SOME PROPERTIES OF WEIGHTED HYPERBOLIC POLYNOMIALS

A new notion of weighted hyperbolic polynomials is introduced and their properties are discussed in this paper.


INTRODUCTION.
Theorems in approximation theory have been studied by a lot of researchers.In [1,2,3,4,5,6,7,  8,12,13] the cases for the weighted polynomials (shorty w-polynomials) in the complex plane were discussed, and in [9,10] the cases for ones in locally compact spaces are also considered.
In the note we will introduce the definition of weighted hyperbolic polynomials (shortly w.h- polynomials) in the unit disk and discuss their properties.Almost all of the theorems with respect to w-h-polynomials in the unit disk are proved by a simple modification or the methods analogous to the cases for weighted ones in the complex plane or locally compact spaces.Therefore we omit the proofbut a few exceptions.
The introduction ofthe definition of w-h-polynomials gives the following advantages: (1) Theorems (propositions) on w,.h-polynomials can be easily obtained in the form similar to ones on usual algebraic polynomials.An example asserts that w-h-capacity, transfinite diameter and Chebyshev constant are equal to each other.
(2) The introduction makes it easy to verify or apply theorems.For example, the sharpness of inequalities can be easily verified.The property will be shown for the case of the finite-infinite-range inequality.
The outline of the note is as follows.In Section 2, definitions of w-h-capacity, transfinite diameter, Chebyshev constant and polynomials are introduced.Furthermore we discuss the definitions of normalized counting measure of zeros of w.h-polynomials and the weak convergence.
In Section 3 we show that some propositions on w-h-polynomials have the same form as the usual ones.The sharpness of some inequality is also discussed In Section 4 a theorem on the normalized counting measure on zeros of w.h-polynomials is presented.We then give a part of proof as the application of proposition in Section 3, which shows that the introduction ofthe definition of w-h-polynomials are useful for the application.2.-DEFINITIONS Let E be a compact set in the unit disk U {]z] < 1} and D be a domain whose boundary consists of 70 { Izl 1} and OE, where OE is the outer boundary of E. The positive, bounded weight function w(z) and E satisfy each of the following assumptions: (1) log w is a continuous function on U.
(2) E has positive w.h-capacity.At first we state the definitions of w-h-capacity and transfinite diameter which is a translation [6].
Let M(E) denote the class ofall positive unit Borel measures whose support is contained in E. Also, for any function f E C, we set IlfllE sp If(z)l.
Next, we introduce a new notion of w-h-polynomials.For each integer n >
Corresponding to the definition of the w-Chebyshev constant introduced in [6], the (new) modified Chebyshev constant is defined by Chh(w,E) lim a,(w,E) 1/'' (2.10)We remark that Mhaskar and Saff have studied the weighted polynomials The notion of the w-h-polynomials is motivated from (2.5), that is, "the w.h-distance between the points zi and zj" must be z,-z w(z,)w(z) and moreover w-h-capacity, transfinite diameter and Chebyshev constant must be equal to each other.
For w-h-polynomials p,,w(z) of degree n, the discrete unit measure defined on compact sets in U with mass (l/n) at each zero of pn.w(z) will be denoted by #,,, tt(p,.w).It will be called the normalized counting measure on the zeros of p,,,,,(z).If pr,,w(Z) has multiple zeros, the obvious modification will be made.
The weak convergence of Un to u as noo is defined by lim f fdu.= f fau (2.11) for every continuous function in the complex plane C with compact support 11 ].
Note that E is ofpositive capacity at each point ifcap({ E E; Iz ffl < 5}) > 0 for every z E E and every 5 > 0. 3.

PROPOSITIONS
As an example, the proposition for the weighted hyperbolic case has the same form as the classical one, we will show the following with respect to w-h-capacity, transfinite diameter and Chebyshev constant.It will be verified by a simple modification ofresults in [6,9].Therefore we omit the proof PROPOSITION 3.1.There hold the equalities Cph(w, E) Trh(w, E) Chh(w, E) The next proposition is of great importance for the study of weighted polynomial approximation theory, which is essentially the same as [6,9].However, we prefer the form different from [6,9], because it is simple and makes it easy to verify the sharpness of some inequality.Furthermore, the form is useful for the application, which will be shown in the next section.for any z in D, where II,=.(z/ll.s the smallest number that is an upper bound for Ip=.(z)l q.e.S,o.q.e. on S,o means that it holds on S if the subset S' ofS where it does not hold is capacity zero.
Since the proof is analogous to that in [6,9], so we omit it.The sharpness of the inequality is easily shown from the well-known fact as follows: For the with the w-h-Fekete points {z'},=, {g,,,o (z)}, and (llp.(/II .} converges locally uniformly to O,(z) in D, and Cph(w, E) respectively.Proposition 3.2 can be restated in the following, which is called a finite-infinite-range inequality.
Considering the above Proposition 3.3, it can be easily shown that c; 1 and c log where h,o(z) exp{/log[.z-t[w(z)w(t)]dv(t)}(3.7) and , is the extremal measure for w-capacity. 4.

APPLICATIONS
Proposition 3.2 in the previous section plays an important role in many theorems.In this section we discuss an example that the proposition can be applied for the proofof some theorems.
We consider the case where lim sup Ip=,(z)l 1/" _< a(z) (4.1) holds for q.. z (5 Sw.This follows from the equality gw(z) Cph(w, E) for q.e.z S,, which was shown in [6,9] for the case of weighted capacity.Since the proof is similar, we omit it.
Combining (4.6) and (4.8), we also obtain Since we intend to show how Proposition 3.2 is applied for the proof, the other part of the proof is omitted.The details will appear in a future paper.
The theorem also shows an example that the theorem on usual algebraic polynomials is translated to one on w.h-polynomials.
We next give a well-known theorem of w-h-polynomials which satisfies the condition of Theorem  Hn [( z-z'l z )w(z)w(z)l (4.9) =1