Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.

Integral Kernels with Reflection Group Invariance

A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations

The integer order differentiation by integration method based on the Jacobi orthogonal polynomials for noisy signals was originally introduced by Mboup, Join and Fliess. We propose to extend this method from the integer order to the fractional order to estimate the fractional order derivatives of noisy signals. Firstly, two fractional order differentiators are deduced from the Jacobi orthogonal polynomial filter, using the Riemann–Liouville and the Caputo fractional order derivative definitions respectively. Exact and simple formulae for these differentiators are given by integral expressions. Hence, they can be used for both continuous-time and discrete-time models in on-line or off-line applications. Secondly, some error bounds are provided for the corresponding estimation errors. These bounds allow to study the design parameters' influence. The noise error contribution due to a large class of stochastic processes is studied in discrete case. The latter shows that the differentiator based on the Caputo fractional order derivative can cope with a class of noises, whose mean value and variance functions are polynomial time-varying. Thanks to the design parameters analysis, the proposed fractional order differentiators are significantly improved by admitting a time-delay. Thirdly, in order to reduce the calculation time for on-line applications, a recursive algorithm is proposed. Finally, the proposed differentiator based on the Riemann–Liouville fractional order derivative is used to estimate the state of a fractional order system and numerical simulations illustrate the accuracy and the robustness with respect to corrupting noises.

Fractional Order Differentiation by Integration and Error Analysis in Noisy Environment

Van Ryzin and Greblicki showed that pattern recognition procedures derived from orthogonal series estimates of a probability density function are Bayes risk consistent. In this note it is proved that these procedures do not lose-under some additional conditions-their asymptotic properties even if the random environment is nonstationary.

On Bayes Risk Consistent Pattern Recognition Procedures in a Quasi-Stationary Environment

Discrete Fourier transform (DFT) implementation requires high computational resources and time; a computational complexity of order O(N2) for a signal of size N. Fast Fourier transform (FFT) algorithm, that uses butterfly structures, has a computational complexity of O(Nlog(N)), a value much less than O(N2). However, in recent years by introducing big data in many applications, FFT calculations still impose serious challenges in terms of computational complexity, time requirement, and energy consumption. Involved data in many of these applications are sparse in the spectral domain. For these data by using Sparse Fast Fourier Transform (SFFT) algorithms with a sub-linear computational and sampling complexity, the problem of computational complexity of Fourier transform has been reduced substantially. Different algorithms and hardware implementations have been introduced and developed for SFFT calculations. This paper presents a categorized review of SFFT, highlights the differences of its various algorithms and implementations, and also reviews the current use of SFFT in different applications. 

A structured review of sparse fast Fourier transform algorithms

Orthogonal Polynomials

Confluent Hypergeometric Functions

Representation Theory

Scattering Theory

Approximate Method for the Helium Atom

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