Abstract
Hölder regularity, isotropic wavelet bases and Hausdorff dimension are not optimal for analyzing directional or anisotropic features for signals on
R
d
with
d
≥
2
. This paper aims to overcome this drawback using rectangular pointwise regularity, hyperbolic wavelets and dimension prints. Rectangular pointwise regularity
L
(
α
1
,
…
,
α
d
)
(
x
1
,
…
,
x
d
)
is defined through local oscillations over rectangular parallelepipeds. It extends the alternative definition of Hölder regularity by means of isotropic local oscillations done by Jaffard. It also substitutes the rectangular pointwise Lipschitz regularity introduced by Ayache et al. and Kamont for
(
α
1
,
…
,
α
d
)
∈
(
0
,
1
)
d
in order to cover any
(
α
1
,
…
,
α
d
)
∈
(
0
,
∞
)
d
. It is also a pointwise version of the Kamont’s anisotropic Besov spaces on the unit cube. We prove characterizations in terms of hyperbolic wavelets. We deduce a new information on the multifractal analysis of rectangular pointwise behaviors, relating the dimension prints of sets of level rectangular pointwise regularities to average quantities expressed in terms of hyperbolic wavelet leaders. We apply our results for some anisotropic selfsimilar cascade wavelet series.