Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights
Hee Sun Jung,1Gou Nakamura,2Ryozi Sakai,3and Noriaki Suzuki3
Academic Editor: W. Yu, B. Ricceri
Received30 Dec 2011
Accepted26 Feb 2012
Published16 May 2012
Abstract
Let and let , where and is an even function. Then we can construct the orthonormal polynomials of degree for . In this paper for an even integer
we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejér interpolation polynomials and related approximation process based at the zeros of . Moreover, for an odd integer , we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros of .
1. Introduction
Let , and let be an even function. Consider the weight , and define, for ,
Suppose that , for all . Then we can construct the orthonormal polynomials of degree for ; that is,
We write by
and denote the zeros of by
Let denote the class of polynomials with degree at most . For we define the higher-order Hermite-Fejér interpolation polynomial based at the zeros as follows:
We note that is the Lagrange interpolation polynomial, is the ordinary Hermite-Fejér interpolation polynomial, and is the Krylov-Stayermann polynomial. For the general cases Kanjin and Sakai [1, 2] started to investigate the so-called Freud-type weights. The fundamental polynomials for the higher-order Hermite-Fejér interpolation polynomial are defined as follows:
Using them, we can write
Furthermore, we extend the operator . Let be a nonnegative integer, and let . For we define the -order Hermite-Fejér interpolation polynomials as follows: for each ,
Especially, is equal to , and for each we see ; that is, for , is the Hermite interpolation polynomial. The fundamental polynomials , , of are defined by
Then we have
For the ordinary Hermite and Hermite-Fejér interpolation polynomial , and the related approximation process, Lubinsky [3] gave some interesting convergence theorems.
Our purpose in this paper is to study and as certain analogies of the Lubinsky theorems in [3] and the related approximation process for the exponential-type weights. Kasuga and Sakai [4–8] investigated the convergence and divergence theorems for the Freud-type weights. Then for an even integer we give the convergence theorems for them; moreover, for an odd integer , we obtain a certain divergence theorem with respect to . In Section 1, we give the preliminaries for these studies, and in Section 2 we write some preliminary description. In Section 3 we report our theorems with some lemmas, and in Section 4 we prove the theorems. Finally, in Section 5, for an odd integer we obtain a certain divergence theorem with respect to .
In what follows we abbreviate several notations as , , , and if there is no confusion. For arbitrary nonzero real-valued functions and , we write if there exist constants independent of such that for all . For arbitrary positive sequences and , we define similarly.
Throughout this paper denote positive constants independent of , or polynomials , and the same symbol does not necessarily denote the same constant in different occurrences.
2. Preliminaries
A function is said to be quasi-increasing if there exists such that for .
Definition 2.1 (see [9]). Let be a continuous even function satisfying the following properties.(a) is continuous in and . (b) exists and is positive in . (c). (d) The function
is quasi-increasing in , with for .(e) There exists such that
Then we say that is in the class . Besides, if there exist a compact subinterval of and such that
then we say that is in the class .
Example 2.2. We present some typical examples of satisfying . If a continuous exponential satisfies
with some constants , , then is called a Freud weight. The class contains the Freud weights. (See [9]) For and a nonnegative integer , we put
where, for ,
and . (See [10]) For with , we put
where for we suppose if ; otherwise. For we suppose and . (See [10]) For , we put
For , we define the Mhaskar-Rakhmanov-Saff number by the equation
We have the following estimates for the coefficients in (1.6) or (1.9).
Lemma 2.3 (see [11, Theorem 2.6]). Let . For each and , we have ,
If we consider the higher-order Hermite-Fejér interpolation polynomial on a certain finite interval, then we can see a remarkable difference between the parity of , for example, the Lagrange interpolation polynomial and the ordinary Hermite-Fejér interpolation polynomial ([12–17]). Also, we can see a similar phenomenon in the case of the infinite intervals ([1, 2, 4–8]). To describe these aspects, however, we need a further strengthened definition for than Definition 2.1.
Definition 2.4. Let , and let be an integer. Assume that is a -times continuously differentiable function on and satisfies the following.(a) exists and , , are positive for . (b)There exist constants such that(c)There exist and such that
Then we say that is in the class . (d)Suppose one of the following.(d-1) is quasi-increasing on a certain positive interval .(d-2) is nondecreasing on a certain positive interval .(d-3)There exist constants and such that on .Then one says that is in the class .
Example 2.5 (cf. [10, Theorem 3.1]). Let be a positive integer, and let be defined in (2.7).Let and be nonnegative even integers with . Then . (a)If , then we see that is quasi-increasing on a certain positive interval and is nondecreasing on . (b)If , then we see that , is nondecreasing on . Hence, . Let . Then , and one has the following. (c)If and , then there exists a constant such that is quasi-increasing on . (d)Let . If , then there exists a constant such that is quasi-increasing on , and, if , then is quasi-decreasing on . (e) Let and , then is nondecreasing on a certain positive interval on .Hence, .
Definition 2.6. One uses the following notation:
Lemma 2.7 (see [18, Corollary 4.5]). Let , . If and , then and, for ,
where
For one has
Remark 2.8. In [19, Theorem 2.2] we see that if is large enough. Therefore Lemma 2.7 holds for all , .
Levin and Lubinsky (see [9, Lemma 3.7]) showed that there exists such that for some and for large enough ,
Lemma 2.9 (see [20, Theorem 1.6]). Let . Let be unbounded. Then, for any , there exists such that, for ,Let be the constant in Definition 2.1(e), that is,
If , then there exists such that
and, if , then for any there exists such that
Remark 2.10. If is bounded, then is called a Freud-type weight, and, if is unbounded, then is called an Erdős-type weight. In (2.20) and (2.21), we set and
Then (2.17) holds. If
then we have (2.21). Note that all the examples in Example 2.5 satisfy this inequality. For the Freud-type exponent , we have
The inequality (2.18) implies
3. Theorems
In the rest of this paper we assume the following for the weight .
Assumption 3.1. Consider the weight , , . (a)(cf. [20, Theorem 1.4]) If is bounded, then we suppose, for in (2.12),
(b)There exist and such that
here, if is bounded, that is, a Freud-type weight, then we set , and if is unbounded, that is, an Erdős-type weight, then we set . Define
Remark 3.2. If is unbounded, then (3.1) holds because of Lemma 2.9 (2.21). (3.2) holds for
In (3.3) we note that as .
We shall state our theorems. Put
Furthermore, we consider the class , and construct the following interpolation polynomial:
Then we have
Define
Here we note that, for some ,
Moreover, we define
Lemma 3.3. Let . Let be fixed. Then one has the following: (a)(See [9, Lemma 3.5(a)]) Uniformly for ,(b)(See [9, Lemma 3.5(b)]) Uniformly for ,Moreover,
(c)(See [9, Lemma 3.11 (3.52)]) Uniformly for ,
(d)(See [9, Lemma 3.4 (3.17), (3.18)]) Uniformly for , one has
Lemma 3.4. For , we have
Proof. Let . By Lemma 3.3(d) we have
So, we have
Now, if , then
So, we have
Let . Then we have
Let denote a positive constant depending only on .
Assumption 3.5. Let and , . Suppose the following.(A-1)Let satisfy that, for a given ,where we suppose . (A-2)Let for a certain . Then we suppose that satisfies
(A-3) Let . Then we suppose that there exists a constant such that
Remark 3.6. In (3.22), we have the following. and
For some positive constant , we have . Hence, from (3.22), it follows that
We have a chain of results under Assumption 3.1.
Proposition 3.7. Let . For satisfying (3.26), one has
Proposition 3.8. Let . For satisfying (3.25), one has
where is defined by (3.3).
Proposition 3.9. Let . For satisfying (3.23), one has
and for satisfying (3.24) one has
Proposition 3.10. Let . Let be fixed. Then one has
Proposition 3.11. Let . Let with be fixed. Then one has
Theorem 3.12. Let . For satisfying (3.22), one has
Theorem 3.13. Let . For satisfying (3.22) and (3.23), one has
Corollary 3.14. Let . For satisfying (3.22), one has
Theorem 3.15. Let . For satisfying (3.22), (3.23) and (3.24), one has
Define
where equals to one of the following:
One also defines
Theorem 3.16. Let . Let satisfy (3.22), (3.23) and (3.24). If
then we have
For example, we can take in the following way. If is unbounded, we have, for ,
(see [10, Lemma 2.1(b)]). Then we set
If is bounded and , , then we take as so that (3.43) can hold. Then defined by (3.43) satisfies (3.40).
4. Proof of Theorems
For constants , the same symbol does not necessarily denote the same constant in different occurrences.
Lemma 4.1. One has the following. (See [19, Theorem 2.3]) Let and . Then, uniformly for one has
and for ,
(See [19, Theorem 2.5(d)]) Let and . Let and . Then
(See [19, Theorem 2.5(c)]) Let and . Then one has
(See [19, Theorem 2.5(a)]) Let and . For we have
Lemma 4.2 (see [19, Theorem 2.2]). Let and . For the zeros , one has the following: For and ,For the minimum positive zero ( is the largest integer ⩽), one hasand for large enough , (See [19, Lemma 4.7]) , where is defined by (1.3).
Lemma 4.3. Let . Then there exist such that
Proof. The first inequality follows from Lemma 3.4. We show the second inequality. Noting (4.8), from Lemma 4.1 (4.1) we have
From (4.2) we see that
Therefore we have the result.
Proof of Proposition 3.7. We recall the definition of :
Using Lemma 2.3, we may estimate, for and ,
Let or . For simplicity, we assume , and let for a fixed small enough. Then we can assume that there exists a constant such that
Assume that . Then we first estimate . By Lemma 4.1(3) and the definition of , we have
and by Remark 3.6
Therefore, we have
Next, we estimate . For , , we have, by Lemma 4.3 and Lemma 4.1(4),
Then, since we know from Remark 3.6 that
we have, by Lemma 3.4,
By the definition of and by Lemma 3.3(a) , we have
By Lemma 4.2, we have
Here, we note from (4.14) and (4.8) that
Thus, we have, for ,
Therefore, we have
because of .Remark 4.4. If we consider the estimate of with (4.13), then we obtain
We continue the proof Proposition 3.7. We need to estimate for . Now, suppose . Then similarly to (4.23), we have
Similarly to (4.21), we have using (4.22)
because of and . Consequently we achieve the result.
Remark 4.5. The above proof implies the following: there exists a constant such that
Proof of Proposition 3.8. We use the same method to prove of Proposition 3.7. So, we let or . For simplicity, we assume , and let for a fixed small enough and there exists a constant such that . Assume that . Since , we may leave out the term with . Hence we consider only the term of . Noting Assumption 3.1, (3.2), and Lemma 2.7, we estimate , where
First we estimate . By Lemma 4.2, (4.16), and the definition of , we have
Since here for some positive constant , we know that
and by (3.25)
Therefore, using (3.3) and (4.15), we have
Noting (4.24), we have
Next, we estimate . Noting (4.19) and (4.22), we have
Then by (4.33) and (3.25) (noting (4.34)), using the notation (3.3) and (4.24), we have
Therefore, since we know from (4.23) that
noting (4.24), we have
Finally, we estimate for . Suppose . Then similar to the above computations, we have
Then, since we know by (4.28) that
we have
Then, since for , we have
Therefore, the result is proved.
Proof of Proposition 3.9. By Lemma 2.3 we see that, for a constant ,
where,
We set
and hereafter we wrote (*) as . Now, we repeat the proof of Proposition 3.7 by exchanging with , , and we note (3.23) (and (3.26)). Then, for we obtain
For , we use the estimate for in the proof of Proposition 3.7; furthermore we use Remark 4.4. Then we have
Consequently, we have
Similarly, we have
Proof of Proposition 3.10. Let be fixed. From Proposition 3.9 we see
Proof of Proposition 3.11. Let with . Then satisfies the condition (A-1). By Proposition 3.8, we see
So, from Propositions 3.10 and 3.8 (noting (3.3)) we have
Proof of Theorem 3.12. Since satisfies (3.22), we see
For a given , there exists a polynomial with such that
In fact, by [21, Theorem 1.4], there exists a polynomial such that
Let . Noting that
and , we have
that is, we have (4.55). Here, we know that
Therefore, by Propositions 3.7, 3.11, and 3.8, we have for large enough
Here we see that, by Proposition 3.7 with (constant depending only on ) and (4.55),
and for large enough, we have
Consequently, noting (3.3), we have
Then we have Theorem 3.12.
Proof of Theorem 3.13. From Theorem 3.12 and Proposition 3.9, we have
Proof of Corollary 3.14. From Theorem 3.12 and Proposition 3.8, we have
Proof of Theorem 3.15. From Theorem 3.13 and Proposition 3.10 we have
Proof of Theorem 3.16. We use Theorems 3.12, 3.13, and Corollary 3.14, and Theorem 3.15. Then we have
5. Divergence Theorem
If is a positive odd integer, then we obtain the unboundedness of . We define
Theorem 5.1 (cf. [1, Theorem 2]). Let be an odd integer. Then there exists a constant and such that for
Let , and let us define
Then we will show that for
Remark 5.2. For the interpolation polynomial , we can see a remarkable difference between the cases of an odd number and of an even number . Let us consider any continuous function . Then we can extend to a continuous function which satisfies (3.22) and , . Then from Theorem 3.12 for an even positive integer , we see
On the other hand, the standard argument (cf, [22, Theorem 4.3]) leads us to the following. Theorem 5.1 means that there exists a certain function such that
that is, for , we see that the interpolation polynomials do not converge to . We also remark that the polynomial interpolates at only . To prove the theorem we use the following lemma.
Lemma 5.3 (see [18, Theorem 11]). For , there is a polynomial of degree such that for , and the following relation holds. Let . Then one has an expression for and :
where for , , , , there exist the constants such that
and satisfies
for with , and for .
Proof of Theorem 5.1. To get a lower bound of , it suffices to consider a lower bound (5.4) of with . Let and we consider the intervals . If we consider large enough, then we have . See the expression
Then we have
It follows from Remark 4.5 that . Hence, it is enough to show that . Let ; then by Lemma 4.2 and the definition of there exists such that
For we consider only such that, for some positive integer ,
Then for each we define
Here, we will see that
Let
Then we have
where and are integers. On the other hand,
Therefore, we have
Now, we take a positive integer such that
Consequently, we have the following. By Lemmas 4.2, Lemma 4.1, , , and Lemma 2.3, we have
Here, using Lemma 5.3 and (5.15), we have
Here, there exists such that . Therefore we see
So we have (5.4), and consequently the proof of Theorem 5.1 is completed.
Acknowledgment
The authors thank the referees for many kind suggestions and comments.
References
Y. Kanjin and R. Sakai, “Pointwise convergence of Hermite-Fejér interpolation of higher order for Freud weights,” The Tohoku Mathematical Journal, vol. 46, no. 2, pp. 181–206, 1994.
Y. Kanjin and R. Sakai, “Convergence of the derivatives of Hermite-Fejér interpolation polynomials of higher order based at the zeros of Freud polynomials,” Journal of Approximation Theory, vol. 80, no. 3, pp. 378–389, 1995.
D. S. Lubinsky, “Hermite and Hermite-Fejér interpolation and associated product integration rules on the real line: the L1 theory,” Journal of Approximation Theory, vol. 70, no. 3, pp. 284–334, 1992.
T. Kasuga and R. Sakai, “Uniform or mean convergence of Hermite-Fejér interpolation of higher order for Freud weights,” Journal of Approximation Theory, vol. 101, no. 2, pp. 330–358, 1999.
T. Kasuga and R. Sakai, “Orthonormal polynomials with generalized Freud-type weights,” Journal of Approximation Theory, vol. 121, no. 1, pp. 13–53, 2003.
T. Kasuga and R. Sakai, “Orthonormal polynomials for generalized Freud-type weights and higher-order Hermite-Fejér interpolation polynomials,” Journal of Approximation Theory, vol. 127, no. 1, pp. 1–38, 2004.
T. Kasuga and R. Sakai, “Orthonormal polynomials for Laguerre-type weights,” Far East Journal of Mathematical Sciences, vol. 15, no. 1, pp. 95–105, 2004.
T. Kasuga and R. Sakai, “Conditions for uniform or mean convergence of higher order Hermite-Fejér interpolation polynomials with generalized Freud-type weights,” Far East Journal of Mathematical Sciences, vol. 19, no. 2, pp. 145–199, 2005.
H. S. Jung and R. Sakai, “The Markov-Bernstein inequality and Hermite-Fejér interpolation for exponential-type weights,” Journal of Approximation Theory, vol. 162, no. 7, pp. 1381–1397, 2010.
R. Sakai, “The degree of approximation of differentiable functions by Hermite interpolation polynomials,” Acta Mathematica Hungarica, vol. 58, no. 1-2, pp. 9–11, 1991.
R. Sakai, “Hermite-Fejér interpolation prescribing higher order derivatives,” in Progress in Approximation Theory, pp. 731–759, Academic Press, Boston, Mass, USA, 1991.
R. Sakai and P. Vértesi, “Hermite-Fejér interpolations of higher order. III,” Studia Scientiarum Mathematicarum Hungarica, vol. 28, no. 1-2, pp. 87–97, 1993.
R. Sakai and P. Vértesi, “Hermite-Fejér interpolations of higher order. IV,” Studia Scientiarum Mathematicarum Hungarica, vol. 28, no. 3-4, pp. 379–386, 1993.
H. S. Jung and R. Sakai, “Derivatives of orthonormal polynomials and coefficients of Hermite-Fejér interpolation polynomials with exponential-type weights,” Journal of Inequalities and Applications, vol. 2010, Article ID 816363, 29 pages, 2010.
H. S. Jung and R. Sakai, “Derivatives of integrating functions for orthonormal polynomials with exponential-type weights,” Journal of Inequalities and Applications, vol. 2009, Article ID 528454, 22 pages, 2009.