238
III - Convergence: Continuous variables
Example 1. Theorem 15 applies to certain Fourier series 23 , for example to
the square wave series cos x - cos3x/3 + cos5x/5 etc. already considered in
nO 9, example 5. On choosing
Un = (_I)n-l cos(2n - I)x,
Vn = I/(2n - 1),
it reduces to showing that the sums
(11.8)
cos x - cos3x + cos5x - ... + (_I)n-l cos(2n - I)x
are bounded. Now the famous formula 2 cos a. cos b = ... shows that
2 cosx. cos x = 1 + cos 2x, 2cosx. cos 3x = cos 2x + cos 4x, etc .. j
on multiplying the sum (8) by 2 cos x one finds, thanks to the alternating
signs, that
whence
(11.9)
cos x - cos 3x + cos 5x - ... + (_I)n-l cos(2n - I)x =
= [1 + (_I)n-l cos2nx] /2 cos x,
Icosx - cos 3x + ... + (_I)n-l cos(2n - I)xl ::; 1/1 cosxl
for any n, of course on condition that cos x i:- 0, i.e. that x is not an odd
multiple of 71"/2. Theorem 15 now shows that the series converges apart from
at these values (and fails to converge at these excluded values ... ) and that
the remainder rp(x) = sex) - sp(x) satisfies, by (2) and vp = I/(2p - 1),
(11.10)
Irp(x)1 ::; 2/(2p -1)1 cosxl·
On a set K where I cos xl remains greater than a fixed number> 0, one then
obtains a bound by I/p up to a factor independent ofx E K: whence uniform
convergence on K, so in particular - but in practice this comes to the same
-, on every interval of the form [-71"/2 + 8, 71" /2 - 8] or [71"/2 + 8, 371"/2 - 8] with
8 > 0. The diagram for example 5 of nO 8 therefore corresponds to reality:
for all 8 and r > 0, the partial sum of order n is, for all sufficiently large n,
everywhere equal up to r to the total sum between ° and 71"/2 - 8, similarly
between 71"/2+8 and 71". But since the total sum passes from 71"/4 to -71"/4 when
one traverses 71"/2, the function is forced, for any n, to decrease precipitously
on the interval [71"/2 - 8,71"/2 + 81 if it is to remain continuous.
One can generalise to series of the form L an cos nx or an sin nx. In the
first case, one observes, thanks again to high school trigonometry, proof that
the latter is not entirely useless, that
23 Hardly surprising. Dirichlet came to Paris in 1822 (he was then 17), and remained
there for four years. Because of this he was up to date with Fourier's work and
Cauchy's first research on the foundations of Analysis. He proved the first general
result on the convergence of Fourier series in 1829.
III - Convergence: Continuous variables
Example 1. Theorem 15 applies to certain Fourier series 23 , for example to
the square wave series cos x - cos3x/3 + cos5x/5 etc. already considered in
nO 9, example 5. On choosing
Un = (_I)n-l cos(2n - I)x,
Vn = I/(2n - 1),
it reduces to showing that the sums
(11.8)
cos x - cos3x + cos5x - ... + (_I)n-l cos(2n - I)x
are bounded. Now the famous formula 2 cos a. cos b = ... shows that
2 cosx. cos x = 1 + cos 2x, 2cosx. cos 3x = cos 2x + cos 4x, etc .. j
on multiplying the sum (8) by 2 cos x one finds, thanks to the alternating
signs, that
whence
(11.9)
cos x - cos 3x + cos 5x - ... + (_I)n-l cos(2n - I)x =
= [1 + (_I)n-l cos2nx] /2 cos x,
Icosx - cos 3x + ... + (_I)n-l cos(2n - I)xl ::; 1/1 cosxl
for any n, of course on condition that cos x i:- 0, i.e. that x is not an odd
multiple of 71"/2. Theorem 15 now shows that the series converges apart from
at these values (and fails to converge at these excluded values ... ) and that
the remainder rp(x) = sex) - sp(x) satisfies, by (2) and vp = I/(2p - 1),
(11.10)
Irp(x)1 ::; 2/(2p -1)1 cosxl·
On a set K where I cos xl remains greater than a fixed number> 0, one then
obtains a bound by I/p up to a factor independent ofx E K: whence uniform
convergence on K, so in particular - but in practice this comes to the same
-, on every interval of the form [-71"/2 + 8, 71" /2 - 8] or [71"/2 + 8, 371"/2 - 8] with
8 > 0. The diagram for example 5 of nO 8 therefore corresponds to reality:
for all 8 and r > 0, the partial sum of order n is, for all sufficiently large n,
everywhere equal up to r to the total sum between ° and 71"/2 - 8, similarly
between 71"/2+8 and 71". But since the total sum passes from 71"/4 to -71"/4 when
one traverses 71"/2, the function is forced, for any n, to decrease precipitously
on the interval [71"/2 - 8,71"/2 + 81 if it is to remain continuous.
One can generalise to series of the form L an cos nx or an sin nx. In the
first case, one observes, thanks again to high school trigonometry, proof that
the latter is not entirely useless, that
23 Hardly surprising. Dirichlet came to Paris in 1822 (he was then 17), and remained
there for four years. Because of this he was up to date with Fourier's work and
Cauchy's first research on the foundations of Analysis. He proved the first general
result on the convergence of Fourier series in 1829.