Page 305 - Matrix Analysis & Applied Linear Algebra

5.4 Orthogonal Vectors 301

where the Fourier coefficients are given by

π
1 1
α 0 = √ f = √ f(t)dt ,
2π 2π −π
π
cos kt 1
α k = √ f = √ f(t) cos kt dt for k =1, 2, 3,... ,
π π −π
π
sin kt 1
β k = √ f = √ f(t) sin kt dt for k =1, 2, 3,... .
π π −π
Substituting these coefficients in (5.4.4) produces the infinite series

∞
a 0
F(t)= + (a n cos nt + b n sin nt) , (5.4.5)
2
n=1
where
π π
1 1
a n = f(t) cos nt dt and b n = f(t) sin nt dt. (5.4.6)
π −π π −π

The series F(t)in (5.4.5) is called the Fourier series expansion for f(t), but,
unlike the situation in finite-dimensional spaces, F(t) need not agree with the
original function f(t). After all, F is periodic, so there is no hope of agreement
when f is not periodic. However, the following statement is true.
• If f(t)isa periodic function with period 2π that is sectionally continu-
41
ous on the interval (−π, π), then the Fourier series F(t) converges to
f(t)at each t ∈ (−π, π), where f is continuous. If f is discontinuous
at t 0 but possesses left-hand and right-hand derivatives at t 0 , then F(t 0 )
converges to the average value

+
f(t )+ f(t )
−
0
0
F(t 0 )= ,
2
+
−
−
where f(t ) and f(t ) denote the one-sided limits f(t )= lim t→t − f(t)
0
0
0
0
+
and f(t )= lim + f(t).
0 t→t
0
For example, the square wave function defined by

−1 when −π< t < 0,
f(t)=
1 when 0 <t<π,
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A function f is sectionally continuous on (a, b) when f has only a finite number of discon-
tinuities in (a, b) and the one-sided limits exist at each point of discontinuity as well as at the
end points a and b.