[chi](x1) = [[f](M1) - f(m1)] + [[f](M2) - [f](m1)] + ... + [[f](M_r)
- [f](m_r)],
then let x1 increase until it is beyond the next maximum M_(r+1), and
then let
[chi](x1) = [[f](M1) - [f](m1)] + [[f](M2) - [f](m1)] + ... +
[[f](M_r) - [f](m_r)] + [[f](M_(r+1)) - [f](x1)]
thus [chi](x1) never diminishes, and is alternately constant and
variable. We see that [psi](x1) - [chi](x1) is continuous as x1
increases from [alpha] to [beta], and that [psi](x1) - [chi](x1) =
[f](x1) - [f]([alpha] + 0), and when x1 reaches [beta], we have
[psi]([beta]) - [chi](x1) = [f]([beta] - 0) - [f]([alpha] + 0). Hence
it is seen that between [alpha] and [beta], [f](x) = [[psi](x) +
[f]([alpha] + 0)] - [chi](x), where [psi](x) + [f]([alpha] + 0),
[chi](x) are continuous and never diminish as x increases; the same
reasoning applies to every continuous portion of [f](x), for which
the functions [psi](x), [chi](x) are formed in the same manner; we now
take [f]1(x)=[psi](x) + [f]([alpha] + 0) + C, [f]2(x) = [chi](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 [f]1(x) nor [f]2(x) diminishes as x
increases through a value for which [f](x) is discontinuous. We thus
see that [f](x) = [f]1(x) - [f]2(x), where [f]1(x), [f]2(x) never
diminish as x increases from a to b, and are discontinuous only where
[f](x) is so. The function [f](x) is a particular case of a class of
functions defined and discussed by Jordan, under the name "functions
with limited variation" (_fonctions a variation bornee_); 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 [f](x) arbitrarily given between the values
-[pi] and +[pi], 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
1/2[pi], 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
_ _
/[pi]/2 sin mz /[pi]/2 [pi]
| ------ dz = | {1 + 2cos 2z + 2cos 4z + ... + 2cos 2nz} dz = ----;
_/ 0 sin z _/ 0 2
this result holds however large the odd integer m may be.
[pi]
2. If 0 < [alpha] < [beta] <= ----,
2
_ _ _
/[beta] sin mz 1 /[gamma] 1 /[beta]
| ------ dz = ----------- | sin mz dz + ---------- | sin mz dz
_/[alpha] sin z sin [alpha]_/[alpha] sin [beta]_/[gamma]
where [alpha] < [gamma] < [beta], hence
_
| /[beta] sin mz | 2 / 1 1 \ 4
| | ------ dz | < -- ( ---------- + --------- ) < -------------;
| _/[alpha] sin z | m \sin[alpha] sin[beta]/ m sin [alpha]
_
| / [beta] sin mz | 4
a precisely similar proof shows that | | ------ dz | < --------,
| _/ [alpha] z | m[alpha]
_ _
/[beta] sin mz /[beta] sin mz
hence the integrals | ------ dz, | ------ dz, converge to
_/[alpha] sin z _/[alpha] z
the limit zero, as m is indefinitely increased.
_
| / [oo] sin [theta] |
3. If [alpha] > 0, | | ----------- d[theta] | cannot exceed
| _/[alpha] [theta] |
1/2[pi]. For by the mean-value theorem
_
| / h sin[theta] | 2 2
| | ---------- d[theta] | < ------- + --,
| _/[alpha] [theta] | [alpha] h
_
| / h sin[theta] | 2
hence | Lh = [oo] | ---------- d[theta] | <= -------;
| _/[alpha] [theta] | [alpha]
_
| /[oo] sin[theta] | 2 [pi]
in particular if [alpha] >= [pi] | | ---------- d[theta] | <= ---- < ----.
| _/[alpha] [theta] | [pi] 2
_
d /[oo] sin[theta] sin[alpha]
Again -------- | ---------- d[theta] = - ----------, [alpha] > 0,
d[alpha]_/[alpha] [theta] [alpha]
_
/[oo] sin[theta]
therefore | ---------- d[theta] increases as [alpha] diminishes,
_/[alpha] [theta]
when [theta] < [alpha] < [pi]; but lim
_ _
/[oo] sin[theta] [pi] | /[oo] sin[theta] | [pi]
| ---------- d[theta] = ----, hence | | ---------- d[theta] | < ----,
[alpha]=0_/[alpha] [theta] 2 | _/[alpha] [theta] | 2
where [alpha] < [pi], and < [pi]/2 where [alpha] >= [pi]. It follows that
_
| /[beta] sin[theta] |
| | ---------- d[theta] | <= [pi], provided 0 <= [alpha] < [beta].
| _/[alpha] [theta] |
_
/[pi]/2 sin mz
To find the limit of | F(z) ------ dz, we observe that it may be
_/ 0 sin z
written in the form
_ _
/[pi]/2 sin mz / [mu] sin mz
F(0) | ------ dz + | {F(z) - F(0)} ------ dz
_/ sin z _/ 0 sin z
_
/[pi]/2 sin mz
+ | {F(z) - F(0)} ------ dz
_/[mu] sin z
where [mu] is a fixed number as small as we please; hence if we use
lemma (1), and apply the second mean-value theorem,
_
/[pi]/2 sin mz [pi]
| F(z) ------ dz - ---- F(0)
_/ 0 sin z 2
_
/[mu] z sin mz
= | {F(z) - F(0)} ----- ------ dz
_/ [0] sin z z
_ _
/[xi]^1 sin mz /[pi]/2 sin mz
+ {F([mu] + 0) - F(0)} | ------ dz + {F (1/2[pi] - 0) - F(0)} | ------ dz
_/ [mu] sin z _/[xi]^1 sin z
when [xi]^1 lies between [mu] and 1/2[pi]. When m is indefinitely
increased, the two last integrals have the limit zero in virtue of
lemma (2). To evaluate the first integral on the right-hand side, let
G/z = {F(z) - F(0)} (sin z/z), and observe that G(z) increases as z
increases from 0 to [mu], hence if we apply the mean value theorem
_ _
| /[mu] sin mz | | /[mu] sin mz |
| | G([mu]) ------ dz| = |G([mu]) | ------ dz|
| _/ 0 z | | _/[xi] z |
_
| /m[mu] sin[theta] |
= |G([mu]) | ---------- d[theta]| < [pi] G([mu]),
| _/m[xi] [theta] |
where 0 < [xi] < [mu], since G(z) has the limit zero when z = 0. If
[epsilon] be an arbitrarily chosen positive number, a fixed value of
[mu] may be so chosen that [pi]G([mu)] < 1/2[epsilon], and thus that
_
| /[mu] sin mz |
| | G(z) ------ dz| < 1/2[epsilon].
| _/0 z |
When [mu] has been so fixed, m may now be so chosen that
_
| /1/2[pi] sin mz [pi] |
| | F(z) ------ dz - ---- F(0)| < [epsilon].
| _/0 sin z 2 |
It has now been shown that when m is indefinitely increased
_
/[pi]/2 sin mz [pi]
| F(z) ------ dz - ---- F(0) has the limit zero.
_/ 0 sin z 2
Returning to the form (4), we now see that the limiting value of
_ _
1 /[pi]/2 sin mz 1 /[pi]/2 sin mz
---- | F(z) ------ dz + ---- | F(-z) ------ dz
[pi]_/ 0 sin z [pi]_/ 0 sin z
1/2{F(+0) + F(-0)}; hence the sum of n + 1 terms of the series
_ _
1 / l 1 / l n[pi](x - x^1)
-- | [f](x) dx + -- [Sigma] | [f](x^1) cos ------------- dx
2l _/-l l _/-l l
converges to the value 1/2 {[f](x + 0) + [f](x - 0)}, or to [f](x) at
a point where [f](x) is continuous, provided [f](x) satisfies
Dirichlet's conditions for the interval from -l to l.
_Proof that Fourier's Series is in General Uniformly Convergent._--To
prove that Fourier's Series converges uniformly to its sum for all
values of x, provided that the immediate neighbourhoods of the points
of discontinuity of [f](x) are excluded, we have
_
| /[pi]/2 sin mz [pi] | 4
| | F(z)------ dz - ---- F(0)| < [pi]G ([mu]) + ---------- {F([mu] + 0) - F(0)}
| _/ sin z 2 | m sin [mu]
4
+ ------------ {F(1/2[pi] - 0) - F(0)}
m sin [xi]^1
[pi][mu] 4
< -------- {[f](x + 2[mu]) - [f](x)} + ---------- {[f](x + 2[mu]) - [f](x)}
sin [mu] m sin [mu]
4
+ ------------ {[f](x + [pi]) - [f](x)}
m sin [xi]^1