Riesz sequence

In mathematics, a sequence of vectors ("x"_"n") in a Hilbert space ("H", ⟨ , ⟩) is called a Riesz sequence if there exist constants

:$cleft(\; sum\_n\; |\; a\_n|^2\; ight)\; leq\; leftVert\; sum\_n\; a\_n\; x\_n\; ightVert^2\; leq\; C\; left(\; sum\_n\; |\; a\_n|^2\; ight)$

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

:$overline\{mathop\{\; m\; span\}\; (x\_n)\}\; =\; H$ .

Theorems

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

Let "φ" be in the "L"^"p" space "L"²(R), let

:$phi\_n(x)\; =\; phi(x-n)$

and let $hat\{varphi\}$ denote the Fourier transform of "φ". Define constants "c" and "C" with

:$1.\; quad\; forall\; (a\_n)\; in\; ell^2,\; cleft(\; sum\_n\; |\; a\_n|^2\; ight)\; leq\; leftVert\; sum\_n\; a\_n\; varphi\_n\; ightVert^2\; leq\; C\; left(\; sum\_n\; |\; a\_n|^2\; ight)$

:$2.\; quad\; cleqsum\_\{n\}left|hat\{varphi\}(omega\; +\; 2pi\; n)\; ight|^2leq\; C$

The first of the above conditions is the definition for ("φ"_"n") to form a Riesz basis for the space it spans.

ee also

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

----