scholarly journals A survey on pseudo-Chebyshev functions

4open ◽  
2020 ◽  
Vol 3 ◽  
pp. 2 ◽  
Author(s):  
Paolo Emilio Ricci
Keyword(s):  
Fourier Series ◽  
Chebyshev Polynomials ◽  
Best Approximation ◽  
Mathematical Objects ◽  
Starting Point ◽  
Singular Matrices ◽  
Chebyshev Functions

In recent articles, by using as a starting point the Grandi (Rhodonea) curves, sets of irrational functions, extending to the fractional degree the 1st, 2nd, 3rd and 4th kind Chebyshev polynomials have been introduced. Therefore, the resulting mathematical objects are called pseudo-Chebyshev functions. In this survey, the results obtained in the above articles are presented in a compact way, in order to make the topic accessible to a wider audience. Applications in the fields of weighted best approximation, roots of 2 × 2 non-singular matrices and Fourier series are derived.

Delving Deeper: The Chebyshev Polynomials: Patterns and Derivation

2004 ◽  
Vol 98 (1) ◽  
pp. 20-25 ◽  
Author(s):  
Benjamin Sinwell
Keyword(s):  
High School ◽  
Number Theory ◽  
Graph Theory ◽  
Fourier Series ◽  
High School Teachers ◽  
Chebyshev Polynomials ◽  
Mathematical Community ◽  
Russian Mathematician ◽  
School Teachers ◽  
Mathematical Exploration

Pafnuty Lvovich Chebyshev, a Russian mathematician, is famous for his work in the area of number theory and for his work on a sequence of polynomials that now bears his name. These Chebyshev polynomials have applications in the fields of polynomial approximation, numerical analysis, graph theory, Fourier series, and many other areas. They can be derived directly from the multiple-angle formulas for sine and cosine. They are relevant in high school and in the broader mathematical community. For this reason, the Chebyshev polynomials were chosen as one of the topics for study at the 2003 High School Teachers Program at the Park City Mathematics Institute (PCMI). The following is a derivation of the Chebyshev polynomials and a mathematical exploration of the patterns that they produce.

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

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 276 ◽  
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.

scholarly journals On the fast approximation of point clouds using Chebyshev polynomials

2021 ◽  
Vol 15 (4) ◽  
pp. 305-317
Author(s):  
Sven Weisbrich ◽  
Georgios Malissiovas ◽  
Frank Neitzel
Keyword(s):  
Chebyshev Polynomials ◽  
Best Approximation ◽  
Point Cloud ◽  
Complex Geometry ◽  
Point Clouds ◽  
Method Of Least Squares ◽  
The Best Approximation ◽  
Chebyshev Points ◽  
Dense Point ◽  
Univariate Function

Abstract Suppose a large and dense point cloud of an object with complex geometry is available that can be approximated by a smooth univariate function. In general, for such point clouds the “best” approximation using the method of least squares is usually hard or sometimes even impossible to compute. In most cases, however, a “near-best” approximation is just as good as the “best”, but usually much easier and faster to calculate. Therefore, a fast approach for the approximation of point clouds using Chebyshev polynomials is described, which is based on an interpolation in the Chebyshev points of the second kind. This allows to calculate the unknown coefficients of the polynomial by means of the Fast Fourier transform (FFT), which can be extremely efficient, especially for high-order polynomials. Thus, the focus of the presented approach is not on sparse point clouds or point clouds which can be approximated by functions with few parameters, but rather on large dense point clouds for whose approximation perhaps even millions of unknown coefficients have to be determined.

A Fourier series kernel based on Chebyshev polynomials

Computing ◽  
1977 ◽  
Vol 18 (1) ◽  
pp. 37-50
Author(s):  
E. Rietsch
Keyword(s):  
Fourier Series ◽  
Chebyshev Polynomials

scholarly journals Fejér and Dirichlet Kernels: Their Associated Polynomials

2009 ◽  
Vol 1 (2) ◽  
pp. 281-284
Author(s):  
M. Galaz-Larios ◽  
R. Garcia-Olivo ◽  
J. López-Bonilla
Keyword(s):  
Fourier Series ◽  
Chebyshev Polynomials ◽  
Fejér Kernel ◽  
Associated Polynomials

We show that the Fejér kernel generates the fifth-kind Chebyshev polynomials.Keywords: Kernels in Fourier series; Chebyshev polynomials.© 2009 JSR Publications.ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. DOI: 10.3329/jsr.v1i2.2282

scholarly journals Regularity and Stability for a Convex Feasibility Problem

2021 ◽  
Author(s):  
Carlo Alberto De Bernardi ◽  
Enrico Miglierina
Keyword(s):  
Best Approximation ◽  
Convex Sets ◽  
Convex Feasibility Problem ◽  
Feasibility Problem ◽  
Convex Feasibility ◽  
Starting Point

AbstractLet us consider two sequences of closed convex sets {An} and {Bn} converging with respect to the Attouch-Wets convergence to A and B, respectively. Given a starting point a0, we consider the sequences of points obtained by projecting onto the “perturbed” sets, i.e., the sequences {an} and {bn} defined inductively by $b_{n}=P_{B_{n}}(a_{n-1})$ b n = P B n ( a n − 1 ) and $a_{n}=P_{A_{n}}(b_{n})$ a n = P A n ( b n ) . Suppose that A ∩ B is bounded, we prove that if the couple (A,B) is (boundedly) regular then the couple (A,B) is d-stable, i.e., for each {an} and {bn} as above we have dist(an,A ∩ B) → 0 and dist(bn,A ∩ B) → 0. Similar results are obtained also in the case A ∩ B = ∅, considering the set of best approximation pairs instead of A ∩ B.

scholarly journals Best approximation of conjugate of a function in generalized Zygmund

2019 ◽  
Vol 50 (4) ◽  
pp. 417-427
Author(s):  
Hare Krishna Nigam
Keyword(s):  
Fourier Series ◽  
Error Estimates ◽  
Best Approximation ◽  
Degree Of Approximation ◽  
Time Study ◽  
Zygmund Class ◽  
Conjugate Fourier Series ◽  
First Time

In this paper, we, for the very first time, study the error estimates of conjugate of a function ~g of g(2-periodic) in generalized Zygmund class Y wz (z 1); by Matix-Euler (TEq) product operatorof conjugate Fourier series. In fact, we establish two theorems on degree of approximation of afunction ~g of g (2-periodic) in generalized Zygmund class Y wz (z 1); by Matix-Euler (TEq)product means of its conjugate Fourier series. Our main theorem generalizes three previouslyknown results. Thus the results of [7], [8] and [26] become the particular cases of our Theorem2.1. Some corollaries are also deduced from our main theorem.

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