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Higher order Hermite-Fejér interpolation polynomials with Laguerre-type weights
Journal of Inequalities and Applications volume 2011, Article number: 122 (2011)
Abstract
Let ℝ+ = [0, ∞) and R : ℝ+ → ℝ+ be a continuous function which is the Laguerre-type exponent, and pn, ρ(x), be the orthonormal polynomials with the weight w ρ (x) = xρ e-R(x). For the zeros of , we consider the higher order Hermite-Fejér interpolation polynomial L n (l, m, f; x) based at the zeros , where 0 ≤ l ≤ m - 1 are positive integers.
2010 Mathematics Subject Classification: 41A10.
1. Introduction and main results
Let ℝ = [-∞, ∞) and ℝ+ = [0, ∞). Let R : ℝ+ → ℝ+ be a continuous, non-negative, and increasing function. Consider the exponential weights w ρ (x) = xρ exp(-R(x)), ρ > -1/2, and then we construct the orthonormal polynomials with the weight w ρ (x). Then, for the zeros of , we obtained various estimations with respect to , k = 1, 2, ..., n, j = 1, 2, ..., ν, in [1]. Hence, in this article, we will investigate the higher order Hermite-Fejér interpolation polynomial L n (l, m, f; x) based at the zeros , using the results from [1], and we will give a divergent theorem. This article is organized as follows. In Section 1, we introduce some notations, the weight classes , with , , and main results. In Section 2, we will introduce the classes and , and then, we will obtain some relations of the factors derived from the classes , and the classes , . Finally, we will prove the main theorems using known results in [1–5], in Section 3.
We say that f : ℝ → ℝ+ is quasi-increasing if there exists C > 0 such that f(x) ≤ Cf(y) for 0 < x < y. The notation f(x) ~ g(x) means that there are positive constants C1, C2 such that for the relevant range of x, C1 ≤ f(x)/g(x) ≤ C2. The similar notation is used for sequences, and sequences of functions. Throughout this article, C, C1, C2, ... denote positive constants independent of n, x, t or polynomials P n (x). The same symbol does not necessarily denote the same constant in different occurrences. We denote the class of polynomials with degree n by .
First, we introduce classes of weights. Levin and Lubinsky [5, 6] introduced the class of weights on ℝ+ as follows. Let I = [0, d), where 0 < d ≤ ∞.
Definition 1.1.[5, 6] We assume that R : I → [0, ∞) has the following properties: Let Q(t) = R(x) and x = t2.
-
(a)
is continuous in I, with limit 0 at 0 and R(0) = 0;
-
(b)
R″(x) exists in (0, d), while Q″ is positive in ;
-
(c)
-
(d)
The function
is quasi-increasing in (0, d), with
-
(e)
There exists C 1 > 0 such that
Then, we write . If there also exist a compact subinterval J* ∋ 0 of and C2 > 0 such that
then we write .
We consider the case d = ∞, that is, the space ℝ+ = [0, ∞), and we strengthen Definition 1.1 slightly.
Definition 1.2. We assume that R : ℝ+ → ℝ+ has the following properties:
-
(a)
R(x), R'(x) are continuous, positive in ℝ+, with R(0) = 0, R'(0) = 0;
-
(b)
R″(x) > 0 exists in ℝ+\{0};
-
(c)
-
(d)
The function
is quasi-increasing in ℝ+\{0}, with
-
(e)
There exists C 1 > 0 such that
There exist a compact subinterval J ∋ 0 of ℝ+ and C2 > 0 such that
then we write .
To obtain estimations of the coefficients of higher order Hermite-Fejér interpolation polynomial based at the zeros , we need to focus on a smaller class of weights.
Definition 1.3. Let and let ν ≥ 2 be an integer. For the exponent R, we assume the following:
-
(a)
R (j)(x) > 0, for 0 ≤ j ≤ ν and x > 0, and R (j)(0) = 0, 0 ≤ j ≤ ν - 1.
-
(b)
There exist positive constants C i > 0, i = 1, 2, ..., ν - 1 such that for i = 1, 2, ..., ν - 1
-
(c)
There exist positive constants C, c 1 > 0 and 0 ≤ δ < 1 such that on x ∈ (0, c 1)
(1.1) -
(d)
There exists c 2 > 0 such that we have one among the following
(d1) is quasi-increasing on (c2, ∞),
(d2) R(ν)(x) is nondecreasing on (c2, ∞).
Then we write .
Example 1.4.[6, 7] Let ν ≥ 2 be a fixed integer. There are some typical examples satisfying all conditions of Definition 1.3 as follows: Let α > 1, l ≥ 1, where l is an integer. Then we define
where exp l (x) = exp(exp(exp ... exp(x)) ...) is the l-th iterated exponential.
-
(1)
If α > ν, .
-
(2)
If α ≤ ν and α is an integer, we define
Then .
In the remainder of this article, we consider the classes and ; Let or . For , we set w ρ (x): = xρw(x). Then we can construct the orthonormal polynomials of degree n with respect to . That is,
Let us denote the zeros of p n,ρ (x) by
The Mhaskar-Rahmanov-Saff numbers a v is defined as follows:
Let l, m be non-negative integers with 0 ≤ l < m ≤ ν. For f ∈ C(l)(ℝ), we define the (l, m)-order Hermite-Fejér interpolation polynomials as follows: For each k = 1, 2, ..., n,
For each , we see L n (m - 1, m, P; x) = P(x). The fundamental polynomials , k = 1, 2, ..., n, of L n (l, m, f; x) are defined by
Here, lk, n, ρ(x) is a fundamental Lagrange interpolation polynomial of degree n - 1 [[8], p. 23] given by
and hs,k, n, ρ(l, m; x) satisfies
Then
In particular, for f ∈ C(ℝ), we define the m-order Hermite-Fejér interpolation polynomials as the (0, m)-order Hermite-Fejér interpolation polynomials L n (0, m, f; x). Then we know that
where e i (m, k, n): = e0,i(0, m, k, n) and
We often denote lk, n(x): = lk, n, ρ(x), hs, k, n(x): = hs, k, n, ρ(x), and xk, n: = xk, n, ρif they do not confuse us.
Theorem 1.5. Letand ρ > -1/2.
-
(a)
For each m ≥ 1 and j = 0, 1, ..., we have
(1.5) -
(b)
For each m ≥ 1 and j = s, ..., m - 1, we have e s, s(l, m, k, n) = 1/s! and
(1.6)
We remark .
Theorem 1.6. Letand ρ > -1/2. Assume that 1 + 2ρ -δ/2 ≥ 0 for ρ < -1/4 and if T(x) is bounded, then assume that
where 0 ≤ δ < 1 is defined in (1.1). Then we have the following:
-
(a)
If j is odd, then we have for m ≥ 1 and j = 0, 1, ..., ν - 1,
(1.8) -
(b)
If j - s is odd, then we have for m ≥ 1 and 0 ≤ s ≤ j ≤ m - 1,
(1.9)
Theorem 1.7. Let 0 < ε < 1/4. Let. Let s be a positive integer with 2 ≤ 2s ≤ ν. Then under the same conditions as the assumptions of Theorem 1.6, there exists μ1(ε, n) > 0 such that
and δ (ε, n) → 0 as n → ∞ and ε → 0.
Theorem 1.8. [4, Lemma 10] Let 0 < ε < 1/4. Let. Let s be a positive integer with 2 ≤ 2s ≤ ν - 1. Suppose the same conditions as the assumptions of Theorem 1.6. Then
-
(a)
for 1 ≤ 2s - 1 ≤ ν - 1,
(1.10)
where δ(ε, n) is defined in Theorem 1.7.
-
(b)
there exists β(n, k) with 0 < D 1 ≤ β(n, k) ≤ D 2 for absolute constants D 1, D 2 such that the following holds:
(1.11)
and |ξ s (m, ε, xk, n, n)| → 0 as n → ∞ and ε → 0.
Theorem 1.9. [4, (4.16)], [9]Let 0 < ε < 1/4. Let. Let s be a positive integer with 2 ≤ 2s ≤ m - 1. Suppose the same conditions as the assumptions of Theorem 1.6. Then for j = 0, 1, 2, ..., there is a polynomial Ψ j (x) of degree j such that (-1) j ψ j (-m) > 0 for m = 1, 3, 5, ... and the following relation holds:
and |η s (m, ε, xk, n, n)| → 0 as n → ∞ and ε → 0.
Theorem 1.10. Let m be an odd positive integer. Suppose the same conditions as the assumptions of Theorem 1.6. Then there is a function f in C(ℝ+) such that for any fixed interval [a, b], a > 0,
2. Preliminaries
Levin and Lubinsky introduced the classes and as analogies of the classes and defined on . They defined the following:
Definition 2.1.[10] We assume that Q : I* → [0, ∞) has the following properties:
-
(a)
Q(t) is continuous in I*, with Q(0) = 0;
-
(b)
Q″(t) exists and is positive in I*\{0};
-
(c)
-
(d)
The function
is quasi-increasing in , with
-
(e)
There exists C 1 > 0 such that
Then we write . If there also exist a compact subinterval J* ∋ 0 of I* and C2 > 0 such that
then we write .
Then we see that and where W(t) = w(x), x = t2, from [6, Lemma 2.2]. In addition, we easily have the following:
Lemma 2.2.[1]Let Q(t) = R(x), x = t2. Then we have
where W(t) = w(x); x = t2.
On ℝ, we can consider the corresponding class to as follows:
Definition 2.3.[11] Let and ν ≥ 2 be an integer. Let Q be a continuous and even function on ℝ. For the exponent Q, we assume the following:
-
(a)
Q (j)(x) > 0, for 0 ≤ j ≤ ν and t ∈ ℝ+/{0}.
-
(b)
There exist positive constants C i > 0 such that for i = 1, 2, ..., ν - 1,
-
(c)
There exist positive constants C, c 1 > 0, and 0 ≤ δ* < 1 such that on t ∈ (0, c 1),
(2.1) -
(d)
There exists c 2 > 0 such that we have one among the following:
(d1) T* (t)/t is quasi-increasing on (c2, ∞),
(d2) Q(ν)(t) is nondecreasing on (c2, ∞).
Then we write .
Let , and ν ≥ 2. For , we set
Then we can construct the orthonormal polynomials of degree n with respect to Wρ*(t). That is,
Let us denote the zeros of Pn, ρ*(t) by
There are many properties of Pn, ρ*(t) = P n (Wρ*; t) with respect to Wρ*(t), of Definition 2.3 in [2, 3, 7, 11–13]. They were obtained by transformations from the results in [5, 6]. Jung and Sakai [2, Theorem 3.3 and 3.6] estimate , k = 1, 2, ..., n, j = 1, 2, ..., ν and Jung and Sakai [1, Theorem 3.2 and 3.3] obtained analogous estimations with respect to , k = 1, 2, ..., n, j = 1, 2, ..., ν. In this article, we consider and pn, ρ(x) = p n (w ρ ; x). In the following, we give the transformation theorems.
Theorem 2.4. [13, Theorem 2.1] Let W(t) = W(x) with x = t2. Then the orthonormal polynomials Pn, ρ*(t) on ℝ can be entirely reduced to the orthonormal polynomials p n , ρ (x) in ℝ+as follows: For n = 0, 1, 2, ...,
In this article, we will use the fact that w ρ (x) = xρ exp(-R(x)) is transformed into W2ρ+1/2(t) = |t|2ρ+1/2exp (-Q(t)) as meaning that
Theorem 2.5. [1, Theorem 2.5] Let Q(t) = R(x), x = t2. Then we have
In particular, we have
where 0 ≤ δ < 1 is defined in (1.1).
For convenience, in the remainder of this article, we set as follows:
and . Then we know that and
In the following, we introduce useful notations:
-
(a)
The Mhaskar-Rahmanov-Saff numbers a v and are defined as the positive roots of the following equations, that is,
and
-
(b)
Let
Then we have the following:
Lemma 2.6. [6, (2.5),(2.7),(2.9)]
To prove main results, we need some lemmas as follows:
Lemma 2.7. [13, Theorem 2.2, Lemma 3.7] For the minimum positive zero, t[n/2],n([n/2] is the largest integer ≤ n/2), we have
and for the maximum zero t1nwe have for large enough n,
Moreover, for some constant 0 < ε ≤ 2 we have
Remark 2.8. (a) Let . Then
(a-1) T(x) is bounded ⇔ T*(t) is bounded.
(a-2) T(x) is unbounded ⇒ a n ≤ C(η)nη for any η > 0.
(a-3) T(a n ) ≤ Cn2-εfor some constant 0 < ε ≤ 2.
-
(b)
Let . Then
(b-1) ρ > -1/2 ⇒ ρ* > -1/2.
(b-2) 1 + 2ρ - δ/2 ≥ 0 for ρ < -1/4 ⇒ 1 + 2ρ* - δ* ≥ 0 for ρ* < 0.
(b-3) .
Proof of Remark 2.8. (a) (a-1) and (a-3) are easily proved from Lemma 2.6. From [11, Theorem 1.6], we know the following: When T*(t) is unbounded, for any η > 0 there exists C(η) > 0 such that
In addition, since T(x) = T*(t)/2 and , we know that (a-2).
-
(b)
Since , we know that and δ* = δ by Theorem 2.5. Then from (2.3) and Lemma 2.6, we have (b-1), (b-2), and (b-3). □
Lemma 2.9. [1, Lemma 3.6] For j = 1, 2, 3, ..., we have
where c j, i > 0(1 ≤ i ≤ j, j = 1, 2, ...) satisfy the following relations: for k = 1, 2, ...,
and for 2 ≤ i ≤ k,
3. Proofs of main results
Our main purpose is to obtain estimations of the coefficients e s, i (l, m, k, n), k = 1, 2, ..., 0 ≤ s ≤ l, s ≤ i ≤ m - 1.
Theorem 3.1. [1, Theorem 1.5] Letand let ρ > -1/2. For each k = 1, 2, ..., n and j = 0, 1, ..., we have
Proof of Theorem 1.5. (a) From Theorem 3.1 we know that
Then, assuming that (a) is true for 1 ≤ m' < m, we have
Therefore, the result is proved by induction with respect to m.
-
(b)
From (2) and (3), we know e s, s (l, m, k, n) = 1/s! and the following recurrence relation: for s + 1 ≤ i ≤ m - 1,
(3.5)
Therefore, we have the result by induction on i and (3.5).
Theorem 3.2. [1, Theorem 1.6] Letand let ρ > -1/2. Suppose the same conditions as the assumptions of Theorem 1.6. For each k = 1, 2, ..., n and j = 1, ..., ν, we have
and in particular, if j is even, then we have
Proof of Theorem 1.6. We use the induction method on m.
-
(a)
For m = 1, we have the result because of
and Theorem 3.2. Now we assume the theorem for 1 ≤ m' < m. Then, we have the following: For 1 ≤ 2s - 1 ≤ ν - 1,
Since
we have
and similarly
Therefore, we have
-
(b)
To prove the result, we proceed by induction on i. From (1.2) and (1.3) we know e s, s (l, m, k, n) = 1/s! and the following recurrence relation: for s + 1 ≤ i ≤ m - 1,
(3.6)
When i - s is odd, we know that
Then, we have (1.9) from (1.5), (1.8), (3.6), and the assumption of induction on i. □
Theorem 3.3. [1, Theorem 1.7] Let 0 < ε < 1/4. Let and let s be a positive integer with 2 ≤ 2s ≤ ν - 1. Suppose the same conditions as the assumptions of Theorem 1.6. Then there exists β(n, k), 0 < D1 ≤ β(n, k) ≤ D2for absolute constants D1, D2such that the following equality holds:
and |ρ s (ε, x k, n , n)| → 0 as n → ∞ and ε → 0.
Lemma 3.4. [3, Theorem 2.5] Letand r = 1, 2, ⋯. Then, uniformly for 1 ≤ k ≤ n,
Lemma 3.5. [2, Theorem 3.3] Let ρ* > -1/2 and, ν ≥ 2. Assume that 1 + 2ρ* - δ* ≥ 0 for ρ* < 0 and if T*(t) is bounded, then assume
where 0 ≤ δ* < 1 is defined in (2.1). Let 0 < α < 1/2. Letand let s be a positive integer with 2 ≤ 2s ≤ ν. Then there exists μ(ε, n) > 0 such that
and μ(ε, n) → 0 as n → ∞ and ε → 0.
Proof of Theorem 1.7. By Lemma 2.9, we have
Since, we have by Lemma 3.5,
and by Lemma 3.4,
we have
where δ(ε, n) = μ(ε, 2n) + ε. □
Here we can estimate the coefficients e i (ν, k, n) of the fundamental polynomials h kn (ν; x).
For j = 0, 1, ..., define ϕ j (1): = (2j + 1)-1 and for k ≥ 2,
Proof of Theorem 1.8. In a manner analogous to the proof of Theorem 1.6 (a), we use mathematical induction with respect to m.
-
(a)
From Theorem 1.7, we know that for 1 ≤ 2s -1 ≤ ν - 1,
From Theorem 1.5, we know that for x k, n ≤ a n /4,
Then, we have by mathematical induction on m,
-
(b)
From Theorem 3.3, we know that for 0 ≤ 2s ≤ ν - 1,
(3.9)
If we let ξ s (1, ε, x k, n , n) = ρ s (ε, x k, n , n), then (1.11) holds for m = 1 because |ξ s (1, ε, x k, n , n)| → 0 as n → ∞ and ε → 0. Now, we split into two terms as follows:
For the second term, we have from (1.10),
For the first term, we let ξ s (m) = ξ s (m, ε, x k, n , n) for convenience. Then we know that
and |ξs-r(1)| → 0 as n → ∞ and ε → 0. By mathematical induction, we assume for 0 ≤ 2r ≤ 2s;
and |ξ r (m - 1)| → 0 as n → ∞ and ε → 0. Then, since
we have for 0 ≤ 2r ≤ 2s, using the definition of (3.7),
Here, we consider (3.10). If we let
then we have
Then, we know that (1.11) holds and |ξ i (j)| → 0 as n → ∞ and ε → 0, using mathematical induction on m. Therefore, we have the result.
We rewrite the relation (3.7) in the form for ν = 1, 2, 3 ...,
and for j = 1, 2, 3 ..., ν = 2, 3, 4, ...,
Now, for every j we will introduce an auxiliary polynomial determined by as the following lemma:
Lemma 3.6. [4, Lemma 11] (i) For j = 0, 1, 2 ..., there exists a unique polynomial Ψ j (y) of degree j such that
-
(ii)
Ψ0(y) = 1 and Ψ j (0) = 0, j = 1, 2, ....
Since Ψ j (y) is a polynomial of degree j, we can replace ϕ j (ν) in (3.7) with Ψ j (y), that is,
for an arbitrary y and j = 0, 1, 2, .... We use the notation F kn (x, y) = (l k, n (x)) y which coincides with if y is an integer. Since l k, n (x k, n ) = 1, we have F kn (x, t) > 0 for x in a neighborhood of xk, nand an arbitrary real number t.
We can show that (∂/∂x) j F kn (x k, n , y) is a polynomial of degree at most j with respect to y for j = 0, 1, 2, ..., where (∂/∂x) j F kn (x k, n , y) is the j th partial derivative of F kn (x, y) with respect to x at (x k, n , y) [14, p. 199]. We prove these facts by induction on j. For j = 0 it is trivial. Suppose that it holds for j ≥ 0. To simplify the notation, let F(x) = F kn (x, y) and l(x) = l k, n (x) for a fixed y. Then F'(x)l(x) = yl'(x)F(x). By Leibniz's rule, we easily see that
which shows that F(j+1)(x k, n ) is a polynomial of degree at most j + 1 with respect to y. Let , j = 0, 1, 2, ... be defined by
Then is a polynomial of degree at most 2j.
By Theorem 1.8 (1.11), we have the following:
Lemma 3.7. [4, Lemma 12] Let j = 0, 1, 2, ..., and M be a positive constant. Let 0 < ε < 1/4, , and |y| ≤ M. Then
-
(a)
there exists κ j (ε, x k, n , n) > 0 such that
(3.13)
and κ j (ε, x k, n , n) → 0 as n → ∞ and ε → 0.
-
(b)
there exists γ j (ε, n) > 0 such that
(3.14)
and γ j (ε, n) → 0 as n → ∞ and ε → 0.
Lemma 3.8. [4, Lemma 13] If y < 0, then for j = 0, 1, 2 ...,
Lemma 3.9. For positive integers s and m with 1 ≤ m ≤ ν,
Proof. If we let , then it suffices to show that C s (m) = 0. For every s,
By (1.11), (3.12) and (3.13), we see that the first sum has the form of
Then, since
we know that as n → ∞ and ε → 0 (see (3.12)). By (3.14) and (3.8), the second sum is bounded by , and we know that as n → ∞ and ε → 0. Therefore, we obtain the following result: for every s,
Theorem 1.9 is important to show a divergence theorem with respect to L n (m, f; x), where m is an odd integer.
Proof of Theorem 1.9. We prove (1.12) by induction on s. Since e0(m, k, n) = 1 and Ψ0(y) = 1, (1.12) holds for s = 0. From (3.6) we write e2s(m, k, n) in the form of
From (1.6), we know that for x k, n ≤ a n /4,
Then, by (1.10) and (3.15), |II| is bounded by . For 0 ≤ i < s, we suppose (1.12). Then, we have for I,
where ξ s-r : = ξ s-r (m, ε, x k, n , n) and η r : = η r (m, ε, x k, n , n) which are defined in (1.11) and (1.12). Then, using Lemma 3.9 and ϕ0 (m) = 1, we have the following form:
Here, since
we see that |η s (m, ε, x k, n , n)| → 0 as n → ∞ and ε → 0 (recall above estimation of |II|). Therefore, we proved the result.
Lemma 3.10. [5, Theorem 1.3] Letand w(x) (C2+). There exists n0such that uniformly for n ≥ n0, we have the following:
-
(a)
For 1 ≤ j ≤ n,
(3.16) -
(b)
For j ≤ n - 1 and x ∈ [x j+1,n, x j,n ],
(3.17) -
(c)
For 0 < a ≤ x k, n ≤ b < ∞,
(3.18) -
(d)
For 0 < a ≤ x k+1,n, x k, n ≤ b < ∞ and x ∈ [(x k+1,n+ x k, n )/2 x k, n+ x k-1,n)/2],
(3.19)
Moreover, for 0 < a ≤ x ≤ b < ∞, there exists a constant C > 0 such that
-
(e)
Uniformly for n ≥ 1 and 1 ≤ j < n,
(3.21) -
(f)
Let Λ be defined in Definition 1.2 (d). There exists C > 0 such that for n ≥ 1,
Proof. (a) and (b) follow from [5, Theorem 1.3]. (e) follows from [5, Theorem 1.4]. We need to prove (c), (d), and (f).
-
(c)
For 0 < a ≤ x k, n ≤ b < ∞, we have (2.11);
so applying (a), we have the result.
-
(d)
Let 0 < a ≤ x k+1, n < x k, n ≤ b < ∞. We take a constant δ > 0 as
Then, by (b) we have
Moreover, by [6, Theorem 1.2] the second inequality holds.
-
(f)
We see
so that by an integration, R(x) ≥ R(1) xΛ for x ≥ 1, and hence we have
Since limn→∞a n = ∞, we can choose n0 such that a n ≥ 2 for all n ≥ n0. Then for some C1 > 0,
Hence, we have the result.
Lemma 3.11. Let the function h kn (m; x) be defined by (1.4) and let 0 < c < a < b < d < ∞. Then we have
Proof. Let c ≤ xk+1,n< x k, n ≤ d. Then by (3.21), we see .
Now, choose α, β > 0 satisfying for all xk+1,n, x k, n ∈ [c, d],
Let x ∈ [a, b] and |x - xj(x),n| = min {|x - x k, n }|; x k, n ∈ [a, b]}, xj(c)+1,n< c ≤ xj(c),n, and xj(d),n≤ d < xj(d)-1,n. Moreover, we take a non-negative integer j k satisfying for each x k, n ∈ [a, b] and k ≠ j(x),
Then we have
Here, by (3.16) and (3.17) we see
and from (3.20), we have |xj(x),n| ≤ b + 1, and so by (1.6) we have
Consequently, we have
Similarly, for c ≤ x k, n ≤ d with x k, n ≠ xj(x),n, we have by (3.18) and (3.20),
and by (1.6) and (3.23),
Therefore, we have for 0 ≤ i ≤ m - 2,
Therefore,
Proof of Theorem 1.10. We use Theorem 1.9 and Lemma 3.11. We find a lower bound for the Lebesgue constants with a positive odd order m and a given interval [a, b], 0 < a < b < ∞. By the expression (1.4) we have
Let c = a/2, d = b + (b - a) and
It follows from Lemma 3.11 that maxa ≤ x ≤ bG n (x) ≤ C with C independent of n. Therefore, it is enough to show that maxa ≤ x ≤ bF n (x) ≥ C log (1 + n). We consider α, β and j k defined in (3.22) and (3.23). Let K (x; [c, d]) be the set of numbers defined as
where j k is a non-negative integer. Then, there exist γ > 0 and C > 0 such that
that is, we see
In fact, from Lemma 3.10 (f), we see a n ≤ c1n1/Λ, Λ > 1/2. By (3.22) and (3.23), we see 0 ∈ K(x; [c, d]) and
Hence, we have
So, we have (3.24). Now, we take an interval [xl+1,n, xl, n] ⊂ [a, b], and we put
By Lemma 3.10 (d), we have
From Lemma 3.10 (e), we know for c ≤ x k, n ≤ d
By Lemma 3.10 (c), we have for c ≤ x k, n ≤ d
Then, we have
Here, we used the following facts:
and
Now we use Theorem 1.9, that is, for c ≤ x k, n ≤ d we have
Therefore, with (3.25) and (3.26), we have
Consequently, the theorem is complete. □
References
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Acknowledgements
The authors thank the referees for many valuable suggestions and comments. Hee Sun Jung was supported by SEOK CHUN Research Fund, Sungkyunkwan University, 2010.
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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment. All authors read and approved the final manuscript.
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Jung, H., Sakai, R. Higher order Hermite-Fejér interpolation polynomials with Laguerre-type weights. J Inequal Appl 2011, 122 (2011). https://doi.org/10.1186/1029-242X-2011-122
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DOI: https://doi.org/10.1186/1029-242X-2011-122