Calculus: Analytic Geometry and Calculus, with Vectors

12.3 Alternating series and Fourier series 613

Such sets are said to be complete. While proofs of such things are so

long and devious that nobody should expect to be able to originate them

in a few days, it can be proved that the set in (12.32) is complete. The

set

(12.365) 17r

(^2) 'r r2 27r 3
1 I
L cos L x, cos x, L cos
Lx,
is also a complete orthonormal set over E when E is the interval 0 <--x
L. The set
(12.366)
(^11) a 1 a 1 2a 1 21r

• cosLx, =sinLx, cos
  I

x,Lsin L

is a complete orthonormal set over E when E is an interval of length 2L.
The world's mathematical storehouse contains many other usefulcom-
plete orthonormal sets, and the above formulas have many important
applications. Henceforth we suppose that q5i, 02, q53, is the trigo-

nometric orthonormal set appearing in (12.32) or (12.365) or (12.366)
and that f has period 2L so that f(x + 2L) = f(x) for each x. Even in
this case, fundamental problems involving validity of the formula

(12.37) f(x) = alol(x) + a202(x) + a4'03(x) +...

remain unsolved. It is, however, known that (12.37) is valid over

-- < x < - provided (i) f has period 2L, (ii) f is bounded and piece-

wise monotone over -L < x <- L, (iii)

(12.371) lim

f(x + h) + f(x - h)_ f(x),

h-*O 2

(iv) f is odd so thatf(-x) = -f(x) in case the orthonormal set is (12.32),

and (v) f is even so that f(-x) = f(x) in case the orthonormal set is

(12.365).

The most illuminating batch of applications of the above ideas involves

the Bernoulli functions Bo(x), B1(x), B2(x), that appeared in Sec-

tion 4.3, Problem 10, and in Section 5.3, Problem 19. These are the

functions of period 1 for which Bo(x) = 1,

(12.381) B'(x) = (n = 1,2,3, )

(12.382) foI B (x) dx = 0 (n = 1,2,3, )

except that (12.381) fails to hold when n is 1 or 2 and x is an integer.

In particular, Bi(x) is the saw-tooth function for which Bi(x) = 0 when

x is an integer and

(12.383) Bi(x) = x - [x] - ji