# $L^{p}$ norms of trigonometric polynomials

Jordan Bell
April 3, 2014

## 1 Introduction

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

 $\sum_{k=-n}^{n}c_{k}e^{ikt},\qquad c_{k}\in\mathbb{C}.$

Using the identity $e^{it}=\cos t+i\sin t$, we can write a trigonometric polynomial of degree $n$ in the form

 $a_{0}+\sum_{k=1}^{n}a_{k}\cos kt+\sum_{k=1}^{n}b_{k}\sin kt,\qquad a_{k},b_{k}% \in\mathbb{C}.$

For $1\leq p<\infty$ and for a $2\pi$-periodic function $f$, we define the $L^{p}$ norm of $f$ by

 $\|f\|_{p}=\left(\frac{1}{2\pi}\int_{0}^{2\pi}|f(t)|^{p}dt\right)^{1/p}.$

For a continuous $2\pi$-periodic function $f$, we define the $L^{\infty}$ norm of $f$ by

 $\|f\|_{\infty}=\max_{0\leq t\leq 2\pi}|f(t)|.$

If $f$ is a continuous $2\pi$-periodic function, then there is a sequence of trigonometric polynomials $f_{n}$ such that $\|f-f_{n}\|_{\infty}\to 0$ as $n\to\infty$ [31, p. 54, Corollary 5.4].

If $1\leq p<\infty$ and $f$ is a continuous $2\pi$-periodic function, then

 $\|f\|_{p}=\left(\frac{1}{2\pi}\int_{0}^{2\pi}|f(t)|^{p}dt\right)^{1/p}\leq% \left(\frac{1}{2\pi}\int_{0}^{2\pi}\|f\|_{\infty}^{p}dt\right)^{1/p}=\|f\|_{% \infty}.$

Jensen’s inequality [16, p. 44, Theorem 2.2] (cf. [30, p. 113, Problem 7.5]) tells us that if $\phi:[0,\infty)\to\mathbb{R}$ is convex, then for any function $h:[0,2\pi]\to[0,\infty)$ we have

 $\phi\left(\frac{1}{2\pi}\int_{0}^{2\pi}h(t)dt\right)\leq\frac{1}{2\pi}\int_{0}% ^{2\pi}\phi(h(t))dt.$

If $1\leq p, then $\phi:[0,\infty)\to\mathbb{R}$ defined by $\phi(x)=x^{q/p}$ is convex. Hence, if $1\leq p then for any $2\pi$-periodic function $f$,

 $\displaystyle\|f\|_{p}$ $\displaystyle=$ $\displaystyle(\phi(\|f\|_{p}^{p}))^{1/q}$ $\displaystyle=$ $\displaystyle\left(\phi\left(\frac{1}{2\pi}\int_{0}^{2\pi}|f(t)|^{p}dt\right)% \right)^{1/q}$ $\displaystyle\leq$ $\displaystyle\left(\frac{1}{2\pi}\int_{0}^{2\pi}\phi(|f(t)|^{p})dt\right)^{1/q}$ $\displaystyle=$ $\displaystyle\left(\frac{1}{2\pi}\int_{0}^{2\pi}|f(t)|^{q}dt\right)^{1/q}$ $\displaystyle=$ $\displaystyle\|f\|_{q}.$

The Dirichlet kernel $D_{n}$ is defined by

 $D_{n}(t)=\sum_{k=-n}^{n}e^{ikt}=1+2\sum_{k=1}^{n}\cos kt.$

One can show [14, p. 71, Exercise 1.1] that

 $\|D_{n}\|_{1}=\frac{4}{\pi^{2}}\cdot\log n+O(1).$

(On the other hand, it can quickly be seen that $\|D_{n}\|_{\infty}=2n+1$, and it follows from Parseval’s identity that $\|D_{n}\|_{2}=\sqrt{2n+1}$.)

Pólya and Szegő [27, Part VI] present various problems about trigonometric polynomials together with solutions to them. A result on $L^{\infty}$ norms of trigonometric polynomials that Pólya and Szegő present is for the sum $A_{n}(t)=\sum_{k=1}^{n}\frac{\sin kt}{k}$. The local maxima and local minima of $A_{n}$ can be explicitly determined [27, p. 74, no. 23], and it can be shown that [27, p. 74, no. 25]

 $\|A_{n}\|_{\infty}\sim\int_{0}^{\pi}\frac{\sin t}{t}dt.$

## 2 Lp norms

If $1\leq p, then [14, p. 123, Exercise 1.8] (cf. [7, p. 102, Theorem 2.6]) there is some $C(p,q)$ such that for any trigonometric polynomial $f$ of degree $n$, we have

 $\|f\|_{q}\leq C(p,q)n^{\frac{1}{p}-\frac{1}{q}}\|f\|_{p}.$

This inequality is sharp [33, p. 230]: for $1\leq p there is some $C^{\prime}(p,q)$ such that if $F_{n}(t)=\frac{1}{n}\sum_{k=0}^{n-1}D_{k}(t)$ ($F_{n}$ is called the Fejér kernel) then

 $\|F_{n}\|_{q}>C^{\prime}(p,q)n^{\frac{1}{p}-\frac{1}{q}}\|F_{n}\|_{p}.$
###### Theorem 1.

Let $1\leq p\leq q\leq\infty$. If $\hat{f}(j)=0$ for $|j|>n+1$ then

 $\|f\|_{q}\leq 5(n+1)^{\frac{1}{p}-\frac{1}{q}}\|f\|_{p}.$
###### Proof.

Let $K_{n}(t)=\sum_{j=-n}^{n}\Big{(}1-\frac{|j|}{n+1}\Big{)}e^{ijt}$, the Fejér kernel. From this expression we get $|K_{n}(t)|\leq K_{n}(0)=n+1$. It’s straightforward to show that $K_{n}(t)=\frac{1}{n+1}\Big{(}\frac{\sin\frac{n+1}{2}t}{\sin\frac{1}{2}t}\Big{)% }^{2}$. Since $\sin\frac{t}{2}>\frac{t}{\pi}$ for $0, we get $|K_{n}(t)|\leq\frac{\pi^{2}}{(n+1)t^{2}}$, and thus we obtain

 $|K_{n}(t)|\leq\min\Big{(}n+1,\frac{\pi^{2}}{(n+1)t^{2}}\Big{)}.$

Then, for any $r\geq 1$,

 $\displaystyle\|K_{n}\|_{r}^{r}$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}|K_{n}(t)|^{r}dt$ $\displaystyle\leq$ $\displaystyle\frac{1}{2\pi}\int_{0}^{\frac{\pi}{n+1}}(n+1)^{r}dt+\frac{1}{2\pi% }\int_{\frac{\pi}{n+1}}^{2\pi}\Big{(}\frac{\pi^{2}}{(n+1)t^{2}}\Big{)}^{r}dt$ $\displaystyle=$ $\displaystyle\frac{(n+1)^{r-1}}{2}+\frac{1}{2}\frac{1}{(n+1)^{r}}\frac{1}{2r-1% }\Big{(}(n+1)^{2r-1}-\frac{1}{2^{2r-1}}\Big{)}$ $\displaystyle\leq$ $\displaystyle\frac{(n+1)^{r-1}}{2}+\frac{1}{2}\frac{1}{(n+1)^{r}}\frac{1}{2r-1% }(n+1)^{2r-1}$ $\displaystyle\leq$ $\displaystyle(n+1)^{r-1}.$

Hence $\|K_{n}\|_{r}\leq(n+1)^{1-\frac{1}{r}}$.

Let $V_{n}(t)=2K_{2n+1}(t)-K_{n}(t)$, the de la Vallée Poussin kernel. Then

 $\|V_{n}\|_{r}\leq 2\|K_{2n+1}\|_{r}+\|K_{n}\|_{r}\leq 2(2n+2)^{1-\frac{1}{r}}+% (n+1)^{1-\frac{1}{r}}\leq 5(n+1)^{1-\frac{1}{r}}.$

For $|j|\leq n+1$ we have $\widehat{V_{n}}(j)=1$, and one thus checks that $V_{n}*f=f$. Take $\frac{1}{q}+1=\frac{1}{p}+\frac{1}{r}$. By Young’s inequality we have

 $\|f\|_{q}=\|V_{n}*f\|_{q}\leq\|V_{n}\|_{r}\|f\|_{p}\leq 5(n+1)^{\frac{1}{p}-% \frac{1}{q}}\|f\|_{p}.$

Let $X_{n}=\{a_{0}+\sum_{k=1}^{n}a_{k}\cos kt+b_{k}\sin kt:a_{k},b_{k}\in\mathbb{R}\}$, the real vector space of real valued trigonometric polynomials of degree $n$, have norm

 $\|f\|_{X_{n}}=\max\{|a_{0}|,|a_{1}|,\ldots,|a_{n}|,|b_{1}|,\ldots,|b_{n}|\}.$

Let $Y_{n,p}$ be the same vector space with the $L^{p}$ norm. Ash and Ganzburg [1] give upper and lower bounds on the operator norm of the map $i:X_{n}\to Y_{n,p}$ defined by $i(f)=f$.

Bernstein’s inequality [14, p. 50, Exercise 7.16] states that for $1\leq p\leq\infty$, if $f$ is a trigonometric polynomial of degree $n$, then

 $\|f^{\prime}\|_{p}\leq n\|f\|_{p}.$

In the other direction, if $f\in C^{1}$ then

 $\begin{split}&\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}f(s)ds+\frac{1}{2\pi}% \int_{0}^{t}sf^{\prime}(s)ds+\frac{1}{2\pi}\int_{t}^{2\pi}(s-2\pi)f^{\prime}(s% )ds\\ \displaystyle=&\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}f(s)ds+\frac{1}{2\pi}% \int_{0}^{2\pi}sf^{\prime}(s)ds-\int_{t}^{2\pi}f^{\prime}(s)ds\\ \displaystyle=&\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}f(s)ds+\frac{1}{2\pi}% sf(s)\Big{|}_{0}^{2\pi}-\frac{1}{2\pi}\int_{0}^{2\pi}f(s)ds-f(s)\Big{|}_{t}^{2% \pi}\\ \displaystyle=&\displaystyle f(t).\end{split}$

Hence

 $\displaystyle|f(t)|$ $\displaystyle\leq$ $\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}|f(s)|ds+\frac{1}{2\pi}\int_{0}^{t}s% |f^{\prime}(s)|ds+\frac{1}{2\pi}\int_{t}^{2\pi}(2\pi-s)|f^{\prime}(s)|ds$ $\displaystyle\leq$ $\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}|f(s)|ds+\int_{0}^{t}|f^{\prime}(s)|% ds+\int_{t}^{2\pi}|f^{\prime}(s)|ds$ $\displaystyle=$ $\displaystyle\|f\|_{1}+2\pi\|f^{\prime}\|_{1},$

so

 $\|f\|_{\infty}\leq\|f\|_{1}+2\pi\|f^{\prime}\|_{1}.$

This is an instance of the Sobolev inequality [26].

It turns out that for a trigonometric polynomial the mass cannot be too concentrated. More precisely, the number of nonzero terms of a trigonometric polynomial restricts how concentrated its mass can be. Let $d\mu=\frac{dt}{2\pi}$. Thus $\mu([0,2\pi])=1$. A result of Turán [20, p. 89, Lemma 1] states that if $\lambda_{1},\ldots,\lambda_{N}\in\mathbb{Z}$ and $T(t)=\sum_{n=1}^{N}b_{n}e^{i\lambda_{n}t}$, $b_{n}\in\mathbb{C}$, then for any closed arc $I\subset[0,2\pi]$,

 $\|T\|_{\infty}\leq\left(\frac{2e}{\mu(I)}\right)^{N-1}\max_{t\in I}|T(t)|.$

Nazarov [11, p. 452] shows that there is some constant $A$ such that if $E$ is a closed subset of $[0,2\pi]$ (not necessarily an arc), then

 $\|\hat{T}\|_{1}\leq\left(\frac{A}{\mu(E)}\right)^{N}\max_{t\in E}|f(T)|.$

Nazarov [23] proves that there exists some constant $C$ such that if $0\leq q\leq 2$ and $\mu(E)\geq\frac{1}{3}$, then

 $\|T\|_{q}\leq e^{C(N-1)\left(1-\frac{\mu(E)}{2\pi}\right)}\left(\frac{1}{2\pi}% \int_{E}|T(t)|^{q}dt\right)^{1/q}.$

These results of Turan and Nazarov are examples of the uncertainty principle [9], which is the general principle that a constrain on the support of the Fourier transform of a function constrains the support of the function itself.

In [10], Hardy and Littlewood present inequalities for norms of $2\pi$-periodic functions in terms of certain series formed from their Fourier coefficients. Let $c_{k}\in\mathbb{C}$, $k\in\mathbb{Z}$, be such that $c_{k}\to 0$ as $k\to\pm\infty$, and define $c_{0}^{*},c_{1}^{*},c_{-1}^{*},c_{2}^{*},c_{-2}^{*},\ldots$ to be the absolute values of the $c_{k}$ ordered in decreasing magnitude. For real $r>1$, define

 $S_{r}^{*}(c)=\left(\sum_{k=-\infty}^{\infty}{c_{k}^{*}}^{r}(|k|+1)^{r-2}\right% )^{1/r}.$

For instance, if $c_{k}=1$ for $-N\leq k\leq N$ and $c_{k}=0$ for $|k|>N$, then $S_{r}^{*}(c)=\left(1+2\sum_{k=2}^{N+1}k^{r-2}\right)^{1/r}$. Hardy and Littlewood state the result [10, p. 164, Theorem 2] that if $1 then there is some constant $A(p)$ such that for any sequence $c$, with $c_{k}\to 0$ as $k\to\pm\infty$, if $f(t)=\sum_{k=-\infty}^{\infty}c_{k}e^{ikt}$ and $\|f\|_{p}<\infty$ then

 $S_{p}^{*}(c)\leq A(p)\|f\|_{p}.$

A proof of this is given in Zygmund [35, vol. II, p. 128, chap. XII, Theorem 6.3]. Asking if this inequality holds for $p=1$ suggests the following question that Hardy and Littlewood pose at the end of their paper [10, p. 168]: Is there a constant $A$ such that for all distinct positive integers $m_{k},k=1,\ldots,N$, we have

 $\|\sum_{k=1}^{N}\cos m_{k}t\|_{1}>A\log N?$

McGehee, Pigno and Smith [18] prove that there is some $K$ such that for all $N$, if $n_{1},\ldots,n_{N}$ are distinct integers and $c_{1},\ldots,c_{N}\in\mathbb{C}$ satisfy $|c_{k}|\geq 1$, then

 $\|\sum_{k=1}^{N}c_{k}e^{in_{k}t}\|_{1}>K\log N.$

Thus

 $\|\sum_{k=1}^{N}\cos m_{k}t\|_{1}=\frac{1}{2}\cdot\|\sum_{k=1}^{N}e^{im_{k}t}+% e^{-im_{k}t}\|_{1}\geq\frac{1}{2}\cdot K\log(2N).$

For $k\geq 2$, define $T_{N}(t)=\sum_{n=1}^{N}e^{in^{k}t}$. Since $\|T_{N}\|_{\infty}=N$, for each $p\geq 1$ we have $\|T_{N}\|_{p}\leq N$. Hua’s lemma [22, p. 116, Theorem 4.6] states that if $\epsilon>0$, then

 $\|T_{N}\|_{2^{k}}=O\left(N^{1-\frac{k}{2^{k}}+\epsilon}\right).$

Hua’s lemma is used in additive number theory. The number of sets of integer solutions of the equation

 $f(x_{1},\ldots,x_{n})=N,\qquad a_{r}\leq x_{r}\leq b_{r}$

is equal to (cf. [12, p. 151])

 $\sum_{a_{1}\leq x_{1}\leq b_{1}}\cdots\sum_{a_{n}\leq x_{n}\leq b_{n}}\int_{0}% ^{1}e^{2\pi i(f(x_{1},\ldots,x_{n})-N)t}dt.$

Borwein and Lockhart [4]: what is the expected $L^{p}$ norm of a trigonometric polynomial of order $n$? Kahane [13, Chapter 6] also presents material on random trigonometric polynomials.

Nursultanov and Tikhonov [25]: the sup on a subset of $\mathbb{T}$ of a trigonometric polynomial $f$ of degree $n$ being lower bounded in terms of $\|f\|_{\infty}$, $n$, and the measure of the subset.

## 3 ℓᵖ norms

For a $2\pi$-periodic function $f$, we define $\hat{f}:\mathbb{Z}\to\mathbb{C}$ by

 $\hat{f}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-ikt}f(t)dt.$

For $1\leq p<\infty$, we define the $\ell^{p}$ norm of $\hat{f}$ by

 $\|\hat{f}\|_{p}=\left(\sum_{k=-\infty}^{\infty}|\hat{f}(k)|^{p}\right)^{1/p},$

and we define the $\ell^{\infty}$ norm of $\hat{f}$ by

 $\|\hat{f}\|_{\infty}=\max_{k\in\mathbb{Z}}|\hat{f}(k)|.$

Parseval’s identity [31, p. 80, Theorem 1.3] states that $\|f\|_{2}=\|\hat{f}\|_{2}$.

If $1\leq p<\infty$, then

 $\|\hat{f}\|_{\infty}\leq\left(\cdots+\|\hat{f}\|_{\infty}^{p}+\cdots\right)^{1% /p}=\|\hat{f}\|_{p}.$

If $1\leq p, then, since for each $k$, $\frac{|\hat{f}(k)|}{\|\hat{f}\|_{q}}\leq 1$,

 $1=\left(\sum_{k=-\infty}^{\infty}\left(\frac{|\hat{f}(k)|}{\|\hat{f}\|_{q}}% \right)^{q}\right)^{1/q}\leq\left(\sum_{k=-\infty}^{\infty}\left(\frac{|\hat{f% }(k)|}{\|\hat{f}\|_{q}}\right)^{p}\right)^{1/q}=\frac{\|\hat{f}\|_{p}^{p/q}}{% \|\hat{f}\|_{q}^{p/q}}.$

Hence for $1\leq p,

 $\|\hat{f}\|_{q}\leq\|\hat{f}\|_{p}.$

For $1\leq p<\infty$, if $f$ is a trigonometric polynomial of degree $n$ then

 $\|\hat{f}\|_{p}=\left(\sum_{k=-n}^{n}|\hat{f}(k)|^{p}\right)^{1/p}\leq\left(% \sum_{k=-n}^{n}\|\hat{f}\|_{\infty}^{p}\right)^{1/p}=(2n+1)^{1/p}\|\hat{f}\|_{% \infty}.$

For $1\leq p, we have [30, p. 123, Problem 8.3] (this is Jensen’s inequality for sums)

 $\left(\sum_{k=-n}^{n}\frac{1}{2n+1}|\hat{f}(k)|^{p}\right)^{1/p}\leq\left(\sum% _{k=-n}^{n}\frac{1}{2n+1}|\hat{f}(k)|^{q}\right)^{1/q},$

i.e.

 $(2n+1)^{-1/p}\|\hat{f}\|_{p}\leq(2n+1)^{-1/q}\|\hat{f}\|_{q}.$

Hence for $1,

 $\|\hat{f}\|_{p}\leq(2n+1)^{\frac{1}{p}-\frac{1}{q}}\|\hat{f}\|_{q}.$

For any $t$,

 $|f(t)|=\left|\sum_{k=-\infty}^{\infty}\hat{f}(k)e^{ikt}\right|\leq\sum_{k=-% \infty}^{\infty}|\hat{f}(k)e^{ikt}|=\sum_{k=-\infty}^{\infty}|\hat{f}(k)|=\|% \hat{f}\|_{1}.$

Hence

 $\|f\|_{\infty}\leq\|\hat{f}\|_{1}.$

For any $k\in\mathbb{Z}$,

 $|\hat{f}(k)|=\left|\frac{1}{2\pi}\int_{0}^{2\pi}e^{-ikt}f(t)dt\right|\leq\frac% {1}{2\pi}\int_{0}^{2\pi}|f(t)|dt=\|f\|_{1}.$

Hence

 $\|\hat{f}\|_{\infty}\leq\|f\|_{1}.$

The Hausdorff-Young inequality [32, p. 57, Corollary 2.4] states that for $1\leq p\leq 2$ and $\frac{1}{p}+\frac{1}{q}=1$, if $f\in L^{p}$ then

 $\|\hat{f}\|_{q}\leq\|f\|_{p}.$

The dual Hausdorff-Young inequality [32, p. 58, Corollary 2.5] states that for $1\leq p\leq 2$ and $\frac{1}{p}+\frac{1}{q}=1$, if $f\in L^{q}$ then

 $\|f\|_{q}\leq\|\hat{f}\|_{q}.$

A survey on the Hausdorff-Young inequality is given in [6])

For $M+1\leq k\leq M+N$, let $a_{k}\in\mathbb{C}$ and let $S(t)=\sum_{k=M+1}^{N+1}a_{k}e^{ikt}$. Let $t_{1},\ldots,t_{R}\in\mathbb{R}$, and let $\delta$ be such that if $r\neq s$ then

 $\|t_{r}-t_{s}\|\geq\delta,$

where $\|t\|=\min_{k}|t-k|$ is the distance from $t$ to a nearest integer. The large sieve [19] is an inequality of the form

 $\sum_{r=1}^{R}|S(2\pi t_{r})|^{2}\leq\Delta(N,\delta)\sum_{k=M+1}^{M+N}|a_{k}|% ^{2}.$

A result of Selberg [19, p. 559, Theorem 3] shows that the large sieve is valid for $\Delta=N-1+\delta^{-1}$.

Kristiansen [15]

Boas [2]

For $F:\mathbb{Z}/n\to\mathbb{C}$, its Fourier transform $\hat{F}:\mathbb{Z}/n\to\mathbb{C}$ (called the discrete Fourier transform) is defined by

 $\hat{F}(k)=\frac{1}{n}\sum_{j=0}^{n-1}F(j)e^{-2\pi ijk/n},\qquad 0\leq k\leq n% -1,$

and one can prove [31, p. 223, Theorem 1.2] that

 $F(j)=\sum_{k=0}^{n-1}\hat{F}(k)e^{2\pi ikj/N},\qquad 0\leq j\leq n-1.$

One can also prove Parseval’s identity for the Fourier transform on $\mathbb{Z}/n$ [31, p. 223, Theorem  1.2]. It states

 $\sum_{k=0}^{n-1}|\hat{F}(k)|^{2}=\frac{1}{n}\sum_{j=0}^{n-1}|F(j)|^{2}.$

Let $P(t)=\sum_{k=0}^{n-1}a_{k}e^{ikt}$. Define $F:\mathbb{Z}/n\to\mathbb{C}$ by

 $F(j)=\sum_{k=0}^{n-1}a_{k}e^{2\pi ikj/n},\qquad 0\leq j\leq n-1.$

(That is, $\hat{F}(k)=a_{k}$.) We then have

 $\sum_{k=0}^{n-1}|a_{k}|^{2}=\frac{1}{n}\sum_{j=0}^{n-1}|F(j)|^{2}=\frac{1}{n}% \sum_{j=0}^{n-1}\big{|}P\Big{(}\frac{2\pi j}{n}\Big{)}\big{|}^{2}.$

Thus

 $\|P\|_{2}=\left(\frac{1}{n}\sum_{j=0}^{n-1}\big{|}P\Big{(}\frac{2\pi j}{n}\Big% {)}\big{|}^{2}\right)^{1/2}.$

The Marcinkiewicz-Zygmund inequalities [35, vol. II, p. 28, chap. X, Theorem 7.5] state that there is a constant $A$ such that for $1\leq p\leq\infty$, if $f$ is a trigonometric polynomial of degree $n$ then

 $\left(\frac{1}{2n+1}\sum_{k=0}^{2n}\big{|}f\Big{(}\frac{2\pi k}{2n+1}\Big{)}% \big{|}^{p}\right)^{1/p}\leq A(2\pi)^{1/p}\|f\|_{p},$

and for each $1 there exists some $A_{p}$ such that if $f$ is a trigonometric polynomial of degree $n$ then

 $\|f\|_{p}\leq A_{p}\left(\frac{1}{2n+1}\sum_{k=0}^{2n}\big{|}f\Big{(}\frac{2% \pi k}{2n+1}\Big{)}\big{|}^{p}\right)^{1/p}.$

Máté and Nevai [17, p. 148, Theorem 6] prove that for $p>0$, if $S_{n}$ is a trigonometric polynomial of degree $n$ then

 $\|S_{n}\|_{\infty}\leq\left(\frac{(1+np)e}{2}\right)^{1/p}\|S_{n}\|_{p}.$

Máté and Nevai [17] prove a version of Bernstein’s inequality for $0, and their result can be sharpened to the following [34]: For $0, if $T_{n}$ is a trigonometric polynomial of order $n$ then

 $\|T_{n}^{\prime}\|_{p}\leq n\|T_{n}\|_{p}.$

Let $\mathop{\mathrm{supp}}\hat{f}=\{k\in\mathbb{Z}:\hat{f}(k)\neq 0\}$. A subset $\Lambda$ of $\mathbb{Z}$ is called a Sidon set [28, p. 121, §5.7.2] if there exists a constant $B$ such that for every trigonometric polynomial $f$ with $\mathop{\mathrm{supp}}\hat{f}\subseteq\Lambda$ we have

 $\|\hat{f}\|_{1}\leq B\|f\|_{\infty}.$

Let $B(\Lambda)$ be the least such $B$. A sequence of positive integers $\lambda_{k}$ is said to be lacunary if there is a constant $\rho$ such that $\lambda_{k+1}>\rho\lambda_{k}$ for all $k$. If $\lambda_{k}$ is a lacunary sequence, then $\{\lambda_{k}\}$ is a Sidon set [21, p. 154, Corollary 6.17]. If $\Lambda\subset\mathbb{Z}$ is a Sidon set, then [28, p. 128, Theorem 5.7.7] (cf. [21, p. 157, Corollary 6.19]) for any $2, for every trigonometric polynomial $f$ with $\mathop{\mathrm{supp}}\hat{f}\subseteq\Lambda$ we have

 $\|f\|_{p}\leq B(\Lambda)\sqrt{p}\|f\|_{2},$

and

 $\|f\|_{2}\leq 2B(\Lambda)\|f\|_{1}.$

Let $0. A subset $E$ of $\mathbb{Z}$ is called a $\Lambda(p)$-set if for every $0 there is some $A(E,p)$ such that for all trigonometric polynomials $f$ with $\mathop{\mathrm{supp}}\hat{f}\subset E$ we have

 $\|f\|_{p}\leq A(E,p)\|f\|_{2}.$

$\Lambda(p)$ sets were introduced by Rudin, and he discusses them in his autobiography [29, Chapter 28]. A modern survey of $\Lambda(p)$-sets is given by Bourgain [5].

Bochkarev [3] proves various lower bounds on the $L^{1}$ norms of certain trigonometric polynomials. Let $c_{k}\in\mathbb{C}$, $k\geq 1$. If there are constants $A$ and $B$ such that

 $A\frac{(\log k)^{s}}{\sqrt{k}}\leq|c_{k}|\leq B\frac{(\log k)^{s}}{\sqrt{k}},% \qquad k\geq 1,$

then [3, p. 58, Theorem 19]

 $\|\sum_{k=1}^{n}c_{k}e^{ik^{2}t}\|_{1}\gg\begin{cases}(\log n)^{s-\frac{1}{2}}% ,&s>\frac{1}{2},\\ \log\log n,&s=\frac{1}{2}.\end{cases}$

If $P(t)=\sum_{k=0}^{n}a_{k}e^{ikt}$ with $a_{k}\in\{-1,1\}$, then by the Cauchy-Schwarz inequality and Parseval’s identity we have

 $\|P\|_{1}=\frac{1}{2\pi}\int_{0}^{2\pi}1\cdot|P(t)|dt\leq\|1\|_{2}\cdot\|P\|_{% 2}=1\cdot\|\hat{P}\|_{2}=\sqrt{n+1}.$

Newman [24] shows that in fact we can do better than what we get using the Cauchy-Schwarz inequality and Parseval’s identity:

 $\|P\|_{1}<\sqrt{n+0.97}.$

A Fekete polynomial is a polynomial of the form $\sum_{k=1}^{l-1}\left(\frac{k}{l}\right)z^{k}$, $l$ prime, where $\left(\frac{k}{l}\right)$ is the Legendre symbol. Let $P_{l}(t)=\sum_{k=1}^{l-1}\left(\frac{k}{l}\right)e^{ikt}$. Erdélyi [8] proves upper and lower bounds on $\left(\frac{1}{|I|}\int_{I}|P_{l}(t)|^{q}dt\right)^{1/q}$, $q>0$, where $I$ is an arc in $[0,2\pi]$.

## References

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