1. Orthogonal Functions and Fourier Series
CHAPTER 12
Orthogonal Functions and Fourier Series

2. Contents 12.1 Orthogonal Functions 12.2 Fourier Series
12.3 Fourier Cosine and Sine Series
12.4 Complex Fourier Series
12.5 Strum-Liouville Problems
12.6 Bessel and Legendre Series

3. 12.1 Orthogonal Functions Inner Product of Function
DEFINITION 12.1
The inner product of two functions f1 and f2 on an
interval [a, b] is the number
Inner Product of Function
Two functions f1 and f2 are said to be orthogonal on an
interval [a, b] if
DEFINITION 12.2
Orthogonal Function

4. Example
The function f1(x) = x2, f2(x) = x3 are orthogonal on the interval [−1, 1] since

5. A set of real-valued functions {0(x), 1(x), 2(x), …}
DEFINITION 12.3
A set of real-valued functions {0(x), 1(x), 2(x), …}
is said to be orthogonal on an interval [a, b] if
(2)
Inner Product of Function

6. Orthonormal Sets
The expression (u, u) = ||u||2 is called the square norm. Thus we can define the square norm of a function as (3) If {n(x)} is an orthogonal set on [a, b] with the property that ||n(x)|| = 1 for all n, then it is called an orthonormal set on [a, b].

7. Example 1
Show that the set {1, cos x, cos 2x, …} is orthogonal on [−, ].
Solution Let 0(x) = 1, n(x) = cos nx, we show that

8. Example 1 (2)
and

9. Example 2
Find the norms of each functions in Example 1.
Solution

10. Vector Analogy
Recalling from the vectors in 3-space that (4) we have (5) Thus we can make an analogy between vectors and functions.

11. Orthogonal Series Expansion
Suppose {n(x)} is an orthogonal set on [a, b]. If f(x) is defined on [a, b], we first write as (6) Then

12. Since {n(x)} is an orthogonal set on [a, b], each term on the right-hand side is zero except m = n. In this case we have

13. In other words, (7) (8) Then (7) becomes (9)

14. A set of real-valued functions {0(x), 1(x), 2(x), …}
DEFINITION 12.4
A set of real-valued functions {0(x), 1(x), 2(x), …}
is said to be orthogonal with respect to a weight
function w(x) on [a, b], if
Orthogonal Set/Weight Function
Under the condition of the above definition, we have (10) (11)

15. Complete Sets
An orthogonal set is complete if the only continuous function orthogonal to each member of the set is the zero function.

16. 12.2 Fourier Series
Trigonometric Series We can show that the set (1) is orthogonal on [−p, p]. Thus a function f defined on [−p, p] can be written as (2)

17. Now we calculate the coefficients
Now we calculate the coefficients (3) Since cos(nx/p) and sin(nx/p) are orthogonal to 1 on this interval, then (3) becomes Thus we have (4)

18. In addition, (5) by orthogonality we have

19. and Thus (5) reduces to and so (6)

20. Finally, if we multiply (2) by sin(mx/p) and use and we find that (7)

21. The Fourier series of a function f defined on the
DEFINITION 12.5
The Fourier series of a function f defined on the
interval (−p, p) is given by (8) where (9) (10) (11)
Fourier Series

22. Example 1 Expand (12) in a Fourier series.
Solution The graph of f is shown in Fig 12.1 with p = .

23. Example 1 (2)
←cos n = (-1)n

24. Example 1 (3)
From (11) we have Therefore (13)

25. Fig 12.1

26. Let f and f’ be piecewise continuous on the interval (−p, p); that
THEOREM 12.1
Let f and f’ be piecewise continuous on the interval (−p, p); that
is, let f and f’ be continuous except at a finite number of points
in the interval and have only finite discontinuous at these points.
Then the Fourier series of f on the interval converge to f(x) at a
point of continuity. At a point of discontinuity, the Fourier
series converges to the average where f(x+) and f(x－) denote the limit of f at x from the right
and from the left, respectively.
Criterion for Convergence

27. Example 2
Referring to Example 1, function f is continuous on (−, ) except at x = 0. Thus the series (13) will converge to at x = 0.

28. Periodic Extension
Fig 12.2 is the periodic extension of the function f in Example 1. Thus the discontinuity at x = 0, 2, 4, …will converge to and at x = , 3, … will converge to

29. Fig 12.2

30. Sequence of Partial Sums
Sequence of Partial Sums Referring to (13), we write the partial sums as See Fig 12.3.

31. Fig 12.3

32. 12.3 Fourier Cosine and Sine Series
Even and Odd Functions
even if f(−x) = f(x)
odd if f(−x) = −f(x)

33. Fig 12.4 Even function

34. Fig 12.5 Odd function

35. (a) The product of two even functions is even.
THEOREM 12.2
(a) The product of two even functions is even.
(b) The product of two odd functions is even.
(c) The product of an even function and an odd function is odd.
(d) The sum (difference) of two even functions is even.
(e) The sum (difference) of two odd functions is odd.
(f) If f is even then
(g) If f is odd then
Properties of Even/Odd Functions

36. Cosine and Sine Series
If f is even on (−p, p) then Similarly, if f is odd on (−p, p) then

37. DEFINITION 12.6
(i) The Fourier series of an even function f on the interval (−p, p) is the cosine series (1) where (2) (3)
Fourier Cosine and Sine Series

38. (continued)
DEFINITION 12.6
(ii) The Fourier series of an odd function f on the interval (−p, p) is the sine series (4) where (5)
Fourier Cosine and Sine Series

39. Example 1 Expand f(x) = x, −2 < x < 2 in a Fourier series.
Solution Inspection of Fig 12.6, we find it is an odd function on (−2, 2) and p = Thus (6) Fig 12.7 is the periodic extension of the function in Example 1.

40. Fig 12.6

41. Fig 12.7

42. Example 2
The function shown in Fig 12.8 is odd on (−, ) with p = . From (5), and so (7)

43. Fig 12.8

44. Gibbs Phenomenon
Fig 12.9 shows the partial sums of (7). We can see there are pronounced spikes near the discontinuities. This overshooting of SN does not smooth out but remains fairly constant even when N is large. This is so-called Gibbs phenomenon.

45. Fig 12.9

46. Half-Range Expansions
If a function f is defined only on 0 < x < L, we can make arbitrary definition of the function on −L < x < 0.
If y = f(x) is defined on 0 < x < L,
reflect the graph about the y-axis onto −L < x < 0; the function is now even. See Fig
reflect the graph through the origin onto −L < x < 0; the function is now odd. See Fig
define f on −L < x < 0 by f(x) = f(x + L). See Fig

47. Fig 12.10

48. Fig 12.11

49. Fig 12.12

50. Example 3
Expand f(x) = x2, 0 < x < L, (a) in a cosine series, (b) in a sine series (c) in a Fourier series.
Solution The graph is shown in Fig

51. Example 3 (2)
(a) Then (8)

52. Example 3 (3)
(b) Hence (9)

53. Example 3 (4)
(c) With p = L/2, n/p = 2n/L, we have Therefore (10) The graph of these periodic extension are shown in Fig

54. Fig 12.14

55. Periodic Driving Force
Consider the following physical system (11) where (12) is a half-range sine expansion.

56. Example 4
Referring to (11), m = 1/16 slug, k = 4 lb/ft, the force f(t) with 2-period is shown in Fig Though f(t) acts on the system for t > 0, we can extend the graph in a 2-periodic manner to the negative t-axis to obtain an odd function. With p = 1, from (5) we have From (11) we have (13)

57. Example 4 (1)
To find a particular solution xp(t), we substitute (12) into (13). Thus Therefore (14)

58. 12.4 Complex Fourier Series
Euler’s formula eix = cos x + i sin x e-ix = cos x  i sin x (1)

59. Complex Fourier Series
From (1), we have (2) Using (2) to replace cos(nx/p) and sin(nx/p), then (3)

60. where c0 = a0/2, cn = (an  ibn)/2, c-n = (an + ibn)/2
where c0 = a0/2, cn = (an  ibn)/2, c-n = (an + ibn)/2. When the function f is real, cn and c-n are complex conjugates. We have (4)

61. (5)

62. (6)

63. The Complex Fourier Series of function f defined on
DEFINITION 12.7
The Complex Fourier Series of function f defined on
an interval (p, p)is given by (7) where (8)
Complex Fourier Series

64. If f satisfies the hypotheses of Theorem 12
If f satisfies the hypotheses of Theorem 12.1, a complex Fourier series converges to f(x) at a point of continuity and to the average at a point of discontinuity.

65. Example 1
Expand f(x) = e-x,  < x <, in a complex Fourier series.
Solution with p = , (8) gives

66. Example 1 (2)
Using Euler’s formula Hence (9)

67. Example 1 (3)
The complex Fourier series is then (10) The series (10) converges to the 2-periodic extension of f.

68. Fundamental Frequency
The fundamental period is T = 2p and then p = T/2. The Fourier series becomes (11) where  = 2/T is called the fundamental angular frequency.

69. Frequency Spectrum
If f is periodic and has fundamental period T, the plot of the points (n, |cn|) is called the frequency spectrum of f.

70. Example 2
In Example 1,  = 1, so that n takes on the values 0, 1, 2, … Using , we see from (9) that See Fig

71. Fig 12.17

72. Example 3
Find the spectrum of the wave shown in Fig The wave is the periodic extension of the function f:

73. Example 3 (2)
Solution Here T = 1 = 2p so p = ½. Since f is 0 on (½, ¼) and (¼, ½), (8) becomes

74. Example 3 (3)
It is easy to check that Fig shows the frequency spectrum of f.

75. Fig 12.19

76. 12.5 Sturm-Liouville Problem
Eigenvalue and Eigenfunctions Recall from Example 2, Sec (1) This equation possesses nontrivial solutions only when  took on the values n = n22/L2, n = 1, 2, 3,…called eigenvalues. The corresponding nontrivial solutions y = c2 sin(nx/L) or simply y = sin(nx/L) are called the eigenfunctions.

77. Example 1
It is left as an exercise to show the three possible cases:  = 0,  = 2 < 0,  = 2 > 0, ( > 0), that the eigenvalues and eigenfunctions for (2) are respectively n = n2 = n22/L2, n = 0, 1, 2, …and y = c1 cos(nx/L), c1  0.

78. Regular Sturm-Liouville Problem
Let p, q, r and r be real-valued functions continuous on [a, b], and let r(x) > 0 and p(x) > 0 for every x in the interval. Then Solve (3) Subject to (4) (5) is said to be a regular Sturm-Liouville problem. The coefficients in (4), (5) are assumed to be real and independent of .

79. THEOREM12.3
There exist an infinite number of real eigenvalues that can be arranged in increasing order 1 < 2 < 3 < … < n < … such that n →  as n → .
For each eigenvalue there is only one eigenfunction (except for nonzero constant multiples).
Eigenfunctions corresponding to different eigenvalues are linearly independent.
The set of eigenfunctions corresponding to the set of eigenvalues is orthogonal with respect to the weight function p(x) on the interval [a, b].
Properties of the Regular
Strum-Liouville Problem

80. Proof of (d) Let ym and yn be eigenfunctions corresponding to eigenvalues m and n. Then (6) (7) From (6)yn  (7)ym we have

81. Integrating the above equation from a to b, then
Integrating the above equation from a to b, then (8) Since all solutions must satisfy the boundary condition (4) and (5), from (4) we have

82. For this system to be satisfied by A1 and B1 not both zero, the determinant must be zero Similarly from (5), we have Thus the right-hand side of (8) is zero. Hence we have the orthogonality relation (9)

83. Example 2
Solve (10)
Solution You should verify that for  = 0 and  < 0, (10) possesses only trivial solution. For  = 2 > 0,  > 0, the general solution is y = c1 cos x + c2 sin x. Now the condition y(0) = 0 implies c1 = 0, thus y = c2 sin x. The second condition y(1) + y(1) = 0 implies c2 sin  + c2 cos. = 0.

84. Example 2 (2)
Choosing c2  0, we have (11) From Fig 12.20, we see there are infinitely many solution for  > 0. It is easy to get the values of  > 0. Thus the eigenvalues are n = n2, n = 1, 2, 3, … and the corresponding eigenfunctions are yn = sin nx.

85. Fig 12.20

86. Singular Sturm-Liouville problem
There are several import conditions of (3)
r(a) = and a boundary condition of the type given in (5) is specified at x = b; (12)
r(b) = 0 and a boundary condition of the type given in (4) is specified at x = a. (13)
r(a) = r(b) = 0 and no boundary condition is specified at either x = a or at x = b; (14)
r(a) = r(b) and boundary conditions y(a) = y(b), y’(a) = y’(b) (15)

87. Notes:
Equation (3) satisfies (12) and (13) is said to be a singular BVP. Equation (3) satisfies (15) is said to be a periodic BVP.

88. By assuming the solutions of (3) are bounded on [a, b], from (8) we have
If r(a) = 0, then the orthogonality relation (9) holds with no boundary condition at x = a; (16)
If r(b) = 0 , then the orthogonality relation (9) holds no boundary condition at x = b; (17)
If r(a) = r(b) = 0, then the orthogonality relation (9) holds with no boundary conditions specified at either x = a or x = b; (18)
If r(a) = r(b), then the orthogonality relation (9) holds wuth peroidic boundary conditions y(a) = y(b), y’(a) = y’(b). (19)

89. Self-Adjoint Form
In fact (3) is the same as (20) Thus we can write the Legendre’s differential equation as (21) Here we find that the coefficient of y is the derivative of the coefficient of y.

90. In addition, if the coefficients are continuous and a(x)  0 for all x in some interval, then any second-order differential equation (22) can be recast into the so-called self-adjoint form (3).
To understand the above fact, we start from a1(x)y + a0(x)y = 0 Let P = a0/a1,  = exp( Pdx),  = P, then y + Py = 0, y + Py = 0, Thus d(y)/dx = 0.

91. Now for (22), let Y = y, and the integrating factor be e [b(x)/a(x)] dx. Then (22) becomes In summary, (22) can become

92. In addition, (23) is the same as (3)

93. Example 3
In Sec 5.3, we saw that the general solution of the parametric Bessel differential equation Dividing the Bessel equation by x2 and multiplying the resulting equation by integrating factor e [(1/x)] dx = eln x = x, we have

94. Example 3 (2)
Now r(0) = 0, and of the two solutions Jn(x) and Yn(x) only Jn(x) is bounded at x = 0. From (16), the set {Jn(ix)}, i = 1, 2, 3, …, is orthogonal with respect to the weight function p(x) = x on [0, b]. Thus (24) provided the i and hence the eigenvalues i = i2 are defined by a boundary condition at x = b of the type given by (5): A2Jn(b) + B2Jn(b) = 0 (25)

95. Example 4
From (21), we identify q(x) = 0, p(x) = 1 and  = n(n + 1). Recall from Sec 5.3 when n = 0, 1, 2, …, Legendre’s DE possesses polynomial solutions Pn(x). We observer that r(−1) = r(1) = 0 together with that fact that Pn(x) are the only solutions of (21) that are bounded on [−1, 1], to conclude that the set {Pn(x)}, n = 0, 1, 2, …, is orthogonal w.s.t. the weight function p(x) = 1 on [−1, 1]. Thus

96. 12.6 Bessel and Legendre Series
Fourier-Bessel Series We have shown that {Jn(ix)}, i = 1, 2, 3, …is orthogonal w.s.t. p(x) = x on [0, b] when the i are defined by (1) This orthogonal series expansion or generalized Fourier series of a function f defined on (0, b) in terms of this orthogonal set is (2) where (3)

97. The square norm of the function Jn(ix) is defined by
The square norm of the function Jn(ix) is defined by (4) The series (2) is called a Fourier-Bessel series.

98. Differential Recurrence Relations
Recalling from (20) and (21) in Sec 5.3, we have the differential recurrence relations as (5) (6)

99. Square Norm
The value of (4) is dependent on i = i2. If y = Jn(x) we have After we multiply by 2xy’, then

100. Integrating by parts on [0, b], then Since y = Jn(x), the lower limit is 0 for n > 0, because Jn(0) = 0. For n = 0, at x = 0. Thus (7) where y = Jn(x).

101. Now we consider three cases for the condition (1).
Case I: If we choose A2 = 1 and B2 = 0, then (1) is (8) There are an infinite number of positive roots xi = ib of (8) (see Fig 5.3), which defines i = xi/b. The eigenvalues are positive and then i = i2 = (xi/b)2. No new eigenvalues result from the negative roots of (8) since Jn(−x) = (−1)nJn(x).

102. The number 0 is not ab eigenvalue for any n since Jn(0) = 0, n= 1, 2, 3, … and J0(0) = 1. When (6) is written as xJn(x) = nJn(x) – xJn+1(x), it follows from (7) and (8) (9)

103. Case II: If we choose A2 = h  0 and B2 = b, then (1) is
Case II: If we choose A2 = h  0 and B2 = b, then (1) is (10) There are an infinite number of positive roots xi = ib for n = 1, 2, 3, …. As before i = i2 = (xi/b)2.  = 0 is not an eigenvalue for n = 1, 2, 3, …. Substituting ibJn(ib) = – hJn(ib) into (7), then (11)

104. Case III: If h = 0 and n = 0 in (10), i are defined from the roots of
Case III: If h = 0 and n = 0 in (10), i are defined from the roots of (12) Though (12) is a special case of (10), it is the only situation fro which  = 0 is an eigenvalue. For n = 0, the result in (6) implies J0(b) = 0 is equivalent to J1(b) = 0.

105. Since x1 = 1b = 0 is a root of the last equation and because J0(0) = 1 is nontrivial, we conclude that from 1 = 12 = (x1/b)2 that 1 is an eigenvalue. But we can not use (11) when 1 = 0, h = 0, n = 0, and n = 0. However from (4) we have (13) For i > we can use (11) with h = 0 and n = 0: (14)

106. The Fourier-Bessel series of a function f defined on
DEFINITION 12.8
The Fourier-Bessel series of a function f defined on
the interval (0, b) is given by
(i)
(15)
(16)
where the i are defined by Jn(b) = 0.
Fourier-Bessel Series

107. (continued)
DEFINITION 12.8
(ii)
(17)
(18)
where the i are defined by hJn(b) + bJ’n(b) = 0.
Fourier-Bessel Series

108. (continued) (iii) (19) (20) where the i are defined by J’0(b) = 0.
DEFINITION 12.8
(iii)
(19)
(20)
where the i are defined by J’0(b) = 0.
Fourier-Bessel Series

109. If f and f’ are piecewise continuous in the open interval
THEOREM 12.4
If f and f’ are piecewise continuous in the open interval
(0, b), then a Fourier-Bessel expansion of f converges
to f(x) at any point where f is continuous ant to the
converge [f(x+) + f(x-)] / 2 at a point where f is
discontinuous.
Conditions for Convergence

110. Example 1
Expand f(x) = x, 0 < x < 3, in a Fourier-Bessel series, using Bessel function of order one satisfying the boundary condition J1(3) = 0.
Solution We use (15) where ci is given by (16) with b = 3:

111. Example 1 (2)
Let t = i x, dx = dt/i, x2 = t2/i2, and use (5) in the form d[t2J2(t)]/dt = t2J1(t):

112. Example 2
If the i in Example 1 are defined by J1(3) + J1(3) = 0, the only thing changing in the expansion is the value of the square norm. Since 3J1(3) + 3J1(3) = 0 matching (10) when h = 3, b = 3 and n = 1. Thus (18) and (17) yield in turn

113. The Fourier-Legendre series of a function f defined
DEFINITION 12.9
The Fourier-Legendre series of a function f defined
on the interval (－1, 1) is given by
(i)
(21)
(22)
where the i are defined by Jn(b) = 0.
Fourier-Legendre Series

114. If f and f’ are piecewise continuous in the open interval
THEOREM 12.5
If f and f’ are piecewise continuous in the open interval
(－1, 1), then a Fourier-Legendre series (21) converges
to f(x) at a point of continuity ant to the converge
[f(x+) + f(x-) / 2 at a point of discontinuous.
Conditions for Convergence

115. Example 3
Write out the first four nonzero terms in the Fourier-Legendre expansion of
Solution From page 269 and (22):

116. Example 3 (2)

117. Fig 12.22

118. Alternative Form of Series
If we let x = cos , the x = 1 implies  = 0, x = −1 implies  = . Since dx = −sin  d, (21) and (22) become, respectively, (23) (24) where f(cos ) has been replaced by F().

119. Thank You !