- Riesz sequence
In
mathematics , asequence of vectors ("x""n") in aHilbert space ("H", ⟨ , ⟩) is called a Riesz sequence if there existconstant s 0such that :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
scalar s ("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 0. Then the following are equivalent: :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 ----
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