Page 351 - Schaum's Outline of Theory and Problems of Advanced Calculus

342 FOURIER SERIES [CHAP. 13

Therefore,
n n
1
X
b n sin x cos t
L L
y ¼
n¼1
with the coefficients b n defined above.

ORTHOGONAL FUNCTIONS
Two vectors A and B are called orthogonal (perpendicular) if A B ¼ 0or A 1 B 1 þ A 2 B 2 þ A 3 B 3 ¼ 0,
where A ¼ A 1 i þ A 2 j þ A 3 k and B ¼ B 1 i þ B 2 j þ B 3 k. Although not geometrically or physically evi-
dent, these ideas can be generalized to include vectors with more than three components. In particular,
we can think of a function, say, AðxÞ,as being a vector with an infinity of components (i.e., an infinite
dimensional vector), the value of each component being specified by substituting a particular value of x in
some interval ða; bÞ.Itis natural in such case to define two functions, AðxÞ and BðxÞ,as orthogonal in
ða; bÞ if

ð b
AðxÞ BðxÞ dx ¼ 0 ð9Þ
a
2
A vector A is called a unit vector or normalized vector if its magnitude is unity, i.e., if A A ¼ A ¼ 1.
Extending the concept, we say that the function AðxÞ is normal or normalized in ða; bÞ if
ð b
2
fAðxÞg dx ¼ 1 ð10Þ
a
From the above it is clear that we can consider a set of functions f k ðxÞg; k ¼ 1; 2; 3; ... ; having the
properties

ð b
m ðxÞ n ðxÞ dx ¼ 0 m 6¼ n ð11Þ
a
ð b
2
f m ðxÞg dx ¼ 1 m ¼ 1; 2; 3; ... ð12Þ
a
In such case, each member of the set is orthogonal to every other member of the set and is also
normalized. We call such a set of functions an orthonormal set.
The equations (11) and (12) can be summarized by writing

ð b
m ðxÞ n ðxÞ dx ¼ mn ð13Þ
a
where mn , called Kronecker’s symbol,is defined as 0 if m 6¼ n and1if m ¼ n.
Just as any vector r in three dimensions can be expanded in a set of mutually orthogonal unit vectors
i; j; k in the form r ¼ c 1 i þ c 2 j þ c 3 k,sowe consider the possibility of expanding a function f ðxÞ in a set
of orthonormal functions, i.e.,

1
X
a @ x @ b
f ðxÞ¼ c n n ðxÞ ð14Þ
n¼1
As we have seen, Fourier series are constructed from orthogonal functions. Generalizations of
Fourier series are of great interest and utility both from theoretical and applied viewpoints.