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Journal of Mathematics
Volume 2013 (2013), Article ID 467351, 5 pages
Interpolation Polynomials of Entire Functions for Erdös-Type Weights
1Science Division, Center for General Education, Aichi Institute of Technology, Yakusa-cho, Toyota 470-0392, Japan
2Department of Mathematics, Meijo University, Nagoya 468-8052, Japan
Received 20 September 2012; Accepted 27 March 2013
Academic Editor: Markos Koutras
Copyright © 2013 Gou Nakamura et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let , and let be an even function. In this paper, we consider some Lagrange interpolation polynomials and the Gauss-Jacobi quadrature formula of entire functions associated with Erdös-type weights , , and we will estimate the error terms.
For a weight , we suppose that for all . Then we define a unique sequence of orthonormal polynomials as follows: where is a polynomial of degree . Furthermore, we denote the zeros of by , . Hence, For a given function , the Lagrange interpolation polynomial with the weight is defined to be a unique polynomial of degree at most which coincides with at , . Then we have the representation where The Gauss-Jacobi quadrature formula to the function is defined by where the coefficients are the Christoffel numbers, which are given by
We denote by a class of all weight functions of the form such that is an even differentiable function on , except possibly at , increasing for , and there exists such that is increasing. In addition, the unique positive sequence determined by satisfies the condition , for some constant .
Theorem 1. Let and be an entire function.(1)There exists a constant , depending on only, such that if satisfies then, uniformly on compact subsets of , (2)If the entire function satisfies then, uniformly on compact subsets of ,
In this paper, we extend the above results to Erdös-type weights. Note that every weight in is not Erdös-type (see Remark 4). In Section 2, we recall the definition of , which is the class of weights we discuss here. Our theorems are stated in Section 3.
For any nonzero real valued functions and , we write if there exists a constant such that for all . Throughout this paper, denotes a positive constant independent of . The same symbol does not necessarily denote the same constant in different occurrences. We say that is quasi-increasing if there exists such that .
First, we recall a class of weights.
Definition 2 (see cf. [6, Definition 1.2]). Let be a continuous even function. We write if satisfies the following properties.(a) is continuous in , with . (b) is nondecreasing in . (c). (d)The function , , is quasi-increasing in and , . (e)There exists such that for , . (f)For any , there exists such that for any ,
Example 3. Let .(1)If is bounded, then the weight is called a Freud-type weight (see ). We see that , , is a Freud-type weight.(2)If is unbounded, then is called an Erdös-type weight. The following s give Erdös-type weights (see [7, Example 1.2], [8, Theorem 3.1]). Let for . Then for , , and , we set Also, for , put .
Remark 4. Every weight in is Freud type. In fact, by definition, . On the other hand, if we assume that is Erdös type, then for any given , there exists a constant such that for . By integration, we see for . Since , ; that is, . If we take sufficiently large, this contradicts the fact that (cf. [7, Lemma 3.2]).
For , we define the Mhaskar-Rakhmanov-Saff number by
For an entire function and , we set Then by the residue theorem, for , we have where is a simply connected domain containing all zeros of and is its boundary. Note that for any , there exists such that . For each we also set Then is a polynomial of degree at most such that holds. By the residue theorem again, we see for .
3. Theorems and Proofs
We state our theorems. Throughout this section, we assume , and let Note that is determined finitely by Lemma 5(1).
Theorem 6. (1) If an entire function satisfies
uniformly on compact subsets; that is, for any compact subset in , there exists a constant such that
for . Moreover, for any ,
holds also uniformly on compact subsets.
(2) If an entire function satisfies then for any , uniformly on compact subsets. Moreover, for any , holds also uniformly on compact subsets.
As for the Gauss-Jacobi quadrature, we have the following theorem.
Theorem 7. Let be an entire function. If holds for some large enough, then for a constant . Further, if holds, then for any ,
Theorem 8. Let be an entire function. Suppose that for every large enough. Then holds uniformly on compact subsets of and for any , also holds uniformly on compact subsets of . Moreover for any , holds true.
Remark 9. The order of an entire function is given by
Also the type of with order is given by
If is an Erdös-type weight and , then (35) holds for every . In fact, by (14),
The following lemma was proved in [2, Lemmas 3.3 and 3.4]. Its proof is available for our case.
Lemma 10. Let , and let be the orthonormal polynomial generated by . Then for all such that , and for all such that . Also we have
Proof of Theorem 6. (1) We estimate the error form given by the formula (17). For a given compact subset , choose a number such that for . We may take the path of integration to be the circle because . By (42), we see . From the definition of and (24), for any , there exists such that for ,
Hence by (43), for large enough,
Here we select such as
Then we have for and any ,
Since , we conclude (26). In the above proof, we only exchange (48) for
Then (27) follows.
(2) Since where , we see . Hence, (28) gives us Thus, (29) and (30) follow from (26) and (27), respectively.
Proof of Theorem 7. First, we show the case . From Lemma 5(3) and (44), we have where is an absolutely constant. Also by Lemma 5(3), we see Now let . Since (see [3, page 131]), (52) gives us Therefore, for , we conclude that We consider the circle , where we will choose large enough later. Now we estimate From Lemma 5(1), (53), (56), and our assumption (31), we have for small enough, Here we select as Then we have which shows that For the case of , as in the proof of (2) of Theorem 6, (33) gives us Hence, (34) follows from (32).
Proof of Theorem 8. In (17), we take the path of integration to be the circle . Then we have In this right-hand side, we can take as large as we like because of our assumption (35). Hence, we have (36). Similarly as above, (37) follows from (36). Also Theorem 7 shows (38).
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