Convergence Phenomenon with Fourier Series of tg(x2)and Alike

The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .

A Review of David Gottlieb’s Work on the Resolution of the Gibbs Phenomenon

Given a piecewise smooth function,it is possible to construct a global expansion in some complete orthogonal basis,such as the Fourier basis.However,the local discontinuities of the function will destroy the convergence of global approximations,even in regions for which the underlying function is analytic.The global expansions are contaminated by the presence of a local discontinuity,and the result is that the partial sums are oscillatory and feature non-uniform convergence.This characteristic behavior is called the Gibbs phenomenon.However,David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing.In this paper we review the history of the Gibbs phenomenon and the story of its resolution.

A Note on the Relationship between the Pearson Product-Moment and the Spearman Rank-Based Coefficients of Correlation

This note derives the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation for the bivariate normal distribution. This new derivation shows the relationship between the two correlation coefficients through an infinite cosine series. A computationally efficient algorithm is also provided to estimate the relationship between the Pearson product-moment coefficient of correlation and the Spearman rank-based coefficient of correlation. The algorithm can be implemented with relative ease using current modern mathematical or statistical software programming languages e.g. R, SAS, Mathematica, Fortran, et al. The algorithm is also available from the author of this article.

Asymptotic Behavior of Gibbs Functions for M-Band Wavelet Expansions

In this paper the asymptotic behavior of Gibbs function for a class of M-band wavelet expansions is given. In particular, the Daubechies’ wavelets are included in this class.