Two-dimensional symmetric and antisymmetric generalizations of exponential and cosine functions
Jiří Hrivnák,Jiří Patera +1 more
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TLDR
The properties of the four families of the recently introduced special functions of two real variables, denoted here by E± and cos±, are studied in this article, where the quality of continuous interpolation, resulting from the discrete expansions, is studied, exemplified, and compared for some model functions.Abstract:
The properties of the four families of the recently introduced special functions of two real variables, denoted here by E± and cos±, 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. The quality of continuous interpolation, resulting from the discrete expansions, is studied, exemplified, and compared for some model functions.read more
Citations
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Weight-lattice discretization of Weyl-orbit functions
Jiří Hrivnák,Mark A. Walton +1 more
TL;DR: In this paper, a discretized Weyl-orbit function has been defined for simple Lie algebras, which permits Fourier-like analysis on the fundamental region of the corresponding affine Weyl group.
Journal ArticleDOI
Two-dimensional symmetric and antisymmetric generalizations of sine functions
TL;DR: In this article, the properties of two-dimensional generalizations of sine functions that are symmetric or antisymmetric with respect to permutations of their two variables are described.
Journal ArticleDOI
Generalized discrete orbit function transforms of affine Weyl groups
Tomasz Czyżycki,Jiří Hrivnák +1 more
TL;DR: In this paper, the affine Weyl groups with their corresponding four types of orbit functions are considered and two independent admissible shifts, which preserve the symmetries of the weight and the dual weight lattices, are classified.
Journal ArticleDOI
Two dimensional symmetric and antisymmetric generalizations of sine functions
TL;DR: In this article, it was shown that sine functions are orthogonal when integrated over a finite region of real Euclidean space, and discretely orthogonality when summed up over a lattice of any density in the real space.
Journal ArticleDOI
Weight-Lattice Discretization of Weyl-Orbit Functions
Jiří Hrivnák,Mark A. Walton +1 more
TL;DR: In this paper, a discretized Weyl-orbit function has been defined for simple Lie algebras, which permits Fourier-like analysis on the fundamental region of the corresponding affine Weyl group.
References
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The Discrete Cosine Transform
TL;DR: A direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are, so this work proves orthog onality in a different way.
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Symmetric convolution and the discrete sine and cosine transforms
TL;DR: The author defines symmetric convolution, relates the DSTs and DCTs to symmetric-periodic sequences, and then uses these principles to develop simple but powerful convolution-multiplication properties for the entire family of DST sine and cosine transforms.
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The Gibbs-Wilbraham phenomenon: An episode in fourier analysis
Edwin Hewitt,Robert E. Hewitt +1 more
TL;DR: In the course of this study, FOLLAND as discussed by the authors uncovered a maze of forgotten results, interesting and difficult generalizations, faulty constants, and some details about the GIBBS phenomenon that have escaped the attention of many writers on the subject.
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Anti)symmetric multivariate trigonometric functions and corresponding Fourier transforms
Anatoliy U. Klimyk,Jiri Patera +1 more
TL;DR: In this article, the symmetric and antisymmetric multivariate sine and cosine functions are studied, which are eigenfunctions of the Laplace operator, satisfying specific conditions at the boundary of a certain domain F of the n-dimensional Euclidean space.
Journal ArticleDOI
The Gibbs phenomenon for Fourier interpolation
TL;DR: The Fourier interpolation polynomials for periodic functions with an isolated jump discontinuity exhibit for growing order a Gibbs phenomenon as discussed by the authors, however, the over-and undershots differ from the ones appearing for the partial sums of the Fourier series and depend on the coincidence of the jump with interpolation nodes.
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