The minimum value and the L 1 norm of the Dirichlet kernel
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1 The miimum value ad the L orm of the Dirichlet kerel For each positive iteger, defie the fuctio D (θ + ( cos θ + cos θ + + cos θ e iθ + + e iθ + e iθ + e + e iθ + e iθ + + e iθ which we call the (th Dirichlet kerel. The Dirichlet kerel is a muchstudied fuctio i aalysis (for example, it arises whe cosiderig partial sums of Fourier series. Note that D (θ is a eve fuctio with period. If f(θ is ay -periodic fuctio, we defie the usual L p orms f(θ p ( /p f(θ p dθ ad if f(θ is also a eve fuctio, we have ( f(θ p f(θ p dθ /p. This ote is cocered with the miimum value ad the L orm of the fuctio D (θ. We prove the followig: mi D (θ C.367 θ where C is the absolute miimum of si t, ad t D (θ log.585 log where log meas atural log, ad f( g( meas f( g(. These results are ot ew (ideed, they probably cout as mathematical folklore but detailed self-cotaied proofs are sometimes hard to fid i the literature, ad such proofs ca be iterestig ad istructive aalysis exercises.
2 We start by otig that D D (θ ca be writte aother way usig the followig maipulatios. D e iθ + + e iθ + eiθ + eiθ + + eiθ eiθ/ D e i( θ + + e iθ/ + eiθ/ + ei3θ/ + + ei(+ θ e iθ/ D e i(+ θ + + e i3θ/ + e iθ/ + eiθ/ + + ei( θ (eiθ/ e iθ/ D ei(+ θ e i(+ θ si( + θ i si( + θ ei(+ θ e i(+ θ D. θ θ eiθ/ e iθ/ i si si If θ [, ], the deomiator si θ is oegative, ad the umerator si( + θ has sig chages whe θ is a iteger multiple of +. We therefore partitio the iterval [, ] ito + subitervals h h ( i h i i h i,,,, where the first subitervals each have width +, ad the last subiterval has width +. For illustrative purposes, the graphs of D (θ ad D5 (θ are give below. I geeral, we have D ( + ad D (θ +. D D theta theta If we just wat to show there exists some egative costat C such that D (θ always dips below C for some θ, we ca accomplish this by simply 3 cosiderig D ( +.
3 If θ 3, we ote the followig. + si θ < θ > si θ θ D (θ si( + θ si θ si θ < θ ( This might cause someoe to cojecture that mi D (θ is asympotically equal to.3, but as metioed before, the true miimum is 3 closer to.367. The first step toward provig this is to compare D (θ to a Riema sum ad a itegral. We observe that we have D (θ + + k k ( cos θ k f where f(x cos(θx. The, the sum k f is a right-edpoit Riema sum for the itegral f(xdx. We therefore have D (θ + + f(x dx + si(θ θ cos(θx dx + si t t which, for a fixed, is miimized whe t miimizes si t. That value of t is t approximately.93, suggestig we should choose θ.93/ (which is a little less tha 3.739/. To make all this more precise, we + have to be more careful about comparig the Riema sum to the itegral. (The fact that the fuctio f(x depeds o makes thigs otrivial. 3
4 The itegral ca be broke ito smaller itegrals f(x dx k k/ (k / f(x dx ad each of these smaller itegrals ca be writte as k/ f(x dx f(x k where x k (k / [ k, k ]. We therefore have k x k. We also have f f(x k k x k f (ξ for some ξ betwee x k ad k. This implies f f(x k k x f (ξ θ si(θξ θ k f f(x k θ k x k θ θ f f(x k θ. That is, f is withi θ of f(x k k/ f(xdx. If we the sum from (k / k to, we get f(x dx θ si(θ θ θ k k + si(θ θ D (θ θ f f(x dx + θ f si(θ + θ θ + si(θ θ + θ. To prove our claim about mi θ D (θ, we have to prove that mi θ D (θ C.367.
5 We will prove this by provig the followig. Propositio: Give ε >, there exists N such that if > N, the (Claim D (θ/ C ε for all θ, (Claim D (θ/ C + ε for some θ. To prove Claim, we use differet argumets for differet values of θ. Claim is trivially true if θ belogs to either of the itervals [, ] or [, 6 ], because si( + θ is oegative there ad hece so is D (θ. Next, we 6 cosider θ [, ]. Usig the fact that si t t o the iterval [, ], + we have Therefore if we have si θ θ si θ θ < + 6 si( +θ D (θ < + si θ si θ 6 D (θ < D (θ < C ε < C < 5 < D (θ < 5. We ow must show that Claim is true for all θ i the iterval [, ], + + ad we must show that D (θ/ C + ε for some θ. This will fiish the proof of the above propositio. From before, we have If θ + + si(θ θ D (θ θ, the this implies D (θ + si(θ θ + si(θ θ θ + C 5 + θ. 8 +
6 which will be above C ε if is large eough. Next, let t be the t that miimizes si t t (so t.93, ad choose θ t /. We the have D (θ + si t + t t + C + t which will be below C + ε if is large eough. This completes our proof that mi θ D (θ C.367. Next, we proceed with our aalysis of D (θ D (θ dθ. Followig our earlier remarks about sig chages of D (θ ad si( + θ, we write k/(+ D (θ D (θ dθ + D (θ dθ. k (k /(+ /(+ We ow assemble some various lemmas that will be useful. Lemma. If a > is real ad k is a iteger, we have k/a (k /a si(aθ dθ a. This lemma is straightforward, ad the proof is omitted. We will use this lemma with a +, i which case it says k/(+ ( + si θ dθ +. (k /(+ Lemma. If < t <, we have where C t si t t + C t 6
7 Proof of Lemma : For positive t, we have which implies t t3 3! si t t t si t t t t t 3 6 t 6t t 3 + t + t 6 t t 6t t 3 which meas that if t the we further have Lemma 3. If we defie si t t + t 6 t t + t 6 ( t + t. H the H log ca be bouded betwee two costats (i fact, it approaches a costat. For example, we have log H log for all. The proof of Lemma 3 is reasoably straightforward ad is hece omitted. Essetially, we regard H as a Riema sum ad compare to a itegral. With these lemmas stated, we are i a positio to estimate D (θ. Recall that we expressed D (θ as a sum of + itegrals D (θ k/(+ D (θ dθ + k (k /(+ /(+ D (θ dθ. For two of those + itegrals, we just use trivial bouds. We kow D (θ +. 7
8 This implies that we have /(+ /(+ D (θ dθ ( + +, D (θ dθ ( + +. Next, we wat upper ad lower bouds for the itegrals of the form k/(+ (k /(+ where k. Note that we have si( + θ D (θ si θ The, for θ i the above iterval, we have si D (θ dθ (k + θ k + (k + si θ si k + ad the applyig Lemma gives us + k si θ si θ si( + θ. si θ si k + si (k + + (k + C (k +. Multiplyig by si( +θ the gives us + ( + ( k si θ + D ( + (θ +C (k si θ (k +. We the itegrate from θ (k to k + ad use Lemma with a. + + This gives us + k k/(+ ( + + D (θ dθ +C (k (k + + (k /(+ 8
9 which simplifies to k/(+ k D (θ dθ (k /(+ (k + C (k ( +. We will get upper ad lower bouds for D (θ if we sum the above from k to (ad use the previously metioed trivial bouds o the other itegrals. This gives us D (θ k D (θ 3 + k, ( k Next, with the help of Lemma 3, we have k k (k + C k (H (log log, (k. ( + (k (H (log( + < (log +. Note that we also have (k C ( + C ( + k (k k C ( ( + < C. This meas that we have upper ad lower bouds for D (θ that are both of the form log ± C. This completes our proof that D (θ log ad i fact, our argumet proves the slightly stroger result that D (θ log + O(. 9
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