Mth 24B Jnury 24, 22 Viktor Grigoryn 5 Convergence of Fourier series Strting from the method of seprtion of vribes for the homogeneous Dirichet nd Neumnn boundry vue probems, we studied the eigenvue probem X = λx with the ssocited boundry conditions. This ed to the sine nd cosine Fourier series respectivey, nd then the fu Fourier series, which corresponds to the periodic boundry conditions. Using the pirwise orthogonity of the eigenfunctions in ech of these cses, we were be to derive formus for the Fourier coefficients. Finy, in the st ecture we demonstrted tht these ides survive for gener boundry conditions for the interv (, b), provided the boundry conditions re symmetric (hermitin). We showed tht in this cse the eigenvues re re, nd the eigenfunctions cn be chosen to be re vued nd pirwise orthogon. One cn so show tht the eigenvues form sequence λ n, s n for the gener symmetric boundry conditions. Notice tht for the eigenvues of the cssic Fourier series, which we computed expicity to be λ n = (nπ/) 2, this property hods. Then for the eigenvues isted s λ λ 2, we wi hve the corresponding eigenfunctions X, X 2,..., which re re vued nd pirwise orthogon. We re interested in expnding ny function f(x) defined on the interv (, b) in terms of these eigenfunctions. Formy writing f(x) = A n X n, () we found the coefficients A n using the pirwise orthogonity of the eigenfunctions to be A n = (f, X b n) (X n, X n ) = f(x)x n(x) dx b X2 n(x) dx. If we now use these formus to compute the coefficients, nd form the series n A nx n, then one shoud mke sure tht this series converges for the equity () to mke ny sense. Since this is series of functions, there re different wys in which the convergence my be understood. We next define three different notions of convergence, foowed by criteri for convergence of the Fourier series for ech of the three notions. 5. Notions of convergence The convergence of the series of functions f n(x) is equivent to the convergence of the prti sums of the series, S, S 2,..., where N S N =. Conversey, the convergence of sequence of functions F (x), F 2 (x),... is equivent to the convergence of the teescoping series f n(x), where f (x) = F (x), nd = F n (x) F n (x), for n = 2, 3,..., since the functions F N (x) re the prti sums of this teescoping series. Definition 5. (Convergence). We sy tht (i) f n(x) converges to f(x) pointwise in (, b), if for ech fixed x (, b), N f(x) s N (equiventy f(x) S N(x) ). Tht is, for ech fixed < x < b the numeric sequence f n(x) converges to the number f(x).
(ii) f n(x) converges to f(x) uniformy in [, b], if N mx x b f(x) s N (equiventy mx f(x) S N(x) ). x b Tht is, the over distnce between the function f(x) nd the prti sums S N (x) converges to zero. Notice tht the endpoints of the interv re incuded in the definition. (iii) f n(x) converges to f(x) in the men-squre (or L 2 sense) in (, b), if N 2 f(x) dx s N (equiventy f(x) S N (x) 2 dx ). Tht is, the the distnce between f(x) nd the prti sums S N (x) in the men-squre sense converges to zero. It is obvious from the definition tht uniform convergence is the strongest notion of the three, since uniformy convergent series wi cery converge pointwise, s we s in L 2 sense (for finite intervs). The converse is not true, since not every pointwise or L 2 convergent series is uniformy convergent. An exmpe is the teescoping series (xn x n ) in the interv (, ) (check tht it converges pointwise nd in L 2 sense, but not uniformy). Between the pointwise nd L 2 convergence, neither is stronger thn the other, since there re series tht converge pointwise, but not in L 2, nd vice vers. An exmpe of pointwise convergent series tht fis to be L 2 convergent is the teescoping series f(x) = (g n(x) g n (x)) in (, ), where g n (x) = { n < x < n n x <, for n =, 2,..., nd g (x). Check tht N f n(x) pointwise, but S N(x) 2 dx = g2 n(x) dx s N. In this exmpe the functions in the series re discontinuous, but one cn cook up simir exmpe with continuous, nd even differentibe (smooth) functions. Finy, n exmpe of series which converges in L 2, but not pointwise, is the teescoping series f n(x) = (g n(x) g n (x)) in (, ), where g n (x) = { ( ) n x = 2 otherwise, for n =,, 2,..., nd g (x). Cery, N f n(x) in the L 2 sense, since S N(x) 2 dx = g2 n(x) dx =. But the numeric series n f n(/2) diverges, so the series f n(x) does not converge pointwise. Notice tht gin the functions in the series re discontinuous. In this cse this is necessry, since for series of continuous functions L 2 convergence impies pointwise convergence, the proof of which is eft s simpe exercise. 5.2 Convergence theorems We wi next ist criteri for convergence of the Fourier series of function f(x) f(x) A A n cos nx + B n sin nx. (2) For convenience, we wi write the series s A nx n (x), which wi incude the fu, s we s sine nd cosine Fourier series. Definition 5.2. A function f(x) is ced piecewise continuous on n interv [, b], if 2
(i) f(x) is continuous on [, b], except for t most finitey mny points x, x 2,..., x k. (ii) t ech of the points x, x 2,..., x k, both the eft-hnd nd right-hnd imits of f(x) exist, f(x i ) = im f(x), f(x i+) = im f(x). x x i x x i + We sy tht the function hs jump discontinuity t such point with the jump equ to f(x i +) f(x i ). At the points x =, b the continuity nd imits re understood to be one-sided. Next we stte the convergence theorems for the Fourier series (2). men-squre convergence We begin with criteri for Theorem 5.3 (L 2 convergence). The Fourier series n A nx n converges to f(x) in the men-squre sense in (, b), if f(x) is squre integrbe over (, b), tht is f(x) 2 dx <. We remrk tht this very wek condition on the function f(x) cn be mde even weker, by repcing the Riemnn integr bove with the Lebesgue integr. The condition then simpy sttes tht f L 2, where L 2 is the spce of squre-integrbe functions in the Lebesgue sense. The bove theorem so hods for generized Fourier series originting from the eigenvue probem X = λx with symmetric boundry conditions. The next criteri is for uniform convergence. Theorem 5.4 (Uniform convergence). The Fourier series converges to f(x) uniformy in [, b], if (i) f(x) is continuous, nd f (x) is piecewise continuous on [, b]. (ii) f(x) stisfies the ssocited boundry conditions. The boundry conditions for the cssic Fourier series wi be the Dirichet conditions for the sine series, Neumnn for the cosine, nd periodic for the fu Fourier series. As the L 2 convergence theorem bove, the uniform convergence theorem cn be extended to hod for the generized Fourier series, in which cse one needs to dd the condition tht f (x) be piecewise continuous on [, b] s we. Finy, we give the criteri for pointwise convergence. Theorem 5.5 (Pointwise convergence). (i) The Fourier series converges to f(x) pointwise in (, b), if f(x) is continuous, nd f (x) is piecewise continuous on [, b]. (ii) More genery, if f(x) is ony piecewise continuous on (, b), s is f (x), then the Fourier series converges t every point x in the interv (, b), nd we hve A n X n (x) = [f(x+) + f(x )], for < x < b. 2 Thus, if function hs jump discontinuity, then its Fourier series converges to the verge of the one-sided imits. In the bove theorems we used the interv (, b), which in the cse of the cssic Fourier series is either (, ), or (, ). It is cer tht the bove convergence theorems wi hod for the periodic extension of the function to the entire re ine s we. Exmpe 5.. We hve seen mny exmpes of Fourier series tht converge pointwise, but fi to be uniformy convergent. One such exmpe is the sine Fourier series of the function f(x) on the interv (, π), = 4 sin nx. nπ n odd 3
This function stisfies the criteri for pointwise convergence, however, the series vnishes t x =, π, but the function is in the neighborhoods of these points. So the series cnnot converge uniformy. Notice tht the second condition of the uniform convergence is not stisfied in this cse, since the function does not stisfy the Dirichet conditions X() = X(π) =. 5.3 Integrtion nd differentition of Fourier series We re interested in using Fourier series to sove boundry vue probems for PDEs, so we woud ike to know under which conditions one cn differentite or integrte the Fourier series of function. The foowing theorems give these necessry conditions, which we stte for 2-periodic functions. It is obvious how the sttements wi chnge for the sine nd cosine series. Theorem 5.6 (Integrtion of Fourier series). Suppose f is piecewise continuous function with the Fourier coefficients n, b n, f(x) n cos nπx + b n sin nπx. Let F (x) = x f(y) dy. If =, then for x in (, b) we hve where A = F (x) = A 2 + n nπx sin nπ/ b n nπ/ nπx cos, (3) F (x) dx. If, then the sum of the series on the right of (3) is F (x) 2 x. The series (3) is obtined by formy integrting the series of f(x) term by term, irrespective of whether this series converges or not. Notice tht if =, then F (x + 2) F (x) = ˆ x+2 x f(y) dy = ˆ f(y) dy = =, so F (x) is 2-periodic. It is so continuous, with piecewise continuous derivtive F (x) = f(x). But then its Fourier series wi converge uniformy, nd the coefficients in (3) cn be obtined by integrtion by prts s foows. A n = ˆ F (x) cos nπx dx = nπx F (x) sin nπ/ ˆ f(x) sin nπx dx = b n nπ/ nπ/, since F (x) sin(nπx/) is 2-periodic. We cn compute B n simiry to get B n = n /(nπ/). If, then the bove rgument cn be ppied to the function f(x) 2. Theorem 5.7 (Differentition of Fourier series). If f is 2-periodic, continuous, with continuous derivtive f (x), nd piecewise continuous second order derivtive f (x), nd hs the Fourier series then we hve f(x) = A A n cos nπx f (x) = nd this series converges uniformy. + B n sin nπx, nπ A n sin nπx + nπ B n cos nπx, 4
Notice tht the series of f (x) is the term by term derivtive of f(x). Indeed, if A n, B n re the Fourier coefficients of f (x), then A n = ˆ f (x) cos nπx dx = f(x) cos nπx + nπ ˆ f(x) sin nπx dx = nπ B n, where we used the 2-periodicity of f(x), which is cruci in this cse. The coefficients B n cn be computed simiry, nd we hve B n = (nπ/)a n. The uniform convergence then foows from Theorem (5.4), since f (x) stisfies the criteri for uniform convergence. 5.4 Concusion After defining three notions of convergence uniform, pointwise nd in men-squre sense we gve the criteri for convergence of the Fourier series for ech of these notions. We wi see in the next ecture tht the spce of squre integrbe functions (L 2 ) is the ntur spce to consider for the Fourier series. For such functions pproximtion by the Fourier series cn be interpreted in terms of orthogon projections, nd expected properties, such s n nog of the Pythgoren theorem, hod. Using these toos we wi give the proof of the uniform convergence theorem (5.4). The proof of the L 2 convergence requires some techniques from the mesure theory, nd is beyond the scope of this css. The proof of the pointwise convergence cn be found in the textbook. 5