1. Introduction

Approximation problem for the classes of functions determined only on the boundary of domain is of great importance alongside with the study of approximation of functions by means of polynomials analytic in the domain and with some conditions on the boundary . Obviously, it is impossible in general to approximate such classes of functions by means of polynomials[12]. Therefore, various kinds of rational functions or so called generalized polynomials are mostly used in this case as an approximation tool[12]. J. I. Mamedkhanov, D. M. Israfilov and I. M. Botchaev investigated the approximation problems of functions determined only on the boundary of domain by means of rational functions of the form for certain classes of curves in terms of uniform metric[1-4].

In this paper, we study the approximation problems of a function from the class by means of a rational function of the form

(1)

where is a point lying strictly inside the considered curve ¹. Without loss of generality, we will assume throughout this paper.

2. Basic Definitions and Notations

1. Let be an almost everywhere finite, non-zero

function measurable on . If the function determined on is measurable, and the function the class . If we define the norm in the class

as , then is integrable on , then we will say that belongs to becomes a Banach space .

2. By we denote the Cauchy singular integral.

3. Denote by a totality of curves for which , where is depending on the point .

Notice that the classes and are equivalent[5].

4. Let be a measurable, almost everywhere finite function on . We will say that belongs to the class if it differs from zero almost everywhere and the operator is bounded in the space . Notice that this class is well studied in the theory of singular operators[5].

5. We will say that the closed curve belongs to the class if where

²

6. We will say that the curve belongs to the class of -curves if the length of the greatest of the arches connecting two arbitrary points and on it has the same order as that of the chord connecting these points[10-11].

Obviously, .

A lot of works have been devoted to the class of curves over the past years[5]. Finally, G. David[6] proved that .

7. By we will denote a function that conformally and univalently maps the exterior (interior) of the curve onto the exterior (interior) of a unit circle normalized by the conditions:

,

and let denote the inverse to function to is a level line of the curve ; for ; for .

8. Let’s consider the following quantities (see[7]):

where .

Obviously, these quantities satisfy all the properties of smoothness modulus on good classes of curves, in particular, on smooth or piecewise-smooth curves. In case of more general classes of curves we will consider the quantitiesand

Proceeding in the same way as in[7], it is easy to see that these quantities satisfy all the properties of smoothness modulus (i.e. , on any curves and they are the best majorants for the functions and , respectively, among all smoothness modulus type functions.

3. Main Result

Now we prove the following:

Theorem. Let and . Then, for every positive integer there exists a rational function of the form (1) such that

.³

4. Proof

Obviously, and imply that . Now, by virtue of we can state that

exists almost everywhere on . It follows that[8], the Cauchy type integral has certain angular values almost everywhere on equal to

.And this in turn implies that

This relation shows that in order to approximate the function given only on the curve in terms of the metric of the space it suffices to approximate the functions and which are analytic inside and outside the given closed curve, respectively, and belong to .

So let us prove that for every positive integer there exist the polynomials and of degree such that

(3)

and

(4)

First we will prove the validity of relation (3). From and it follows that if

,

then . This allows to state that the singular integral exists almost everywhere on in the sense of principal value. The last statement enables us to approximate the function by the Jackson-Dzyadyk polynomials[9] represented in the following form:

(5)

where and is a kernel that represents a trigonometric polynomial of at most -th degree and satisfies the conditions

(6)

(7)

(8)

Furthermore, from fulfillment of conditions (6) - (8) we directly get

(9)

Now, taking into account relation (6), we represent the function as follows:

(10)

Further, by virtue of relations (5) and (10), we estimate the difference in the sense of metric.

Obviously, we have

We apply Minkovski inequality, and then Minkovski’s generalized inequality to find that

(11)

where

As , the latter relation yields

It follows from (11) that

(12)

Therefore, using classical technique we get (3).

To complete the proof, we only have to show the validity of relation (4). To this end, we map the plane onto the plane by means of the function

(13)

Obviously, the contour is mapped into some contour and the functions and are transformed into the functions and , respectively.

Let us prove the validity of the following statements:

(14)

and

Validity of (14) is obvious.

Further, combining relation

(16)

with relation (14) and the fact that lies strictly outside , we get relation (15).

At the same time, we notice that for is the interior of the curve ) the function on the plane takes the form

.

If we take into accountthen we get:

Obviously, the function in the plane corresponds to the function in the plane . Hence, by virtue of relation (3) we get

(17)

Further, taking into account that , and the point lies inside , we get

(18)

To complete the proof of relation (4) we use the obvious relation

(19)

Thus, from relation (17), by virtue of (18) and (19), we get the required relation (4).

Now, to complete the proof of the theorem it suffices to make use of relations (3) and (4) and for ,from whence, by virtue of relation (2), the statement of theorem follows.

Notes

1. By and we will denote an interior and an exterior of the curve , respectively.

2. denotes various positive constants depending only on explicit parameters. In addition, we will use the notations and, if .

3. Signs ≼ and ≍ define an ordinal relation. Namely, means . And means .