thus χ(x1) never diminishes, and is alternately constant and variable. We see that ψ(x1) − χ(x1) is continuous as x1 increases from α to β, and that ψ(x1) − χ(x1) = ƒ(x1) − ƒ(α + 0), and when x1 reaches β, we have ψ(β) − χ(x1) = ƒ(β − 0) − ƒ(α + 0). Hence it is seen that between α and β, ƒ(x) = [ψ(x) + ƒ(α + 0)] − χ(x), where ψ(x) + ƒ(α + 0), χ(x) are continuous and never diminish as x increases; the same reasoning applies to every continuous portion of ƒ(x), for which the functions ψ(x), χ(x) are formed in the same manner; we now take ƒ1(x) = ψ(x) + ƒ(α + 0) + C, ƒ2(x) = χ(x) + C, where C is constant between consecutive discontinuities, but may have different values in the next interval between discontinuities; the C can be so chosen that neither ƒ1(x) nor ƒ2(x) diminishes as x increases through a value for which ƒ(x) is discontinuous. We thus see that ƒ(x) = ƒ1(x) − ƒ2(x), where ƒ1(x), ƒ2(x) never diminish as x increases from a to b, and are discontinuous only where ƒ(x) is so. The function ƒ(x) is a particular case of a class of functions defined and discussed by Jordan, under the name “functions with limited variation” (fonctions à variation bornée); in general such functions have not necessarily only a finite number of maxima and minima.
Proof of the Convergence of Fourier’s Series.—It will now be assumed that a function ƒ(x) arbitrarily given between the values −π and +π, has the following properties:—
(a) The function is everywhere numerically less than some fixed positive number, and continuous except for a finite number of values of the variable, for which it may be ordinarily discontinuous.
(b) The function only changes from increasing to diminishing or vice versa, a finite number of times within the interval; this is usually expressed by saying that the number of maxima and minima is finite.
These limitations on the nature of the function are known as Dirichlet’s conditions; it follows from them that the function is integrable throughout the interval.
On these assumptions, we can investigate the limiting value of Dirichlet’s integral; it will be necessary to consider only the case of a function F(z) which does not diminish as z increases from 0 to ½ π, since it has been shown that in the general case the difference of two such functions may be taken. The following lemmas will be required:
1. Since
| ∫ π/2 0 | sin mz | dz = ∫ π/2 0 {1 + 2cos 2z + 2cos 4z + ... + 2cos 2nz} dz = | π | ; |
| sin z | 2 |
this result holds however large the odd integer m may be.
2. If 0 < α < β ≦ π/2,