Department of Mathematics, Faculty of Art and Science, Balikesir University, 10145 Balikesir, Turkey
Academic Editor: V. M.Β Kokilashvili
Let and be, respectively, bounded and unbounded components of a plane curve satisfying Dini's smoothness condition. In addition to partial sum of Faber series of belonging to weighted Smirnov-Orlicz space (), we prove that interpolating polynomials and Poisson polynomials are near best approximant for . Also considering a weighted fractional moduli of smoothness, we obtain direct and converse theorems of trigonometric polynomial approximation in Orlicz spaces with Muckenhoupt weights. On the bases of these approximation theorems, we prove direct and converse theorems of approximation, respectively, by algebraic polynomials and rational functions in weighted Smirnov-Orlicz spaces and .
1. Introduction
Let and be, respectively, bounded and unbounded components of a closed rectifiable curve of complex plane . Without loss of generality we may suppose that . By Riemann conformal mapping theorem [1, page 26], if is connected Jordan curve that consists of more than one point, there exists a conformal mapping of complex unit disc onto . Let for a given . We denote by , , Smirnov’s classes of analytic functions satisfying
where positive constant is independent of .
It is well known that for every and every function has a nontangential boundary values a.e. on , the boundary function belongs to Lebesgue space on . If , then is a Banach space with the norm
Smirnov classes , , of analytic functions can be defined similarly and are fulfilling the same above properties to that of .
A smooth Jordan curve will be called Dini-smooth, if the function , the angle between the tangent line and the positive real axis expressed as a function of arclength , has modulus of continuity satisfying the Dini condition
A Jordan curve will be called Radon curve, if has bounded variation and it does not contain cusp point.
Main approximation problems in the spaces , , were dealt with by several mathematicians so far. Walsh and Russell gave [2] results in , , for algebraic polynomial approximation orders in case of analytic boundary. Al’per proved [3] direct and converse approximation theorems by algebraic polynomials in , , for Dini-smooth boundary. Kokilashvili improved [4] to Al’per’s direct and converse results of algebraic polynomial approximation, and then considering Regular curves that Cauchy’s Singular Integral Operator is bounded (corners are permitted), he obtained [5] improved direct and converse approximation theorems in Smirnov spaces , . Andersson proved [6] that Kokilashvili’s results also holds in . When the boundary is a regular curve, approximation of functions of , , by partial sum of Faber series was obtained by Israfilov in [7, 8]. These results are generalized to Muckenhoupt weighted Smirnov’s spaces in [9–12]. Approximation properties of Faber series in so-called weighted and unweighted Smirnov-Orlicz spaces are investigated in [13–20]. Most of the above results use the partial sums of Faber series as approximation tool. Interpolating polynomials [16] and Poisson polynomials [21] can be also considered as an approximating polynomial. In the present paper we obtain that in addition to partial sums of Faber series of belonging to weighted Smirnov-Orlicz space , interpolating polynomials and Poisson polynomials are near best approximant for . Also considering a weighted fractional moduli of smoothness, we obtain in Section 2 direct and converse theorems of trigonometric polynomial approximation in Orlicz spaces with Muckenhoupt weights. On the bases of these approximation theorems we prove in Section 3 direct and converse theorems of approximation, respectively, by algebraic polynomials and rational functions in weighted Smirnov-Orlicz spaces and .
Throughout the work, we will denote by , the constants that are different in different places.
2. Approximation Theorems in Weighted Orlicz Space
A function is called Young function if is even, continuous, nonnegative in , increasing on such that
A Young function is said to satisfy condition () if there is a constant such that
for all .
Two Young functions and are said to be equivalent if there are such that
A function is said to be quasiconvex if there exist a convex Young function and a constant such that
holds.
A nonnegative function defined on will be called weight if is measurable and a.e. positive. Let be a quasiconvex Young function. We denote by the class of Lebesgue measurable functions satisfying the condition
The linear span of the weighted Orlicz class , denoted by , becomes a normed space with the Orlicz norm
where , is the complementary function of .
If is quasiconvex and is its complementary function, then Young’s inequality holds
For a quasiconvex function we define the indice of as
The indice was first defined and used by Gogatishvili and Kokilashvili in [22] to obtain weighted inequalities for maximal function. We note that the indice is much more convenient than Gustavsson and Peetre’s lower index and Boyd’s upper index. If , then it can be easily seen that and becomes a Banach space with the Orlicz norm. The Banach space is called weighted Orlicz space.
We define the Luxemburg functional as
There exist [23, page 23] constants such that
For a weight we denote by the class of measurable functions on such that belongs to Lebesgue space on . We set for .
A 2-periodic weight function belongs to the Muckenhoupt class , , if
with a finite constant independent of , where is any subinterval of and denotes the length of .
We will denote by a class of functions satisfying condition such that is quasiconvex for some .
In the present section we consider the trigonometric polynomial approximation problems for functions and its fractional derivatives in the spaces , , where . We prove a Jackson type direct theorem and a converse theorem of trigonometric approximation with respect to the fractional order moduli of smoothness in weighted Orlicz spaces with Muckenhoupt weights. In the particular case, we obtain a constructive characterization of Lipschitz class in these spaces.
In weighted Lebesgue and Orlicz spaces with Muckenhoupt weights, these results were investigated in [24–29]. For more general doubling weights, some of these problems were investigated in [30]. Jackson and converse inequalities were proved for Lebesgue spaces with Freud weight in [31]. For a general discussion of weighted polynomial approximation, we can refer to the books [32, 33].
Let , , , ,
be the Fourier and the conjugate Fourier series of , respectively. Putting in (2.12), we define for
For a given , assuming
we define th fractional integral of as [34, v.2, page 134]
where
as principal value.
Let be given. We define fractional derivative of a function , satisfying (2.15), as
provided the right hand side exists.
Setting , , , , and , we define
where for and are Binomial coefficients, is Steklov’s mean operator, and is identity operator.
Theorem A (see [23, page 278, Theorem 6.7.1]). One suppose that is anyone of the operators , , and . If , , and , then there exists a constant such that
holds.
Since modular inequality implies the norm inequality, under the conditions of Theorem A, we obtain from (2.20) that
with a constant independent of .
By [35, page 14, (1.51)], there exists a constant depending only on such that
we have
and therefore
provided , , where .
Let . For , we define the fractional modulus of smoothness of index for , as
where denotes the integer part of a real number .
Since the operator is bounded in , , where , we have by (2.24) that
where the constant , dependent only on and .
Remark 2.1. The modulus of smoothness , where , , , has the following properties:(i) is nonnegative, nondecreasing function of and subadditive,(ii).
For formulations of our results, we need several lemmas.
Lemma A (see [36]). For , we suppose that(i),
(ii),
be two series in a Banach space . Let
for . Then,
for some if and only if there exists a such that
where and are constants depending only on one another.
If , , and , then from Theorem A(ii) and Abel’s transformation we get
and therefore from (2.14) and(2.30)
From the property
it is known that
for , .
Lemma 2.2. Let , and . If , then there exists a constant independent of such that
holds.
Proof. Without loss of generality one can assume that . Since
we have by (2.30) and Theorem A(iii) that
and from Lemma A
Hence from (2.33) and (2.31), we find
General case follows immediately from this.
Let . We denote by , , , the linear space of -periodic real valued functions such that .
Lemma 2.3. Let . If with and , then for , there is a constant dependent only on and such that
holds.
Proof. If , then from boundedness (see (2.21)) of the operator we get that
Let , . Since
we have
From (2.21) we get the boundedness of in and we have
From Lemma 2.2 we get
Now we have
Since
we get
Now we show that
For this we set
For given and , by Lemma 3 of [37], there exists a trigonometric polynomial such that
which by (2.7) this implies that
and hence we obtain
In this case from (2.40) we have
in norm. If , then
Hence,
Therefore,
Since
we obtain
Consequently,
and (2.48) holds. Now (2.47) and (2.48) imply the result.
Lemma 2.4. Let , , , and . If , then
hold, where the constant is dependent only on and .
Proof. First we prove that if , then
It is easily seen that if , then (2.61) holds. Now, we assume . In this case, putting , we have
Then,
and hence (2.61) holds. We note that if taking , for the remaining cases or or , it can easily be obtained from the last inequality that the required inequality (2.61) holds. Now we will show that if , , then
Putting
we have
Therefore,
Since
we obtain that
Using (2.61), (2.64), and Lemma 2.2, we get
which is the required result (2.60) for . On the other hand in case of the inequality (2.60) can be obtained by Marcinkiewicz Multiplier Theorem for where and .
Definition 2.5. For , and , the Peetre-functional is defined as
Proposition 2.6. Let , , and . Then the functional in (2.71) and the modulus , are equivalent.
Proof. If , then we have
Putting
we have
and hence
On the other hand, we find
Now, let . Then and
Since
we get
Taking into account
by a recursive procedure, we obtain
Now we can formulate the results.
Theorem 2.7. Let and . If with , then there is a constant dependent only on and such that for
holds.
Proof. We put , . From Remark 2.1(i), (2.64), (2.71), Proposition 2.6, and (2.61), we get for every and
Theorem 2.8. Let and . If with , then there is a constant dependent only on and such that for
holds.
Proof. Let be the best approximating polynomial of and let . Then,
By Lemma 2.4 we have
Since
we get
Fractional Bernstein inequality of Lemma 2.2 gives
Hence,
It is easily seen that
where
Therefore,
If we choose , then
Last two inequalities complete the proof.
From Theorems 2.7 and 2.8 we have the following corollaries.
Corollary 2.9. Let and . If with and
then
hold.
Definition 2.10. Let and . If and then for we set .
Corollary 2.11. Let and . If , , and , then .
Corollary 2.12. Let and let , , where . Then the following conditions are equivalent:
Theorem 2.13. Let , , where . If and
then
hold where the constant is dependent only on and .
Proof of Theorem 2.13. The condition (2.98) and Lemma 2.3 implies that exist and . Since
we have for
On the other hand, we find
and Theorem 2.13 is proved.
As a corollary of Theorems 2.7, 2.8, and 2.13 we have the following.
Corollary 2.14. Let , , , and
for some . In this case for , there exists a constant dependent only on , , and such that
hold.
3. Near Best Approximants in Weighted Smirnov-Orlicz Space
Let and be the conformal mappings of and onto the complement of , normalized by the conditions
respectively. We denote by and the inverse mappings of and , respectively, and . These mappings and have in some deleted neighborhood of the representations
Therefore, the functions
are analytic in and have, respectively, simple zero and zero of order 2 at . Hence they have expansions
where and are, respectively, Faber Polynomials of degree for continuums and , with the integral representations [38, pp. 35, 255]
We put
and correspond the series
with the function , that is,
This series is called the Faber-Laurent series of the function and the coefficients and are said to be the Faber-Laurent coefficients of . For further information about the Faber polynomials and Faber Laurent series, we refer to monographs [39, Chapter I, Section 6], [40, Chapter II], and [38].
It is well known that, using the Faber polynomials, approximating polynomials can be constructed [3]. The interpolating polynomials can also be used for this aim. In their work [41] under the assumption , , Shen and Zhong obtain a series of interpolation nodes in and show that interpolating polynomials and best approximating polynomial in , , have the same order of convergence. In [42] considering and choosing the interpolation nodes as the zeros of the Faber polynomials, Zhu obtain similar result.
In the above-cited works, does not admit corners, whereas many domains in the complex plain may have corners. When is a piecewise Vanishing Rotation curve [43] Zhong and Zhu show that the interpolating polynomials based on the zeros of the Faber polynomials converge to in the , norm.
A function is called a weight on , if is measurable and has measure zero. We denote by the linear space of Lebesgue measurable functions satisfying the condition
for some .
The space becomes a Banach space with the Orlicz norm
where is the complementary function of and
The Banach space is called weighted Orlicz space on .
For and let . For fixed , the set of all weights satisfying the relation
is denoted by .
We denote by the set of all measurable functions such that belongs to Lebesgue space , , on .
Definition 3.1. Let be a weight on and let , , . The classes of functions and will be called weighted Smirnov-Orlicz classes with respect to domains and , respectively.
In this chapter, we prove that the convergence rate of the interpolating polynomials based on the zeros of the is the same with the best approximating algebraic polynomials in the weighted Smirnov-Orlicz class under the assumption that is a closed Radon curve. This means that interpolating polynomials based on the zeros of the Faber polynomials are near best approximant of belonging to weighted Smirnov-Orlicz class .
In the case that all of the zeros of the Faber polynomial are in , we denote by the interpolating polynomial for based on the zeros of .
Let . Then the functions and defined by
are analytic in and , respectively, and .
We denote by
the minimal error of approximation by polynomials of , where is the set of algebraic polynomials of degree not greater than .
Let be a rectifiable Jordan curve, , and let
be Cauchy’s singular integral of at the point . The linear operator is called the Cauchy singular operator.
If one of the functions or has the nontangential limits a.e. on , then exists a.e. on and also the other one has the nontangential limits a.e. on . Conversely, if exists a.e. on , then both functions and have the nontangential limits a.e. on . In both cases, the formulae
hold, and hence
holds a.e. on (see, e.g., [1, page 431]).
Lemma 3.2. f is a regular curve, and , then for every one has
where the constant depends only on and .
Proof. Assertion (3.20) immediately follows from modular inequality
given in (7.5.13) of [23].
Theorem 3.3. If is a closed Radon curve, and , then for every one has
where the constant depends only on and .
Proof. First of all we know [16] that all zeros of the Faber polynomials are in . Since interpolating operator is linear and corresponds by a polynomial of degree not more than , we need only to show that, for large values of , is uniformly bounded in weighted Smirnov-Orlicz class . We suppose that is the best approximating algebraic polynomial for in . In this case we have
Since we assumed the interpolation nodes as the zeros of the Faber polynomials , using [39, page 59], we have
and consequently
By Lemma 3.2, we get
We set , where is the exterior angle of the point . By the Radon assumption on we get . Then one can find for
and therefore
From the last inequality we obtain
Since
we obtain that is uniformly bounded in , namely,
Therefore, we conclude that
and interpolating polynomial is near best approximant for .
If is Dini-smooth, then [44] there exist constants and such that
Similar inequalities hold also for and , in case of and , respectively.
We define Poisson polynomial for function
Theorem 3.4. If is a Dini-smooth curve, and , then for every one has
where the constant depends only on and .
Proof. From (3.8) and (3.5), we have
where and
If is near best approximant for , we get
Using
we find
Taking in the last inequality, the nontangential boundary values from inside of , and using (3.18), we have
Since is analytic in , we have
and taking nontangential limit in (3.42) we get
and hence by transformation we obtain
Since one has
we can write
From equality
we have
On the other hand,
We denote by a subarc of with the center such that it has arc lenght . In this case
and, by (1.3),
Hence,
Inequalities (3.46), (3.48), and (3.52) imply that
For every , one has
and therefore we get the required inequality of Theorem 3.4.
Theorem 3.4 signifies that Poisson polynomial is near best approximant for .
For , we set
If and , then by Theorem A(ii) we have
and consequently for any .
Definition 3.5. Let , , and . The function
is called th modulus of smoothness of .
It can easily be verified that the function is continuous, nonnegative, subadditive and satisfy for .
Let be a Dini-smooth curve and be a weight on . We associate with the following two weights defined on by
and let for . Then from (3.33), we have and for . Using the nontangential boundary values of and on , we define
for .
We set
where , , and is the set of rational functions of the form .
Now we can give several applications of approximation theorems of Section 2.
Theorem 3.6. Let be a Dini-smooth curve, and with . Then there is a constant such that for any natural number
where and is the th partial sum of the Faber-Laurent series of .
Corollary 3.7. Let be a Dini-smooth curve, and with . Then there is a constant such that for every natural number
where is the th partial sum of the Faber series of .
Corollary 3.8. Let be a Dini-smooth curve, and with . Then there is a constant such that for every natural number
where is as in Theorem 3.6.
Theorem 3.9. Let be a Dini-smooth curve, and with . Then for there exists a constant such that
hold.
Corollary 3.10. Under the conditions of Corollary 3.7, if
then for and
Definition 3.11. Let and . If , then for we set
Corollary 3.12. Let and . If , , and , then .
By Corollaries 3.7 and 3.10 we have the constructive characterization of the class .
Corollary 3.13. Let and , where , be fulfilled. Then the following conditions are equivalent:(a). (b).
The inverse theorem for unbounded domains has the following form.
Theorem 3.14. Let be a Dini-smooth curve, and with . Then there is a constant such that for every natural number
holds.
By the similar way to that of , we obtain the following corollaries.
Corollary 3.15. Under the conditions of Corollary 3.8, if
then for and
Corollary 3.16. Under the conditions of Theorem 3.14, if
then .
By Corollaries 3.8 and 3.15, we have the following.
Corollary 3.17. Let and the conditions of Theorem 3.14 be fulfilled. Then the following conditions are equivalent,(a),
(b).
Before the proofs, we need some auxiliary lemmas.
Lemma 3.18. Let be a Dini-smooth curve, and with . Then, and for every .
Proof. Using , we can find a such that , where the inclusion maps being continuous (see, e.g., Lemma 2.13 of [20]). Since by [9], we get and . Using and boundedness of operator in , we obtain from (3.18) that
Lemma 3.19. Let and . Then there exists a constant such that for every natural number
where and is th partial sum of the Taylor series of at the origin.
Proof. Using Theorem 2.7 this lemma can be proved by the same method of Theorem 3 of [45].
Let be the set of all polynomials (with no restrictions on the degree), and let be the set of traces of members of on . We define the operators and defined on as
Then it is readily seen that
If , then
which, by (3.18), implies that
a.e. on .
Similarly taking from outside of the nontangential limit in the relation
we get
a.e. on .
Since is bounded in , we have the following result.
Lemma 3.20. Let be a Dini-smooth curve, and with . Then the linear operators
are bounded.
The set of trigonometric polynomials is dense in , which implies density of the algebraic polynomials in . Consequently, from Lemma 3.20, we can extend the operators and from to the spaces and as linear and bounded operators, respectively, and for the extensions and , we have the representations
Lemma 3.21. Let and with . Then,
where
and is the Poisson kernel.
Proof. There are numbers and such that
Since [46, Theorem 10] is a bounded operator in for every , we have by Marcinkiewicz Interpolation Theorem
From density of trigonometric polynomials in , we have density of the set of continuous functions on in . Consequently, there is a continuous function on such that, for given and ,
On the other hand, since the Poisson integral of a continuous function converges to it uniformly on [47, page 239], we have by (2.7) and
for . Then, from (3.85), (3.86), and (3.87), we conclude that
This completes the proof.
Theorem 3.22. Let be a Dini-smooth curve, and with . Then the linear operators
are one-to-one and onto.
Proof. The proof we give, only for the operator . For the operator is the proof goes similarly. Let with the Taylor expansion
Since is a Dini-smooth curve, the conditions , and are equivalent.
Let . Since is the Poisson integral of its boundary function [48, page 41], we have
and using Lemma 3.21, we get , as .
Therefore, the boundedness of the operator implies that
Since is uniformly convergent on , one has
From the last equality and Lemma 3 of [39, page 43] we have
and therefore
On the other hand, applying (3.33), (2.7), and weighted version of Hölder’s inequality
we obtain
because .
Using here the relation (3.92), we get
and then by (3.95), for . If , then , and therefore . This means that the operator is one-to-one.
Now we take a function and consider the function . The Cauchy type integral
represents analytic functions and in and , respectively. Since , by Lemma 3.18, we have
and moreover
a.e. on . Since and , we have
which proves that the coefficients , also become the Taylor coefficients of the function at the origin, that is,
and also
Hence the functions and have the same Faber coefficients , and therefore . This proves that the operator is onto.
Proof of Theorem 3.6. We prove that the rational function
satisfies the required inequality of Theorem 3.6. This inequality is true if we can show that