Hilbert frames are overcomplete and stable families of functions from a separable
Hilbert space which provide (not necessarily unique) series representations for each
element of the space. The sibling frame case is our motivation for the development
of boundedness criteria for operators associated to certain function families.
After a short introduction of the main notions in frame theory we present the localization
concept for function families in the sense of Frazier/Jawerth/Meyer. In
the context of non–stationary wavelet frames an appropriate separation condition
on the index set of the function family will be posed and results on the boundedness
of the operator associated with the function family will be given.
We present an extension of Meyer’s ”vaguelettes family” to the non–stationary case
with compact support. We prove that every non–stationary vaguelettes family is a
Bessel family and give an explicit Bessel bound. The proof makes use of Schur’s
Lemma.
Our goal is to construct locally supported frames and corresponding sibling duals
which are defined from a non–stationary MRA - in general - and especially from
the non–stationary B–spline MRA of given order m ≥ 2, on a bounded interval
I. The main results in this paper may be used to verify the two Bessel conditions
which appear in the definition of sibling frames. In this way one gets explicit frame
bounds.
This work was motivated by and is intended to supplement the theory of non–
stationary tight wavelet frames of C.K. Chui, W. He and J. Stöckler.