87 research outputs found
Multivariate Wavelet Frames
We proved that for any matrix dilation and for any positive integer ,
there exists a compactly supported tight wavelet frame with approximation order
. Explicit methods for construction of dual and tight wavelet frames with a
given number of vanishing moments are suggested.Comment: LaTex file, 28 page
Multivariate exact and falsified sampling approximation
Approximation properties of the expansions , where is a matrix dilation, is either the
sampled value of a signal at or the integral average of near
(falsified sampled value), are studied. Error estimations in
-norm, , are given in terms of the Fourier transform of
. The approximation order depends on how smooth is , on the order of
Strang-Fix condition for and on . Some special properties of
are
required. To estimate the approximation order of falsified sampling
expansions we compare them with a differential expansions , where is an appropriate
differential operator. Some concrete functions applicable for
implementations are constructed. In particular, compactly supported splines and
band-limited functions can be taken as . Some of these functions provide
expansions interpolating a signal at the points .Comment: 23 page
Approximation by multivariate Kantorovich-Kotelnikov operators
Approximation properties of multivariate Kantorovich-Kotelnikov type
operators generated by different band-limited functions are studied. In
particular, a wide class of functions with discontinuous Fourier transform is
considered. The -rate of convergence for these operators is given in terms
of the classical moduli of smoothness. Several examples of the
Kantorovich-Kotelnikov operators generated by the -function and its
linear combinations are provided
Quasi-projection operators in the weighted spaces
Approximation properties of multivariate quasi-projection operators are
studied in the paper. Wide classes of such operators are considered, including
the sampling and the Kantorovich-Kotelnikov type operators generated by
different band-limited functions.The rate of convergence in the weighted
-spaces for these operators is investigated. The results allow to estimate
the error for reconstruction of signals (approximated functions) whose decay is
not enough to be in
Walsh and wavelet methods for differential equations on the Cantor group
Ordinary and partial differential equation for unknown functions defined on
the Cantor dyadic group are studied. We consider two types of equations:
related to the Gibbs derivatives and to the fractional modified Gibbs
derivatives (or pseudo differential-operators). We find solutions in classes of
distributions and study under what assumptions these solutions are regular
functions with some "good" properties.Comment: 24 page
On orthogonal -adic wavelet bases
A variety of different orthogonal wavelet bases has been found in L_2(R) for
the last three decades. It appeared that similar constructions also exist for
functions defined on some other algebraic structures, such as the Cantor and
Vilenkin groups and local fields of positive characteristic. In the present
paper we show that the situation is quite different for the field of -adic
numbers. Namely, it is proved that any orthogonal wavelet basis consisting of
band-limited (periodic) functions is a modification of Haar basis. This is a
little bit unexpected because from the wavelet theory point of view, the
additive group of -adic numbers looks very similar to the Vilenkin group
where analogs of the Daubechies wavelets (and even band-limited ones) do exist.
We note that all -adic wavelet bases and frames appeared in the literature
consist of Schwartz-Bruhat functions (i.e., band-limited and compactly
supported ones)
Differential and falsified sampling expansions
Differential and falsified sampling expansions , where is a matrix dilation, are studied. In
the case of differential expansions, , where is an
appropriate differential operator. For a large class of functions , the
approximation order of differential expansions was recently studied. Some
smoothness of the Fourier transform of from this class is required. In
the present paper, we obtain similar results for a class of band-limited
functions with the discontinuous Fourier transform. In the case of
falsified expansions, is the mathematical expectation of random integral
average of a signal near the point . To estimate the approximation
order of the falsified sampling expansions we compare them with the
differential expansions. Error estimations in -norm are given in terms of
the Fourier transform of .Comment: arXiv admin note: substantial text overlap with arXiv:1407.032
-Adic Haar multiresolution analysis and pseudo-differential operators
The notion of {\em -adic multiresolution analysis (MRA)} is introduced. We
discuss a ``natural'' refinement equation whose solution (a refinable function)
is the characteristic function of the unit disc. This equation reflects the
fact that the characteristic function of the unit disc is a sum of
characteristic functions of mutually disjoint discs of radius . This
refinement equation generates a MRA. The case is studied in detail. Our
MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real
setting, the refinable function generating our Haar MRA is 1-periodic, which
never holds for real refinable functions. This fact implies that there exist
infinity many different 2-adic orthonormal wavelet bases in {\cL}^2(\bQ_2)
generated by the same Haar MRA. All of these bases are described. We also
constructed multidimensional 2-adic Haar orthonormal bases for
{\cL}^2(\bQ_2^n) by means of the tensor product of one-dimensional MRAs. A
criterion for a multidimensional -adic wavelet to be an eigenfunction for a
pseudo-differential operator is derived. We proved also that these wavelets are
eigenfunctions of the Taibleson multidimensional fractional operator. These
facts create the necessary prerequisites for intensive using our bases in
applications
-Adic multiresolution analysis and wavelet frames
We study -adic multiresolution analyses (MRAs). A complete
characterisation of test functions generating MRAs (scaling functions) is
given. We prove that only 1-periodic test functions may be taken as orthogonal
scaling functions. We also suggest a method for the construction of wavelet
functions and prove that any wavelet function generates a -adic wavelet
frame.Comment: 16 page
Uniform approximation by multivariate quasi-projection operators
Approximation properties of quasi-projection operators are studied. Such an operator is associated with a
function satisfying the Strang-Fix conditions and a tempered
distribution such that compatibility conditions with
hold. Error estimates in the uniform norm are obtained for a wide
class of quasi-projection operators defined on the space of uniformly
continuous functions and on the anisotropic Besov spaces. Under additional
assumptions on and , two-sided estimates in
terms of realizations of the -functional are also obtained.Comment: arXiv admin note: text overlap with arXiv:2006.1327
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