Lecture Introduction to Some Convergence theorems Fridy 4, 005 Lecturer: Nti Linil Notes: Mukund Nrsimhn nd Chris Ré. Recp Recll tht for f : T C, we hd defined ˆf(r) = π T f(t)e irt dt nd we were trying to reconstruct f from ˆf. The clssicl theory tries to determine if/when the following is true (for n pproprite definition of equlity). f(t)?? = r Z ˆf(r)e irt In the lst lecture, we proved Fejér s theorem f k n f where the denotes convolution nd k n (Fejér kernels) re trignometric polynomils tht stisfy. k n 0. T k n = 3. k n (s) 0 uniformly s n outside [ δ, δ] for ny δ > 0. If is finite belin group, then the spce of ll functions f : C forms n lgebr with the opertions (+, ) where + is the usul pointwise sum nd is convolution. If insted of finite belin group, we tke to be T then there is no unit in this lgebr (i.e., no element h with the property tht h f = f for ll f). However the k n behve s pproximte units nd ply n importnt role in this theory. If we let S n (f, t) = ˆf(r)e irt r= n Then S n (f, t) = f D n, where D n is the Dirichlet kernel tht is given by D n (x) = sin ( n + ) s sin s The Dirichlet kernel does not hve ll the nice properties of the the Fejér kernel. In prticulr, 8
. D n chnges sign.. D n does not converge uniformly to 0 outside rbitrrily smll [ δ, δ] intervls. Remrk. The choice of n pproprite kernel cn simplify pplictions nd proofs tremendously.. The Clssicl Theory Let G be loclly compct belin group. Definition.. A chrcter on G is homomorphism χ : G T. Nmely mpping stisfyin χ(g +g ) = χ(g )χ(g ) for ll g, g G. If χ, χ re ny two chrcters of G, then it is esily verified tht χ χ is lso chrcter of G, nd so the set of chrcters of G forms commuttive group under multipliction. An importnt role is plyed by Ĝ, the group of ll continuous chrcters. For exmple, ˆT = Z nd ˆR = R. For ny function f : G C, ssocite with it function ˆf : Ĝ C where ˆf(χ) = f, χ. For exmple, if G = T then χ r (t) = e irt for r Z. Then we hve ˆf(χ r ) = ˆf(r). We cll ˆf : Ĝ C the Fourier trnsform of f. Now Ĝ is lso loclly compct belin group nd we cn ply the sme gme bckwrds to construct ˆf. Pontrygin s theorem sserts tht Ĝ = G nd so we cn sk the question: Does ˆf = f? While in theory Fejér nswered the question of when ˆf uniquely determines f, this question is still left unnswered. For the generl theory, we will lso require normlized nonnegtive mesure µ on G tht is trnsltion invrint: µ(s) = µ( + S) = µ ({ + s s S }) for every S G nd G. There exists unique such mesure which is clled the Hr mesure..3 L p spces Definition.. If (, Ω, µ) is mesure spce, then L p (, Ω, µ) is the spce of ll mesureble functions f : R such tht [ ] f p = f p p dµ < For exmple, if = N, Ω is the set of ll finite subsets of, nd µ is the counting mesure, then (x, x,..., x n,... ) p = ( x i p ) p. For p =, we define x = sup x i i N Symmetriztion is technique tht we will find useful. Loosely, the ide is tht we re verging over ll the group elements. Given function f : G C, we symmetrize it by defining g : G C s follows. g(x) = f(x + ) dµ() G 9
We will use this concept in the proof of the following result. Proposition.. If G is loclly compct belin group, with normlized Hr mesure µ, nd if χ, χ Ĝ re two distinct chrcters then χ, χ = 0. i.e., { 0 χ χ I = χ (x)χ (x) dµ(x) = δ χ,χ = χ = χ Proof. For ny fixed G, I = χ (x)χ (x) dµ(x) = χ (x + )χ (x + ) dµ(x). Therefore, I = χ (x + )χ (x + ) dµ(x) = χ (x)χ ()χ (x)χ () dµ(x) = χ ()χ () χ (x)χ (x) dµ(x) = χ ()χ ()I This cn only be true if either I = 0 or χ () = χ (). If χ χ, then there is t lest one such tht χ () χ (). It follows tht either χ = χ or I = 0. G By letting χ be the chrcter tht is identiclly, we conclude tht χ χ(x) dµ(x) = 0. Ĝ with χ for ny.4 Approximtion Theory Weierstrss s theorem sttes tht the polynomils re dense in L [, b] C[, b] Fejér s theorem is bout pproximting functions using trignometric polynomils. Proposition.. cos nx cn be expressed s degree n polynomil in cos x. Proof. Use the identity cos(u + v) + cos(u v) = cos u cos v nd induction on n. The polynomil T n (x) where T n (cos x) = cos(nx) is clled n th Chebyshev s polynomil. It cn be seen tht T 0 (s) =, T (s) = s, T (s) = s nd in generl T n (s) = n s n plus some lower order terms. Theorem.3 (Chebyshev). The normlized degree n polynomil p(x) = x n +... tht pproximtes the function f(x) = 0 (on [, ]) s well s possible in the L [, ] norm sense is given by n T n (x). i.e., min p normlized polynomil This theorem cn be proved using liner progrmming. mx p(x) = x n This nottion is intended to imply tht the norm on this spce is the sup-norm (clerly C[, b] L [, b]) 0
.4. Moment Problems Suppose tht is rndom vrible. The simplest informtion bout re its moments. These re expressions of the form µ r = f(x)x r dx, where f is the probbility distribution function of. A moment problem sks: Suppose I know ll (or some of) the moments {µ r } r N. Do I know the distribution of? Theorem.4 (Husdorff Moment Theorem). If f, g : [, b] C re two continuous functions nd if for ll r = 0,,,..., we hve b f(x)x r dx = b g(x)x r dx then f = g. Equivlently, if h : [, b] C is continuous function with b h(x)xr dx = 0 for ll r N, then h 0. Proof. By Weierstrss s theorem, we know tht for ll ɛ > 0, there is polynomil P such tht h P < ɛ. If b h(x)xr dx = 0 for ll r N, then it follows tht b h(x)q(x) dx = 0 for every polynomil Q(x), nd so in prticulr, b h(x)p (x) dx. Therefore, Therefore, 0 = b h(x)p (x) dx = b h, h = h(x)h(x) dx + b b ( ) h(x) P (x) h(x) dx ( ) h(x) P (x) h(x) dx Since ( h is continuous, it is bounded on [, b] by some constnt c nd so on [, b] we hve h(x) P (x) h(x)) c ɛ b. Therefore, for ny δ > 0 we cn pick ɛ > 0 so tht so tht h δ. Hence h 0..4. A little Ergodic Theory Theorem.5. Let f : T C be continuous nd γ be irrtionl. Then lim f ( e πir) = f(t) dt n n r= Proof. We show tht this result holds when f(t) = e ist. Using Fejér s theorem, it will follow tht the result holds for ny continuous function. Now, clerly π T eist dt = 0. Therefore, e πirsγ e ist dt n π r= T = e πirsγ n r= = n eπisγ e πinsγ e πisγ T n ( e πisγ ) Since γ is irrtionl, e πisγ is bounded wy from 0. Therefore, this quntity goes to zero, nd hence the result follows.
Figure.: Probbility of Property v. p This result hs pplictions in the evlutions of integrls, volume of convex bodies. Is is lso used in the proof of the following result. Theorem.6 (Weyl). Let γ be n irrtionl number. For x R, we denote by x = x [x] the frctionl prt of x. For ny 0 < < b <, we hve { r n : rγ < b} lim = b n n Proof. We would like to use Theorem.5 with the function f = [,b]. However, this function is not continuous. To get round this, we define functions f + [,b] f s shown in the following digrm. f + nd f re continuous functions pproximting f. We let let them pproch f nd pss to the limit. This is relted to more generl ergodic theorem by Birkhoff. Theorem.7 (Birkhoff, 93). Let (Ω, F, p) be probbility mesure nd T : Ω Ω be mesure preserving trnsformtion. Let L (Ω, F, p) be rndom vrible. Then Where I is the σ-field of T -invrint sets. n T k E [; I] k=.5 Some Convergence Theorems We seek conditions under which S n (f, t) f(t) (preferbly uniformly). Some history: DuBois Rymond gve n exmple of continuous function such tht lim sup S n (f, 0) =. Kolmogorov [] found Lebesgue mesureble function f : T R such tht for ll t, lim sup S n (f, t) =.
Crleson [] showed tht if f : T C is continuous function (even Riemnn integrble), then S n (f, t) f(t) lmost everywhere. Khne nd Ktznelson [3] showed tht for every E T with µ(e) = 0, there exists continuous function f : T C such tht S n (f, t) f(t) if nd only if t E. Definition.3. l p = L p (N, Finite sets, counting mesure). = {x (x 0,... ) p < }. Theorem.8. Let f : T C be continuous nd suppose tht r Z ˆf(r) < (so ˆf l ). Then S n (f, t) f uniformly on T. Proof. See lecture 3, theorem 3...6 The L theory The fct tht e(t) = e ist is n orthonorml fmily of functions llows to develop very stisfctory theory. Given function f, the best coefficients λ, λ,..., λ n so tht f n λ je j is minimized is given by λ j = f, e j. This nswer pplies just s well in ny inner product normed spce (Hilbert spce) whenever {e j } forms n orthonorml system. Theorem.9 (Bessel s Inequlity). For every λ, λ,..., λ n, with equlity when λ i = f, e i f λ i e i f f, e i Proof. We offer proof here for the rel cse, in the next lecture the complex cse will be done s well. f λ i e i = (f f, e i e i ) + ( f, e i e i λ i e i ) = (f f, e i e i ) + ( f, e i e i λ i e i ) cross terms = f f, e i e i, f, e i e i λ i e i + cross terms Observe tht the terms in the cross terms re orthogonl to one nother since i f f, e i e i, e i = 0. We write f, e i f f, e j e j, e i λ i f f, e j e i, e i j= Observe tht ech innter product term is 0. Since if i = j, then we pply i f f, e i e i, e i = 0. If i j, then they re orthogonl bsis vectors. i j= 3
We wnt to mke this s smll s possible nd hve only control over the λ i s. Since this term is squred nd therefore non-negtive, the sum is minimized when we set i λ i = f, e i. With this choice, f λ i e i = f λ i e i, f = f, f = f λ i e i λ i f, e i + f, e i where the lst inequlity is obtined by setting λ i = f, e i. λ i References [] A. N. Kolmogorov, Une série de Fourier-Lebesgue divergente prtout, CRAS Pris, 83, pp. 37-38, 96. [] L. Crleson, Convergence nd growth of prtil sums of Fourier series, Act Mth. 6, pp. 35-57, 964. [3] J-P Khne nd Y. Ktznelson, Sur les ensembles de divergence des séries trignométriques, Studi Mthemtic, 6 pp. 305-306, 966 4