Teaching of Fourier series expansions in undergraduate education

Abstract.We construct covering maps from infinite blowups of the$n$-dimensional Sierpinski gasket$S{{G}_{n}}$to certain compact fractafolds based on$S{{G}_{n}}$. These maps are fractal analogs of the usual covering maps fromthe line to the circle. The construction extends work of the second author in the case$n=2$, but a differentmethod of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give a characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.

Download Full-text

Fourier Coefficients of a Class of Eta Quotients of Weight 4

International Journal of Advanced Mathematical Sciences ◽

10.14419/ijams.v4i1.5737 ◽

2016 ◽

Vol 4 (1) ◽

pp. 4

Author(s):

Barış Kendirli

Keyword(s):

Fourier Series ◽

Fourier Coefficients ◽

Series Expansions ◽

Explicit Formulas

<p>Recently, there have been several works on the coefficients of the Fourier series expansions of a class of eta quotients by Williams, Yao, Xia and Jin, Kendirli, and Alaca. Some important explicit formulas have been discovered. Williams expressed all coefficients of one hundred and twenty-six eta quotients in terms of and and Yao, Xia and Jin, following the method of proof of Williams, expressed only even coefficients of one hundred and four eta quotients in terms of and The author has expressed the even and odd coefficients of the Fourier series expansions of a class of eta quotients in terms of and for Meanwhile, Alaca has obtained the coefficients of the Fourier series expansions of a class of eta quotients in in terms of and Here, we will express the coefficients of the Fourier series expansions of a class of eta quotients in in terms of and Fourier coefficients of the four eta quotients.</p>

Download Full-text

Fourier Series for Functions Related to Chebyshev Polynomials of the First Kind and Lucas Polynomials

Mathematics ◽

10.3390/math6120276 ◽

2018 ◽

Vol 6 (12) ◽

pp. 276 ◽

Cited By ~ 7

Author(s):

Taekyun Kim ◽

Dae Kim ◽

Lee-Chae Jang ◽

Gwan-Woo Jang

Keyword(s):

Fourier Series ◽

Chebyshev Polynomials ◽

Bernoulli Polynomials ◽

Series Expansions ◽

Linear Combinations ◽

Lucas Polynomials ◽

Finite Products ◽

Derive Fourier Series

In this paper, we derive Fourier series expansions for functions related to sums of finite products of Chebyshev polynomials of the first kind and of Lucas polynomials. From the Fourier series expansions, we are able to express those two kinds of sums of finite products of polynomials as linear combinations of Bernoulli polynomials.

Download Full-text