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Exact wavelets on the ball
 IEEE Trans. Sig. Proc
"... Abstract—We develop an exact wavelet transform on the threedimensional ball (i.e. on the solid sphere), which we name the flaglet transform. For this purpose we first construct an exact transform on the radial halfline using damped Laguerre polynomials and develop a corresponding quadrature rule. ..."
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Abstract—We develop an exact wavelet transform on the threedimensional ball (i.e. on the solid sphere), which we name the flaglet transform. For this purpose we first construct an exact transform on the radial halfline using damped Laguerre polynomials and develop a corresponding quadrature rule. Combined with the spherical harmonic transform, this approach leads to a sampling theorem on the ball and a novel threedimensional decomposition which we call the FourierLaguerre transform. We relate this new transform to the wellknown FourierBessel decomposition and show that bandlimitedness in the FourierLaguerre basis is a sufficient condition to compute the FourierBessel decomposition exactly. We then construct the flaglet transform on the ball through a harmonic tiling, which is exact thanks to the exactness of the FourierLaguerre transform (from which the name flaglets is coined). The corresponding wavelet kernels are well localised in real and FourierLaguerre spaces and their angular aperture is invariant under radial translation. We introduce a multiresolution algorithm to perform the flaglet transform rapidly, while capturing all information at each wavelet scale in the minimal number of samples on the ball. Our implementation of these new tools achieves floatingpoint precision and is made publicly available. We perform numerical experiments demonstrating the speed and accuracy of these libraries and illustrate their capabilities on a simple denoising example. Index Terms—Harmonic analysis, wavelets, ball. I.
THE STRANGE HISTORY OF B FUNCTIONS OR HOW THEORETICAL CHEMISTS AND MATHEMATICIANS DO (NOT) INTERACT
, 811
"... B functions are a class of relatively complicated exponentially decaying basis functions. Since the molecular multicenter integrals of the much simpler Slatertype functions are notoriously difficult, it is not at all obvious why B functions should offer any advantages. However, B functions have Fou ..."
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B functions are a class of relatively complicated exponentially decaying basis functions. Since the molecular multicenter integrals of the much simpler Slatertype functions are notoriously difficult, it is not at all obvious why B functions should offer any advantages. However, B functions have Fourier transforms of exceptional simplicity, which greatly simplifies many of their molecular multicenter integrals. This article discusses the historical development of B functions from the perspective of the interaction between mathematics and theoretical chemistry, which traditionally has not been very good. Nevertheless, future progress in theoretical chemistry depends very much on a fertile interaction with neighboring disciplines.
Miana: C0semigroups and resolvent operators approximated by Laguerre expansions. arXiv:1311.7542
, 2013
"... ar ..."
Acceleration of generalized hypergeometric functions through precise remainder asymptotics, preprint arxiv.org/1102.3003
"... Abstract. We express the asymptotics of the remainders of the partial sums {sn} of the generalized hypergeometric function q+1Fq α1,...,αq+1 β1,...,βq ∣∣ ∣ z) through an inverse power series znnλ ∑ ck nk, where the exponent λ and the asymptotic coefficients {ck} may be recursively computed to any de ..."
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Abstract. We express the asymptotics of the remainders of the partial sums {sn} of the generalized hypergeometric function q+1Fq α1,...,αq+1 β1,...,βq ∣∣ ∣ z) through an inverse power series znnλ ∑ ck nk, where the exponent λ and the asymptotic coefficients {ck} may be recursively computed to any desired order from the hypergeometric parameters and argument. From this we derive a new series acceleration technique that can be applied to any such function, even with complex parameters and at the branch point z = 1. For moderate parameters (up to approximately ten) a C implementation at fixed precision is very effective at computing these functions; for larger parameters an implementation in higher than machine precision would be needed. Even for larger parameters, however, our C implementation is able to correctly determine whether or not it has converged; and when it converges, its estimate of its error is accurate. 1.
CFSPML IMPLEMENTATION FOR THE UNCONDI TIONALLY STABLE FDLTD METHOD
"... Abstract—This paper introduces the implementation of complex frequency shifted perfectly matched layer (CFSPML) absorbing boundary conditions for the unconditionally stable finitedifference Laguerre timedomain (FDLTD) method. It has been shown that the relative performance of the CFSPML implemen ..."
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Abstract—This paper introduces the implementation of complex frequency shifted perfectly matched layer (CFSPML) absorbing boundary conditions for the unconditionally stable finitedifference Laguerre timedomain (FDLTD) method. It has been shown that the relative performance of the CFSPML implementations is superior to the PML and Mur ABCs performance by an example. 1.
Counting words with Laguerre polynomials
"... Abstract. We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of formal sums of Laguerre polynomials. We use this method to find the generating function for kary words avoiding any vincular pattern that has only ones. We als ..."
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Abstract. We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of formal sums of Laguerre polynomials. We use this method to find the generating function for kary words avoiding any vincular pattern that has only ones. We also give generating functions for kary words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length, as well as the analogous results for compositions. Résumé. Nous développons une méthode pour compter des mots satisfaisants certaines restrictions en établissant une interprétation combinatoire utile d’un produit de sommes formelles de polynômes de Laguerre. Nous utilisons cette méthode pour trouver la série génératrice pour les mots kaires évitant les motifs vinculars consistant uniquement de uns. Nous présentons en suite les séries génératrices pour les mots kaires évitant de façon cyclique les motifs vinculars consistant uniquement de uns et dont chaque série de uns entre deux tirets est de la même longueur. Nous présentons aussi les résultats analogues pour les compositions.
SLEPIAN SPATIALSPECTRAL CONCENTRATION ON THE BALL
"... Abstract. We formulate and solve the Slepian spatialspectral concentration problem on the threedimensional ball. Both the standard FourierBessel and also the FourierLaguerre spectral domains are considered since the latter exhibits a number of practical advantages (spectral decoupling and exact ..."
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Abstract. We formulate and solve the Slepian spatialspectral concentration problem on the threedimensional ball. Both the standard FourierBessel and also the FourierLaguerre spectral domains are considered since the latter exhibits a number of practical advantages (spectral decoupling and exact computation). The Slepian spatial and spectral concentration problems are formulated as eigenvalue problems, the eigenfunctions of which form an orthogonal family of concentrated functions. Equivalence between the spatial and spectral problems is shown. The spherical Shannon number on the ball is derived, which acts as the analog of the spacebandwidth product in the Euclidean setting, giving an estimate of the number of concentrated eigenfunctions and thus the dimension of the space of functions that can be concentrated in both the spatial and spectral domains simultaneously. Various symmetries of the spatial region are considered that reduce considerably the computational burden of recovering eigenfunctions, either by decoupling the problem into smaller subproblems or by affording analytic calculations. The family of concentrated eigenfunctions forms a Slepian basis that can be used be represent concentrated signals efficiently. We illustrate our results with numerical examples and show that the Slepian basis indeeds permits a sparse representation of concentrated signals. Key words. Slepian concentration problem, bandlimited function, eigenvalue problem, harmonic analysis, ball