L1 norms of products of sines

Jordan Bell
January 2, 2014

1 Introduction

A trigonometric polynomial of degree n is an expression of the form


Using the identity eit=cost+isint, we can write a trigonometric polynomial of degree n in the form


The trigonometric functions coskt and sinkt, k, are the building blocks for 2π-periodic functions (cf. [6]). To formalize the idea of the size of a 2π-periodic function and to formalize the idea of approximating 2π-periodic functions using trigonometric polynomials, we introduce Lp norms.

For 1p< and for a 2π-periodic function f, we define the Lp norm of f by


For a continuous 2π-periodic function f, we define the L norm of f by


If f is a continuous 2π-periodic function, then there is a sequence of trigonometric polynomials fn such that f-fn0 as n [11, p. 54, Corollary 5.4].

The Dirichlet kernel Dn is defined by


One can show [5, p. 71, Exercise 1.1] that


(On the other hand, it can quickly be seen that Dn=2n+1, and it follows immediately from Parseval’s identity that Dn2=2n+1.)

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 An(t)=k=1nsinktk. The local maxima and local minima of An can be explicitly determined [8, p. 74, no. 23], and it can be shown that [8, p. 74, no. 25]


In [2, p. 532, Theorem 2], the author proves the following.

Theorem 1.

Let Fn(t)=k=1nsin(kt), let M be the maximum value of


for w(0,π), let A=eM, and let B=4eM(1-e2M)-14. We have


We compute that M=-0.49452 and A=0.60985.

When I was working on this problem, I first found simpler weaker estimates that apply to a larger class of products.

For k1, let ak be a positive integer, and let


In this paper we show that we can use simpler methods to obtain nontrivial upper and lower bounds on Fna1. The results are substantially weaker than Theorem 1, but hold for any sequence a. As well, their proofs can be more readily understood. We present an asymptotic result showing that the L1 norm of sin(t)sin(qmt)sin(qm(n-1)t) approaches (2π)n as m, for q2 an integer. We present inequalities for the norms of trigonometric polynomials.

In Figure 1 we plot k=18|sin(kt)| for 0t2π. In Figure 2 we plot k=14|sin(pkt)| for 0t2π, where pk is the kth prime. In Figure 3 we plot k=110|sin(kt)| for 0t2π, where x is the least integer x. We want upper and lower bounds on the areas under these graphs.

Figure 1: k=18|sin(kt)| for 0t2π
Figure 2: k=14|sin(pkt)| for 0t2π, where pk is the kth prime
Figure 3: k=110|sin(kt)| for 0t2π

2 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 2.

For any sequence a, we have


Hölder’s inequality [7, p. 45, Theorem 2.3] (cf. [10, p. 151, Exercise 9.9]) states that if k=1n1pk=1 then


As 1=k=1n1n, this implies that


For each k,

02π|sin(akt)|n𝑑t = 1ak02πak|sint|n𝑑t
= 1akj=1ak2π(j-1)2πj|sint|n𝑑t
= 1akj=1ak02π|sin(t+2π(j-1))|n𝑑t
= 1akj=1ak02π|sint|n𝑑t
= 02π|sint|n𝑑t
= 20πsinntdt.

Let Gn=0πsinntdt. Doing integration by parts, for n2 we have

Gn = 0πsinn-1tsintdt
= -sinn-1tcost|0π+(n-1)0πsinn-2tcos2tdt
= (n-1)0πsinn-2t(1-sin2t)𝑑t
= (n-1)Gn-2-(n-1)Gn.



Say n=2m+1, m1. For m=1, we have G2m+1=G3=23G1=43. Assume that for some m1 we have

G2m+1=22m+1m!m!(2m+1)!. (1)


G2m+3 = 2m+22m+3G2m+1
= 2m+22m+322m+1m!m!(2m+1)!
= 22m+3(m+1)!(m+1)!(2m+3)!.

Therefore, by induction (1) holds for all m1. By applying Stirling’s approximation to (1) we get

G2m+1 22m+12πm(me)m2πm(me)m2π(2m+1)(2m+1e)2m+1
= 22m+12π2π(2m+1)(mm+12)2m+1122m+1e2m+1e2m
= e2π2m+1(mm+12)2m+1
< e2π2m+1.



Say n=2m, m1. For m=1, we have G2m=G2=12G0=π2. Assume that for some m1 we have

G2m=π(2m)!22mm!m!. (2)


G2m+2 = 2m+12m+2G2m
= 2m+12m+2π(2m)!22mm!m!
= π(2m+2)!22m+2(m+1)!(m+1)!.

Therefore, by induction (2) holds for all m1. Like for n=2m+1, by applying Stirling’s approximation to (2) we get G2m2π2m, and so


Hence Gn=O(1n). It follows that

Fna1 k=1n(12π02π|sin(akt)|n𝑑t)1/n
= k=1n(Gnπ)1/n
= Gnπ
= O(1n).

In the proof of Theorem 2, we saw that for any ak1, we have


We showed that


We can check, by taking logarithms and using L’Hospital’s rule, that


Therefore, there is some C1>0 such that for all m1 we have


We also showed that


and hence there is some C2>0 such that for all m1 we have


If C=min{C1,C2}, then for all n2 we have


It follows that for any positive integer a, if ak=a for all k1 then


In other words, if all the terms in the sequence a are the same then the inequality given by Theorem 2 is sharp.

In the above theorem we gave an upper bound on Fna1, and in the following theorem we give a lower bound on Fna1.

Theorem 3.

For any sequence a, we have


Since -log is a convex function on (0,), by Jensen’s inequality [7, p. 44, Theorem 2.2] we have for any nonnegative function f with f1< that


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 Fna is constant,

log(12π02π|Fna(t)|𝑑t)>12π02πlog(|Fna(t)|)𝑑t. (3)

The left-hand side of (3) is logFna1, and the the right-hand side is equal to


But for each k,

02πlog|sin(akt)|dt = 1ak02πaklog|sint|dt
= 1akj=1ak2π(j-1)2πjlog|sint|dt
= 1akj=1ak02πlog|sin(t+2π(j-1))|dt
= 1akj=1ak02πlog|sint|dt
= 02πlog|sint|dt.

Hence (3) is


We calculate 02πlog|sint|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(1+x1-x), and the power series expansion of xcot(x), and the second using the Fourier series of log|sint|.) First,


We have

0π2logsintdt = -π20log(sin(t+π2))𝑑t
= -π20log(sintcosπ2+sinπ2cost)𝑑t
= -π20logcostdt
= 0π2logcostdt.


20π2logsintdt = 0π2logsintdt+0π2logcostdt
= 0π2log(2sintcost)-log2dt
= 0π2logsin(2t)dt-π2log2
= 120πlogsintdt-π2log2.

Because 0πlogsintdt=20π2logsintdt, we have


and so




Therefore we have


and thus


In the above theorem we gave a lower bound for Fna1. In the following theorem we give another lower bound for Fna1, and we then construct examples where one lower bound is better than the other.

Theorem 4.

For any sequence a, if




If 0tπ2 then sint2πt [13]. In words, if 0<t<π2, then (t,sint) lies above the line joining (0,0) and (π2,1).

02π|Fna(t)|𝑑t = 02πk=1n|sin(akt)|dt
> 0π2Ank=1nsin(akt)dt
= (2π)n(k=1nak)0π2Antn𝑑t
= (2π)n(k=1nak)1n+1(π2An)n+1
= π21n+11Ann+1k=1nak.

If, for instance, ak=2k, the above inequality is


which is worse than (i.e. less than) the inequality given by Theorem 3.

If ak=k, the inequality is


and applying Stirling’s approximation we get


which is also worse than the inequality given by Theorem 3.

But if ak=k, the inequality is


Taking logarithms and using L’Hospital’s rule, we get


Then using Stirling’s approximation we obtain


and since e1/2<2, this lower bound is better than (i.e. greater than) the lower bound 2-n in Theorem 3.

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. 21, Definition 3.6], we say that a measure preserving map T:XX is r-fold mixing if for all g,f1,,frLr+1(X) we have

limm1,,mrXg(t)k=1rfk(Tj=1kmj(t))dμ(t)=(Xg(t)𝑑μ(t))k=1r(Xfk(t)𝑑μ(t)). (4)

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 [0,1]. Let q2 be an integer, and define Tq:[0,1][0,1] by Tq(t)=R(qt); R(x)=x-[x], where [x] is the greatest integer x. Tq is mixing of all orders. This can be proved by first showing that the dynamical system ([0,1],λ,Tq) is isomorphic to a Bernoulli shift (cf. [3, p. 17, Example 2.8]). This implies that if the Bernoulli shift is r-fold mixing then Tq is r-fold mixing. One then shows that a Bernoulli shift is mixing of all orders [3, p. 53, Exercise 2.7.9]. Using that Tq is mixing of all orders gets us the following result.

Theorem 5.

Let q2 be an integer. For each n1 we have


Define g(t)=f1(t)==fn(t)=|sin(2πt)|. For any nonzero integer N we have


and it follows from (4), using m=max{m1,,mn}, that


In other words, if for k1 we set ak(m)=q(k-1)m, then for each n we have


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 Tq is mixing of all orders. In the following we explain a more understandable derivation of the case n=1 of Theorem 5. For a measure space X with probability measure μ, we say that a measure preserving transformation T:XX is mixing (in other words, 1-fold mixing), if for all f,gL2(X) we have


Stein and Shakarchi [12, p. 305] prove that T2:[0,1][0,1] (for Tq(t)=R(qt)) is mixing, and their argument works to show that Tq:[0,1][0,1] is mixing for q2 an integer. Hence, taking f(t)=g(t)=|sin(2πt)|, we get


4 Sequences of powers

If Sn=k=1nak, then

Fna(t) = k=1nsin(akt)
= k=1neiakt-e-iakt2i
= k=1n(-e-iakt2i)(1-e2iakt)
= 12nexp(inπ2-iSnt)k=1n(1-e2iakt).

Thus |Fna(t)|=2-nk=1n|1-e2iakt|. Define Pna(t)=k=1n(1-eiakt). Bell, Borwein and Richmond [1] estimate Pna when ak is a power of k or is quadratic in k. They prove that if ak=km, with m2 an integer, then there exists a constant 1<c<2 such that Pna>cn for all sufficiently large n. The product Pna(t)=k=1n(1-eiakt) can be written as a sum,


for Sn=k=1nak. We have






It follows that


As Sn=k=1nkm=O(nm+1), the above lower bound on Fna1 is better than the one given by Theorem 3.


  • [1] J. P. Bell, P. B. Borwein, and L. B. Richmond (1998) Growth of the product j=1n(1-xaj). Acta Arith. 86 (2), pp. 155–170. External Links: ISSN 0065-1036, MathReview (Mihail N. Kolountzakis) Cited by: §4.
  • [2] J. Bell (2013) Estimates for the norms of products of sines and cosines. J. Math. Anal. Appl. 405 (2), pp. 530–545. External Links: ISSN 0022-247X, Document, Link Cited by: §1.
  • [3] M. Einsiedler and T. Ward (2010) Ergodic theory: with a view towards number theory. Graduate Texts in Mathematics, Vol. 259, Springer. Cited by: §3.
  • [4] L. Euler (1770) De summis serierum numeros Bernoullianos involventium. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae 14, pp. 129–167. Note: E393, Opera omnia I.15, pp. 91–130 Cited by: §2.
  • [5] Y. Katznelson (2004) An introduction to harmonic analysis. third edition, Cambridge Mathematical Library, Cambridge University Press. Cited by: §1.
  • [6] I. Kleiner and A. Shenitzer (1993) Mathematical building blocks. Math. Mag. 66 (1), pp. 3–13. External Links: ISSN 0025-570X, Document, Link, MathReview (B. Artmann) Cited by: §1.
  • [7] E. H. Lieb and M. Loss (2001) Analysis. second edition, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI. Cited by: §2, §2.
  • [8] G. Pólya and G. Szegő (1976) Problems and theorems in analysis, volume II. Die Grundlehren der mathematischen Wissenschaften, Vol. 216, Springer. Note: Translated from the German by C. E. Billigheimer Cited by: §1.
  • [9] Ya. G. Sinai (Ed.) (2000) Dynamical systems, ergodic theory and applications. second edition, Encyclopaedia of Mathematical Sciences, Vol. 100, Springer. Cited by: §3.
  • [10] J. M. Steele (2004) The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities. MAA Problem Books Series, Cambridge University Press. Cited by: §2.
  • [11] E. M. Stein and R. Shakarchi (2003) Fourier analysis: an introduction. Princeton Lectures in Analysis, Vol. I, Princeton University Press. Cited by: §1.
  • [12] E. M. Stein and R. Shakarchi (2005) Real analysis: measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, Vol. III, Princeton University Press. Cited by: §3.
  • [13] F. Yuefeng (1996) Proof without words: Jordan’s inequality 2xπsinxx, 0xπ2. Math. Mag. 69 (2), pp. 126. Cited by: §2.