Lecture 3: Convergence of Fourier Series

Transcription

1 Lecture 3: Covergece of Fourier Series Himashu Tyagi Let f be a absolutely itegrable fuctio o T : [ π,π], i.e., f L (T). For,,... defie ˆf() f(θ)e i θ dθ. π T The series ˆf()e i θ is called the Fourier Series (FS) of f. The FS of f does ot coverge to f poit-wise a.e.. If we restrict to cotiuous fuctios, the poit-wise covergece of FS ca ot be guarateed. I this lecture, we defie weaker but importat criteria for covergece uder which the FS of cotiuous f does coverge to f. A. The defiitio of covolutio give i Lecture ca be easily modified for fuctios i L (T). For f,g L (T) the covolutio f g is defied as f g(x) π π f(x y)g(y)dy, where f(x) f(x mod T). Also, Theorem of Lecture ad property (v) of covolutio hold. Therefore, we ca mimic the proof therei to show there is o idetity for covolutio i L (T), provided we prove the Riema- Lebesgue Lemma. Here we give a alterate proof of Riema-Lebesgue Lemma usig approximate idetity. Defiitio. A sequece {K } of cotiuous fuctios o R is a summability kerel if for all N (i) R K (x)dx, (ii) R K (x) dx M, (iii) For each 0 < δ < π, lim K (x) dx 0. x >δ This lecture is take from the lecture otes of the course Fourier Series orgaized by Prof. Rajedra Bhatia at the Idia Statistical Istitute, New Delhi, Idia. It was also published as: R. Bhatia, Fourier series(secod Editio), Hidusta Book Agecy (Idia), 003.

2 Note here that limits of itegral i property three represets the uio of regios [ π, δ) ad (δ,π]. As was show i the last lecture, a summability kerel is a bouded approximate idetity for L (T). Also, the Fejer kerel K (t) j π ( j + ) e ijt ( + )π ) is a summability kerel. Therefore we have the followig lemma. ( si + t si t Lemma. For f L (T) ad,,..., σ (f) : K f. The σ (f) f i L (T). Note that σ (f) is a fuctio of form N a e it. Such a fuctio is called a trigoometric polyomial of degree N. The Uiqueess Theorem for FS follows from this Lemma. The Lemma above says that for all fuctios f i L (T) ad all positive ǫ, we ca fid a trigoometric polyomial P such that f P L < ǫ. It ca be simply verified that for trigoometric polyomial P of degree N, ˆP() 0 for all greater tha N. The Riema-Lebesgue Lemma ow follows. Theorem. Let f L (T), the lim ˆf() 0. Proof: Let ǫ > 0 ad let P be a trigoometric polyomial o T such that f P L < ǫ. If > degree of P, the ˆf() ˆ (f P)() f P L < ǫ. B. Now we restrict to cotiuous fuctios f o T. Sice T is compact, f is bouded. Therefore we get followig stregtheig of Theorem 5 of Lecture. Theorem 3. For a summability kerel {K } N ad a cotiuous fuctio f o T, K f coverges to f poit-wise. This proof is take from Y. Katzelso, Itroductio to harmoic aalysis, third editio, Cambridge Uiversity Press, 004.

3 Proof: Let ǫ > 0. Choose δ > 0 such that if t δ, the f f (t) L < (ǫ/m). This ca be doe sice f is cotiuous ad T is compact imply f is uiformly cotiuous. The for t i T we have K f(t) f(t) K (y) f(t y) f(t) dy [Usig K (x)dx ] R R K (y) f(t y) f(t) dy + K (y) f(t y) f(t) dy y δ ǫ + f L ǫ, for sufficietly large. y >δ K (y) dy y >δ C. The Theorem will be istrumetal i the proof of covergece of FS. We begi by defiig the covergece criteria we shall be iterested i. Defiitio. A series x with complex terms x for,,..., is said to be Abel summable to L if the followig holds: (i) For every 0 < r <, the series r x coverges to a limit L(r). (ii) L(r) coverges to L as r. Defiitio 3. For a series x with complex terms x for,,..., defie s N x ad σ s +...+s. The the series x is said to be Cesáro summable to L [also called (C,)-summable to L] if the sequece σ coverges to L as. We state the ext result without proof. Theorem 4. A series x with complex terms x for,,... coverges to L implies it is Cesáro summable to L. Furthermore, x is Cesáro summable to L implies it is Abel summable to L. Cosider the FS of a cotiuous fuctio f o T. Defie the partial sums s N (f;θ) ˆf()e i θ, N 3

4 N 0,,,... Also, for e (t) e it, f e (θ) π ˆf()e i θ. Therefore, s N (f;θ) f D N (θ) where D N (θ) π N e it. D N (t) is called the Dirichlet kerel. If the Dirichlet kerel was a summability kerel we could have show the poit-wise covergece of FS of f to f. However this is ot true. To show that FS of f is Cesáro summable to f, we eed to show that N+ N + Next we show that N+ N + s (f;θ) coverges to f(θ). Note that s (f;θ) f N + D (θ). D is the Fejer kerel! First ote that D (t) Multiplyig ad dividig [ + Therefore, + (N + )π (N + )π k [ + e ikt cos kt]. k cos kt] with si t/ we get k cos kt si t/ [si t/ + cos kt si t/] k N + D (t) si( + /)t. si t/ (N + )π k si( + /)t si t/ 4

5 Agai, multiplyig ad dividig by sit/ we get N + D (t) (N + )π (N + )π (N + )π si( + /)t si t/ (si t/) si( + /)t si t/ (si t/) ( ) si (N+)t si t. Sice Fejer kerel is a summability kerel, we have the followig result. Theorem 5. The FS of f is Cesáro summable to f. Note that this, usig Theorem 4 also implies that FS of f is Abel summable to f. This result ca also be obtaied directly. The proof etails aalysis of what is kow as the Poisso kerel. [You might have ecoutered this already i partial differetial equatios] 5