Riesz sequence explained

(H,\langle ⋅ , ⋅ \rangle) is called a Riesz sequence if there exist constants 0 <c\le C< +inf ty such thatc\left( \sum _{n} | a ^{2} _{n|} \right) \leq \left\Vert \sum _{n} a _{n} x _{n} \right\Vert ^{2} \leq C \left( \sum _{n} | a ^{2} _{n|} \right)

for all sequences of scalars (a_n) in the ℓ^p space ℓ². A Riesz sequence is called a Riesz basis if

\overline{ \rmspan (x _{n)}} = H .

Alternatively, one can define the Riesz basis as a family of the form

\left\{x _{n} \right\} ^{inf ty} _{n =1} = \left\{ Ue _{n} \right\} ^{inf ty} _{n =1} , where \left\{e _{n} \right\} ^{inf ty} _{n =1} is an orthonormal basis for H and U : H → 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

Riesz sequence should not be confused with Paley-Wiener theorem.

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 λ \sqrt{\sum |a _{i} | ^{2} }

for some constant

λ , 0 \leq λ < 1 , and arbitrary scalars a _{1} , ... c, a _{n} (n = 1 , 2 , 3 , ... c) . 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 <c\le C< +inf ty . Then the following are equivalent:1 . \forall (a _{n)} \in \ell ^{2, c\left(} \sum _{n} | a ^{2} _{n|} \right) \leq \left\Vert \sum _{n} a _{n} \varphi _{n} \right\Vert ^{2} \leq C \left( \sum _{n} | a ^{2} _{n|} \right) 2 . 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.

See also

• Orthonormal basis
• Hilbert space
• Frame of a vector space

References

• Antoine . J.-P. . Balazs . P. . Frames, Semi-Frames, and Hilbert Scales . Numerical Functional Analysis and Optimization . 33 . 7-9 . 2012 . 0163-0563 . 10.1080/01630563.2012.682128. 1203.0506 .
  • • Book: Paley, Raymond E. A. C. . Raymond Paley . Norbert Wiener. Wiener . Norbert . Fourier Transforms in the Complex Domain . American Mathematical Soc. . Providence, RI . 1934 . 978-0-8218-1019-4.
• Book: Young, Robert M. . An Introduction to Non-Harmonic Fourier Series, Revised Edition, 93 . Academic Press . 2001 . 978-0-12-772955-8.