Principles of Fourier Analysis, 2nd Edition

10

Fourier Series over Finite Intervals (Sine and Cosine Series)

In practice, the functions of interest are often only defined over finite intervals, not the entire real line. For example, if f (t) is the temperature at time t of the coffee in a person’s cup, then f (t) is only defined for α < t < β where α is the time the coffee is poured into the cup, and β is the time the cup is finally emptied.

For the rest of this chapter, L will denote some positive real number, and f will denote some piecewise continuous function that is defined only on the interval (0, L) (as illustrated in figure 10.1). As in chapter 8, our interest is in deriving an expression for f in terms of sines and cosines. This time, however, we are only interested in this ...