124 Chapter 1 Fourier Series and Integrals
5.Usethesoftwaretoapproximatethefunctionf(t)=e−t^2 by the Sampling
Theorem. Try=4,N=2.
6.Simplify the final formula for sampling tof(t)=sin(t)∑∞
−∞f(nπ
) (− 1 )n
t−nπ.1.12 Comments and References
The first use of trigonometric series occurred in the middle of the eighteenth
century. Euler seems to have originated the use of orthogonality for the de-
termination of coefficients. In the early nineteenth century Fourier made ex-
tensive use of trigonometric series in studying problems of heat conduction
(see Chapter 2). His claim, that an arbitrary function could be represented as
a trigonometric series, led to an extensive reexamination of the foundations
of calculus. Fourier seems to have been among the first to recognize that a
function might have different analytical expressions in different places.
Dirichlet established sufficient conditions (similar to those of our conver-
gence theorem) for the convergence of Fourier series around 1830. Later, Rie-
mann was led to redefine the integral as part of his attempt to discover condi-
tions on a function necessary and sufficient for the convergence of its Fourier
series. This problem has never been solved. Many other great mathematicians
have founded important theories (the theory of sets, for one) in the course of
studying Fourier series, and they continue to be a subject of active research. An
entertaining and readable account of the history and uses of Fourier series is
inThe Mathematical Experience, by Davis and Hersh. (See the Bibliography.)
Historical interest aside, Fourier series and integrals are extremely impor-
tant in applied mathematics, physics, and engineering, and they merit further
study. A superbly written and organized book is Tolstov’sFourier Series.Its
mathematical prerequisites are not too high.Fourier Series and Boundary Value
Problemsby Churchill and Brown is a standard text for some engineering ap-
plications.
About 1960 it became clear that the numerical computation of Fourier co-
efficients could be rearranged to achieve dramatic reductions in the amount
of arithmetic required. The result, called thefast Fourier transform,orFFT,has
revolutionized the use of Fourier series in applications. SeeThe Fast Fourier
Transformby James S. Walker.
The sampling theorem mentioned in the last section has become bread and
butter in communications engineering. For extensive information on this as
well as the FFT, seeIntegral and Discrete Transforms with Applications and Error
Analysis, by A.J. Jerri.