Approximation by Faber–Laurent Rational Functions in Variable Exponent Morrey Spaces

Abstract

Let G be a finite Jordan domain bounded by a Dini-smooth curve $\Gamma$ in the complex plane ${\mathbb {C}}$. In this work, approximation properties of the Faber–Laurent rational series expansions in variable exponent Morrey spaces $L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$ are studied. Also, direct theorems of approximation theory in variable exponent Morrey–Smirnov classes, defined in domains with a Dini-smooth boundary, are proved.

1 Introduction, Some Auxiliary Results and Main Results

Let J denote the interval $[0,2\pi ]$ or a Jordan rectifiable curve $\Gamma \subset {\mathbb {C}}$. Let us denote by $\wp$ the class of Lebesgue measurable functions $p(\cdot ):\Gamma \rightarrow [0,\infty )$ such that

$$\begin{aligned} 1<p_{*}:= {\text {essinf}}_{z\in J} p(z)\le p^{*}:={\text {esssup}}_{z\in J} p(z)<\infty . \end{aligned}$$

(1.1)

Let $\left| J\right|$ be the Lebesgue measure of J. We suppose that the function $p(\cdot )$ satisfies the condition

$$\begin{aligned} |p(z_1)-p(z_2)|\ln \!\left( \frac{|J|}{|z_1-z_2|}\right) \le c, \quad \text{ for } \text{ all } z_1,z_2\in J, \end{aligned}$$

(1.2)

where the constant c is independent of $z_1$ and $z_2$. A function $p(\cdot ) \in \wp$ is said to belong to the class $\wp ^{\log }(J)$, if the condition (1.2) is satisfied.

For $p(\cdot )\in \wp ^{\log }(\Gamma )$, we define a class $L^{p(\cdot )}(\Gamma )$ of Lebesgue measurable functions $f(\cdot ):\Gamma \rightarrow {\mathbb {R}}$ satisfying the condition

$$\begin{aligned} \int _\Gamma |f(z)|^{p(z)}\,|dz| <\infty . \end{aligned}$$

This class $L^{p(\cdot )}(\Gamma )$ is a Banach space with respect to the norm

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot )}(\Gamma )}:= \inf \left\{ \lambda >0:\int _\Gamma \left| \frac{f(x)}{\lambda }\right| ^{p(z)}\,|dz| \le 1 \right\} . \end{aligned}$$

Let G be a finite domain in the complex plane ${{\mathbb {C}}}$, bounded by the rectifiable Jordan curve $\Gamma$. Without loss of generality we assume $0\in {\text {Int}}\Gamma$. Let $G^-: ={\text {Ext}}\Gamma$. Let also ${\mathbb {T}}:=\{w\in {\mathbb {C}}:|w|=1\}$, ${\mathbb {D}}={\text {Int}}{\mathbb {T}}$ and ${\mathbb {D}}^-={\text {Ext}}{\mathbb {T}}$. We recall that if for a given analytic function $f(\cdot )$ on G, there exists a sequence of rectifiable Jordan curves $(\Gamma _n)$ in G tending to the boundary $\Gamma$ in the sense that $\Gamma _n$ eventually surrounds each compact subdomain of G such that

$$\begin{aligned} \int _{\Gamma _n}|f(z)|^p\,|dz|\le M<\infty , \end{aligned}$$

then we say that $f(\cdot )$ belongs to the Smirnov class $E^{p}(G^-)$, $1\le p<\infty$. Each function $f(\cdot )\in E^{p}(G)$ has non-tangential limits almost everywhere (a.e.) on $\Gamma$ and the boundary function belongs to $L^p(\Gamma )$.

We denote by $\varphi (\cdot )$ the conformal mapping of $G^-$ onto ${\mathbb {D}}^-$ normalized by

$$\begin{aligned} \varphi (\infty ) =\infty , \quad \lim _{z\rightarrow \infty }\frac{\varphi (z)}{z}>0. \end{aligned}$$

Let $\psi (\cdot )$ be the inverse of $\varphi (\cdot )$. The functions $\varphi (\cdot )$ and $\psi (\cdot )$ have continuous extensions to $\Gamma$ and ${\mathbb {T}}$, their derivatives $\varphi '(\cdot )$ and $\psi '(\cdot )$ have definite non-tangential limit values on $\Gamma$ and ${\mathbb {T}}$ a.e., and they are integrable with respect to the Lebesgue measure on $\Gamma$ and ${\mathbb {T}}$, respectively. It is known that $\varphi '(\cdot )\in E^1(G^-)$ and $\psi '(\cdot )\in E^1({\mathbb {D}}^-)$. Note that the general information about Smirnov classes can be found in [14, pp. 168–185], [22, pp. 438–453].

Let $\Gamma$ be a rectifiable Jordan curve in the complex plane. We denote $\Gamma (t,r)=\Gamma \cap B(t,r)$, $t\subset \Gamma$, $r>0$, where $B(t,r) =\{z\in {\mathbb {C}}:|z-t|<r\}$. The Morrey spaces $L^{p,\lambda }(\Gamma )$ for a given $0\le \lambda \le 1$ and $p\ge 1$, are defined as the set of functions $f(\cdot )\in L_{loc}^{p}(\Gamma )$ such that

$$\begin{aligned} \Vert f\Vert _{L^{p,\lambda }(\Gamma )}:= \sup _{z\in \Gamma ,\,0<r<L} r^{-\lambda /p}\Vert f\Vert _{L^p(\Gamma (t,r))}<\infty , \end{aligned}$$

where L is the length of the curve $\Gamma$.

Note that $L^{p,0}(\Gamma ) =L^p(\Gamma )$, and if $\lambda <0$ or $\lambda >1$, then $L^{p,\lambda }(\Gamma )=\Theta$, where $\Theta$ is the set of all functions equivalent to 0 on $\Gamma$.

Let $G:={\text {Int}}\Gamma$ and $L^{p,\lambda }(\Gamma )$, $0<\lambda \le 1$ and $1<p<\infty$, be a Morrey space defined on $\Gamma$. We also define the Morrey-Smirnov classes $E^{p,\lambda }(G)$ as

$$\begin{aligned} E^{p,\lambda }(G) := \{f(\cdot )\in E_1(G) :f(\cdot )\in L^{p,\lambda }(\Gamma )\} . \end{aligned}$$

Hence for $f(\cdot )\in E^{p,\lambda }(G)$ we can define the $E^{p,\lambda }(G)$ norm as

$$\begin{aligned} \Vert f\Vert _{E^{p,\lambda }(G)}:=\Vert f\Vert _{L^{p,\lambda }(\Gamma )}. \end{aligned}$$

Let $p(\cdot ):\Gamma \rightarrow [1,+\infty ]$ be a Lebesgue measurable function satisfying condition (1.1) and $\lambda (\cdot ):\Gamma \rightarrow [0,1]$ be a measurable function. We define the variable exponent Morrey spaces $L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$ as the set of Lebesgue measurable functions $f(\cdot )$ defined on $\Gamma$, such that

$$\begin{aligned} S_{p(\cdot ),\lambda (\cdot )}(f)= \sup _{t\in \Gamma ,\,0<r<L}r^{-\lambda (x)} \int _{\Gamma (t,r)}|f(s)|^{p(s)}\,ds<\infty . \end{aligned}$$

The norm in $L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$ is defined as follows

$$\begin{aligned} \Vert f\Vert _{L^{p(\cdot ),\lambda (\cdot )}(\Gamma )}:= \inf \left\{ \nu >0:S_{p(\cdot ),\lambda (\cdot )}\left( \frac{f}{\nu }\right) <1\right\} . \end{aligned}$$

It is known that $L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$ is a Banach space. Note that the properties of classical Morrey spaces and variable exponent Morrey spaces have been investigated by several authors (see, for example, [3, 16,17,18,19, 30, 40, 42, 46,47,48, 50, 51, 54]).

We define also the variable exponent Morrey-Smirnov class $E^{p(\cdot ),\lambda (\cdot )}(G)$ as

$$\begin{aligned} E^{p(\cdot ),\lambda (\cdot )}(G) :=\left\{ f(\cdot )\in E^{1}(G):f(\cdot )\in L^{p(\cdot ),\lambda (\cdot )}(\Gamma ) \right\} . \end{aligned}$$

Note that $E^{p(\cdot ),\lambda (\cdot )}(G)$ is a Banach space with respect to the norm

$$\begin{aligned} \Vert f\Vert _{E^{p(\cdot ),\lambda (\cdot )}(G)}:= \Vert f\Vert _{L^{p(\cdot ),\lambda (\cdot )}(\Gamma )}. \end{aligned}$$

Let $p(\cdot ):{\mathbb {T}}\rightarrow [1,+\infty ]$ and $\lambda (\cdot ):{\mathbb {T}}\rightarrow [0,1]$ be measurable functions such that $0\le \lambda _*\le \lambda ^*<1$. Also assume that $p(\cdot )\in \wp ^{\log }$. For $f(\cdot )\in L^{p(\cdot ),\lambda (\cdot )}({\mathbb {T}})$ we define the operator

$$\begin{aligned} (\nu _{h_{i}}f)(\omega ) := \frac{1}{h} \int _0^{h}f(\omega e^{it}) dt,\omega \in {\mathbb {T}}, \quad 0<h<\pi . \end{aligned}$$

It is clear that the operator $\nu _h$ is a bounded linear operator on $L^{p(\cdot )\lambda (\cdot )}({\mathbb {T}})$ [21]:

$$\begin{aligned} \Vert \nu _{h}(f)\Vert _{L^{p(\cdot )}({\mathbb {T}})}\le c_1\Vert f\Vert _{L^{p(\cdot )}({\mathbb {T}})}. \end{aligned}$$

The function

$$\begin{aligned} \Omega (f,\delta )_{p(\cdot ),\lambda (\cdot )}:= \sup _{0<h\le \delta }\Vert f(\cdot )-\nu _hf(\cdot )\Vert _{L^{p(\cdot ),\lambda (\cdot )}({\mathbb {T}}) },\quad \delta >0, \end{aligned}$$

is called the modulus of smoothness of $f(\cdot )\in L^{p(\cdot )\lambda (\cdot )}({\mathbb {T}}\mathbf {)}$.

It can easily be shown that $\Omega (f,\cdot )_{p(\cdot ),\lambda (\cdot )}$ is a continuous, non-negative and non-decreasing function satisfying the conditions

$$\begin{aligned} \lim _{\delta \rightarrow 0}\Omega (f,\delta )_{p(\cdot ),\lambda (\cdot )}&=0,\\ \Omega (f+g,\delta )_{p(\cdot ),\lambda (\cdot )}&\le \Omega (f,\delta )_{p(\cdot ),\lambda (\cdot )}+ \Omega (g,\delta )_{p(\cdot ),\lambda (\cdot )},\quad \delta >0, \end{aligned}$$

for $f(\cdot ),g(\cdot )\in L^{p(\cdot ),\lambda (\cdot )}({\mathbb {T}}\mathbf {)}$.

We denote by $w=\phi (z)$ the conformal mapping of $G^-$ onto the domain ${\mathbb {D}}:=\{ w\in {\mathbb {C}}:|w|>1\}$ normalized by the conditions

$$\begin{aligned} \phi (\infty )=\infty ,\qquad \lim _{z\rightarrow \infty }\frac{\phi (z)}{z}>0 \end{aligned}$$

and let $\psi (\cdot )$ be the inverse mapping of $\phi (\cdot )$.

We denote by $w=\phi _1(z)$ the conformal mapping of G onto the domain ${\mathbb {D}}=\{ w\in {\mathbb {C}}:|w|>1\}$, normalized by the conditions

$$\begin{aligned} \phi _1(0) =\infty ,\qquad \lim _{z\rightarrow 0}(z\phi _1(z))>0, \end{aligned}$$

and let $\psi _1(\cdot )$ be the inverse mapping of $\phi _1(\cdot )$.

The functions $\psi (\cdot )$ and $\psi _1(\cdot )$ have in some deleted neighborhood of the point $w=\infty$ the representations

$$\begin{aligned} \psi (w)=\gamma w+\gamma _0+\frac{\gamma _1}{w}+\frac{\gamma _2}{w^2}+\cdots , \qquad \gamma >0, \end{aligned}$$

and

$$\begin{aligned} \psi _1(w) =\frac{\alpha _1}{w}+\frac{\alpha _2}{w^2}+ \cdots +\frac{\alpha _k}{w^k}+\cdots , \qquad \alpha _1>0. \end{aligned}$$

The following expansions hold [10, 14, 41, 49]:

$$\begin{aligned} \frac{\psi '(w)}{\psi (w)-z}= \sum _{k=0}^{\infty }\frac{\Phi _k(z)}{w^{k+1}}, \qquad z\in G\text { and }w\in {\mathbb {D}}^-, \end{aligned}$$

(1.3)

and

$$\begin{aligned} \frac{\psi _1'(w)}{\psi _1(w)-z} = \sum _{k=0}^{\infty }-\frac{F\left( \frac{1}{z}\right) }{w^{k+1}}, \qquad z\in G^-\text { and }w\in {\mathbb {D}}^-, \end{aligned}$$

(1.4)

where $\Phi _k(z)$ and $F_k(1/z)$ are the Faber polynomials of degree k with respect to z and 1/z for the continuums ${\overline{G}}$ and $\overline{{\mathbb {C}}}\backslash G$, respectively. Also, for the Faber polynomials $\Phi _k(z)$ and rational functions $F_k(1/z)$ the integral representations

$$\begin{aligned} \Phi _k(z)= & {} [\phi (z)]^k+ \frac{1}{2\pi i}\int _{\Gamma }\frac{[\phi (\zeta )]^n}{\zeta -z}d\zeta , \qquad k=0,1,2,\ldots ,\,z\in G, \end{aligned}$$

(1.5)

$$\begin{aligned} F_k\left( \frac{1}{z}\right)= & {} [\phi _1(z)]^k- \frac{1}{2\pi i}\int _{\Gamma }\frac{[\phi _1(\zeta )]^n}{\zeta -z}d\zeta , \qquad k=0,1,2,\ldots ,\,z\in G \end{aligned}$$

(1.6)

hold [10, 49].

Let also $\chi (\cdot )$ be a continuous function on $2\pi$. Its modulus of continuity is defined by

$$\begin{aligned} \omega (t,\chi ):= \sup _{t_1,t_2\in [0,2\pi ],|t_1-t_2|<t}|\chi (t_1)-\chi (t_2)|, \qquad t\ge 0. \end{aligned}$$

The curve $\Gamma$ is called Dini-smooth if it has the parametrization

$$\begin{aligned} \Gamma :\chi (t),\qquad 0\le t\le 2\pi , \end{aligned}$$

such that $\chi '(t)$ is Dini-continuous, i.e.

$$\begin{aligned} \int _0^{\pi }\frac{\omega (t,\chi ')}{t}dt<\infty \end{aligned}$$

and

$$\begin{aligned} \chi '(t) \ne 0 \end{aligned}$$

[45, p. 48]

Let $f(\cdot )\in L_1(\Gamma )$. Then the functions $f^+(\cdot )$ and $f^-(\cdot )$ defined by

$$\begin{aligned} f^+(z)= \frac{1}{2\pi i}\int _{\Gamma }\frac{f(\zeta )}{\zeta -z}d\zeta = \frac{1}{2\pi i}\int _{{\mathbb {T}}}\frac{f(\psi (w))\psi '(w)}{\psi (w)-z}dw, \qquad z\in G \end{aligned}$$

(1.7)

and

$$\begin{aligned} f^-(z)= \frac{1}{2\pi i}\int _{\Gamma }\frac{f(\zeta )}{\zeta -z}d\zeta = \frac{1}{2\pi i}\int _{{\mathbb {T}}}\frac{f(\psi _1(w))\psi _1'(w)}{\psi _1(w)-z}dw,\qquad z\in G^- \end{aligned}$$

(1.8)

are analytic in G and $G^-$, respectively, and $f^-(\infty )=0$. Thus the limit

$$\begin{aligned} S_\Gamma (f)(z):=\lim _{\varepsilon \rightarrow \infty } \frac{1}{2\pi i} \int _{\Gamma \cap \{\zeta :|\zeta -z|>\varepsilon \}} \frac{f(\zeta )}{\zeta -z}d\zeta \end{aligned}$$

exists and is finite for almost all $z\in \Gamma$.

The quantity $S_\Gamma (f)(z)$ is called the Cauchy singular integral of $f(\cdot )$ at $z\in \Gamma$. According to the Privalov theorem [22, p. 431], if one of the functions $f^+(\cdot )$ or $f^-(\cdot )$ has non-tangential limits a.e. on $\Gamma$, then $S_\Gamma (f)(z)$ exists a.e. on $\Gamma$ and also the other one has non-tangential limits a.e. on $\Gamma$. Conversely, if $S_\Gamma (f)(z)$ exists a.e. on $\Gamma$, then the functions $f^+(\cdot )$ and $f^-(\cdot )$ have non-tangential limits a.e. on $\Gamma$. In both cases, the formulae

$$\begin{aligned} f^+(z)=S_{\Gamma }(f)(z)+\frac{1}{2}f(z),\qquad f^-(z)=S_{\Gamma }(f)(z)-\frac{1}{2}f(z) \end{aligned}$$

(1.9)

and hence

$$\begin{aligned} f(z)=f^+(z)-f^-(z) \end{aligned}$$

(1.10)

hold a.e. on $\Gamma$. From the results in [39] , it follows that if $\Gamma$ is a Dini-smooth curve $S_{\Gamma }$ is bounded on $L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$. Note that some properties of the Cauchy singular integral in the different spaces were investigated in [8, 13, 15, 20, 34,35,36, 38].

Let $f(\cdot )\in L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$: Using (1.3), (1.4), (1.7), (1.8) and (1.10) we can associate the Faber-Laurent series

$$\begin{aligned} f(z)\backsim \sum _{k=0}^{\infty }a_k\Phi _k(z) + \sum _{k=1}^{\infty }b_kF_k\left( \frac{1}{z}\right) , \end{aligned}$$

where the coefficients $a_k$ and $b_k$ are defined by

$$\begin{aligned} a_k:= \frac{1}{2\pi i}\int _{{\mathbb {T}}}\frac{f[\psi (w)]}{w^{k+1}}d\omega , \qquad k=0,1,2,\ldots \end{aligned}$$

and

$$\begin{aligned} b_k:= \frac{1}{2\pi i}\int _{{\mathbb {T}}}\frac{f[\psi _1(w)]}{w^{k+1}}dw, \qquad k=0,1,2,\ldots . \end{aligned}$$

The coefficients $a_k$ and $b_k$ are said to be the Faber-Laurent coefficients of $f(\cdot )$.

If $\Gamma$ is a Dini-smooth curve, then from the results in [53], it follows that

$$\begin{aligned} \left. \begin{array}{c} 0<c_2<|\phi '(w)|<c_3<\infty , \quad 0<c_4<|\phi _1'(w)|<c_5<\infty \\ 0<c_6<|\psi '(w)|<c_7<\infty , \quad 0<c_8<|\psi _1'(\omega )|<c_9<\infty \end{array} \right\} \end{aligned}$$

(1.11)

where the constants $c_2,c_3,c_4,c_5$ and $c_6,c_7,c_8,c_9$ are independent of $z\in {\bar{G}}^-$ and $|w|\ge 1$, respectively.

Let $\Gamma$ be a Dini-smooth curve and let $f_0(w):=f[\psi (w)]$ for $f(\cdot )\in L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$, $p_0(w):=p(\psi (w))$ and let $f_1(w):=f[\psi _1(w)]$ for $f(\cdot )\in L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$, $p_1(w):=p(\psi _1(w))$. Then using (1.11) and the method applied for the proof of a similar result in [29, Lem. 1], we obtain $f_0(\cdot )\in L^{p_0(\cdot ),\lambda (\cdot )}({\mathbb {T}})$ and $f_1(\cdot )\in L^{p_1(\cdot ),\lambda (\cdot )}({\mathbb {T}})$.

Moreover, $f_0^-(\infty )=f_1^-(\infty )=0$ and by (1.10)

$$\begin{aligned} \left. \begin{array}{l} f_0(w)=f_0^+(w)-f_0^-(w) \\ f_1(w)=f_1^+(w)-f_1^-(w) \end{array} \right\} \end{aligned}$$

(1.12)

a.e. on ${\mathbb {T}}$.

Note that the density of polynomials is an indispensable condition in approximation problems. Therefore, the polynomials are dense in the spaces $L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$, $E^{p(\cdot )\lambda (\cdot )}(G)$ and $E^{p(\cdot )\lambda (\cdot )}(G^-)$.

Using [21, Thm. 6.1] and the method applied for the proof of a similar result in [10] we can prove the following Lemma:

Lemma 1.1

Let $p(\cdot ):~{\mathbb {T}}\rightarrow [1,+\infty ]$ and $\lambda (\cdot ):{\mathbb {T}}\rightarrow [0,1]$ be measurable functions. Let $g(\cdot )\in E^{p(\cdot ),\lambda (\cdot )}(D)$ with $p(\cdot )\in \wp ^{\log }({\mathbb {T}}), 0\le \lambda _*\le \lambda ^*<1$. If $\sum _{k=0}^nd_k(g)w^k$ is the nth partial sum of the Taylor series of $g(\cdot )$ at the origin, then

$$\begin{aligned} \left\| g(w)-\sum _{k=0}^nd_kw^k \right\| _{L^{p(\cdot ),\lambda (\cdot )}({\mathbb {T}})}\le c_{10}(p)\,\Omega \!\left( g,\frac{1}{n}\right) _{p(\cdot ),\lambda (\cdot )}, \qquad \text {for all } n\in {\mathbb {N}} \end{aligned}$$

with some constant $c_{10}(p)>0$ independent of n.

Lemma 1.2

Let $p(\cdot ):~{\mathbb {T}}\rightarrow [1,+\infty ]$ and $\lambda (\cdot ):{\mathbb {T}}\rightarrow [0,1]$ be measurable functions. Let $g(\cdot )\in L^{p(\cdot ),\lambda (\cdot )}({\mathbb {T}})$ with $p(\cdot )\in \wp ^{\log }({\mathbb {T}})$, $0\le \lambda _*\le \lambda ^*<1$. Then the inequality

$$\begin{aligned} \Omega (g^+,\cdot )_{p(\cdot ),\lambda (\cdot )}\le c_{11}\Omega (g,\cdot )_{p(\cdot ),\lambda (\cdot )} \end{aligned}$$

(1.13)

holds.

Proof of Lemma 1.2

It is clear that the equality

$$\begin{aligned} g^+=S_{{\mathbb {T}}}(g)+\frac{1}{2}g \end{aligned}$$

(1.14)

holds a.e. on ${\mathbb {T}}$. Using the method of proof of [10, Lem. 3.3] (see also, [29, Lem. 2] and the boundedness of the singular operator $S_{{\mathbb {T}}}(g)$ in $L^{p(\cdot ),\lambda (\cdot )}({\mathbb {T}})$ we can prove that

$$\begin{aligned} \Omega (S_T(g),\cdot )_{p(\cdot ),\lambda (\cdot )}\le c_{12}\Omega (g,\cdot )_{p(\cdot ),\lambda (\cdot )}. \end{aligned}$$

(1.15)

Then using the subadditivity of the modulus of smoothness $\Omega (g^+,\cdot )_{p(\cdot ),\lambda (\cdot )}$, (1.14) and (1.15) we obtain inequality (1.13) of Lemma 1.2. $\square$

We set

$$\begin{aligned} R_n(f,z):= \sum _{k=0}^na_k\Phi _k(z)+\sum _{k=0}^nb_kF_k\left( \frac{1}{z}\right) . \end{aligned}$$

The rational function $R_n(f,z)$ is called the Faber-Laurent rational function of degree n of $f(\cdot )$.

The problems of approximation of the functions in classical Morrey spaces and variable exponent Morrey spaces were investigated in [1, 2, 9, 11, 12, 21, 26, 27]. In this work the approximation of the functions by Faber-Laurent rational functions in the variable exponent Morrey classes defined on the Dini-smooth curve are investigated. Similar problems of approximation of the functions by Faber-Laurent rational functions in different spaces were studied in [6, 7, 10, 23, 25, 28, 29, 31,32,33, 43, 44, 55].

Our main results are as follows.

Theorem 1.1

Let $\Gamma$ be a Dini-smooth curve. Let $p(\cdot ):\Gamma \rightarrow [1,+\infty ]$ and $\lambda (\cdot ):\Gamma \rightarrow [0,1]$ be measurable functions. If $p(\cdot )\in \wp ^{\log }(\Gamma )$, $0\le \lambda _*\le \lambda ^*<1$ and $f(\cdot )\in L^{p(\cdot )\lambda (\cdot )}(\Gamma )$, then for every natural number n there are a constant $c_{10}>0$ and rational function

$$\begin{aligned} R_n(z,f):=\sum _{k=-n}^na_k^{(n)}z^k \end{aligned}$$

such that

$$\begin{aligned} \Vert f-R_n(\cdot ,f)\Vert _{L^{p(\cdot ),\lambda (\cdot )}(\Gamma )}\le c_{13}\left[ \Omega \left( f_0,\frac{1}{n}\right) _{p_0(\cdot ),\lambda (\cdot )}+ \Omega \left( f_1,\frac{1}{n}\right) _{p_1(\cdot ),\lambda (\cdot )}\right] , \end{aligned}$$

where $R_n(\cdot ,f)$ is the n-th partial sum of the Faber-Laurent series of $f(\cdot )$.

Theorem 1.2

Let $\Gamma$ be a Dini-smooth curve. Let $p(\cdot ):\Gamma \rightarrow [1,+\infty ]$ and $\lambda (\cdot ):\Gamma \rightarrow [0,1]$ be measurable functions. If $p(\cdot )\in \wp ^{\log }(\Gamma )$, $0\le \lambda _*\le \lambda ^*<1$ and $f(\cdot )\in E^{p(\cdot )\lambda \cdot )}(G)$, then for every natural number n the inequality

$$\begin{aligned} \left\| f(z)-\sum _{k=0}^na_k\Phi _k(z) \right\| _{L^{p(\cdot ),\lambda (\cdot )}(\Gamma )}\le c_{14}\Omega \left( f_0,\frac{1}{n}\right) _{p_0(\cdot ),\lambda (\cdot )} \end{aligned}$$

(1.16)

holds with a constant $c_{14}>0$ independent of n.

Note that the order of polynomial approximation in $E^p(G)$, $p\ge 1$ has been investigated by several authors. In [52] Walsh an Rusel gave results when $\Gamma$ is an analytic curve. When $\Gamma$ is a Dini-smooth curve direct and inverse theorems were proved by S. Y. Alper [4], These results were later extended to domains with regular boundary for $p>1$ by Kokilashvili [37] and for $p\ge 1$ by Andersson [5]. For domains with a regular boundary the approximation directly as the nth partial sums of p-Faber polynomial of $f(\cdot )\in E^p(G)$ have been constructed in [23]. The approximation properties of the p-Faber series expansions in the $\omega$-weighted Smirnov class $E^p(G,\omega )$ of analytic functions in G whose boundary is a regular Jordan curve are investigated in [24].

Theorem 1.3

Let $\Gamma$ be a Dini-smooth curve. Let $p(\cdot ):\Gamma \rightarrow [1,+\infty ]$ and $\lambda (\cdot ):\Gamma \rightarrow [0,1]$ be measurable functions. If $p(\cdot )\in \wp ^{\log }(\Gamma )$, $0\le \lambda _*\le \lambda ^*<1$ and $f(\cdot )\in E^{p(\cdot )\lambda (\cdot )}(G^-)$, then for every natural number n the inequality

$$\begin{aligned} \left\| f- f(\infty )- \sum _{k=0}^n-b_kF_{k}\left( \frac{1}{z}\right) \right\| _{L^p(\cdot ),\lambda (\cdot )(\Gamma )}\le c_{15}\Omega \left( f_1,\frac{1}{n}\right) _{p_1(\cdot ),\lambda (\cdot )} \end{aligned}$$

(1.17)

holds, with a constant $c_{15}>0$ independent of n.

2 Proof of the Main Result

Proof of Theorem 1.1

Let $f(\cdot )\in L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$. Then from (1.11), we have $f_0(\cdot )\in L^{p_0(\cdot ),\lambda (\cdot )}({\mathbb {T}})$, $f_1(\cdot )\in L^{p_1(\cdot ),\lambda (\cdot )}({\mathbb {T}})$. According to(1.12) we obtain that

$$\begin{aligned} f(\zeta )= f_0^+(\phi (\zeta ))-f_0^-(\phi (\zeta )), \qquad f(\xi )=f_1^+(\phi _1(\xi ))-f_1^-(\phi _1(\xi )). \end{aligned}$$

(2.1)

a.e. on $\Gamma$.

We prove that the rational function

$$\begin{aligned} f(z)= \sum _{k=0}^{n} a_k\Phi _k(z) + \sum _{k=1}^{n}b_kF_k\left( \frac{1}{z}\right) \end{aligned}$$

satisfies the condition of Theorem 1.1.

Let $z^*\in G^-$. Using the method of proof in [28], we can prove that $f_0^-(\phi (\zeta )) \in E^{p(\cdot ),\lambda (\cdot )}(G^-)\in E^1(G^-)$. Then it is clear that

$$\begin{aligned} \frac{1}{2\pi i}\int _{\Gamma } \frac{f_0^-(\phi (\zeta ))}{\zeta -z^*}d\zeta = -f_0^-(\phi (z^*)). \end{aligned}$$

Then from last equality, (1.5) and (2.1) we have

$$\begin{aligned} \sum _{k=0}^na_k\Phi _k(z^*)= & {} \sum _{k=0}^na_k[\phi (z^*)]^k+ \frac{1}{2\pi i}\int _\Gamma \frac{1}{\zeta -z^*}\sum _{k=0}^na_k[\phi (\zeta )]^k d\zeta \nonumber \\= & {} \sum _{k=0}^na_k[\phi (z^*)]^k+ \frac{1}{2\pi i}\int _{\Gamma } \frac{1}{\zeta -z^*}\sum _{k=0}^n a_k[\phi (\zeta )]^k-f_0^+[\phi (\zeta )] d\zeta \nonumber \\&+\frac{1}{2\pi i} \int _\Gamma \frac{f(\zeta )}{\zeta -z^*}d\zeta - f_0^-[\phi (z^*)]. \end{aligned}$$

(2.2)

Use of(1.8) and (2.2) gives us

$$\begin{aligned} \sum _{k=0}^na_k\Phi _k(z^*)= & {} \sum _{k=0}^na_k[\phi (z^*)]^k+ \frac{1}{2\pi i}\int _\Gamma \frac{1}{\zeta -z^*}\sum _{k=0}^na_k[\phi (\zeta )]^k d\zeta \nonumber \\= & {} \sum _{k=0}^na_k[\phi (z^*)]^k+ \frac{1}{2\pi i}\int _\Gamma \frac{1}{\zeta -z^*} \sum _{k=0}^na_k[\phi (\zeta )]^k-f_0^+[\phi (\zeta )] d\zeta \nonumber \\&+f^-(z^*)-f_0^-[\phi (z^*)]. \end{aligned}$$

(2.3)

Taking the limit as $z^*\rightarrow z\in \Gamma$ along all non-tangential paths outside $\Gamma$ and considering (1.9), (1.10), (2.1) and (2.3) we obtain

$$\begin{aligned} f^+(z)-\sum _{k=0}^na_k\Phi _k(z^*)= & {} \frac{1}{2}\left[ f_0^+[\phi (z^*)] - \sum _{k=0}^na_k[\phi (z^*)]^k \right] \nonumber \\&+S_\Gamma \left( \left[ f_0^+[\phi (z^*)] - \sum _{k=0}^na_k[\phi (z^*)]^k \right] \right) . \end{aligned}$$

(2.4)

According to [39] the singular operator $S_\Gamma :L^{p(\cdot ),\lambda (\cdot )}(\Gamma )\rightarrow L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$ is bounded. Then using (2.1), Minkowski’s inequality, Lemma 1.1 and 1.2 we reach

$$\begin{aligned}&\left\| f^+(z)-\sum _{k=0}^na_k\Phi _k(z^*) \right\| _{L^{p(\cdot ),\lambda (\cdot )}(\Gamma )} \nonumber \\&\quad \le \frac{1}{2}\left\| f_0^+[\phi (z^*)] - \sum _{k=0}^na_k[\phi (z^*)]^k \right\| _{L^{p(\cdot ),\lambda (\cdot )}(\Gamma )} \nonumber \\&\qquad +\left\| S_\Gamma \left( \left[ f_0^+[\phi (z^*)] - \sum _{k=0}^na_k[\phi (z^*)]^k \right] \right) \right\| _{L^{p(\cdot ),\lambda (\cdot )}(\Gamma )} \nonumber \\&\quad \le \frac{1}{2}\left\| f_0^+(w)-\sum _{k=0}^na_kw^k \right\| _{L^{p_0(\cdot ),\lambda (\cdot )}({\mathbb {T}})}+ c_{16}\left\| f_0^+(w)-\sum _{k=0}^na_kw^k \right\| _{L^{p_0(\cdot ),\lambda (\cdot )}({\mathbb {T}})}\nonumber \\&\quad \le c_{17}\left\| f_0^+(w)-\sum _{k=0}^na_kw^k \right\| _{L^{p_0(\cdot ),\lambda (\cdot )}(T)}\nonumber \\&\quad \le c_{18}\left\| f_0^+(w)-\sum _{k=0}^n\alpha _k(f_0^+)w^k \right\| _{L^{p_0(\cdot ),\lambda (\cdot )}(T)}\nonumber \\&\quad \le c_{19}\Omega \left( f_0^+,\frac{1}{n}\right) _{p_0(\cdot ),\lambda (\cdot )}\le \Omega _{20}\left( f_0,\frac{1}{n}\right) _{p_0(\cdot ),\lambda (\cdot )}. \end{aligned}$$

(2.5)

Let $z^*\in G$. Using the method of proof in [28] we can prove that $f_1^-(\phi _1(\zeta ))\in E^{pt(\cdot ),\lambda (\cdot )}(G^-) \in E^1(G^-)$. Therefore,

$$\begin{aligned} \frac{1}{2\pi i}\int _\Gamma \frac{f_1^-(\phi _1(\zeta ))}{\zeta -z^*}d\zeta = f_1^{-}(\phi _1(z^*)). \end{aligned}$$

Then, using the last equality, (1.6) and (2.1) we have

$$\begin{aligned}&\sum _{k=1}^nb_kF_k\left( \frac{1}{z^*}\right) \\&\quad =\sum _{k=1}^nb_k[\phi _1(z^*)]^k- \frac{1}{2\pi i}\int _\Gamma \frac{1}{\xi -z^*}\sum _{k=1}^nb_k[\phi _1(\xi )]^k d\xi \\&\quad = \sum _{k=1}^nb_k[\phi _1(z^*)]^k- \frac{1}{2\pi i}\int _\Gamma \frac{1}{\xi -z^*} \left( \sum _{k=1}^nb_k[\phi _1(\xi )]^k-f_1^+[\phi _1(\xi )] \right) d\xi \\&\qquad - \frac{1}{2\pi i}\int _\Gamma \frac{f(\xi )}{\xi -z^*}d\xi - \frac{1}{2\pi i}\int _\Gamma \frac{f_1^-(\phi _1(\zeta ))}{\zeta -z^*}d\zeta \\&\quad = \sum _{k=1}^nb_k[\phi _1(z^*)]^k- \frac{1}{2\pi i}\int _\Gamma \frac{1}{\xi -z^*} \left( \sum _{k=1}^nb_k[\phi _1(\xi )]^k-f_1^+[\phi _1(\xi )] \right) d\xi \\&\qquad - f^+(z^*)-f_1^-[\phi _1(z^*)]. \end{aligned}$$

Taking the limit as $z^*\rightarrow z$ along all non-tangential paths inside of $\Gamma$ we have

$$\begin{aligned}&\sum _{k=1}^nb_kF_k\left( \frac{1}{z}\right) \\&\quad = \sum _{k=1}^nb_k[\phi _1(z)]^k- \frac{1}{2}\left( \sum _{k=1}^nb_k[\phi _1(z)]^k-f_1^+[\phi _1(z)] \right) \\&\qquad - S_\Gamma \left( \sum _{k=1}^nb_k[\phi _1(z)]^k-f_1^+[\phi _1(z)] \right) -f^+(z)-f_1^-[\phi _1(z)] \end{aligned}$$

a.e. on $\Gamma$. Use of(1.10) and (2.1) gives

$$\begin{aligned}&f^-(z)+\sum _{k=1}^nb_kF_k\left( \frac{1}{z}\right) \nonumber \\&\quad = \frac{1}{2}\left( \sum _{k=1}^nb_k[\phi _1(z)]^k-f_1^+[\phi _1(z)] \right) \nonumber \\&\qquad - S_\Gamma \left( \sum _{k=1}^nb_k[\phi _1(z)]^k-f_1^+[\phi _1(z)] \right) . \end{aligned}$$

(2.6)

Consideration of (2.6), Minkowski’s inequality and the boundedness of $S_\Gamma$ in $L^{p(\cdot ),\lambda (\cdot )}(\Gamma )$, Lemma 1.1 and 1.2 gives rise to

$$\begin{aligned}&\left\| f^-(z) +\sum _{k=1}^nb_kF_k\left( \frac{1}{z}\right) \right\| _{L^{p(\cdot )\lambda (\cdot )}(\Gamma )}\nonumber \\&\quad \le \left\| \frac{1}{2}\left( \sum _{k=1}^nb_k[\phi _1(z)]^k-f_1^+[\phi _1(z)] \right) \right\| _{L^{p(\cdot )\lambda (\cdot )}(\Gamma )}\nonumber \\&\qquad + \left\| S_\Gamma \left( \sum _{k=1}^nb_k[\phi _1(z)]^k-f_1^+[\phi _1(z)] \right) \right\| _{L^{p(\cdot ),\lambda (\cdot )}(\Gamma )}\nonumber \\&\quad \le \frac{1}{2}\left\| \sum _{k=1}^nb_kw^k-f_1^+(w) \right\| _{L^{p_1(\cdot ),\lambda (\cdot )}({\mathbb {T}})}\nonumber \\&\qquad + c_{21}\left\| \sum _{k=1}^nb_kw^k-f_1^+(w) \right\| _{L^{p_1(\cdot ),\lambda (\cdot )}({\mathbb {T}})}\nonumber \\&\quad \le c_{22}\left\| \sum _{k=1}^nb_kw^k-f_1^+(w) \right\| _{L^{p_1(\cdot ),\lambda (\cdot )}({\mathbb {T}})}\nonumber \\&\quad = c_{22}\left\| \sum _{k=1}^n\beta _k(f_1^+)w^k-f_1^+(w) \right\| _{L^{p_1(\cdot ),\lambda (\cdot )}({\mathbb {T}})}\nonumber \\&\quad \le c_{23}\Omega \left( f_1^+,\frac{1}{n}\right) _{p_1(\cdot ),\lambda (\cdot )} \nonumber \\&\quad \le c_{24}\Omega \left( f_1,\frac{1}{n}\right) _{p_1(\cdot ),\lambda (\cdot )} \end{aligned}$$

(2.7)

Now combining (1.9), (2.5) and (2.7) we obtain

$$\begin{aligned} \Vert f-R_n(\cdot ,f)\Vert _{L^{p(\cdot )}(\Gamma )} \le c_{25}(p)\left[ \Omega \left( f_0,\frac{1}{n}\right) _{p_0(\cdot ),\lambda (\cdot )}+ \Omega \left( f_1,\frac{1}{n}\right) _{p_1(\cdot ),\lambda (\cdot )} \right] . \end{aligned}$$

The proof of Theorem 1.1 is completed. $\square$

Proof of Theorem 1.2

Let $z^*\in G^-$. If $f(\cdot )\in E^{p(\cdot ),\lambda (\cdot )}(G)$, then $f(\cdot )\in E^p(G)$ and $f(\zeta )/(\zeta -z^*)\in E^p(G)$. Therefore, $\int _{\Gamma } f(\zeta )/(\zeta -z^*)d\zeta =0$. That is $f^-(z)=0$ a.e. on $\Gamma$. Then taking into account (1.10),

$$\begin{aligned} \left\| f_0^+(w)-\sum _{k=0}^na_kw^k \right\| _{L^{p(\cdot ),\lambda (\cdot )}({\mathbb {T}})}\le & {} c_{26}(p)\Omega \left( f_0,\frac{1}{n}\right) _{p(\cdot ),\lambda (\cdot )} \quad \text{ for } \text{ all } n\in {\mathbb {N}},\\ \left\| f_0^+(z)-\sum _{k=0}^na_k\Phi _k(z) \right\| _{L^{p(\cdot ),\lambda (\cdot )}(\Gamma )}\le & {} c_{27}\left\| f_0^+(w)-\sum _{k=0}^na_kw^k \right\| _{L^{p(\cdot ),\lambda (\cdot )}({\mathbb {T}})} \end{aligned}$$

we have the inequality (1.16) of Theorem 1.2. $\square$

Proof of Theorem 1.3

Let $z^*\in G$ and $f(\cdot )\in E^{p(\cdot ),\lambda (\cdot )}(G^-)$. It is clear that $\int _\Gamma f(\zeta )/(\zeta -z^*)=f(\infty )$. Then we have $f^+(z)=f(\infty$) a.e. on $\Gamma$. Now combining (1.10),

$$\begin{aligned} \left\| f_1^+(w)-\sum _{k=0}^nb_kw^k \right\| _{L^{p(\cdot ),\lambda (\cdot )}({\mathbb {T}})}\le & {} c_{28}(p)\Omega \left( f_1,\frac{1}{n}\right) _{p(\cdot ),\lambda (\cdot )} \quad \text{ for } \text{ all } n\in {\mathbb {N}}, \\ \left\| f^-(z)-\sum _{k=0}^nb_kF_k\left( \frac{1}{z}\right) \right\| _{L^{p(\cdot ),\lambda (\cdot )}(\Gamma )}\le & {} c_{29}\left\| f_1^+(w)-\sum _{k=0}^nb_kw^k \right\| _{L^{p(\cdot ),\lambda (\cdot )}({\mathbb {T}})} \end{aligned}$$

we obtain the inequality (1.17) of Theorem 1.3. $\square$