#### Glasnik Matematicki, Vol. 39, No.1 (2004), 111-138.

### NONSEPARABLE WALSH-TYPE FUNCTIONS ON
**R**^{d}

### Morten Nielsen

Department of Mathematical Sciences, Aalborg University,
Fr. Bajers Vej 7G, DK-9220 Aalborg East, Denmark

*e-mail:* `mnielsen@math.auc.dk`

**Abstract.** We study wavelet packets in the setting of a
multiresolution analysis of
*L*^{2}(**R**^{d}) generated by an arbitrary
dilation matrix *A* satisfying |det *A*| = 2.
In particular, we consider
the wavelet packets associated with a multiresolution analysis
with a scaling function given by the characteristic function of
some set (called a tile) in
**R**^{d}.
The functions in this class
of wavelet packets are called generalized Walsh functions, and it
is proved that the new functions share two major convergence
properties with the Walsh system defined on [0,1).
The functions constitute a Schauder basis for
*L*^{p}(**R**^{d}),
1 < *p* < ∞,
and the expansion of *L*^{p}-functions converge
pointwise almost everywhere. Finally, we introduce a family of compactly supported wavelet packets in
**R**^{2} of class
*C*^{r}(**R**^{2}),
1 ≤ *r* <
∞, modeled after the
generalized Walsh function. It is proved that this class of smooth
wavelet packets has the same convergence properties as the generalized
Walsh functions.

**2000 Mathematics Subject Classification.**
42C10, 42C40.

**Key words and phrases.** Walsh functions, nonstationary
wavelet packets, nonseparable wavelet systems,
pointwise convergence a.e.

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