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Euler‐Maclaurin Summation of Trigonometric Fourier Series
Author(s) -
MacDonald D. A.
Publication year - 1994
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1994923213
Subject(s) - mathematics , trigonometric polynomial , fourier series , series (stratigraphy) , discrete fourier series , polynomial , mathematical analysis , function series , trigonometric integral , euler's formula , conjugate fourier series , fourier analysis , trigonometric series , taylor series , fourier transform , trigonometric functions , trigonometric substitution , trigonometry , geometry , fractional fourier transform , short time fourier transform , paleontology , linear interpolation , bicubic interpolation , biology
Trigonometric Fourier series are, in general, difficult to sum to high accuracy. An example is given by the series in which α and β (>0) are rational numbers satisfying 0< β / α ≤1, where λ is an independent variable and j is a positive integer or zero. This paper presents a method for the efficient evaluation of the sum of such series. Fourier series which are the real or the imaginary part of , but which are not explicitly expressible as simple polynomials in λ , are obtained as the sum of a logarithic term and an infinite series in powers of λ , whose expansion is valid when 0< λ ≤(2π/ α ) and is exact . When the Fourier series is expressible as a polynomial in λ , the method identifies that polynomial.