New Trends in Applied Harmonic Analysis by Akram Aldroubi Carlos Cabrelli Stephane Jaffard & Ursula Molter

Author:Akram Aldroubi, Carlos Cabrelli, Stephane Jaffard & Ursula Molter
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

6.2 Frame Theory

Hilbert space frame theory is the tool which is used to connect many of the equivalent forms of the Paving Conjecture. So we start with an introduction to this area. A family of vectors in a Hilbert space is a Riesz basic sequence if there are constants A, B > 0 so that for all scalars we have

We call A, B the lower and upper Riesz basis bounds for . If the Riesz basic sequence spans , we call it a Riesz basis for . So is a Riesz basis for means there is an orthonormal basis so that the operator T(e i ) = f i is invertible. In particular, each Riesz basis is bounded. That is, .

Hilbert space frames were introduced by Duffin and Schaeffer [25] to address some very deep problems in nonharmonic Fourier series (see [44]). A family of elements of a (finite or infinite dimensional) Hilbert space is called a frame for if there are constants (called the lower and upper frame bounds, respectively) so that for all

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