Multi-resolution analysis of multiplicity \(d\): Applications to dyadic interpolation. (English) Zbl 0814.42017

Hervé, Loïc

Appl. Comput. Harmon. Anal. 1, No. 4, 299-315 (1994).

Summary: This paper studies the multi-resolution analyses of multiplicity \(d\) \((d\in \mathbb{N}^*)\), that is, the families \((V_ n)_{n\in \mathbb{Z}}\) of closed subspaces in \(\mathbb{L}^ 2(\mathbb{R})\) such that \(V_ n\subset V_{n+ 1}\), \(V_{n+1}= DV_ n\), where \(Df(x)= f(2x)\), and such that there exists a Riesz basis for \(V_ 0\) of the form \(\{\phi_ i(\cdot- k)\), \(i= 1,\dots, d\), \(k\in \mathbb{Z}\}\), with \(\phi_ 1,\dots, \phi_ d\in V_ 0\). Using the Fourier transform, we prove that \[ \widehat\Phi(\lambda)= ^ t[\widehat\phi_ 1(\lambda),\dots, \widehat\phi_ d (\lambda)]= H(\lambda/2) \widehat\Phi(\lambda/2), \] where \(H\) is in the set \({\mathcal M}_ d\) of continuous one-periodic functions taking values in \({\mathcal M}(d, \mathbb{C})\). If \(d= 1\), the definition corresponds to the standard multi-resolution analyses and one can characterize the regular one- periodic complex-valued functions \(H\) (called, then, scaling filters) which yield a multi-resolution analysis.
In this paper, we generalize this study to \(d\geq 2\) by giving conditions on \(H\in {\mathcal M}_ d\) so that there exists \(\widehat\Phi= ^ t[\widehat \phi_ 1, \dots, \widehat\phi_ d]\) in \(\mathbb{L}^ 2(\mathbb{R},\mathbb{C}^ d)\) solution of \(\widehat\Phi(\lambda)= H(\lambda/2) \widehat\Phi(\lambda/2)\) and so that the integer translates of \(\phi_ 1,\dots, \phi_ d\) form a Riesz family. Then, the latter span the space \(V_ 0\) of a multi-resolution analysis of multiplicity \(d\). We show that the conditions on \(H\) focus on the zeros of \(\text{det }H(\cdot)\) and on simple spectral hypotheses for the operator \(P_ H\) defined on \({\mathcal M}_ d\) by \[ P_ H F(\lambda)= H(\lambda/2) F(\lambda/2) H(\lambda/2)^*+ H(\lambda/2+ 1/2) F(\lambda/2+ 1/2) H(\lambda/2+ 1/2)^*. \] Finally, we explore connections with the order \(r\) dyadic interpolation schemes, where \(r\in \mathbb{N}^*\).

MSC:

42C15	General harmonic expansions, frames
46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Keywords:

scaling matrix filters; wavelets; multi-resolution analyses; closed subspaces; Riesz basis; Fourier transform; spectral hypotheses; dyadic interpolation schemes