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Limit points of eigenvalues of truncated unbounded tridiagonal operators

100%
Ifantis E. K., Kokologiannaki C. G., Petropoulou E.
Open Mathematics
|
2007
|
tom 5
|
nr 2
335-344
EN
Let T be a self-adjoint tridiagonal operator in a Hilbert space H with the orthonormal basis {e n}n=1∞, σ(T) be the spectrum of T and Λ(T) be the set of all the limit points of eigenvalues of the truncated operator T N. We give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T) and we connect this problem with an old problem in analysis.
2
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An Extension of 3D Zernike Moments for Shape Description and Retrieval of Maps Defined in Rectangular Solids

100%
Sit A., Mitchell J. C., Phillips G. N., Wright S. J.
Molecular Based Mathematical Biology
|
2013
|
tom 1
75-89
EN
Zernike polynomials have been widely used in the description and shape retrieval of 3D objects. These orthonormal polynomials allow for efficient description and reconstruction of objects that can be scaled to fit within the unit ball. However, maps defined within box-shaped regions ¶ for example, rectangular prisms or cubes ¶ are not well suited to representation by Zernike polynomials, because these functions are not orthogonal over such regions. In particular, the representations require many expansion terms to describe object features along the edges and corners of the region. We overcome this problem by applying a Gram-Schmidt process to re-orthogonalize the Zernike polynomials so that they recover the orthonormality property over a specified box-shaped domain. We compare the shape retrieval performance of these new polynomial bases to that of the classical Zernike unit-ball polynomials.
3
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A general construction of nonseparable multivariate orthonormal wavelet bases

100%
Karoui A.
Open Mathematics
|
2008
|
tom 6
|
nr 4
504-525
EN
The construction of nonseparable and compactly supported orthonormal wavelet bases of L 2(R n); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating multidimensional multiwavelet bases.
4
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Connections between real polynomial solutions of hypergeometric-type differential equations with Rodrigues formula

84%
Weber H.
Open Mathematics
|
2007
|
tom 5
|
nr 2
415-427
EN
Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas. Applications to the classical polynomials are given.
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