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

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.)
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