4,379 research outputs found
(Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms
Four families of special functions, depending on n variables, are studied. We
call them symmetric and antisymmetric multivariate sine and cosine functions.
They are given as determinants or antideterminants of matrices, whose matrix
elements are sine or cosine functions of one variable each. These functions are
eigenfunctions of the Laplace operator, satisfying specific conditions at the
boundary of a certain domain F of the n-dimensional Euclidean space. Discrete
and continuous orthogonality on F of the functions within each family, allows
one to introduce symmetrized and antisymmetrized multivariate Fourier-like
transforms, involving the symmetric and antisymmetric multivariate sine and
cosine functions.Comment: 25 pages, no figures; LaTaX; corrected typo
Two-dimensional symmetric and antisymmetric generalizations of exponential and cosine functions
Properties of the four families of recently introduced special functions of
two real variables, denoted here by , and , are studied. The
superscripts and refer to the symmetric and antisymmetric functions
respectively. The functions are considered in all details required for their
exploitation in Fourier expansions of digital data, sampled on square grids of
any density and for general position of the grid in the real plane relative to
the lattice defined by the underlying group theory. Quality of continuous
interpolation, resulting from the discrete expansions, is studied, exemplified
and compared for some model functions.Comment: 22 pages, 10 figure
Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization
A versatile method is described for the practical computation of the discrete
Fourier transforms (DFT) of a continuous function given by its values
at the points of a uniform grid generated by conjugacy classes
of elements of finite adjoint order in the fundamental region of
compact semisimple Lie groups. The present implementation of the method is for
the groups SU(2), when is reduced to a one-dimensional segment, and for
in multidimensional cases. This simplest case
turns out to result in a transform known as discrete cosine transform (DCT),
which is often considered to be simply a specific type of the standard DFT.
Here we show that the DCT is very different from the standard DFT when the
properties of the continuous extensions of these two discrete transforms from
the discrete grid points to all points are
considered. (A) Unlike the continuous extension of the DFT, the continuous
extension of (the inverse) DCT, called CEDCT, closely approximates
between the grid points . (B) For increasing , the derivative of CEDCT
converges to the derivative of . And (C), for CEDCT the principle of
locality is valid. Finally, we use the continuous extension of 2-dimensional
DCT to illustrate its potential for interpolation, as well as for the data
compression of 2D images.Comment: submitted to JMP on April 3, 2003; still waiting for the referee's
Repor
Three variable exponential functions of the alternating group
New class of special functions of three real variables, based on the
alternating subgroup of the permutation group , is studied. These
functions are used for Fourier-like expansion of digital data given on lattice
of any density and general position. Such functions have only trivial analogs
in one and two variables; a connection to the functions of is shown.
Continuous interpolation of the three dimensional data is studied and
exemplified.Comment: 10 pages, 3 figure
Signal Flow Graph Approach to Efficient DST I-IV Algorithms
In this paper, fast and efficient discrete sine transformation (DST)
algorithms are presented based on the factorization of sparse, scaled
orthogonal, rotation, rotation-reflection, and butterfly matrices. These
algorithms are completely recursive and solely based on DST I-IV. The presented
algorithms have low arithmetic cost compared to the known fast DST algorithms.
Furthermore, the language of signal flow graph representation of digital
structures is used to describe these efficient and recursive DST algorithms
having points signal flow graph for DST-I and points signal flow
graphs for DST II-IV
Two dimensional symmetric and antisymmetric generalizations of sine functions
Properties of 2-dimensional generalizations of sine functions that are
symmetric or antisymmetric with respect to permutation of their two variables
are described. It is shown that the functions are orthogonal when integrated
over a finite region of the real Euclidean space, and that they are
discretely orthogonal when summed up over a lattice of any density in .
Decomposability of the products of functions into their sums is shown by
explicitly decomposing products of all types. The formalism is set up for
Fourier-like expansions of digital data over 2-dimensional lattices in .
Continuous interpolation of digital data is studied.Comment: 12 pages, 5 figure
Some extremal functions in Fourier analysis, II
We obtain extremal majorants and minorants of exponential type for a class of
even functions on which includes and , where . We also give periodic versions of these results in which the
majorants and minorants are trigonometric polynomials of bounded degree. As
applications we obtain optimal estimates for certain Hermitian forms, which
include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev
inequalities. A further application provides an Erd\"{o}s-Tur\'{a}n-type
inequality that estimates the sup norm of algebraic polynomials on the unit
disc in terms of power sums in the roots of the polynomials.Comment: 40 pages. Accepted for publication in Trans. Amer. Math. So
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