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SECTION 5.2 209 §5.2 Generalized Fourier Series in Chapters 3 and 4

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SECTION 5.2 209 §5.2 Generalized Fourier Series in Chapters 3 and 4 SECTION 5.2 209 §5.2 Generalized Fourier Series In Chapters 3 and 4 we learned how to express functions f(x), which are piecewise smooth on the interval 0 ≤ x ≤ L, in the form of Fourier sine series ∞ L f(x+)+f(x-) nπx 2 nπx = X b sin where b = Z f(x) sin dx. (5.8) 2 n L n L L n=1 0 We regard the Fourier coefficients bn as the components of the function f(x) with respect to the basis functions {sin (nπx/L)}. In Section 5.1 we discovered that the sin (nπx/L) are eigenfunctions of Sturm-Liouville system 5.1, and it has be- come our practice to replace eigenfunctions with normalized eigenfunctions, namely p2/L sin (nπx/L). Representation 5.8 can easily be replaced by an equivalent ex- pression in terms of these normalized eigenfunctions, ∞ L f(x+)+f(x-) r 2 nπx! r 2 nπx ! = X c sin where c = Z f(x) sin dx. (5.9) 2 n L L n L L n=1 0 Constants cn are the components of f(x) with respect to the orthonormal basis {p2/L sin (nπx/L)}. Equation 5.8 should be compared with equation 3.3 in Section 3.1, together with the fact that the length of sin (nπx/L)ispL/2. Equation 5.9 is analogous to equation 3.1. The same function f(x) can be represented by a Fourier cosine series in terms of normalized eigenfunctions of system 5.2, ∞ f(x+)+f(x-) c r 2 nπx! = √0 + X c cos (5.10a) 2 n L L L n=1 where L L Z 1 Z r 2 nπx ! c0 = f(x) √ dx and cn = f(x) cos dx, n > 0. (5.10b) 0 L 0 L L A natural question to ask now is the following: Given a function f(x), defined on the interval a ≤ x ≤ b, and given a Sturm-Liouville system on the same interval, is it always possible to express f(x) in terms of the orthonormal eigenfunctions of the Sturm-Liouville system? It is still not clear that every Sturm-Liouville system has an infinity of eigenfunctions, but, as we shall see, this is indeed the case. We wish then to investigate the possibility of finding coefficients cn such that on a ≤ x ≤ b, ∞ f(x)=X cnyn(x), (5.11) n=1 where yn(x) are the orthonormal eigenfunctions of Sturm-Liouville system 5.3. If we formally multiply equation 5.11 by p(x)ym(x), and integrate term-by-term between x = a and x = b, b ∞ b Z Z p(x)f(x)ym(x) dx = X cn p(x)yn(x)ym(x) dx. a n=1 a 210 SECTION 5.2 Because of the orthogonality of eigenfunctions, only the mth term in the series does not vanish, and therefore b Z p(x)f(x)ym(x) dx = cm. (5.12) a This has been strictly a formal procedure. It has illustrated that if f(x) can be represented in form 5.11, and if the series is suitably convergent, coefficients cn can be calculated according to formula 5.12. What we must answer is the converse ques- tion: If coefficients cn are calculated according to 5.12, where yn(x) are orthonormal eigenfunctions of a Sturm-Liouville system, does series 5.11 converge to f(x)? This question is answered in the following theorem. Theorem 5.2 Let p, q, r, r0, and (pr)00 be real and continuous functions of x for a ≤ x ≤ b, and let p>0 and r>0 for a ≤ x ≤ b. Let l1, l2, h1, and h2 be real constants independent of λ. Then Sturm-Liouville system 5.3 has a countable infinity of simple eigenvalues λ1 <λ2 <λ3 < ··· (all real), not more than a finite number of which are negative, and limn→∞ λn = ∞. Corresponding orthonormal eigenfunctions yn(x) are such 0 −1/2 0 that yn(x) and yn (x) are continuous and |yn(x)| and |λn yn (x)| are uniformly bounded with respect to x and n.Iff(x) is piecewise smooth on a ≤ x ≤ b, then for any x in a<x<b, ∞ b f(x+)+f(x-) = X c y (x), where c = Z p(x)f(x)y (x) dx. (5.13) 2 n n n n n=1 a Series 5.13 is called the generalized Fourier series for f(x) with respect to the eigenfunctions yn(x), and the cn are the generalized Fourier coefficients. They are the components of f(x) with respect to the orthonormal basis of eigen- functions {yn(x)}. Notice the similarity between this theorem and Theorem 3.2 in Section 3.1 for Fourier series. Both guarantee pointwise convergence of Fourier series for a piecewise smooth function to the value of the function at a point of continuity of the function, and to the average value of right- and left-hand limits at a point of discontinuity. Because the eigenfunctions in Theorem 3.2 of Section 3.1 are periodic, convergence is also assured at the end points of the interval 0 ≤ x ≤ 2L. This is not the case in Theorem 5.2 above. Eigenfunctions are not generally periodic, and convergence at x = a and x = b is not guaranteed. It should be clear, however, that when l1 = 0 (in which case yn(a) = 0) convergence of the series in 5.13 at x = a can be expected only if f(a) = 0 also. A similar statement can be made at x = b. Because series 5.13 is a representation of the function f(x) in terms of normal- ized eigenfunctions of a regular Sturm-Liouville system, it is also called an eigen- function expansion of f(x). We use both terms, namely, generalized Fourier series and eigenfunction expansion, freely and interchangeably. We say that the normalized eigenfunctions of Sturm-Liouville system 5.3 form a complete set for the space of piecewise smooth functions on the interval a ≤ x ≤ b; this means that every piecewise smooth function can be expressed in a convergent series of the eigenfunctions. When a regular Sturm-Liouville system satisfies the conditions of this theorem as well as the conditions that q(x) ≥ 0 for a ≤ x ≤ b, and l1h1 ≥ 0, l2h2 ≥ 0, it is said to be a proper Sturm-Liouville system. For such a system we shall take l1, l2, h1, and h2 all nonnegative, in which case we can prove the following corollary. SECTION 5.2 211 Corollary All eigenvalues of a proper Sturm-Liouville system are nonnegative. Furthermore, zero is an eigenvalue of a proper Sturm-Liouville system only when q(x) ≡ 0 and h1 = h2 =0. Proof Let λ and y(x) be an eigenpair of a regular Sturm-Liouville system. Mul- tiplication of differential equation 5.3a by y(x) and integration from x = a to x = b gives b b b λ Z p(x)y2(x) dx = Z q(x)y2(x) dx − Z y(x)[r(x)y0(x)]0 dx a a a b b b = Z q(x)y2(x) dx − nr(x)y(x)y0(x)o + Z r(x)[y0(x)]2 dx. a a a When we solve boundary conditions 5.3b,c for y0(b) and y0(a) and substitute into the second term on the right, we obtain b b b λ Z p(x)y2(x) dx = Z q(x)y2(x) dx + Z r(x)[y0(x)]2 dx a a a h h + 2 r(b)y2(b)+ 1 r(a)y2(a). l2 l1 When the Sturm-Liouville system is proper, every term on the right is nonnegative, as is the integral on the left, and therefore λ ≥ 0. (If either l1 =0orl2 = 0, the corresponding terms in the above equation are absent and the result is the same.) Furthermore, if λ = 0 is an eigenvalue, then each of the four terms on the right side of the above equation must vanish separately. The first requires that q(x) ≡ 0 and the second that y0(x) = 0. But the fact that y(x) is constant implies that the last two terms can vanish only if h1 = h2 =0. Since eigenvalues of a proper Sturm-Liouville system must be nonnegative, we may replace λ by λ2 in differential equation 5.3a whenever it is convenient to do so, d dy r(x) +[λ2p(x) − q(x)]y =0, a<x<b. dx dx This often has the advantage of eliminating square roots in calculations. Example 5.4 Expand the function f(x)=L − x in terms of normalized eigenfunctions of the Sturm-Liouville system of Example 5.1. Solution According to Example 5.1, eigenfunctions of the Sturm-Liouville system (2n − 1)πx are sin . Because 2L 2 L 2 (2n − 1)πx Z (2n − 1)πx L sin = sin dx = , 2L 2L 2 0 r 2 (2n − 1)πx normalized eigenfunctions are X (x)= sin . In terms of these eigen- n L 2L functions, the generalized Fourier series for f(x)=L − x is ∞ L − x = X cnXn(x), n=1 212 SECTION 5.2 where √ L r 2 (2n − 1)πx 2 2L3/2 π 2(−1)n Z − cn = (L x) sin dx = 2 + 2 . 0 L 2L π 2n − 1 (2n − 1) Thus, √ ∞ 2 2L3/2 π 2(−1)n r 2 (2n − 1)πx L − x = X + sin . π2 2n − 1 (2n − 1)2 L 2L n=1 Theorem 5.2 guarantees convergence of the series to L − x for 0 <x<L.It obviously does not converge to L − x at x = 0, but it does converge to L − x at x = L.
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