infty}\in \ell^{2} . A Riesz sequence is called a Riesz basis if \overline{\mathop{\rm span} (x_n)} = H. Equivalently, a Riesz basis for H is 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. Subsequently, there exist constants 0 such that 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. 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 L 2 (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. 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}, & n>0 \\ 0, & n=0 \\ n+\frac{1}{4}, & n When \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.

See also

Orthonormal basis Hilbert space Frame of a vector space

References

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