Research Article | Open Access
K. Itoh, R. Sakai, N. Suzuki, "The de la Vallée Poussin Mean and Polynomial Approximation for Exponential Weight", International Journal of Analysis, vol. 2015, Article ID 706930, 8 pages, 2015. https://doi.org/10.1155/2015/706930
The de la Vallée Poussin Mean and Polynomial Approximation for Exponential Weight
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 .
Let and = . We consider an exponential weighton , 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 byIf 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 thatFor , we denote by the usual space on . For a function with , we setfor (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 satisfiesfor some . Then there exists a constant such that when , thenand 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  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 (, 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 setwhere (-times). Then is an Erdős-type weight, which belongs to with (see ).
We recall some notation which we use later (cf. ). By definition, the partial sum of Fourier series is given bywhereIt is known that, by the Christoffel-Darboux formula,where is the leading coefficient of ; that is, .
The Christoffel function is defined byand type Christoffel function is defined byIf , 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
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 thatand henceholds 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 thatfor every .
Lemma 6 (see [10, Theorem 3.2]). If is an Erdős-type weight, then for any there exists a constant such thatfor every .
Lemma 7 (see [9, Lemma 3.5(a), (b) and Lemma ]). Let . There exists a constant such thatfor every .
4. Proof of Theorem 1 for
We begin with the following proposition.
Proposition 8. Let . Then there exists a constant such thatfor every and every .
Theorem 9. Let . Then there exists a constant such thatfor any and .
Proof. Let and with . We setwhere is the characteristic function of a set . ThenWe first consider an estimate of . By the Schwarz inequalityand since (26) and (30) give us so thatNext, for an estimate of , we setand denote by the Fourier coefficients of ; that is,Using the Christoffel-Darboux formula (19), we haveBy (24) and the Schwarz inequality, we have Since (26) and (30) implyThis 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 thatfor any .
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 usThis together with (38) and (7) impliesSince as , we may assume thatholds for every if we take larger as needed, where is the constant in Lemma 5. Thenand 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 impliesNext, we estimate . Sinceas before, where , Proposition 8 gives usBy Lemma 5, we can take small enough such that for . Then we have Also since and , we haveThese 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 seeand henceTherefore, using Corollary 10, we haveSince the operator normsfor and are bounded, the Riesz-Thorin interpolation theorem gives usfor 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 operatorwe havefor every . This implies (9).
We begin with the following corollary.
Corollary 12. Let and satisfy (7). Then there exists a constant such that, for every and ,
To discuss polynomial approximations, we define the degree of weighted polynomial approximation for byWe quote two results from our previous papers.
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