§2. Uniform convergence
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X i-+ X, does not converge uniformly. It is clear that the fn are not equicontinuous at the point x = 0 since, to satisfy the inequality Ifn(x) - fn(O)1 < r,
it is necessary to suppose Ixl < r/n (or to confine oneself to a neighbourhood of x = 1, which is not an answer to the problem). For a given r there
is no r' > 0 which works for all the functions fn on a neighbourhood of x = O.
Uniform convergence features in many very general theorems on approximation of functions by much simpler functions. Some of these results can
be stated very easily, but their proofs are not accessible at this stage of the
exposition.
First there is Weierstrass' famous approximation theorem: every real function f defined and continuous on a compact interval I is the limit of a sequence of polynomials which converges to f uniformly on I. In other words,
for every r > 0 there exists a polynomial p( x) such that If (x) - p( x) I < r
for all x E I. Weierstrass preferred to state his theorem in terms of a series
of polynomials, but this clearly comes to the same. We will establish it in
Chap. V, n° 28.
To state the second result, also due to Weierstrass, let us call any function
of the form
p(x) = Co + al cos x + b1 sin x + ... + an cosnx + bn sin nx
(where co, ... ,bn are a finite set of complex constants) a trigonometric polynomial of period 27f. Then every function defined, continuous, and of period
27f on JR., is the uniform limit of a sequence of trigonometric polynomials on
JR6.
We remark that this second result immediately provides another, analogous to the first. Consider a function f defined and continuous on a compact
interval I = [a, b], and suppose b < a + 27f (strict inequality). Consider a
continuous function g, defined on the interval [b, a + 27f], equal to feb) for
x = b and to f(a) for x = a + 27f; we may define a periodic function h on JR.
by requiring it to be equal to f on I and to 9 between b and a + 27f (figure 7).
The function h is clearly continuous at all points of JR.. If one applies
the preceding theorem to h, and then confines oneself to examining what
happens on I, one sees that for every r > 0 there exists a trigonometric
polynomialp such that If(x)-p(x)1 < r for all x E I. The hypothesis imposed
on I is essential since the trigonometric polynomials have period 27f; if, for
example, I is the interval [a, a + 37f], one will have p( x + 27f) = p( x) for
a S; x S; a + 7f, so that every function which is the uniform, or even simple,
limit of trigonometric polynomials on I must possess the same property,
which is clearly not the case in general.
Likewise, Weierstrass' first theorem does not apply in the case of a noncompact interval I. If I is not bounded, then every uniform limit f of poly6 This result does not say that every periodic continuous function is the sum of
its Fourier series; this is false.
209
X i-+ X, does not converge uniformly. It is clear that the fn are not equicontinuous at the point x = 0 since, to satisfy the inequality Ifn(x) - fn(O)1 < r,
it is necessary to suppose Ixl < r/n (or to confine oneself to a neighbourhood of x = 1, which is not an answer to the problem). For a given r there
is no r' > 0 which works for all the functions fn on a neighbourhood of x = O.
Uniform convergence features in many very general theorems on approximation of functions by much simpler functions. Some of these results can
be stated very easily, but their proofs are not accessible at this stage of the
exposition.
First there is Weierstrass' famous approximation theorem: every real function f defined and continuous on a compact interval I is the limit of a sequence of polynomials which converges to f uniformly on I. In other words,
for every r > 0 there exists a polynomial p( x) such that If (x) - p( x) I < r
for all x E I. Weierstrass preferred to state his theorem in terms of a series
of polynomials, but this clearly comes to the same. We will establish it in
Chap. V, n° 28.
To state the second result, also due to Weierstrass, let us call any function
of the form
p(x) = Co + al cos x + b1 sin x + ... + an cosnx + bn sin nx
(where co, ... ,bn are a finite set of complex constants) a trigonometric polynomial of period 27f. Then every function defined, continuous, and of period
27f on JR., is the uniform limit of a sequence of trigonometric polynomials on
JR6.
We remark that this second result immediately provides another, analogous to the first. Consider a function f defined and continuous on a compact
interval I = [a, b], and suppose b < a + 27f (strict inequality). Consider a
continuous function g, defined on the interval [b, a + 27f], equal to feb) for
x = b and to f(a) for x = a + 27f; we may define a periodic function h on JR.
by requiring it to be equal to f on I and to 9 between b and a + 27f (figure 7).
The function h is clearly continuous at all points of JR.. If one applies
the preceding theorem to h, and then confines oneself to examining what
happens on I, one sees that for every r > 0 there exists a trigonometric
polynomialp such that If(x)-p(x)1 < r for all x E I. The hypothesis imposed
on I is essential since the trigonometric polynomials have period 27f; if, for
example, I is the interval [a, a + 37f], one will have p( x + 27f) = p( x) for
a S; x S; a + 7f, so that every function which is the uniform, or even simple,
limit of trigonometric polynomials on I must possess the same property,
which is clearly not the case in general.
Likewise, Weierstrass' first theorem does not apply in the case of a noncompact interval I. If I is not bounded, then every uniform limit f of poly6 This result does not say that every periodic continuous function is the sum of
its Fourier series; this is false.