L NORMS OF PRODUCTS OF SINES JORDAN BELL Abstract Product of sines Summary of paper and motivate it Partly a survey of L p norms of trigonometric polynomials and exponential sums There are no introductory surveys of norms of trigonometric polynomials and we do this here while focusing on a particular class A trigonometric polynomial of degree n is an expression of the form n c k e ikt, c k C k n Using the identity e it cos t + i sin t, we can write a trigonometric polynomial of degree n in the form n n a + cos kt + b k sin kt,, b k C The trigonometric functions cos kt and sin kt, k Z, are the building blocks for -periodic functions cf [6] To formalize the idea of the size of a -periodic function and to formalize the idea of approximating -periodic functions using trigonometric polynomials, we introduce L p norms For p < and for a -periodic function f, we define the L p norm of f by /p f p ft dt p For a continuous -periodic function f, we define the L norm of f by f max ft t If f is a continuous -periodic function, then there is a sequence of trigonometric polynomials f n such that f f n as n [, p 54, Corollary 54] The Dirichlet kernel D n is defined by n n D n t e ikt + cos kt k n One can show [5, p 7, Exercise ] that D n 4 log n + O On the other hand, it can quickly be seen that D n n +, and it follows immediately from Parseval s identity that D n n + Date: January, 4 Key words and phrases Fourier series, Hölder s inequality, Jensen s inequality, L p mixing, q-series, Stirling s approximation, trigonometric polynomials norms,
JORDAN BELL Pólya and Szegő [8, Part VI] present various problems about trigonometric polynomials together with solutions to them A result on L norms of trigonometric polynomials that Pólya and Szegő present is for the sum A n t n sin kt k The local maxima and local minima of A n can be explicitly determined [8, p 74, no 3], and it can be shown that [8, p 74, no 5] A n sin t dt t In [, p 53, Theorem ], the author proves the following Theorem Let F n t n sinkt, let M be the maximum value of w w log sin tdt for w,, let A e M, and let B 4e M e M 4 We have F n B n An We compute that M 4945 and A 6985 When I was working on this problem, I first found simpler weaker estimates that apply to a larger class of products For k, let be a positive integer, and let Fn a t sin t, a In this paper we show that we can use simpler methods to obtain nontrivial upper and lower bounds on Fn a The results are substantially weaker than Theorem, but hold for any sequence a As well, their proofs can be more readily understood We present an asymptotic result showing that the L norm of sint sinq m t sinq mn t approaches n as m, for q an integer We present inequalities for the norms of trigonometric polynomials In Figure we plot 8 sinkt for t In Figure we plot 4 sinp kt for t, where p k is the kth prime In Figure 3 we plot sin k t for t, where x is the least integer x We want upper and lower bounds on the areas under these graphs Upper and lower bounds Hölder s inequality is the first tool for which we reach when we want to bound the norm of a product Theorem For any sequence a, we have Fn a O n
L NORMS OF PRODUCTS OF SINES 3 7 6 5 4 3 4 4 Π 4 Π Figure 8 sinkt for t 3 4 4 Π 4 Π Figure 4 sinp kt for t, where p k is the kth prime Proof Hölder s inequality [7, p 45, Theorem 3] cf [, p 5, Exercise 99] states that if n p k then f f pk
4 JORDAN BELL 3 5 5 5 4 4 Π 4 Π Figure 3 sin k t for t As n n, this implies that Fn a sin t n For each k, sin t n dt ak sin t n dt j j j j j sin t n dt sin n tdt sin t n dt sint + j n dt sin t n dt
L NORMS OF PRODUCTS OF SINES 5 Let G n sinn tdt Doing integration by parts, for n we have Thus, G n sin n t sin tdt sin n t cos t + n sin n t cos tdt n sin n t sin tdt n G n n G n G n n n G n Say n m +, m For m, we have G m+ G 3 3 G 4 3 Assume that for some m we have G m+ m+ m!m! m +! Then G m+3 m + m + 3 G m+ m + m + 3 m+ m!m! m +! m+3 m +!m +! m + 3! Therefore, by induction holds for all m By applying Stirling s approximation to we get m+ m m m m m m e e G m+ m+ m + < m+ e m+ m m + e m m + m + e m + m+ em+ m + m+ e m m+ Thus, G m+ O m + Say n m, m For m, we have G m G G Assume that for some m we have G m m! m m!m!
6 JORDAN BELL Then G m+ m + m + G m m + m + m! m m!m! m +! m+ m +!m +! Therefore, by induction holds for all m Like for n m +, by applying Stirling s approximation to we get G m m, and so G m O m Hence G n O n It follows that F a n /n Gn G n O n /n sin t n dt In the proof of Theorem, we saw that for any, we have sin t n dt G n, G n sin n tdt, We showed that G m+ e m+ m m + m + We can check, by taking logarithms and using L Hospital s rule, that m+ m lim m m + e Therefore, there is some C > such that for all m we have G m+ C m + We also showed that G m m, and hence there is some C > such that for all m we have G m C m
L NORMS OF PRODUCTS OF SINES 7 If C min{c, C }, then for all n we have G n C n It follows that for any positive integer a, if a for all k then Fn a G n C n In other words, if all the terms in the sequence a are the same then the inequality given by Theorem is sharp In the above theorem we gave an upper bound on F a n, and in the following theorem we give a lower bound on F a n Theorem 3 For any sequence a, we have F a n > n Proof Since log is a convex function on,, by Jensen s inequality [7, p 44, Theorem ] we have for any nonnegative function f with f < that log ftdt logftdt, and the two sides are equal if and only if f is constant almost everywhere for continuous f this is equivalent to f being constant Hence, as there is no sequence a of positive integers such that Fn a is constant, 3 log Fn a t dt > log F a n t dt The left-hand side of 3 is log Fn a, and the the right-hand side is equal to n log sin t dt n log sin t dt But for each k, Hence 3 is log sin t dt ak log sin t dt j j j j j log F a n > n log sin t dt log sin t dt log sint + j dt log sin t dt log sin t dt
8 JORDAN BELL We calculate log sin t dt in the following way The earliest evaluation of this integral of which the author is aware is by Euler [4], who gives two derivations, the first using the Euler-Maclaurin summation formula, the power series expansion for log +x x, and the power series expansion of x cotx, and the second using the Fourier series of log sin t First, We have Therefore, log sin tdt log sin t dt 4 log sin tdt Because log sin tdt and so Thus Therefore we have and thus log sin tdt log sin t + dt log sin t cos + sin cos t dt log cos tdt log cos tdt log sin tdt, we have log sin tdt log sin tdt + log cos tdt log sin t cos t log dt log sintdt log log sin tdt log log sin tdt log, log sin tdt log log sin t dt log log F a n > n log n log log n, F a n > n
L NORMS OF PRODUCTS OF SINES 9 In the above theorem we gave a lower bound for F a n In the following theorem we give another lower bound for F a n, and we then construct examples where one lower bound is better than the other Theorem 4 For any sequence a, if then F a n > 4 A n max k n, n + A n+ n Proof If t then sin t t [3] In words, if < t <, then t, sin t lies above the line joining, and, Fn a t dt sin t dt > An An sin tdt t dt n n n n n + A n+ n An n + t n dt A n n+ If, for instance, k, the above inequality is Fn a > 4 n + nn+ nn+ 4 n + nn+, which is worse than ie less than the inequality given by Theorem 3 If k, the inequality is Fn a > 4 n + n!, nn+ and applying Stirling s approximation we get F a n n + n e n, which is also worse than the inequality given by Theorem 3 But if k, the inequality is Fn a > 4 n + k 4 n + n + n+ n n+ + n n+ k
JORDAN BELL Taking logarithms and using L Hospital s rule, we get lim + n+ e n n Then using Stirling s approximation we obtain F a n n 5 4 e n, and since e / <, this lower bound is better than ie greater than the lower bound n in Theorem 3 Mixing This section talks about measure spaces and mixing These topics take repeated exposure to become comfortable with, but Theorem 5 is a pretty result whose statement can be understood without understanding its proof The notion of mixing is related to independent random variables, for which the expectation of their product is equal to the product of their expectations Let X be a measure space with probability measure µ Following [9, p, Definition 36], we say that a measure preserving map T : X X is r-fold mixing if for all g, f,, f r L r+ X we have 4 lim m,,m r X r gtdµt X gt X r f k T k j mj t dµt f k tdµt If for each r the map T is r-fold mixing, we say that T is mixing of all orders Let λ be Lebesgue measure on [, ] Let q be an integer, and define T q : [, ] [, ] by T q t Rqt; Rx x [x], where [x] is the greatest integer x T q is mixing of all orders This can be proved by first showing that the dynamical system [, ], λ, T q is isomorphic to a Bernoulli shift cf [3, p 7, Example 8] This implies that if the Bernoulli shift is r-fold mixing then T q is r-fold mixing One then shows that a Bernoulli shift is mixing of all orders [3, p 53, Exercise 79] Using that T q is mixing of all orders gets us the following result Theorem 5 Let q be an integer For each n we have sint sin q km t n+ dt lim m Proof Define gt f t f n t sint For any nonzero integer N we have sinnt dt, and it follows from 4, using m max{m,, m n }, that sint sin q km t dt lim m n+
L NORMS OF PRODUCTS OF SINES In other words, if for k we set m q k m, then for each n we have n am lim Fn, am m m It would be overwhelming for a reader without experience in ergodic theory to work out the details of the reasoning that we indicated above Theorem 5 for why T q is mixing of all orders In the following we explain a more understandable derivation of the case n of Theorem 5 For a measure space X with probability measure µ, we say that a measure preserving transformation T : X X is mixing in other words, -fold mixing, if for all f, g L X we have lim m X f T m t gtdµt X ftdµt X gtdµt Stein and Shakarchi [, p 35] prove that T : [, ] [, ] for T q t Rqt is mixing, and their argument works to show that T q : [, ] [, ] is mixing for q an integer Hence, taking ft gt sint, we get lim m If S n n, then F a n t sinq m t sint dt 3 Sequences of powers sin t e iakt e it i e i t e i t i in n exp is n nt e i t Thus Fn a t n n e i t Define P a n t n e i t Bell, Borwein and Richmond [] estimate Pn a when is a power of k or is quadratic in k They prove that if k m, with m an integer, then there exists a constant < c < such that Pn a > c n for all sufficiently large n The product Pn a t n eiakt can be written as a sum, S n e iakt e ikt, e ikt Pn a tdt, k for S n n We have Pn a S n S n S n t e ikt e ikt But k k e ikt Pn a t dt k P a n t dt P a n
JORDAN BELL Hence Pn a S n + Pn a It follows that Fn a n P n a n P n a S n + > c n S n + As S n n km On m+, the above lower bound on Fn a is better than the one given by Theorem 3 References J P Bell, P B Borwein, and L B Richmond, Growth of the product n j xa j, Acta Arith 86 998, no, 55 7 Jordan Bell, Estimates for the norms of products of sines and cosines, J Math Anal Appl 45 3, no, 53 545 3 Manfred Einsiedler and Thomas Ward, Ergodic theory: with a view towards number theory, Graduate Texts in Mathematics, vol 59, Springer, 4 Leonhard Euler, De summis serierum numeros Bernoullianos involventium, Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 4 77, 9 67, E393, Opera omnia I5, pp 9 3 5 Yitzhak Katznelson, An introduction to harmonic analysis, third ed, Cambridge Mathematical Library, Cambridge University Press, 4 6 I Kleiner and A Shenitzer, Mathematical building blocks, Math Mag 66 993, no, 3 3 7 Elliott H Lieb and Michael Loss, Analysis, second ed, Graduate Studies in Mathematics, vol 4, American Mathematical Society, Providence, RI, 8 George Pólya and Gábor Szegő, Problems and theorems in analysis, volume II, Die Grundlehren der mathematischen Wissenschaften, vol 6, Springer, 976, Translated from the German by C E Billigheimer 9 Ya G Sinai ed, Dynamical systems, ergodic theory and applications, second ed, Encyclopaedia of Mathematical Sciences, vol, Springer, J Michael Steele, The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities, MAA Problem Books Series, Cambridge University Press, 4 Elias M Stein and Rami Shakarchi, Fourier analysis: an introduction, Princeton Lectures in Analysis, vol I, Princeton University Press, 3, Real analysis: measure theory, integration, and Hilbert spaces, Princeton Lectures in Analysis, vol III, Princeton University Press, 5 3 Feng Yuefeng, Proof without words: Jordan s inequality x sin x x, x, Math Mag 69 996, no, 6 E-mail address: jordanbell@gmailcom Department of Mathematics, University of Toronto, Toronto, Ontario, Canada