Trigonometric Series

Richard Courant and Fritz John
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Richard Courant: New York University, Courant Institute of Mathematical Sciences
Fritz John: New York University, Courant Institute of Mathematical Sciences

Chapter 8 in Introduction to Calculus and Analysis, 1989, pp 571-632 from Springer

Abstract: Abstract The functions represented by power series, or as Lagrange called them, the “analytic functions,” play indeed a central role in analysis. But the class of analytic functions is too restricted in many instances. It was therefore an event of major importance for all of mathematics and for a great variety of applications when Fourier in his “Théorie analytique de la chaleur”1 observed and illustrated by many examples the fact that convergent trigonometric series of the form 1 $$f(x)=\frac{{{a_0}}}{2}+\sum\limits_{v = 1}^\infty {({a_v}\cos{\text{}}vx+{b_v}\sin{\text{}}vx)}$$ with constant coefficients a v , b v are capable of representing a wide class of “arbitrary” functions f(x), a class which includes essentially every function of specific interest, whether defined geometrically by mechanical means, or in any other way: even functions possessing jump discontinuities, or obeying different laws of formation in different intervals, can thus be expressed.

Keywords: Fourier Series; Periodic Function; Fourier Coefficient; Fourier Expansion; Trigonometric Series (search for similar items in EconPapers)
Date: 1989
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DOI: 10.1007/978-1-4613-8955-2_8

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