We construct orthogonal and biorthogonal wavelets on a given closed subset of the real line. We also study wavelets satisfying certain types of boundary conditions. We introduce the concept of \wavelet prob-ing", which is closely related to our construction of wavelets. This technique allows us to very quickly perform a number of diierent numerical tasks associated with wavelets. x1. Introduction Wavelets and multiscale analysis have emerged in a number of diierent elds, from harmonic analysis and partial diierential equations in pure mathematics to signal and image processing in computer science and electrical engineering. Typically a general function, signal, or image is broken up into linear combinations of translated and scaled versions of some simple, basic building blocks. Multiscale analysis comes with a natural hierarchical structure obtained by only considering the linear combinations of building blocks up to a certain scale. This hierarchical structure is particularly suited for fast numerical implementations; the underlying idea being that functions on a certain scale only need to be sampled at a rate approximately given by the scale they live on. To discuss this in more concrete terms, let us consider a standard, orthogonal wavelet decomposition of a general function f on the line. This is a representation in terms of linear combinations of translated dilates of a single function : f(x) =
[1]
B. Jawerth,et al.
A discrete transform and decompositions of distribution spaces
,
1990
.
[2]
Stéphane Mallat,et al.
Multifrequency channel decompositions of images and wavelet models
,
1989,
IEEE Trans. Acoust. Speech Signal Process..
[3]
G. Weiss,et al.
Littlewood-Paley Theory and the Study of Function Spaces
,
1991
.
[4]
I. Daubechies.
Orthonormal bases of compactly supported wavelets
,
1988
.
[5]
I. Daubechies,et al.
Multiresolution analysis, wavelets and fast algorithms on an interval
,
1993
.
[6]
P. Auscher.
Wavelets with boundary conditions on the interval
,
1993
.
[7]
DaubechiesIngrid.
Orthonormal bases of compactly supported wavelets II
,
1993
.
[8]
Yves Meyer,et al.
Size properties of wavelet packets
,
1992
.
[9]
P. Franklin.
A set of continuous orthogonal functions
,
1928
.
[10]
Walter Gautschi,et al.
Norm estimates for inverses of Vandermonde matrices
,
1974
.