§2. Uniform convergence
221
A consequence, which will be generalised in nO 13:
Theorem 9. The sum of a normally convergent series of continuous functions is continuous 15 •
Example 1. The sum of a trigonometric series
00
00
(8.6)
Co + Lan. cosnx + L bn . sinnx
n=l
n=l
whose coefficients satisfy
is a continuous function on lR [this is by no means the case for the square wave
series of nO 2 which we shall reexamine in nO 11]. We shall prove (Chap. VII)
that the Fourier series of periodic functions having an everywhere continuous
derivative are of this kind, though this condition is not necessary16.
Example 2. Consider the Riemann series
((s) = L l/n s
for s > 1, not necessarily an integer. Since as is an increasing function of s
for all a > 1,
l/n s ::; l/n u for s 2: 0'
and since the series L l/n u converges for a > 1, one deduces that the series
converges normally on every interval [0', +oo[ with 0' > 1, strict inequality.
15 The definition of normal convergence, and Theorem 9, extend in an obvious way
to sums indexed by any countable set of indices I. See nO 18.
16 The theory, whose more elementary aspects we shall expound in Chap. VII,
associates to each regulated periodic function a trigonometric series whose coefficients satisfy a definitely weaker relation, namely E(lan l 2 + Ibn l 2 ) < +00, as
for example in the case of the square wave series. The complete theory, valid for
"square integrable" functions in the sense of the Lebesgue integral, does not go
much further. To go beyond this type of series one has to use the "distributions"
of L. Schwartz (Chap. V), generalised functions to which one can assign Fourier
series whose coefficients increase no faster than a power of n, as we shall see in
Chap. VII.
Let us warn the reader: beside very simple elementary aspects, the theory of
trigonometric series presents very subtle aspects, far outside the framework of
absolutely or uniformly convergent series; happily one does not use them in most
domains of analysis, even though there are always specialists interested in them
and they form the topic of much interesting work.
221
A consequence, which will be generalised in nO 13:
Theorem 9. The sum of a normally convergent series of continuous functions is continuous 15 •
Example 1. The sum of a trigonometric series
00
00
(8.6)
Co + Lan. cosnx + L bn . sinnx
n=l
n=l
whose coefficients satisfy
is a continuous function on lR [this is by no means the case for the square wave
series of nO 2 which we shall reexamine in nO 11]. We shall prove (Chap. VII)
that the Fourier series of periodic functions having an everywhere continuous
derivative are of this kind, though this condition is not necessary16.
Example 2. Consider the Riemann series
((s) = L l/n s
for s > 1, not necessarily an integer. Since as is an increasing function of s
for all a > 1,
l/n s ::; l/n u for s 2: 0'
and since the series L l/n u converges for a > 1, one deduces that the series
converges normally on every interval [0', +oo[ with 0' > 1, strict inequality.
15 The definition of normal convergence, and Theorem 9, extend in an obvious way
to sums indexed by any countable set of indices I. See nO 18.
16 The theory, whose more elementary aspects we shall expound in Chap. VII,
associates to each regulated periodic function a trigonometric series whose coefficients satisfy a definitely weaker relation, namely E(lan l 2 + Ibn l 2 ) < +00, as
for example in the case of the square wave series. The complete theory, valid for
"square integrable" functions in the sense of the Lebesgue integral, does not go
much further. To go beyond this type of series one has to use the "distributions"
of L. Schwartz (Chap. V), generalised functions to which one can assign Fourier
series whose coefficients increase no faster than a power of n, as we shall see in
Chap. VII.
Let us warn the reader: beside very simple elementary aspects, the theory of
trigonometric series presents very subtle aspects, far outside the framework of
absolutely or uniformly convergent series; happily one does not use them in most
domains of analysis, even though there are always specialists interested in them
and they form the topic of much interesting work.