12.1 Orthogonal Functions
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12.1 Orthogonal Functions
Introduction In this section we shall see how the two vector concepts of inner or dot product and orthogonality of vectors can be extended to functions
Inner Product Recall that if u = u1 i + u2 j + u3 k and v = v1 i + v2 j + v3 k then u v = u1v1 + u2v2 + u3v3.
The inner product of functions is defined below
12.1 DEFINITION Inner Product of Functions
The inner product of two functions f1 and f2 on the interval [a, b] is the number b (f1, f2) = f (x) f (x)dx. a 1 2
12.2 DEFINITION Orthogonal Functions
Two functions f1 and f2 are said to be orthogonal on the interval [a, b] if b (f1, f2) = f (x) f (x)dx 0. a 1 2
2 3 For example, f1 = 2x and f2 = x /3 are orthogonal on [–3, 3] since 2 2 2 2 3 2 5 1 6 2 (f1, f2) = x x dx x dx x 0 3 2 3 2 9 -2
12.3 DEFINITION Orthogonal Sets
A set of real-valued functions {0 ,1 ,2 ,} is said to be orthogonal on the interval [a, b] if b ( , )= (x) (x)dx 0, m n. m n a m n
Example 1* (p. 654). Orthogonal Set of Functions Determine which of these sets: {1, sinh x, sinh 2x, …} and {sin x, sin 3x, sin 5x, …} is (are) orthogonal on the interval [–3, 3].
1 Orthonormal Sets Note that for a vector u 2 2 2 2 (u, u) = u1 u2 u3 u and so u (u,u) . Similarly, we define b (x) 2 ( , ) 2 (x)dx n n n a n and so b (x) ( , ) 2 (x)dx . n n n a n
If {n (x)} is an orthogonal set of functions on an interval [a, b] with the additional property that: n (x) 1for all n = 0, 1, 2, then {n (x)} is an orthonormal set on the interval.
0 1 2 Note that if {0 ,1 ,2 ,} is orthogonal then { , , , } is always 0 1 2 orthonormal.
Example 2* (p. 654). Norms Find the norms of all the functions in Example 1*.
2 Orthogonal Series Expansion Suppose {n (x)} is an infinite orthogonal set of functions on an interval [a, b]. We ask: If y = f(x) is a function defined on the interval [a, b], is it possible to determine a set of coefficients cn, n = 0, 1, 2, …, for which
f(x) = c00 (x) c11 (x) c22 (x) cnn (x) ? (1) Yes, we can find the coefficients by utilizing the inner product.
3 12.4 DEFINITION Orthogonal Set/Weight Function
A set of real-valued functions {0 ,1 ,2 ,} is said to be orthogonal with respect to a weight function w(x) on an interval [a, b] if b w(x) (x) (x)dx 0, m n. a m n
Usually we assume: w(x) > 0 on the interval of orthogonality [a, b].
The series (2) is said to be an orthogonal series expansion of f or a generalized Fourier series.
4 12.2 Fourier Series
Introduction In this section we shall learn how to expand functions in terms of a special orthogonal set of trigonometric functions.
Trigonometric Series
12.5 DEFINITION Fourier Series The Fourier Series of a function f defined on the interval (-p, p) is given by a n n 0 f (x) an cos x bn sin x, 2 n1 p p where 1 p a f (x)dx, 0 p p 1 p n a f (x)cos xdx, n p p p
1 p n b f (x)sin xdx. n p p p
The coefficients an, bn are called Fourier coefficients of f.
Example 1* (p. 660). (#5, p. 662) Expansion in a Fourier Series
5 Convergence of a Fourier Series The following theorem gives sufficient conditions for convergence of a Fourier series at a point.
12.1 THEOREM Conditions for Convergence 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 discontinuities at these points. Then the Fourier series of f on the interval converges to f(x) at a point of continuity. At a point of discontinuity, the Fourier series converges to the average f (x) f (x) , 2 where f(x+) and f(x–) denote the limit of f at x from the right and from the left respectively.
Example 2 (p. 660). Convergence of a Point of Discontinuity
Periodic Extension When f is piecewise continuous and the right- and left-hand derivatives exist at x = –p and p, respectively, then its Fourier series converges to the average [f(p+) + f(p–)]/2 at these endpoints and to this value extended periodically to 3p, 5p, 7 p, and so on.
Problem # 17 (p. 662)
Problem # 18 (p. 662)
6 12.3 Fourier Cosine and Sine Series
Review The effort expended in the evaluation of coefficients a0, an and bn in expanding a function f in a Fourier series is reduced significantly when f is either an even or an odd function.
Even and Odd Functions A function f is said to be even if f(x) = f(–x) and odd if f(x) = – f(–x). For example,
12.2 THEOREM Properties of Even/Odd Functions (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 even. (d) The sum (difference) of two even functions is even. (e) The sum (difference) of two odd functions is odd. a a (f) If f is even, then f (x)dx 2 f (x)dx a 0 a (g) If f is odd, then f (x)dx 0. a
7 Cosine and Sine Series If f is an even function on (–p, p), then the coefficients a0, an and bn from Section 12.2 become 1 p 2 p a f (x)dx f (x)dx, 0 p p p 0
1 p n 2 p n a f (x)cos xdx f (x)cos xdx, n p p p p 0 p
1 p n 2 p n b f (x)sin xdx f (x)sin xdx 0. n p p p p 0 p Similarly, when f is odd on the interval (–p, p), 2 p n a = 0, n = 0, 1, 2, …, bn f (x)sin xdx. n p 0 p We summarize the results in the following definition.
12.6 DEFINITION Fourier Cosine and Sine Series (i) The Fourier Series of an even function f defined on the interval (-p, p) is the cosine series a0 n f (x) an cos x, 2 n1 p where 1 p a f (x)dx, 0 p p
2 p n a f (x)cos xdx. n p 0 p (ii) The Fourier Series of an odd function f defined on the interval (-p, p) is the sine series n f (x) bn sin x, n1 p where 2 p n b f (x)sin xdx. n p 0 p
Example 1* (pp.664-5) Expansion in a Cosine Series
8 Half-Range Expansions. If y = f(x) is defined on 0 < x < L it can still be expanded as a Fourier series by first supplying an 'arbitrary' definition on the interval –L < x < 0, that is, extending the domain of definition of the function to –L < x < L. The three most important cases are: (i) reflect the graph of the function about the y- axis onto –L < x < 0; the function is now even on –L < x < L; or (ii) reflect the graph of the function through the origin onto –L < x < 0; the function is now odd on –L < x < L; or (iii) define f on –L < x < 0 by f(x) = f(x + L) The expansions obtained in this manner are called half-range expansions.
Remark. In the first two cases the expansion is exactly as in the cosine and sine series expansion. In the last cases we have to consider the expansion on the interval (–L/2, L/2).
Example 3* (pp.666-7) Expansion in Three Series Expand f(x) = sin x, 0 < x < L, (a) in a cosine series; (b) in a sine series; and (c) in a Fourier series.
9 12.4 Complex Fourier Series
Introduction In this section we are going to use Euler's formula: eix cos x isin x which implies eix cos x isin x to recast the Fourier series in Definition 12.5 into a complex form or exponential form.
Complex Fourier Series
12.7 DEFINITION Complex Fourier Series The complex Fourier Series of a function f defined on an interval (-p, p) is given by inx / p f (x) cne , n
1 p inx / p where cn f (x)e xdx, n 0,1, 2, 2 p p
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 f (x) f (x) , 2 at a point of discontinuity.
Example * (pp.671-2) Complex Fourier Series
10 12.5 Sturm-Liouville Theorem
Review Some Ordinary Differential Equations (ODEs)
Linear DEs General Solution
x y'y 0, y c1e 2 y'' y 0, 0 y c1 cosx c2 sinx y c ex c ex , or y'' 2 y 0, 0 1 2 y c1 coshx c2 sinhx
Cauchy-Euler DEs General Solution x > 0
y c x c x , or x 2 y''xy' 2 y 0, 1 2 y c1 c2 ln x
Parametric Bessel DEs General Solution x > 0
2 2 x y''xy' y 0, 0 y c1 J 0 (x) c2Y0 (x)
Legendre's DE (n = 0, 1, 2, … ) Particular Solutions are polynomials
2 (1 x )y''2xy'n(n 1)y 0, y P0 (x) 1,
y P1 (x) x, 1 2 y P2 (x) 2 (3x 1),
Eigenvalues and Eigenfunctions An orthogonal set of functions can be generated by solving a two-point boundary-value problem (BVP) involving a second-order DE containing a parameter say, .
Example 2 (pp. 168-9) Nontrivial Solutions of a BVP Solve the BVP: y''y 0, y(0) = 0, y(L) = 0.
Solution. We consider THREE cases: 0, 0 and 0.
11 Case I. For 0, the general solution of the DE is y c1 x c2 .
Applying the boundary conditions in turn yields c1 0 c2 . Hence we only have the trivial solution: y = 0.
Case II. For 0, it is convenient to write 2 , where 0. 2 2 Hence the auxiliary equation is m 0, which has roots m1 and m2 . Since the interval under consideration is [0, L], we choose the hyperbolic expression for the general solution of the DE, i. e., y c1 coshx c2 sinhx.. Applying the boundary conditions in turn yields c1 0 c2 . Once again, we only have the trivial solution: y = 0.
Case III. For 0, it is convenient to write 2 , where 0. The 2 2 auxiliary equation is m 0, which has complex roots m1 i and m2 i and so the general solution of the DE is y c1 cosx c2 sinx. As before y(0) = 0 yields c1 0 and so y c2 sinx. Next y(L) = 0 implies c2 sinL 0 . If c2 = 0 then necessarily y = 0. But this time we can require c2 0 since sinL 0 is satisfied by any integer multiple of . 2 n 2 n L n or or , n = 1, 2, 3, … L n n L
Therefore for any real nonzero c2, y c2 sin(nx / L) is a solution of the DE for each n. since the DE is homogeneous, any constant multiple of a solution is also a solution. Thus we may, if desired, simply take c2 = 1. In other words, for each number in the sequence 2 4 2 9 2 , , , … 1 L2 2 L2 3 L2 the corresponding function in the sequence 2 3 y sin x , y sin x , y sin x , … 1 L 2 L 3 L is a nontrivial solution of the original problem.
2 2 2 Definitions The numbers n n / L , n = 1, 2, 3, …for which the BVP above has a nontrivial solution are known as characteristic values or, more commonly, eigenvalues. The solutions depending on theses values of n , yn c2 sin(nx / L) or simply yn sin(nx / L) are called characteristic functions or, more commonly, eigenfunctions.
12 Example 1 (p. 675) Eigenvalues and Eigenfunctions Solve the BVP: y''y 0, y'(0) = 0, y'(L) = 0.
Regular Sturm-Liouville Problem Let p, q, r and r' be real-valued functions continuous on an interval [a, b], and let r(x) > 0 and p(x) > 0 for every x in the interval. Then d Solve: [r(x)y'] (q(x) p(x))y 0 dx
Subject to: A1 y(a) B1 y'(a) 0
A2 y(b) B2 y'(b) 0 is said to be a regular Sturm-Liouville problem. The coefficients in the boundary conditions are assumed to be real and independent of .
12.3 THEOREM Properties of the Regular Sturm-Liouville Problem (a) There exist an infinite number of real eigenvalues that can be arranged in increasing order 1 2 3 ⋯ n ⋯ such that
n as n . (b) For each eigenvalue there is only one eigenfunction (except for nonzero constant multiples). (c) Eigenfunctions corresponding to different eigenvalues are linearly independent. (d) 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], b i. e., p(x)y y dx 0, m n. a m n
13 Read the proof of (d) on page 676.
Example 2* (p.677) A Regular Sturm-Liouville Problem
14 d Self-Adjoint Form If we carry out the differentiation [r(x)y'], the self- dx adjoint differential equation d [r(x)y'] (q(x) p(x))y 0 (1) dx becomes r(x)y''r'(x)y'(q(x) p(x))y 0. For example, Legendre's DE: (1 x 2 )y''2xy'n(n 1)y 0 is of the above form with r(x) = 1 – x2 and so r'(x) = –2x. In other words, another way of writing Legendre's DE is d [(1 x 2 )y'] n(n 1)y 0. dx Moreover, Bessel's equation, Cauchy-Euler equations and DE's with constant coefficients all have the above form. In fact, if the coefficients are continuous and a(x) 0 for all x in some interval, then any second order DE a(x)y''b(x)y'(c(x) d(x))y 0 can be recast into the so-called self-adjoint form. Read how to do this from pages 678 – 9.
Remark. We use the form in (1) to determine the weight function p(x) needed in the orthogonality relation.
15 Bessel Equation: x 2 y''xy'(x 2 2 )y 0, where is an integer has solution on (0, ) n (1) 2n J x / 2 (2) n0 n!( n)! called Bessel's function of the first kind of order . Letting t x, 0 in x 2 y''xy'( 2 x 2 2 )y 0, (3) then by the Chain Rule 2 2 dy dy dt dy d y d dy dt 2 d y and . dx dt dx dt dx 2 dt dx dx dt 2 Accordingly, (3) becomes 2 2 t 2 d y t dy 2 2 (t )y 0 dt 2 dt or d 2 y dy t 2 t (t 2 2 )y 0. dt 2 dt
Example 3 (p.677) Parametric Bessel DE
Example 3* (p.677) Laguerre's and Hermite's DEs
16 12.6 Bessel and Legendre Series
12.6.1 Fourier-Bessel Series We saw in Example 3 of Section 12.5 that for a fixed value of n the set of Bessel functions {J n ( i x)}, i = 1, 2, 3, … is orthogonal WRT the weight function p(x) = x on an interval [0, b] when the i s are defined by means of a boundary condition of the form
A2 J n (b) B2J n '(b) 0 . (1) The eigenvalues of the corresponding Sturm-Liouville problem are 2 i i . The orthogonal series expansion or generalized Fourier series of a function f defined on the interval (0, b) (see Section 12.1) in terms of this orthogonal set is
f (x) ci J n ( i x), i1 b xJ ( x) f (x)dx 0 n i where ci 2 . J n ( i x) This is known as Fourier-Bessel series. The square norm of the function
J n ( i x) is defined by (see Section 12.1) b J ( x) 2 xJ 2 ( x)dx n i 0 n i
Differential Recurrence Relations The following differential recurrence relations are often useful in the evaluation of the coefficients ci: d [x n J (x)] x n J (x) dx n n1 d [x n J (x)] x n J (x). dx n n1
17 12.8 DEFINITION Fourier-Bessel Series The Fourier-Bessel Series of a function f defined on the interval (0, b) is given by
(i) f (x) ci J n ( i x), i1 2 b c xJ ( x) f (x)dx i 2 2 0 n i b J n1 ( ib) where the i s are defined by J n (b) 0 .
(ii) f (x) ci J n ( i x), i1 2 2 b c i xJ ( x) f (x)dx i 2 2 2 2 2 0 n i ( i b n h )J n ( ib) where the i s are defined by hJ n (b) bJ n '(b) 0 .
(iii) f (x) c1 ci J 0 ( i x), i2 2 b 2 b c xf (x)dx, c xJ ( x) f (x)dx 1 2 0 i 2 2 0 0 i b b J 0 ( ib) where the i s are defined by J 0 '(b) 0.
12.4 THEOREM Conditions for Convergence If f and f ' be piecewise continuous on the open interval (0, b), then the Fourier-Bessel expansion of f on the interval converges to f(x) at any point of continuity. At a point of discontinuity it converges to the average f (x) f (x) . 2
Example 1* (p.684) Expansion in a Fourier-Bessel Series
Example 2* (p.685) Expansion in a Fourier-Bessel Series
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