Riesz sequence

In mathematics, a sequence of vectors (x_n) in a Hilbert space [math]\displaystyle{ (H,\langle\cdot,\cdot\rangle) }[/math] is called a Riesz sequence if there exist constants [math]\displaystyle{ 0\lt c\le C\lt +\infty }[/math] such that

[math]\displaystyle{ 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) }[/math]

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

[math]\displaystyle{ \overline{\mathop{\rm span} (x_n)} = H }[/math].

Alternatively, one can define the Riesz basis as a family of the form [math]\displaystyle{ \left\{x_{n} \right\}_{n=1}^{\infty} = \left\{ Ue_{n} \right\}_{n=1}^{\infty} }[/math], where [math]\displaystyle{ \left\{e_{n} \right\}_{n=1}^{\infty} }[/math] is an orthonormal basis for [math]\displaystyle{ H }[/math] and [math]\displaystyle{ U : H \rightarrow H }[/math] is a bounded bijective operator.

Paley-Wiener criterion

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

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

for some constant [math]\displaystyle{ \lambda }[/math], [math]\displaystyle{ 0 \leq \lambda \lt 1 }[/math], and arbitrary scalars [math]\displaystyle{ a_{1},\dotsc, a_{n} }[/math] [math]\displaystyle{ (n = 1,2,3,\dotsc) }[/math] . Then [math]\displaystyle{ \{x_{n}\} }[/math] is a Riesz basis for [math]\displaystyle{ H }[/math]. Hence, Riesz bases need not be orthonormal.

Theorems

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

Let [math]\displaystyle{ \varphi }[/math] be in the L^p space L²(R), let

[math]\displaystyle{ \varphi_n(x) = \varphi(x-n) }[/math]

and let [math]\displaystyle{ \hat{\varphi} }[/math] denote the Fourier transform of [math]\displaystyle{ {\varphi} }[/math]. Define constants c and C with [math]\displaystyle{ 0\lt c\le C\lt +\infty }[/math]. Then the following are equivalent:

[math]\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) }[/math]

[math]\displaystyle{ 2. \quad c\leq\sum_{n}\left|\hat{\varphi}(\omega + 2\pi n)\right|^2\leq C }[/math]

The first of the above conditions is the definition for ([math]\displaystyle{ {\varphi_n} }[/math]) to form a Riesz basis for the space it spans.

See also

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

References

• Christensen, Ole (2001), "Frames, Riesz bases, and Discrete Gabor/Wavelet expansions", Bulletin of the American Mathematical Society, New Series 38 (3): 273–291, doi:10.1090/S0273-0979-01-00903-X, https://www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00903-X/S0273-0979-01-00903-X.pdf 
• Mallat, Stéphane (2008), A Wavelet Tour of Signal Processing: The Sparse Way (3rd ed.), pp. 46–47, ISBN 9780123743701, https://www.di.ens.fr/~mallat/papiers/WaveletTourChap1-2-3.pdf