The de la Vallée Poussin Mean and Polynomial Approximation for Exponential Weight

Abstract

We study boundedness of the de la Vallée Poussin means for exponential weight on . Our main result is for every and every , where . As an application, we obtain for .

1. Introduction

Let and = . We consider an exponential weight on , where is an even and nonnegative function on . Throughout this paper we always assume that belongs to a relevant class (see Section 2). We consider a function defined by If is bounded, then is called a Freud-type weight, and otherwise, is called an Erdős-type weight.

Let . By definition, , so that is an increasing and continuous function on with and . By the intermediate value theorem, there exists a unique such that for every ; that is, holds. This is called the Mhaskar-Rakhmanov-Saff number (MRS number). In this sequel, notation always stands for the MRS number for a weight defined by (3) (see [1, page 11], [2, page 180]).

Let be orthogonal polynomials for a weight ; that is, is the polynomial of degree such that For , we denote by the usual space on . For a function with , we set for (the partial sum of Fourier series). The de la Vallée Poussin mean of is defined by

The main result of this paper is the following theorem.

Theorem 1. Let . We assume that satisfies for some . Then there exists a constant such that when , then and when , then

For Freud-type weights, the following inequality is known: (cf. [3, Lemma 6], [2, Theorem ]). Note that when is bounded, then (7) holds evidently, so (10) follows from (8) or (9) immediately. As for Erdős-type weights, Lubinsky and Mthembu [4] proved (8) for , (9) for , and for under the assumption for every . (They discussed on Cesaro means ( means) mainly, but it easily ensures the result on de la Vallée Poussin means.) We note that their assumption implies our condition (7) (see Remark 16 in Section 7). Hence our results (8) and (9) improve their result. For more general weights, see [5–7].

Using (9), we have the following result. There exists a constant such that, for any , where is the set of all polynomials of degree at most . It is known that the right-hand side of (11) attains some ; however, it is not easy to determine such explicitly. Equation (11) means that the de la Vallée Poussin mean takes the place of in some sense; that is, is a good concrete approximation polynomial for given function .

This paper is organized as follows. The definition of class is given in Section 2. Recalling some estimates for exponential weights in Section 3, we will give a proof of Theorem 1 for the case of in Section 4. Basic method of our proof is classic and well known ([3], see also [4, 6]), but we repeat it in order to make a role of clear. The complete proof of Theorem 1 is given in Section 5. The estimate (11) is shown in Section 6 together with another application. We discuss the condition (7) in Section 7.

Throughout this paper will denote a positive constant whose value is not necessarily the same at each occurrence; it may vary even within a line. When we write , then is a constant which depends on only.

2. Definitions and Notation

We say that an exponential weight belongs to class , when is a continuous and even function and satisfies the following conditions.(a) is continuous in with .(b) exists and is positive in .(c).(d)The function defined in (2) is quasi-increasing in (i.e., there exists such that whenever ), and there exists such that (e)There exists such that  and there also exist a compact subinterval of and such that

Let . Suppose that and . If there exist and such that for , then we write . Note that if is a Freud-type, the last inequality holds with .

A typical example of Freud-type weight is with . Note that the Hermite polynomials are the orthogonal polynomials for the weight . Let ,  ,  , and  , and we set where (-times). Then is an Erdős-type weight, which belongs to with (see [8]).

We recall some notation which we use later (cf. [9]). By definition, the partial sum of Fourier series is given by where It is known that, by the Christoffel-Darboux formula, where is the leading coefficient of ; that is, .

The Christoffel function is defined by and type Christoffel function is defined by If , we know

The MRS numbers for weight are monotonically increasing, and we see easily Moreover, for , there exists independent of , such that (see [9, Lemma 13.9]). We also use the following functions: where and

3. Lemmas

We recall some basic estimates.

Lemma 2 (infinite-finite range inequality [9, Theorem 1.9(a)]). Let , and . Then

Lemma 3 (see [10, Lemma 3.4]). Let . There exists a constant such that and hence holds for every and .

Lemma 4 (see [9, Theorem 9.3]). Let , and . Then there exists a constant such that, for every and , and when , one has

Lemma 5. Let . Then there exist and such that for every .

Proof. It is known that there exist and such that for every [9, Theorem 3.2(e)]. If , then and hence (32) holds evidently. If , then Since is quasi-increasing, (33) shows (32).

Lemma 6 (see [10, Theorem 3.2]). If is an Erdős-type weight, then for any there exists a constant such that for every .

Lemma 7 (see [9, Lemma 3.5(a), (b) and Lemma ]). Let . There exists a constant such that for every .

4. Proof of Theorem 1 for

We begin with the following proposition.

Proposition 8. Let . Then there exists a constant such that for every and every .

Proof. From Lemmas 3 and 4 for , the proposition follows immediately.

The following estimate is known essentially (see [6, Theorem 4], [7, Theorem 1.2] and [5, Lemma 1]), but we give a proof for the sake of completeness.

Theorem 9. Let . Then there exists a constant such that for any and .

Proof. Let and with . We set where is the characteristic function of a set . Then We first consider an estimate of . By the Schwarz inequality and since (26) and (30) give us so that Next, for an estimate of , we set and denote by the Fourier coefficients of ; that is, Using the Christoffel-Darboux formula (19), we have By (24) and the Schwarz inequality, we have Since (26) and (30) imply This together with (47) gives us which completes the proof of (41).

The following corollary contains (9) for .

Corollary 10. Let . Then there exists a constant such that for any .

Proof. From Theorem 9 and Lemma 3, this corollary follows immediately.

We give a proof of (8) for .

Theorem 11. Let such that with some . Then there exists a constant such that if , one has

Proof. Let and such that . As in the proof of Theorem 9, we set ,  . Then . Since , by Lemma 2, we may suppose that .
We first prove that, for any , whenever and . In fact, we now denote by a constant in Lemma 7. Since is quasi-increasing, there exists a constant such that whenever . Then by (7), there exists such that holds for every , so that Lemma 7 gives us This together with (38) and (7) implies Since as , we may assume that holds for every if we take larger as needed, where is the constant in Lemma 5. Then and hence Lemma 5 gives us When , then for , so that (57) holds immediately.
Now we estimate . By the Schwarz inequality, (57), and Proposition 8, which implies Next, we estimate . Since as before, where , Proposition 8 gives us By Lemma 5, we can take small enough such that for . Then we have Also since and , we have These estimates show and hence follows. This together with (64) implies (56).

5. Proof of Theorem 1 for

In this section we complete the proof of Theorem 1.

Proof of (8). The -norm case is Theorem 11. We prove the -norm case. By the duality of -norm, Since , we see and hence Therefore, using Corollary 10, we have Since the operator norms for and are bounded, the Riesz-Thorin interpolation theorem gives us for every with . This implies (8).

Proof of (9). The -norm case is Corollary 10. Now, we show -norm case. Similar as above, Since by (56), we see Hence by the Riesz-Thorin interpolation theorem for the operator we have for every . This implies (9).

6. Corollaries

We begin with the following corollary.

Corollary 12. Let and satisfy (7). Then there exists a constant such that, for every and ,

Proof. By Lemma 2 and (9) in Theorem 1, holds for every .

To discuss polynomial approximations, we define the degree of weighted polynomial approximation for by We quote two results from our previous papers.

Proposition 13 (see [11, Theorem 1] and [10, Theorem 6.1]). Let and . Then there exists a constant such that for every , if is absolutely continuous and , then and if , then holds true.

Corollary 14. Let and satisfy (7). Then there exists a constant such that, for every and every , and when , Moreover if is absolutely continuous and , then

Proof. Since for every , we have , and hence (9) gives us which shows (87). By using (82) and (8), we also obtain (88) and (89), respectively. Favard-type inequality (90) follows from (85) and (87).

Combining (8) with (86), we obtain the following estimate of the derivative of .

Corollary 15. Let and satisfy (7). Then there exists a constant such that for every

7. Remarks

In this final section we make three remarks. First one is concerning the condition (7) in Theorem 1.

Remark 16. Let .
(1) Let . If satisfies for some . Then the condition (7) holds true. In particular, all satisfy (7).
(2) If for some , then (93) holds for some . In particular, all the weights discussed in [4] satisfy (7).
In fact, if is Freud-type, then (7) holds clearly, so that we may assume that is Erdős-type. In [10, Lemma 3.2], we showed that when satisfies (93), then holds for some . Hence this and (36) imply (7). Since , satisfies (93).
(3) The condition (7) is equivalent to with some constant , because (see [1, Lemma ]).

Next we remark on the degree of weighted polynomial approximation .

Remark 17. Let . It is known that if , then for (when we further assume that (e.g., [1, Theorem 1.4]). Hence (87) implies This is a concrete polynomial approximation for a given function . Similar argument can not apply to (89), because may not belong to . To overcome this difficulty, we use a mollification of a weight ([10, Theorem 4.1]): let with . Then we can construct a new weight which satisfies , , and , where and are the MRS number and a function defined (2) with respect to , respectively. Hence and (89) show that if with and if , then (when we further assume that .

Remark 18. Equation (92) suggests that the following inequality would be true: We will discuss this estimate elsewhere.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.