Abstract
Let G be a doubly connected domain in the complex plane \(\mathbb {C}\) , bounded by Ahlfors 1-regular curves. In this study the approximation of the functions by Faber–Laurent rational functions in the \(\omega \)-weighted generalized grand Smirnov classes \(\mathcal {E}^{p),\theta }(G,\omega )\) in the term of the rth\(,~r=1,2\ldots ,\) mean modulus of smoothness are investigated.
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1 Introduction
Let \(\Gamma \subset \mathbb {C} ~~\)be a Jordan rectifiable curve and let \(\omega :\Gamma \rightarrow \left[ 0,\infty \right] \) be a weight function, that is a positive almost everywhere (a.e.) and integrable function on \(\Gamma .~\)For \(1<p<\infty ~\)we define a class \(L^{p}(\Gamma ,\omega )~\)of Lebesgue measurable functions f on \(\Gamma ~\)satisfying the condition
where \(\left| \Gamma \right| ~\) is the length of \(\ \Gamma .~\) We denote by \(L^{p),\theta }(\Gamma ,\omega \mathbf {)},\) \(\theta \ge 0,\) the Lebesgue space of all measurable functions f on \(\Gamma ,~\)that is, the space of all such functions for which
The space \(L^{p),\theta }(\ \Gamma ,\omega \mathbf {)}\) is called the generalized grand Lebesgue space. \(L^{p),\theta }(\Gamma ,\omega \mathbf {)~}\) is Banach function space, nonreflexive and nonseparable. The grand and generalized grand Lebesgue space were introduceed in the works [13, 26], respectively. If \(\ \theta _{1}<\theta _{2}\) then for \(0<\varepsilon <p-1~\)the embeddings:
hold. Note that the information about properties and applications of the grand Lebesgue spaces can be found in [11, 13, 26, 33, 35, 36].
A Jordan curve \(\Gamma ~\)is called Ahlfors 1-regular [37], if there exists a number \(c>0~\)such that for every \(r>0,~\sup \left\{ \left| \Gamma \cap D(z,r)\right| :z\in \Gamma \right\} \le cr,~\), where D(z, r) is an open disk with radius r and centered at z and \( \left| \Gamma \cap D(z,r)\right| ~\)is the length of the set \(\Gamma \cap D(z,r).\)
Let \(\omega ~\)be a weight function on \(\Gamma .~\omega ~\)is said to satisfy Muckenhoupt’s \(A_{p}\)-condition on \(\Gamma ~\)if
Let us further assume that B is a simply connected domain with a rectifiable Jordan boundary \(\Gamma \) and \(B^{-}:=\mathrm{ext}\Gamma \). Without loss of generality we assume that \(0\in B.~\)Let
Also, \(\phi ^{*}\) stand for the conformal mapping of \(B^{-}\) onto \(D^{-}\) normalized by
and
and let \(\psi ^{*}\) be the inverse of \(\phi ^{*}\). Let \(\phi _{1}^{*}\) be the conformal mapping of B onto \(D^{-},\) normalized by
and
The inverse mapping of \(\phi _{1}^{*}\) will be denoted by \(\psi _{1}^{*}.\)
Note that the mappings \(\psi ^{*}\) and \(\psi _{1}^{*}~\)have in some deleted neighborhood of \(\infty ~\)representations:
and
For \(1<p<\infty \) and \(0<\varepsilon <p-1~\)the functions:
and
are analytic in the domain \(D^{-}.~\)The following expansions hold:
and
where \(\Phi _{k,p-\varepsilon }(z)\) and \(F_{k,p-\varepsilon }(\frac{1}{z})\) are the \(p-\varepsilon ~~\)Faber polynomials of degree k with respect to z and \(\frac{1}{z}~\)for the continuums \(\overline{B\text {~}}\) and \( \overline{B}\backslash B,\) respectively (see also [5, 20, 23] and ([34], pp. 255–257).
Let \(E^{1}(B)~\)be a classical Smirnov class of analytic functions in B. The set \(\ E^{p),\theta }(B,\omega ):=\left\{ f\in E^{1}(B):f\in L^{p),\theta }(\Gamma ,\omega )\right\} \) is called the \(\omega \)-weighted generalized grand Smirnov class in B.
Let \(\omega \in A_{p}(\mathbb {T})\). For \(f\in L^{p),\theta }(\Gamma ,\omega )\) we define the operator
where
If \(\omega \in A_{p}(\mathbb {T})\mathbb {~}\)and \(f\in L^{p}({\mathbb {T}}, ~\omega \mathbf {),~}\)then the operator \(\nu _{h}\) is a bounded on \( L^{p),\theta }\left( \mathbb {T},\omega \right) \) [24]:
Let \(1<p<\infty ,~\) \(\omega \in A_{p}(\mathbb {T})\) and \(f\in L^{p),\theta }( \mathbb {T}\),\(~\omega \mathbf {)},\theta >0.\) The function
is called the r-th mean modulus of \(f\in L^{p),~\theta }(\mathbb {T}\),\(~\omega \mathbf {)}\).
It can be easily shown that \(\Omega _{p),\theta ,\omega }^{r}\left( f,\cdot \right) \) is a continuous, non-negative and nondecreasing function satisfying the conditions:
for \(f,g\in L^{p),\theta }(\mathbb {T}\), \(\omega \mathbf {)}\).
Let G be a doubly connected domain in the complex plane \({\mathbb { C}}\), bounded by the rectifiable Jordan curves \(\Gamma _{1}\) and \(\Gamma _{2} \) (the closed curve \(\Gamma _{2}\) is in the closed curve \(\Gamma _{1}\)). Without loss of generality we assume \(0\in \) int\(\Gamma _{2}\). Let \( G_{1}^{0} \): = int\(\Gamma _{1}\), \(G_{1}^{\infty }\): =ext\(\Gamma _{1}\), \( G_{2}^{0}\): =int\(\Gamma _{2}\), \(G_{2}^{\infty }\):=ext\(\Gamma _{2}\).
We denote by \(w=\phi \left( z\right) \) the conformal mapping of \( G_{1}^{\infty }\) onto domain \(D^{-}\) normalized by the conditions:
and let \(\psi \) be the inverse mapping of \(\phi \).
We denote by \(w=\phi _{1}\left( z\right) \) the conformal mapping of \( G_{2}^{0}\) onto domain \(\,D^{-}\,\) normalized by the conditions:
and let \(\psi _{1}\) be the inverse mapping of \(\phi _{1}.\)
Let us take
For \(\Phi _{k,p-\varepsilon }\left( z\right) \) and \( F_{k,p-\varepsilon }\left( \frac{1}{z}\right) \) the following integral representations hold [5, 20, 23, 34], pp. 255–257:
-
(1)
If \(z\in intC_{\rho _{0}},\) then
$$\begin{aligned} \Phi _{k,p-\varepsilon }\left( z\right) =\frac{1}{2\pi i}\int \limits _{C_{ \rho _{0}}}\frac{\left[ \phi \left( \zeta \right) \right] ^{k}\left( \phi ^{\prime }(\zeta )\right) ^{\frac{1}{p-\varepsilon }}}{\zeta -z}\mathrm{d}\zeta . \end{aligned}$$(1.1) -
(2)
If \(z\in \mathrm{ext}C_{\rho _{0}}\), then
$$\begin{aligned}&\Phi _{k,p-\varepsilon }\left( z\right) \nonumber \\&\quad =\left[ \phi \left( z\right) \right] ^{k}\left( \phi ^{\prime }(z)\right) ^{\frac{1}{p-\varepsilon }}+\frac{1}{2\pi i}\int \limits _{C_{\rho _{0}}}\frac{ \left[ \phi \left( \zeta \right) \right] ^{k}\left( \phi ^{\prime }(\zeta )\right) ^{ \frac{1}{p-\varepsilon }}}{\zeta -z}\mathrm{d}\zeta .\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{aligned}$$(1.2) -
(3)
If \(z\in intC_{r_{0}}\), then
$$\begin{aligned}&F_{k,p-\varepsilon }(\frac{1}{z}) \nonumber \\&\quad =\left[ \phi _{1}\left( z\right) \right] ^{k-\frac{2}{p-\varepsilon } }\left( \phi ^{\prime }(z)\right) ^{\frac{1}{p-\varepsilon }}-\frac{1}{2\pi i }\int \limits _{C_{r_{0}}}\frac{\left[ \phi _{1}\left( \zeta \right) \right] ^{k-\frac{2}{p-\varepsilon }}(\phi _{1}^{\prime }(\zeta ))^{\frac{1}{ p-\varepsilon }}}{\zeta -z}\mathrm{d}\zeta .\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{aligned}$$(1.3) -
(4)
If \(z\in \mathrm{ext}C_{r_{0}},~\) then
$$\begin{aligned} F_{k,p-\varepsilon }\left( \frac{1}{z}\right) =-\frac{1}{2\pi i} \int \limits _{C_{r_{0}}}\frac{\left[ \phi _{1}\left( \zeta \right) \right] ^{k-\frac{2}{p-\varepsilon }}\left( \phi _{1}^{\prime }(\zeta )\right) ^{\frac{1}{ p-\varepsilon }}}{\zeta -z}\mathrm{d}\zeta . \end{aligned}$$(1.4)If a function \(f\left( z\right) \) is analytic in the doubly connected domain bounded by the curves \(C_{\rho _{0}}\) and \(\Gamma _{r_{0}}\), then the following series expansion holds:
$$\begin{aligned} f\left( z\right) =\sum \limits _{k=0}^{\infty }a_{k}\Phi _{k,p-\varepsilon }\left( z\right) +\sum \limits _{k=1}^{\infty }b_{k}F_{k,p-\varepsilon }\left( \frac{1}{z}\right) , \end{aligned}$$(1.5)where
and
The series (1.5) is called the \(p-\varepsilon ~\) Faber–Laurent series of f, and the coefficients \(a_{k}\) and \(b_{k}\) are said to be the \( p-\varepsilon ~\) Faber–Laurent coefficients of f . For \(z\in G\) by Cauchy’s integral formulae we have
If \(z\in \mathrm{int}\Gamma _{2} \) and \(z\in \mathrm{ext}\Gamma _{1} \), then
Let us consider
The function \(I_{1}(z) \) determines the functions \(I_{1}^{+}(z) \) and \( I_{1}^{-}(z) \), while the function \(I_{2}(z) \) determines the functions \( I_{2}^{+}(z) \) and \(I_{2}^{-}(z) \). The functions \(I_{1}^{+}(z) \) and \( I_{1}^{-}(z) \) are analytic in \(\mathrm{int}\Gamma _{1} \) and \(\mathrm{ext}\Gamma _{1} \), respectively. The functions \(I_{2}^{+}(z) \) and \(I_{2}^{-}(z) \) are analytic in \(\mathrm{int}\Gamma _{2} \) and \(\mathrm{ext}\Gamma _{2}\), respectively.
Let B be a finite domain in the complex plane bounded by a rectifiable Jordan curve \(\Gamma \) and \(f\in L_{1}\left( \Gamma \right) \). Then the functions \(f^{+}\) and \(f^{-}\) defined by
and
are analytic in B and \(B^{-} \)respectively, and \(f^{-} \left( \infty \right) =0.\) Thus the limit
exists and is finite for almost all \(z\in \Gamma \).
The quantity \(S_{\Gamma } (f)(z)\) is called the Cauchy singular integral of f at \(z\in \Gamma \).
According to the Privalov theorem ([12], p. 431), if one of the functions \( f^{+}\) or \(f^{-}\) has the non-tangential limits a.e. on \(\Gamma \), then \( S_{\Gamma }(f)(z)\) exists a.e. on \(\Gamma \) and also the other one has the non-tangential limits a.e. on \(\Gamma \). Conversely, if \(S_{\Gamma }(f)(z)\) exists a.e. on \(\Gamma \), then the functions \(f^{+}\left( z\right) \) and \( f^{-}\left( z\right) \) have non-tangential limits a.e. on \(\Gamma \). In both cases, the formulae
and hence
holds a.e. on \(\Gamma \). From the results given in [33], it follows that if \( \ \Gamma \) is an Ahlfors 1- regular curve, then \(S_{\Gamma }\) is bounded on \( L^{p),\theta }(\Gamma ,\omega ).\)
We will say that the doubly connected domain G is bounded by the Ahlfors 1-regular curve if the domains \(G_{1}^{0}\) and \(G_{2}^{0}\) are bounded by the closed Ahlfors 1-regular curves.
Let \(\Gamma _{i}\) \((i=1,2)\) be a regular curve and let \(f_{0}:=f\left[ \psi \left( w\right) \right] \psi ^{\prime }(w)^{\frac{1}{p-\varepsilon }}\) for \( f\in L^{p),\theta }(\Gamma _{1},\omega )\) and let \(f_{1}(w):=f\left[ \psi _{1}(w)\right] \) \((\psi _{1}^{\prime }(w))^{\frac{1}{p-\varepsilon }}w^{ \frac{2}{p-\varepsilon }~\ }\)for \(f\in L^{p),\theta }(\Gamma _{2},\omega ).\ \)We also set \(\omega _{0}(w):=\omega \left[ \psi (w)\right] \) , \(\omega _{1}(w):=\omega \left[ \psi _{1}(w)\right] .~\)Then , if \(f\in L^{p),\theta }(\Gamma _{1},\omega )\) and \(f\in L^{p),\theta }(\Gamma _{2},\omega )~\)we obtain \(f_{0}\in L^{p),\theta }(\mathbb {T\,}\),\(~\omega _{0})\) and \(f_{1}\in L^{p),\theta }(\mathbb {T}\),\(~\omega _{1})\).
Moreover, \(f_{0}^{-}(\infty )=~f_{1}^{-}(\infty )=0\) and by (1.7)
a.e. on \(\mathbb {T}\).
Now, in the doubly connected domain we define the \(\omega \)-weighted generalized grand Smirnov class . Let \(E^{1}(G)~\)be a classical Smirnov class of analytic functions in G. The set \(E^{p),\theta }(G,\omega ):=\left\{ f\in E^{1}(G): f\in L^{p),\theta }(\Gamma ,\omega )\right\} \) is called the \(\omega \)-weighted generalized grand Smirnov class in \(G.\ \)We denote by \(\mathcal {E}^{p),\theta }\left( G,\omega \right) ~\)the closure of Smirnov class \(E^{p}(G,\omega )\) in the space \(E^{p),\theta }(G,\omega ).\)
Lemma 1.1
[23, 24]. Let \(g\in \mathcal {E}^{p),\theta }(D,\omega )\) , \(\omega \in A_{p}(\mathbb {T} ),~1<p<\infty ~\)and \(\theta >0.~\)If\(~\sum \nolimits _{k=0}^{n}d_{k}(g)w^{k}\) is the nth partial sum of the Taylor series of g at the origin, then there exists a constant \(c_{2}>0\) such that
for every natural number n.
We set
The rational function \(R_{n}(f,z)\) is called the \(p-\varepsilon ~\) Faber–Laurent rational function of degree n of f.
Since series of Faber polynomials are a generalization of Taylor series to the case of a simply connected domain, it is natural to consider the construction of a similar generalization of Laurent series to the case of a doubly-connected domain.
The problems of approximation of the functions in the non-weighted and weighted grand Lebesgue spaces were investigated in [6,7,8,9,10, 23, 24]. In this study the approximation problems of the functions by Faber–Laurent rational functions in the weighted generalized grand Smirnov classes \(\mathcal {E}^{p),\theta }\left( G,\omega \right) \), \(\theta >0\), defined in the doubly connected domains with the regular boundaries are studied. Similar problems in the different spaces were investigated by several authors (see for example, [1,2,3,4,5, 14,15,16,17,18,19,20,21,22,23, 25, 27,28,32, 38, 39]).
Our main result can be formulated as following.
Theorem 1.2
Let G be a finite doubly connected domain with the Ahlfors 1-regular boundary \(\Gamma =\Gamma _{1}\cup \Gamma _{2}.\) If \(\ \omega \in A_{p}(\Gamma ),\omega _{0},\omega _{1}\in A_{p}(\mathbb {T}),~1<p<\infty ~\)and \(f\in \mathcal {E}^{p),\theta }\left( G,\omega \right) \), \(\theta >0,~\) then there is a constant \(c_{3}\) \(>0~\)such that for any \(n=1,2,3,\ldots \)
where \(R_{n}\left( .,f\right) \) is the \(p-\varepsilon ~\) Faber–Laurent rational function of degree n of f.
2 Proof of main result
Proof of Theorem 1.1
We take the curves \(\Gamma _{1}\), \(\Gamma _{2}\) and \(\mathbb {T}:=\left\{ w\in {\mathbb {C}}:\,\left| w\right| =1\right\} \) as the curves of integration in the formulas (1.2)–(1.5) and (1.6), respectively. (This is possible due to the conditions of Theorem 1.2). Let \(f\in \mathcal {E}^{p),\theta }(G,\omega ).\) Then \(f_{0}\in L^{p),\theta }(\mathbb {T},\omega _{0}),~f_{1}\in L^{p),\theta }(\mathbb {T},\omega _{1}).\) According to (1.8)
Let \(z\in \mathrm{ext}\Gamma _{1}.\) Using (1.2) and (2.1) we have
For \(z\in \mathrm{ext}\Gamma _{2}\), the relations(1.4) and (2.1) imply that
For \(z\in \mathrm{ext}\Gamma _{1}\), by virtue (2.2), (2.3) we obtain
Taking limit as \(z\rightarrow z^{*}\in \Gamma _{1}\) along all non-tangential paths outside \(\Gamma _{1}\), it appears that
a.e. on \(\Gamma _{1} \).
Now using (2.4), Minkowski’s inequality and the boundedness of \(S_{\Gamma _{1}}\) in \({ L}^{p),\theta }(\Gamma _{1},\omega )~\ \) [33] we get
That is, the Faber–Laurent coefficients \(a_{k}\) and \(b_{k}\) of the function f are the Taylor coefficients of the functions \( f_{0}^{+}\) and \(f_{1}^{+}\), respectively. Then by (2.5), Lemma 1 and [23] we obtain
Let \(\,z\in \mathrm{int}\Gamma _{2}.\) Then from (1.3) and (2.1) we have
For \(z\in \mathrm{int}\Gamma _{1},\) using (1.1) and (2.1) we obtain
Now, by virtue of (2.6) and (2.7) for \(z\in \mathrm{int}\Gamma _{2}\) , we conclude that
Taking the limit as \(z\rightarrow z^{*}\in \Gamma _{2}\) along all non-tangential paths inside \(\Gamma _{2}\), we reach
a.e. on \(\Gamma _{2} \).
Using (2.8), Minkowski’s inequality and the boundedness of \(S_{\Gamma _{2}}\) in \({ L}^{p),\theta }(\Gamma _{2},\omega )\) [33] we get
Use of (2.9), Lemma 1.1 and [23] leads to
The proof is complete. \(\square \)
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Jafarov, S.Z. On approximation of functions by rational functions in weighted generalized grand Smirnov classes. Arab. J. Math. 11, 293–302 (2022). https://doi.org/10.1007/s40065-021-00358-6
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DOI: https://doi.org/10.1007/s40065-021-00358-6