TITLE:
Convergence Phenomenon with Fourier Series of
tg(
x
2
)
and Alike
AUTHORS:
Alfred Wünsche
KEYWORDS:
Gibbs Phenomenon, Generalized Functions, Weak Convergence, Chebyshev Polynomials of First and Second Kind, Even and Odd Generating Functions for Chebyshev Polynomials, Polylogarithms, Completeness Relations
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.14 No.7,
July
29,
2024
ABSTRACT: The Fourier series of the 2π-periodic functions
tg(
x
2
)
and
1
sin(
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
cos
n
(
x
)
for the symmetric part of functions and
sin(
x
)
cos
n−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.