On the mean summability of series by nonlinear basis

TL;DR

This paper investigates the approximation and summability properties of series using nonlinear Fourier bases, demonstrating their effectiveness in signal processing and establishing Bernstein's inequalities for nonlinear trigonometric polynomials.

Contribution

It introduces new approximation results and summability methods for nonlinear Fourier series, extending classical analysis to nonlinear bases in signal processing.

Findings

Partial sums and Cesàro means are effective for nonlinear Fourier series in Lp spaces.

Bernstein's inequalities are proved for nonlinear trigonometric polynomials.

Results enhance understanding of nonlinear basis approximation in signal analysis.

Abstract

The nonlinear signal processing has achieved a rapid process in the recent years. A family of nonlinear Fourier bases, as a typical family of mono-component signals, has been constructed and applied to signal processing. In this paper, the approximation properties of the partial sums and Ces?aro summability of series by the nonlinear Fourier basis are investigated in the Lp(T). Furthermore, these results are applied to the prove of Bernstein's inequalities for nonlinear trigonometric polynomials.

Equations

$${\left(\frac{1}{2\pi }{\int }_{\mathbb{T}}\mid f(x){\mid }^{p}dx\right)}^{1/p}<\infty$$

$$\parallel f{\parallel }_p={\left(\frac{1}{2\pi }{\int }_{\mathbb{T}}\mid f(x){\mid }^{p}dx\right)}^{1/p}.$$

$$\parallel f{\parallel }_{\infty }=\inf \{M\mid \mid f(x)\mid \le Ma.e.in\mathbb{T}\}.$$

$$\omega {\left(f;t\right)}_p:=\underset{0\le h\le t}{\sup }\parallel f(.+h)-f(.){\parallel }_p$$

$$\omega (0)=0,\omega (t_1+t_2)\le \omega (t_1)+\omega (t_2).$$

$${\tau }_a=\frac{z-a}{1-\overline{a}z}$$

$$e^{i{\theta }_a(t)}:={\tau }_a(e^{it})=\frac{e^{it}-a}{1-\overline{a}{e}^{it}}$$

$${\theta }_a(t+2\pi )={\theta }_a(t)+2\pi$$

$${\theta }_a^{\prime }(t)=p_a(t)=\frac{1-{\left|a\right|}^2}{1-2\left|a\right|\cos (t-t_a)+{\left|a\right|}^2}$$

$$0<\frac{1-\left|a\right|}{1+\left|a\right|}\le {\theta }_a^{\prime }(t)\le \frac{1+\left|a\right|}{1-\left|a\right|}$$

$$\sum _{k\in \mathbb{Z}}\mid c_k{\mid }^2=\frac{1}{2\pi }{\int }_{\mathbb{T}}\sum _{k\in \mathbb{Z}}\mid c_k{e}^{ik{\theta }_a(x)}{\mid }^{2}dx=\frac{1}{2\pi }{\int }_{\mathbb{T}}\sum _{k\in \mathbb{Z}}\mid c_k{e}^{ik{\theta }_a(t)}{\mid }^2{p}_a(t)dt$$

$${\tau }_n^a:=span\left\{e^{ik{\theta }_a(t)}:\mid k\mid \le n\right\}.$$

$$E_n^a(f{)}_p=\underset{T\in {\tau }_n^a}{\inf }\parallel f-T{\parallel }_{L^p(\mathbb{T})}.$$

$$E_n^a(f{)}_{\infty }\le \frac{24}{1-\mid a\mid }\omega {\left(f,\frac{1}{n}\right)}_{\infty }.$$

$$\frac{1-\mid a\mid }{2}\omega (f,t{)}_{\infty }\le \omega (f\circ {\theta }_a^{-1},t{)}_{\infty }\le \frac{2}{1-\mid a\mid }\omega (f,t{)}_{\infty }.$$

$$f(x)\sim \sum _{k=-\infty }^{\infty }{c}_k{e}^{ik{\theta }_a(x)}$$

$$c_k=c_k(f)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(t)e^{-ik{\theta }_a(t)}{p}_a(t)dt,k\in \mathbb{Z}.$$

$$f(x)\sim \sum _{k=-\infty }^{\infty }{c}_k{e}^{ik{\theta }_a(x)},g(x)\sim \sum _{k=-\infty }^{\infty }{d}_k{e}^{ik{\theta }_a(x)}.$$

$$(Af+Bg)(x)\sim \sum _{k=-\infty }^{\infty }(A{c}_k+B{d}_k)e^{ik{\theta }_a(x)}$$

$$(f\ast g)(x)\sim \sum _{k=-\infty }^{\infty }\left(\frac{1+\mid a\mid }{1-\mid a\mid }\right)(c_k{d}_k)e^{ik{\theta }_a(x)}$$

$$W_r^p(\mathbb{T})=\left\{f\in X^p(\mathbb{T}):f,...,f^{(r-1)}\in \mathcal{A}\mathcal{C}(\mathbb{T}),f^{(r)}\in X^p(\mathbb{T})\right\}.$$

$$f^{\prime }(x)\sim \sum _{k=-\infty }^{\infty }\left(\frac{1+\mid a\mid }{1-\mid a\mid }\right)(ik{c}_k)e^{ik{\theta }_a(x)}$$

$$S_n^a(f)(x)=\sum _{k=-n}^n{c}_k{e}^{ik{\theta }_a(x)}$$

$$c_k=c_k(f)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(t)e^{-ik{\theta }_a(t)}{p}_a(t)dt.$$

$$S_n^a(f)(x)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(t)D_n({\theta }_a(x)-{\theta }_a(t))p_a(t)dt.$$

$$=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }F({\theta }_a(x)+t)D_n(t)dt,$$

$$=\frac{1}{2\pi }{\int }_0^{\pi }\left\{F({\theta }_a(x)-t)+F({\theta }_a(x)+t)\right\}{D}_n(t)dt.$$

$$S_n^a(f)(x)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }F({\theta }_a(x)+y)D_n(y)dy.$$

$${\int }_{-\pi }^{0}F({\theta }_a(x)+t)D_n(t)dt={\int }_0^{\pi }F({\theta }_a(x)+t)D_n(t)dt,$$

$$\parallel F\parallel_{p}\leq\left\{\begin{array}[]{ll}\parallel f\parallel_{p},&\hbox{$p=1,\infty,f\in C(\mathbb{T})\quad if\quad p=\infty$;}\\ \left(\frac{1+\left|a\right|}{1-\left|a\right|}\right)^{1/p}\parallel f\parallel_{p}&\hbox{$0<p<\infty$;}\end{array}\right.$$

Full text

1. Introduction

Let $0<p<\infty$ and $\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}$. Let $f$ is a periodic function on $\mathbb{T}$, then we denote $L^{p}(\mathbb{T})$ the set of Lebesgue measurable functions $f:\mathbb{T}\rightarrow\mathbb{R}$ (or $\mathbb{C})$) such that

[TABLE]

where the integral is a Lebesgue integral, and we identify functions that differ on a a set of measure zero. We define $L^{p}$-norm of $f$ by

[TABLE]

For $p=\infty$ the space $L^{\infty}(\mathbb{T})$ consists of the Lebesgue measurable functions $f:\mathbb{T}\rightarrow\mathbb{R}$ (or $\mathbb{C})$ that are essentially bounded on $\mathbb{T}$,meaning that $f$ is bounded on a subset of $\mathbb{T}$ whose complement has measure zero. The norm on is essential supremum

[TABLE]

Note that $\parallel f\parallel_{p}$ may take the value $\infty$.

The another important concept is the modulus of smoothness is defined by

[TABLE]

If $p=\infty$, then it is called modulus of contiunity. And this nondecreasing continuous function on the interval $[0,2\pi]$ having properties:

[TABLE]

A family of nonlinear Fourier bases as the extension of the classical Fourier basis, have been constructed and applied to signal processing [1, 2, 7, 9, 8]. For any complex number $a=\left|a\right|e^{it_{a}},\left|a\right|<1$, the nonlinear phase function $\theta_{a}(t)$ is defined by the radical boundary value of the Möbuis transformation

[TABLE]

that is,

[TABLE]

It is easily seen that

[TABLE]

and its derivative is the Poisson kernel

[TABLE]

which satisfies

[TABLE]

Hence, $\theta_{a}(t)$ is a strictly monotonic increasing function, which makes $cos\theta_{a}(t)$ be a special mono-component signal [7, 8]. It has been shown that for any sequence $\left\{c_{k}\right\}_{k\in\mathbb{Z}}$ of finite nonzero terms, there holds

[TABLE]

which combining with (1.2) implies that the so-called nonlinear Fourier basis $\left\{e^{in\theta_{a}(t)}\right\}_{n\in\mathbb{Z}}$ forms a Riesz basis for $L^{2}(\mathbb{T})$ with the upper bound $\sqrt{\frac{1+\left|a\right|}{1-\left|a\right|}}$ and the lower bound $\sqrt{\frac{1-\left|a\right|}{1+\left|a\right|}}$. When $a=0$, $\left\{e^{in\theta_{a}(t)}\right\}_{n\in\mathbb{Z}}$ is simply the Fourier basis $\left\{e^{int}\right\}_{n\in\mathbb{Z}}$.

Let $\tau^{a}_{n}$ be the space of all the nonlinear trigonometric polynomials of degree less than or equal to $n$, that is,

[TABLE]

The approximation error of $f\in L^{p}(\mathbb{T})$,

[TABLE]

Let us recall some known lemmas (see [5]) which will be used in the sequel of paper.

Lemma 1**.**

Let $f\in C(\mathbb{T})$. We have

[TABLE]

Lemma 2**.**

Let $f\in C(\mathbb{T})$. We have

[TABLE]

In the present paper first we deal with some properties of nonlinear Fourier series. Then we discuss $n$-th partial sums and Cesàro sum of nonlinear Fourier series. Also we prove the necessary and sufficient condition for nonlinear Fourier series which governs the $(C,1)$ summability in $L^{p}(\mathbb{T})$ for arbitrary function $f$ from $L^{p}(\mathbb{T})$. This result is applied to the prove of Bernstein’s inequality for nonlinear trigonometric polynomials.

2. Convergence of Nonlinear Fourier Series

Let $f\in L^{1}(\mathbb{T})$. Then $1\leq p\leq\infty$ by nonlinear basis denoted by

[TABLE]

be its series by nonlinear Fourier basis where

[TABLE]

For simplicity throughtout the present paper we write nonlinear Fourier series as series by nonlinear Fourier basis. Now we can begin to give some properties of nonlinear Fourier series:

Assume $f,g\in L^{1}(\mathbb{T})$ and

[TABLE]

For linearity and convolution we have the following properties respectively.

[TABLE]

[TABLE]

Furthermore remember Sobolev space i.e.

[TABLE]

Assume that $f\in W^{p}_{1}(\mathbb{T})$. It can be easily seen that property i.e.

[TABLE]

holds.

We wish to examine the convergence of nonlinear Fourier series. To discuss the convergence, pointwise or uniform, of nonlinear Fourier series, we need to discuss the convergence of the sequence $\{S^{a}_{n}\}$ of partial sums. We have

[TABLE]

where

[TABLE]

Proposition 1**.**

Let $S_{n}^{a}(f)$ be the sequence of partial sums of the nonlinear Fourier series of $f$. Let $f\in L^{1}(\mathbb{T})$ which is $2\pi$-periodic. Then

[TABLE]

[TABLE]

[TABLE]

Proof.

By the expression for the $c_{k}$ and considering the equality $S_{n}^{a}(f)(x)=S_{n}(F)(\theta_{a}(x))$ where $F=f\circ\theta_{a}^{-1}$ we have already established (2.1). By a change of variable $u-\theta_{a}(x)=y$ and the (1.1) equality of $\theta_{a}(x)$ phase function’s $2\pi$-periodicity, it follows that the integral does not change as long as the length of the interval of integration is $2\pi$ we get

[TABLE]

This proves (2.2). Finally, we split the integral in (2.2) as the sum of integrals over $[-\pi,0]$ and $[0,\pi]$. Now

[TABLE]

using the change of variable $y=-t$ and the evenness of $D_{n}$. This proves (2.3). ∎

We can begin with properties of the operators $S^{a}_{n}$ of partial sums of nonlinear Fourier series. For this first we need following lemmas.

Lemma 3**.**

Let $F:=f\circ\theta_{a}^{-1}$. Then we have

[TABLE]

Proof.

Let $f\in\mathcal{C}(\mathbb{T})$ ve $F:=f\circ\theta_{a}^{-1}$. Therefore for $p=1,\infty$ we have

[TABLE]

And for $0<p<\infty$ the following inequality holds.

[TABLE]

By change of variable $\theta_{a}(t)=u$ and using phase function’s property which is giving in (1.2), we have the following equality.

[TABLE]

Therefore we have result that we wanted. ∎

For the spaces $L^{1}(\mathbb{T})$, $C(\mathbb{T})$, one can evaluate the norms $\parallel S_{n}^{a}\parallel$ by direct computation.

Theorem 1**.**

One has

[TABLE]

Proof.

Let consider $S_{n}^{a}$ be an operator $C(\mathbb{T})$ to $C(\mathbb{T})$. By using Lemma 2.1 and equality (2.2), we see that each of the norms (2.2) is equal to

[TABLE]

[TABLE]

[TABLE]

From the Lebesgue constant’s definition (see in [4])

[TABLE]

holds. ∎

Theorem 2**.**

For all $f\in C(\mathbb{T})$,

[TABLE]

holds.

Proof.

Let $f\in C(\mathbb{T})$. Consider linear operator $S_{n}^{a}:\mathcal{C}(\mathbb{T})\rightarrow\mathcal{C}(\mathbb{T})$, $n=0,1,...$. The best approximation by $t\in\tau_{n}^{a}$ is

[TABLE]

Thus we can write the following equality.

[TABLE]

[TABLE]

holds. Therefore from Theorem 2.1

[TABLE]

holds. Hence by using theorem for Lebesgue constant (see in [4]) we have

[TABLE]

This completes the proof. ∎

Theorem 3**.**

Let consider $S_{n}^{a}:L^{p}(\mathbb{T})\rightarrow L^{p}(\mathbb{T})$ linear operator sequence, where $n=0,1,...$ and $1\leq p<\infty$. Then

[TABLE]

holds.

Proof.

Let $1\leq p<\infty$. Consider operator $S_{n}^{a}:L^{p}(\mathbb{T})\rightarrow L^{p}(\mathbb{T})$, $n=0,1,...$. Thus if we consider the inequality (2.2)

[TABLE]

[TABLE]

holds. Hence by using Minkowski integration inequality (see in [10]), we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Therefore considering Lemma 2.2 and Dirichlet kernel’s definition (see in [3]) is giving result that we wanted. ∎

Now let examine the convergence of partial sum of the nonlinear Fourier series in $L^{p}(\mathbb{T})$ space.

Theorem 4**.**

Let $1\leq p<\infty$. Then for all $f\in L^{p}(\mathbb{T})$

[TABLE]

holds if and only if, there exist a $M>0$ constant that only depend on $p$ such that

[TABLE]

Proof.

For necessity let $f\in L^{p}(\mathbb{T})$ and $\parallel f-S_{n}^{a}(f)\parallel_{p}\rightarrow 0$, $n\rightarrow\infty$. Thus $\left\{S_{n}^{a}(f)\right\}$ is bounded since it converges in $L^{p}(\mathbb{T})$ norm space for $f\in L^{p}(\mathbb{T})$. Therefore there exists $M_{f}>0$ for all $f\in L^{p}(\mathbb{T})$. Such that $\parallel S_{n}^{a}(f)\parallel_{p}<M_{f}$. Thus considering uniform bounded principle (e.g. [6]) we have

[TABLE]

If we consider Theorem 2.3 and choose

[TABLE]

for $f\in L^{p}(\mathbb{T})$, then we can write for (2.5) inequality.

For sufficiency let $\parallel S_{n}^{a}(f)\parallel_{p}\leq M_{p}^{a}\parallel f\parallel_{p}$. For

[TABLE]

where $f\in L^{p}(\mathbb{T})$ and $n=0,1,2,...$ . Therefore we can prove (2.4) with the help of the nonlinear polynomials $t_{n}^{a}$ of best approximation of the function:

[TABLE]

[TABLE]

Since

[TABLE]

this completes the proof. ∎

Corollary 1**.**

The norms of the operators $S_{n}^{a}(f)$ are bounded in each space $f\in L^{p}(\mathbb{T})$, $1<p<\infty$.

From Theorem 2.2, Theorem 2.3 and Theorem 2.4 we now derive

[TABLE]

We see that the partial sums $S_{n}^{a}$ approximate almost as well as its polynomial of best approximation. This is true even for $f\in C(\mathbb{T})$, if the factor $\log n$ is not essential for the problem considered.

For the partial sums $S_{n}^{a}$ of the Fourier series of $f$ we do not have $S_{n}^{a}(f)\rightarrow f$ in $C(\mathbb{T})$ for each $f$. But we do have fast convergence of $S_{n}^{a}$ for smooth functions $f$. Actually, the convergence $\parallel S_{n}^{a}(f)-f\parallel_{\infty}\rightarrow 0$ can be arbitrarily fast for some $f$ (that are not trigonometric polynomials): The following theorems shows that it is sufficient to take $f(t)=\sum_{k=0}^{\infty}[a_{k}cos(k\theta_{a}(x))+b_{k}sin(k\theta_{a}(x))]$, where $a_{k}$, $b_{k}$ converge to zero sufficiently fast without being zero.

Theorem 5**.**

For $f\in L^{1}(\mathbb{T})$

[TABLE]

holds.

Proof.

Let $\varepsilon>0$. In this case there exists a $t\in\tau^{a}_{n}$ such that

[TABLE]

In this inequality let $t$ has degree $N$. Therefore we can write

[TABLE]

Thus for $\mid k\mid>N$,

[TABLE]

[TABLE]

[TABLE]

holds. Hence we obtain the result from following inequality.

[TABLE]

∎

Corollary 2**.**

Let $f\in L^{1}(\mathbb{T})$. Then for all $f$,

[TABLE]

holds.

Proof.

By writing $a^{a}_{n}(f)=c^{a}_{n}(f)+c^{a}_{-n}(f)$ ve $b^{a}_{n}(f)=\frac{c^{a}_{n}(f)+c^{a}_{-n}(f)}{i}$ in Theorem 2.5, we can complete the proof. ∎

Theorem 6**.**

Let $f\in L^{1}(\mathbb{T})$. If

[TABLE]

then for $n\rightarrow\infty$, $S^{a}_{n}(f)(x)\rightarrow f(x)$ holds.

Proof.

Let assume $f\in L^{1}(\mathbb{T})$. By using (2.3) equality, we can write

[TABLE]

Therefore considering Dirichlet kernel’s property (see in [10]), we obtain the following.

[TABLE]

[TABLE]

Now let think the $\phi^{a}_{x}(t)$ as following.

[TABLE]

In this case the last equality written as

[TABLE]

From hypothesis $\phi^{a}_{x}(t)\in L^{1}(\mathbb{T})$. Hence

[TABLE]

holds and by using Jordan inequality (see [10]), we obtain

[TABLE]

Therefore

[TABLE]

[TABLE]

[TABLE]

holds. Here if we say that

[TABLE]

We find that $g^{a}_{x}(t)\in L^{1}(\mathbb{T})$. In this case, considering the following equalities

[TABLE]

[TABLE]

we see that

[TABLE]

holds. Thus by considering Corollary 2.2 and Riemann-Lebesgue Lemma (see in [10]) for $\mid n\mid\rightarrow\infty$, we have

[TABLE]

So we obtain $S^{a}_{n}(f)(x)\rightarrow f(x)$ as $n\rightarrow\infty$. This completes the proof. ∎

Corollary 3**.**

Let $f\in L^{1}(\mathbb{T})$. If $\mid F(x)-F(y)\mid\leq M\mid x-y\mid^{\alpha}$, then we have

[TABLE]

Proof.

Let assume that $f\in L^{1}(\mathbb{T})$. Therefore

[TABLE]

[TABLE]

[TABLE]

holds. Thus if we consider $\int^{\pi}_{0}t^{\alpha-1}dt=\frac{\pi^{\alpha}}{\alpha}$ equality, we have the following by using comparison test.

[TABLE]

Therefore from Theorem 2.6, we obtain $S^{a}_{n}(f)(x)\rightarrow f(x),\quad n\rightarrow\infty$. ∎

Corollary 4**.**

Let $f\in L^{1}(\mathbb{T})$. If $f$ is differentable and $F:=f\circ\theta_{a}^{-1}$ for all $x\in\mathbb{T}$, then we have

[TABLE]

Proof.

Let $f\in L^{1}(\mathbb{T})$. Hence

[TABLE]

holds. For $\exists\quad\delta>0\ni 0<\mid t\mid<\delta$

[TABLE]

holds. Thus we have

[TABLE]

Therefore

[TABLE]

holds for $\mid t\mid>\delta$. Hence we have the following inequality.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

So by considering Theorem 2.6, we obtain

[TABLE]

∎

3. Boundness and Cesàro Mean Summability for Nonlinear Fourier Series

Precisely, we prove the necessary and sufficient condition for nonlinear Fourier series which governs the $(C,1)$ summability in $L^{p}$ for arbitrary function $f$ from $L^{p}$. This result is applied to the prove of Bernstein s inequality for nonlinear trigonometric polynomials. Let $f\in L^{1}(\mathbb{T})$ and $S_{n}^{a}(f)$ is partial sum of nonlinear Fourier series. Define $n$-th Fejér (Cesàro) for nonlinear Fourier series defined by

[TABLE]

Now let find new expressions for Cesàro mean which are very useful.

Proposition 2**.**

Let $\sigma_{n}^{a}(f)$ be the sequence of partial sums of the nonlinear Fourier series of $f$. Let $f\in L^{1}(\mathbb{T})$ which is $2\pi$-periodic. Then

[TABLE]

[TABLE]

[TABLE]

Proof.

By the expression for the $c_{k}$, phase function $\theta_{a}(x)$ and considering $\sigma_{n}^{a}(f)(x):=\frac{1}{n+1}\sum^{n}_{k=1}\sigma^{a}_{k}(f)(x)$, we have already established (3.1). By a change of variable $y=\theta_{a}(t)-\theta_{a}(x)$ and the (1.1) equality of $\theta_{a}(x)$’s $2\pi$-periodicity, it follows that the integral does not change as long as the length of the interval of integration is $2\pi$ we get

[TABLE]

This proves (3.2). Finally, we split the integral in (3.2) as the sum of integrals over $[-\pi,0]$ and $[0,\pi]$. Now

[TABLE]

using the change of variable $y=-t$ and the evenness of $K_{n}$. This proves (3.3). ∎

Theorem 7**.**

Let $\sigma_{n}^{a}:\mathcal{C}(\mathbb{T})\rightarrow\mathcal{C}(\mathbb{T})$ for $n=0,1,...$ be linear operator sequence. Then we have

[TABLE]

for all $f\in\mathcal{C}(\mathbb{T})$.

Proof.

Let consider $\sigma_{n}^{a}:\mathcal{C}(\mathbb{T})\rightarrow\mathcal{C}(\mathbb{T})$ as a linear operator for $n=0,1,...$. Thus by using (3.2) inequality and Fejér kernel’s positivity (see in [3]), we have

[TABLE]

[TABLE]

[TABLE]

If we consider Lemma 2.1 for last equality, we obtain that

[TABLE]

Finally by using Fejér kernel’s property (see in [10]) we obtain the result.

[TABLE]

∎

Theorem 8**.**

Let $f\in\mathcal{C}(\mathbb{T})$. Then

[TABLE]

Proof.

Let $\sigma_{n}^{a}:\mathcal{C}(\mathbb{T})\rightarrow\mathcal{C}(\mathbb{T})$, $n=0,1,...$. Then we have

[TABLE]

[TABLE]

[TABLE]

by (3.2) equality. Therefore since $f\in C(\mathbb{T})$ and $\theta_{a}$ continuous, we can write

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Now if we use Lemma 2.1 and Fejér kernel’s property (see in [10]), we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since there exists $\exists N\in\mathbb{N}:\quad n\geq N$

[TABLE]

as $n\rightarrow\infty$, we obtain

[TABLE]

Therefore

[TABLE]

holds for $\exists N\in\mathbb{N}:n\geq N$ ve $\forall x\in\mathbb{T}$. Thus $\sigma_{n}^{a}(f)\rightarrow f$ uniformly as $n\rightarrow\infty$ ∎

Theorem 9**.**

Let consider $\sigma_{n}^{a}:L^{p}(\mathbb{T})\rightarrow L^{p}(\mathbb{T})$, $1\leq p<\infty$ and $n=0,1,...$ linear operator sequence. Then

[TABLE]

holds for all $f\in L^{p}(\mathbb{T})$.

Proof.

Let consider $\sigma_{n}^{a}:L^{p}(\mathbb{T})\rightarrow L^{p}(\mathbb{T})$,$1\leq p<\infty$ and $n=0,1,...$ linear operator sequence. Then we have

[TABLE]

Hence by (3.2) equality, we have

[TABLE]

Now by using Minkowski integral inequality (see in [10]),

[TABLE]

[TABLE]

holds. Therefore we obtain the following result by using Lemma 2.1.

[TABLE]

∎

Theorem 10**.**

Let $1\leq p<\infty$.Then we have

[TABLE]

for all $f\in L^{p}(\mathbb{T})$.

Proof.

Let $\sigma_{n}^{a}:L_{p}(\mathbb{T})\rightarrow L_{p}(\mathbb{T})$, $n=0,1,...$ linear operator and $f\in L_{p}(\mathbb{T})$. Take $\varepsilon>0$.Then there exists $\exists\quad g\in\mathcal{C}(\mathbb{T})$ such that

[TABLE]

Let $g\in\mathcal{C}(\mathbb{T})$. Then for $n\rightarrow\infty$, we have $\sigma_{n}^{a}(g)\rightarrow g$ converges uniformly by Theorem 3.2. Thus by using the Lusin Theorem (see in [3]), there exists $\exists\quad n\geq N$ such that $\parallel\sigma_{n}^{a}(g)-g\parallel_{p}<\frac{\varepsilon}{3}$.

[TABLE]

[TABLE]

Therefore by Theorem 3.3, we have

[TABLE]

[TABLE]

This completes the proof. ∎

4. Bernstein’s inequality for Nonlinear Fourier Series

Applying the inequalities for the Cesàro means of nonlinear trigonometric series derived in the previous section, we can prove the nonlinear version of the well-known Bernstein’s inequality. For any trigonometric polynomial $t_{n}(x)$ of order $\leq n$, for every $1\leq p\leq\infty$, we have

[TABLE]

The last inequality is known as integral Bernstein’s inequality. The following extension of (4.1) is true.

Theorem 11**.**

Let $1<p<\infty$ and assume that $t_{a}\in\tau_{n}^{a}$. Then the inequality

[TABLE]

holds.

Proof.

It is well known from the (2.1) equality that

[TABLE]

where

[TABLE]

is the Dirichlet’s kernel of order $n$. Let $T=t_{n,a}\circ\theta_{a}^{-1}$. By the derivation, we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $K_{n-1}$ is the Fejér’s kernel of order $n-1$. By taking the absolute values, we get

[TABLE]

[TABLE]

[TABLE]

If we use Theorem 3.3, we get that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From this we obtain

[TABLE]

and this completes the proof. ∎