I(a0+∆a0, . . . , bn+∆bn) =I(a0, . . . , bn) +∆I, (6.4.16) where ∆I stands for the remaining terms. Since the first derivatives, all mixed second derivatives, and all remaining higher derivatives vanish, we obtain
∆I = 1 2!
∂2I
∂a20∆a20+ n k=1
∂2I
∂a2k∆a2k+∂2I
∂b2k ∆b2k
. (6.4.17) By virtue of equations (6.4.14) and (6.4.15),∆I is positive. Hence, forIto have a minimum value, the coefficientsa0,ak,bkmust be given by equations (6.4.11), (6.4.12), and (6.4.13) respectively. These coefficients are called the Fourier coefficientsoff(x) and the series in (6.4.1) is said to be theFourier seriescorresponding to f(x), where its coefficients a0,ak andbk are given by (6.4.11), (6.4.12), and (6.4.13) respectively. Thus, the correspondence (6.4.1) asserts nothing about the convergence or divergence of the formally constructed Fourier series. The question arises whether it is possible to represent all continuous functions by Fourier series. The investigation of the sufficient conditions for such a representation to be possible turns out to be a difficult problem.
We remark that the possibility of representing the given functionf(x) by a Fourier series does not imply that the Fourier series converges to the functionf(x). If the Fourier series of a continuous function converges uniformly, then it represents the function. As a matter of fact, there exist Fourier series which diverge. A convergent trigonometric series need not be a Fourier series. For instance, the trigonometric series
∞ n=2
sinnx logn,
which is convergent for all values ofx, is not a Fourier series, for there is no integrable function corresponding to this series.
6.5 Convergence of Fourier Series
We introduce three kinds of convergence of a Fouriers Series: (i) Point- wise Convergence, (ii) Uniform Convergence, and (iii) Mean-Square Con- vergence.
Definition 6.5.1.(Pointwise Convergence). An infinite series2∞
n=1fn(x) is called pointwise convergentin a < x < btof(x)if it converges to f(x) for eachxina < x < b. In other words, for each xin a < x < b, we have
|f(x)−sn(x)| →0 as n→ ∞, wheresn(x)is thenth partial sum defined by sn(x) =2n
k=1fk(x).
Definition 6.5.2.(Uniform Convergence). The series2∞
n=1fn(x)is said to converge uniformlytof(x)ina≤x≤b if
amax≤x≤b|f(x)−sn(x)| →0 as n→ ∞.
Evidently, uniform convergence implies pointwise convergence, but the con- verse is not necessarily true.
Definition 6.5.3.(Mean-Square Convergence). The series 2∞
n=1fn(x) converges in the mean-square(orL2) sense tof(x)ina≤x≤b if
b
a |f(x)−sn(x)|2dx→0 as n→ ∞.
It is noted that uniform convergence is stronger than both pointwise convergence and mean-square convergence.
The study of convergence of Fourier series has a long and complex his- tory. The fundamental question is whether the Fourier series of a periodic functionf converge tof. The answer is certainlynot obvious. Iff(x) is 2π- periodic continuous function, then the Fourier series (6.4.1) may converge to f for a givenxin −π≤x≤π, butnot for all xin −π≤x≤π. This leads to the questions oflocal convergenceor the behavior off near a given point x, and of global convergenceor the overall behavior of a function f over the entire interval [−π, π].
There is another question that deals with the mean-square convergence of the Fourier series to f(x) in (−π, π), that is, if f(x) is integrable on (−π, π), then
1 2π
π
−π|f(x)−sn(x)|2dx→0 as n→ ∞.
This is known as the mean-square convergence theorem which does not provide any insight into the problem of pointwise convergence. Indeed, the mean-square convergence theorem does not guarantee the convergence of the Fourier series for anyx. On the other hand, iff(x) is 2π-periodic and piecewise smooth onR, then the Fourier series (6.4.1) of the functionfcon- verges for everyxin−π≤x≤π. It has been known since 1876 that there are periodic continuous functions whose Fourier series diverge at certain points. It was an open question for a period of a century whether a Fourier series of a continuous function converges at any point. In 1966, Lennart Carleson (1966) provided an affirmative answer with a deep theorem which states that the Fourier series of any square integrable functionf converges tof at almost every point.
Let f(x) be piecewise continuous and periodic with period 2π. It is obvious that
6.5 Convergence of Fourier Series 175 π
−π
[f(x)−sn(x)]2dx≥0, (6.5.1) Expanding (6.5.1) gives
π
−π
[f(x)−sn(x)]2dx= π
−π
[f(x)]2dx−2 π
−π
f(x)sn(x)dx +
π
−π
[sn(x)]2dx.
But, by the definitions of the Fourier coefficients (6.4.11), (6.4.12), and (6.4.13) and by the orthogonal relations for the trigonometric series (6.3.4), we have
π
−π
f(x)sn(x)dx= π
−π
f(x) a0
2 + n k=1
(akcoskx+bksinkx)
dx
= πa20 2 +π
n k=1
a2k+b2k , (6.5.2)
and
π
−π
s2n(x)dx= π
−π
a0
2 + n k=1
(akcoskx+bksinkx) 2
dx
= πa20 2 +π
n k=1
a2k+b2k . (6.5.3)
Consequently, π
−π
[f(x)−sn(x)]2dx= π
−π
f2(x)dx− πa20
2 +π n k=1
a2k+b2k
≥0.
(6.5.4) It follows from (6.5.4) that
a20 2 +
n k=1
a2k+b2k ≤ 1 π
π
−π
f2(x)dx (6.5.5) for all values ofn. Since the right hand of equation (6.5.5) is independent ofn, we obtain
a20 2 +
∞ k=1
a2k+b2k ≤ 1 π
π
−π
f2(x)dx. (6.5.6) This is known asBessel’s inequality.
We see that the left side is nondecreasing and is bounded above, and therefore, the series
a20 2 +
∞ k=1
a2k+b2k , (6.5.7)
converges. Thus, the necessary condition for the convergence of series (6.5.7) is that
klim→∞ak = 0, lim
k→∞bk = 0. (6.5.8)
The Fourier series is said toconverge in the mean tof(x) when
nlim→∞
π
−π
f(x)−
a0
2 + n k=1
akcoskx+bksinkx 2
dx= 0. (6.5.9) If the Fourier series converges in the mean tof(x), then
a20 2 +
∞ k=1
a2k+b2k = 1 π
π
−π
f2(x)dx. (6.5.10) This is called Parseval’s relation and is one of the central results in the theory of Fourier series. This relation is frequently used to derive the sum of many important numerical infinite series. Furthermore, if the relation (6.5.9) holds true, the set of trigonometric functions 1, cosx, sinx, cos 2x, sin 2x,. . .is said to be complete.
The Parseval relation (6.5.10) can formally be derived from the conver- gence of Fourier series tof(x) in [−π, π]. In other words, if
f(x) = 1 2a0+
∞ k=1
(akcoskx+bksinkx), (6.5.11) wherea0,akandbkare given by (6.4.11), (6.4.12) and (6.4.13) respectively, we multiply ( 6.4.11) byπ1f(x) and integrate the resulting expression from
−πto πto obtain 1
π π
k=1
f2(x)dx= a0
2π π
−π
f(x)dx+ ∞ k=1
ak
π π
−π
f(x) coskx dx + bk
π π
−π
f(x) sinkx dx
.(6.5.12) Replacing all integrals on the right hand side of (6.5.12) by the Fourier coefficients gives the Parseval relation (6.5.10).