# Sets of Fourier coefficients using numerical quadrature

### Description

One approach to the calculation of Fourier trigonometric coefficients f(r) of a given function f(x) is to apply the trapezoidal quadrature rule to the integral representation f(r) = {line_integral}{sub 0}{sup 1} f(x)e{sup -2{pi}irx}dx. Some of the difficulties in this approach are discussed. A possible way of overcoming many of these is by means of a subtraction function. Thus, one sets f(x) = h{sub p-1}(x) + g{sub p}(x), where h{sub -1}(x) is an algebraic polynomial of degree p-1, specified in such a way that the Fourier series of g{sub p}(x) converges more rapidly than that of f(x). To obtain the Fourier ... continued below

24 pages

### Creation Information

Lyness, J. N. October 10, 2001.

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## What

### Description

One approach to the calculation of Fourier trigonometric coefficients f(r) of a given function f(x) is to apply the trapezoidal quadrature rule to the integral representation f(r) = {line_integral}{sub 0}{sup 1} f(x)e{sup -2{pi}irx}dx. Some of the difficulties in this approach are discussed. A possible way of overcoming many of these is by means of a subtraction function. Thus, one sets f(x) = h{sub p-1}(x) + g{sub p}(x), where h{sub -1}(x) is an algebraic polynomial of degree p-1, specified in such a way that the Fourier series of g{sub p}(x) converges more rapidly than that of f(x). To obtain the Fourier coefficients of f(x), one uses an analytic expression for those of h{sub p-1}(x) and numerical quadrature to approximately those of g{sub p}(x).

24 pages

### Source

• International Conference on Numerical Algorithms, Marrakesh (MA), 10/01/2001--10/05/2001

### Identifier

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• Report No.: ANL/MCS/CP-105987
• Grant Number: W-31-109-ENG-38
• Office of Scientific & Technical Information Report Number: 792107

### Collections

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## When

### Creation Date

• October 10, 2001

### Added to The UNT Digital Library

• Oct. 19, 2015, 7:39 p.m.

### Description Last Updated

• March 30, 2016, 4:34 p.m.

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