A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an orthonormal system which is complete in L 2 ((−1, 1), |x| 2α+1 dx). This orthonormal system is a generalization of the classical exponential system defining Fourier series.
We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials Bn(x; λ) in detail. The starting point is their Fourier series on [0, 1] which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze… (More)
In this paper we introduce a process we have called " Gauss-Seidelization " for solving nonlinear equations. We have used this name because the process is inspired by the well-known Gauss-Seidel method to numerically solve a system of linear equations. Together with some convergence results, we present several numerical experiments in order to emphasize how… (More)
The concept of k-coherence of two positive measures µ 1 and µ 2 is useful in the study of the Sobolev orthogonal polynomials. If µ 1 or µ 2 are compactly supported on Êthen any 0-coherent pair or symmetrically 1-coherent pair (µ 1 , µ 2) must contain a Jacobi measure (up to affine transformation). Here examples of k-coherent pairs (k ≥ 1) when neither µ 1… (More)
Computer algebra systems are a great help for mathematical research but sometimes unexpected errors in the software can also badly affect it. As an example, we show how we have detected an error of Mathematica computing determinants of matrices of integer numbers: not only it computes the determinants wrongly, but also it produces different results if one… (More)
General expressions are found for the orthonormal polyno-mials and the kernels relative to measures on the real line of the form µ+M δ c , in terms of those of the measures dµ and (x−c) 2 dµ. In particular , these relations allow us to obtain that Nevai's class M (0, 1) is closed for adding a mass point, as well as several bounds for the polynomials and… (More)
Hurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive… (More)
In this paper we describe some advances in the knowledge of the behavior of aliquot sequences starting with a number less than 10000. For some starting values, it is shown for the first time that the sequence terminates. The current record for the maximum of a terminating sequence is located in the one starting at 4170; it converges to 1 after 869… (More)