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Fourier series in orthogonal polynomials with respect to a measure ν on [−1, 1] are studied when ν is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in [−1, 1]. We prove some weighted norm inequalities for the partial sum operators Sn, their maximal operator S * and the commutator [M b , Sn], where M b denotes the(More)
Let Jµ denote the Bessel function of order µ. The functions x −α/2−β/2−1/2 J α+β+2n+1 (x 1/2), n = 0, 1, 2,. .. , form an orthogonal system in L 2 ((0, ∞), x α+β dx) when α + β > −1. In this paper we analyze the range of p, α and β for which the Fourier series with respect to this system converges in the L p ((0, ∞), x α dx)-norm. Also, we describe the(More)
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)
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)
We study some problems related to convergence and divergence a.e. for Fourier series in systems {(pk} , where {(¡>k} is either a system of orthonor-mal polynomials with respect to a measure dp on [-1, 1] or a Bessel system on [0,1]. We obtain boundedness in weighted LP spaces for the maximal operators associated to Fourier-Jacobi and Fourier-Bessel series.(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)