Higher order derivatives of approximation polynomials on $\mathbb{R}$

D. Leviatan has investigated the behavior of the higher order derivatives of approximation polynomials of the differentiable function $f$ on $[-1,1]$. Especially, when $P_n$ is the best approximation of $f$, he estimates the differences $\|f^{(k)}-P_n^{(k)}\|_{L_\infty([-1,1])}$, $k=0,1,2,...$. In this paper, we give the analogies for them with respect to the differentiable functions on $\mathbb{R}$, and we apply the result to the monotone approximation.


Introduction
Let R = (−∞, ∞) and R + = [0, ∞). We say that f : R → R + is quasiincreasing 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 C 1 , C 2 such that for the relevant range of x, C 1 f (x)/g(x) C 2 . The similar notation is used for sequences and sequences of functions. Throughout C, C 1 , C 2 , ... denote positive constants independent of n, x, t. The same symbol does not necessarily denote the same constant in different occurrences. We denote the class of polynomials with degree n by P n .
First, we introduce some classes of weights. Levin and Lubinsky [8] introduced the class of weights on R as follows.
For convenience, we denote T instead of T w , if there is no confusion. Next, we give some typical examples of F (C 2 +).
In this paper, we estimate f (k) (x) − P (k) n;f (x) w(x) , x ∈ R, k = 0, 1, ..., r for f ∈ C (r) (R) and for some exponential type weight w in L p (R)-space, 1 < p ≤ ∞, where P n;f ∈ P n is the best approximation of f . Furthermore, we give an application for a monotone approximation with linear differential operators. In Section 2 we write the theorems in the space L ∞ (R), then we also denote a certain assumption and some notations which need to state the theorems. In Section 3 we give some lemmas and the proofs of theorems. In Section 4, we consider the similar problem in L p (R)-space, 1 < p < ∞. In Section 5, we give a simple application of the result to the monotone approximation.

Theorems and Preliminaries
First, we introduce some well-known notations. If f is a continuous function on R, then we define f w L∞(R) := sup t∈R |f (t)w(t)|, and for 1 ≤ p < ∞ we denote we suppose that f ∈ C(R) and lim |x|→∞ |w(x)f (x)| = 0. We denote the rate of approximation of f by The Mhaskar-Rakhmanov-Saff numbers a x is defined as follows: To write our theorems we need some preliminaries. We need further assumptions.
Applying Theorem 2.3 with w or w −1/4 , we have the following corollary.

Proof of Theorems
Throughout this section we suppose w ∈ F (C 2 +). We give the proofs of theorems. First, we give some lemmas to prove the theorems. We construct the orthonormal polynomials p n (x) = p n (w 2 , x) of degree n for w 2 (x), that is, We denote the partial sum off (x) by

Moreover, we define the de la Vallée Poussin means by
, and so and so When w ∈ F * , we can replace w 1/4 with w.
where C j , j = 1, 2, 3, do not depend on f and n.
Proof. Damelin and Lubinsky [3] or Damelin [2] have treated a certain class E 1 of weights containing the conditions (a)-(d) in Definition 1.1 and where C i , i = 1, 2, 3 > 0 are some constants, and they obtain this Proposition for w ∈ E 1 . Therefore, we may show F (C 2 +) ⊂ E 1 . In fact, from Definition 1.1 (d) and (e), we have for y ≥ x > 0,

is a function having bounded variation on any compact interval and if
then there exists a constant C > 0 such that for every t > 0, and so (2) Let us suppose that f is continuous and To prove this theorem we need the following lemma.
(2) If t = a u /u, u > 0 large enough and then there exist C 1 , C 2 > 0 such that Hence we have Moreover, we see Here we see In fact, from Lemma 3.2 (2), for t = au On the other hand, we have Hence we have Therefore, using (3.4), (3.5) and (3.6), we have Consequently, by (3.3) and (3.7) we have Hence, setting t = C 2 an n , if we use Proposition 3.3, then (2) Given ε > 0, and let us take L = L(ε) > 0 large enough as Then we have Now, there exists ε > 0 small enough such that because if we put t = a u /u, then we see σ(t) = a u and |x| ≤ σ(2t) < a u . Hence, noting [8,Lemma 3.7], that is, for some ε > 0, and for large enough t, and if w is the Erdös-type weight, then from Lemma 3.2 (4), we have .
If w ∈ F * , we also have (3.8), because for some δ > 0 and u > 0 large enough, Therefore, using Lemma 3.2 (3), Lemma 3.5 and the assumption On the other hand, Therefore, we have the result. Then we have ≤ C a n n gw L∞(R) .
Proof. We let then we have for arbitrary P n ∈ P n , Therefore, we have Here, from Theorem 3.4 we see that Therefore, we have (3.10). Next we show (3.11). Since and for any P ∈ P n , j ≥ n + 1, Using (3.10) and (3.1), we have (3.11).

Now we set
then there exists S 2n ∈ P 2n such that w (F − S 2n ) L∞(R) ≤ C a n n E n w 1/4 , f ′ , and wS ′ 2n L∞(R) ≤ CE n−1 w 1/4 , f ′ . When w ∈ F * , we also have same results replacing w 1/4 with w.

Proof of Theorem 2.3.
We prove the theorem only in case of unbounded T (x), in the case of Freud case F * we can prove it similarly. We show that for k = 0, 1, ..., r, n;f,w w(x) ≤ CT k/2 (x)E n−k w 1/4 , f (k) . If r = 0, then (3.15) is trivial. For some r ≥ 0 we suppose that (3.15) holds, and let f ∈ C (r+1) (R). Then f ′ ∈ C (r) (R). Let q n−1 ∈ P n−1 be the polynomial of best approximation of f ′ with respect to the weight w. Then, from our assumption we have for 0 ≤ k ≤ r, As (3.14) we set S 2n = x 0 (v n (f ′ )(t) − q n−1 (t))dt + f (0), then from Lemma 3.7 (3.18) (F − S 2n ) w L∞(R) ≤ C a n n E n w 1/4 , f ′ , Here we apply Theorem 3.9 with the weight w −(k−1)/2 . In fact, by Theorem 2.2 we have w −(k−1)/2 ∈ F λ (C r+2 +). Then, noting a 2n ∼ a n from Lemma 3.2 (1), we see that is, Let R n ∈ P n denote the polynomial of best approximation of F with w. By Theorem 3.9 with w − k 2 again, for 0 ≤ k ≤ r + 1 we have and by (3.18) ≤ C E n (w, F ) + a n n E n w 1/4 , f ′ ≤ C a n n E n−1 (w, f ′ ) + a n n E n−1 (w 1/4 , f ′ ) ≤ C a n n E n−1 w 1/4 , f ′ .
≤ C n T (x) a n k a n n E n−1 w 1/4 , f ′ .
Since E n (F, w) = E n (w, f ) and (3.17)), we know that P n;f,w := Q n + R n is the polynomial of best approximation of f with w. Now, from (3.16), (3.17) and (3.23) we have for 1 ≤ k ≤ r + 1, For k = 0 it is trivial. Consequently, we have (3.15) for all r ≥ 0. Moreover, using Theorem 3.8, we conclude Theorem 2.3.
Proof of Corollary 2.4. It follows from Theroem 2.3.
Proof of Corollary 2.5. Applying Theorem 2.3 with w k/2 , we have for 0 ≤ j ≤ r Especially, when j = k, we obtain In this section we will give an analogy of Theorem 2.3 in L p (R)-space (1 ≤ p ≤ ∞) and we will prove it using the same method as the proof of Theorem 2.3. Let 1 ≤ p ≤ ∞. Let w = exp(−Q) ∈ F λ (C 3 +), 0 < λ < 3/2, and let β > 1 be fixed. Then we set w ♯ and w ♭ as follows; w(x) Theorem 4.1. Let r ≥ 0 be an integer. Let w = exp(−Q) ∈ F λ (C r+2 +), 0 < λ < (r + 2)/(r + 1), and let β > 1 be fixed. Suppose that T 1/4 f (r) w ∈ L p (R). Let P p,n;f,w ∈ P n be the best approximation of f with respect to the weight w in L p (R)-space, that is, Then there exists an absolute constant C r > 0 which depends only on r such that for 0 ≤ k ≤ r and x ∈ R, When w ∈ F * , we can replace w 1/4 and w ♯ −k/2 with w and w ♯ , respectively in the above.
Especially when p = ∞, we can refer to w ♯ or w ♭ as w. In this case, we can note that Corollary 4.2 and Corollary 4.3 imply Corollary 2.4, and Corollary 2.5, respectively.
To prove Theorem 4.1 we need to prepare some notations and lemmas. . Let w ∈ F (C 2 +) and let 1 ≤ p ≤ ∞. If g : R → R is absolutely continuous, g(0) = 0, and wg ′ ∈ L p (R), then . Let w ∈ F (C 2 +) and let 1 ≤ p ≤ ∞. If wf ′ ∈ L p (R), then E p,n (w, f ) ≤ Cω p f, w, a n n ≤ C a n n wf ′ Lp(R) .
Proof. The first inequality follows from Proposition 3.3. We show the second inequality. By [9,Lemma 7] we have Hence we see Then we have from (3.5), Here, from Lemma 4.4 we have From ≤ C a n n gw Lp(R) .
Proof. For arbitrary P n ∈ P n , we have by (3.13) and Hölder inequality where φ is defined in (3.12). Then, we obtain by Lemma 4.5, Here, for p = 1 we may consider Hence, we have ≤ C a n n gw Lp(R) .
< ∞, and let q n−1 ∈ P n be the best approximation of f ′ with respect to the weight w on L p (R) space, that is, Using q n−1 , define F (x) and S 2n as (3.17) and (3.14). Then we have and wS ′ 2n Lp(R) ≤ CE p,n−1 w 1/4 , f ′ . When w ∈ F * , we also have same results replacing w 1/4 with w.
Proof. By Lemma 4.6 (4.6), we have the result using the same method as the proof of Lemma 3.7.
Proof of Theorem 4.1. We will prove it similarly to the proof of Theorem 2.3. First, let q n−1 ∈ P n−1 be the polynomial of best approximation of f ′ with respect to the weight w on L p (R) space. Then using q n−1 , we define F (x) and S 2n in the same method as (3.17) and (3.14). Then we have using Lemma 4.7 Lp(R) ≤ C a n n E p,n w 1/4 , f ′ and (4.7) S ′ 2n w Lp(R) ≤ CE p,n−1 w 1/4 , f ′ . Then we see from Theorem 3.9 and (4.7), and using w ♯ ≤ w and Theorem 3.8 where R n ∈ P n denotes the polynomial of best approximation of F with w on L p (R) space(by the similar calculation as (3.20) and (3.21)). Then, we see w ♯ −k/2 (x) ≤ w −(k−1)/2 (x). By (4.8) and (4.9) and Theorem 3.8, we have (4.10) ≤ C n a n k−1 E p,n−1 w 1/4 , f ′ ≤ CE p,n−k w 1/4 , f (k) .
By the same reason to (3.24), we know that P p,n;f,w := Q n + R n is the polynomial of best approximation of f with w on L p (R) space. Therefore, using P p,n;f,w , (4.10) and the method of mathematical induction, we have for 1 ≤ k ≤ r + 1, Proof of Corollary 4.2. It follows from Theorem 4.1.
Proof of Corollary 4.3. If we apply Theorem 4.1 with w k/2 and w ♭ k/2 , then we can obtain the results.

Monotone Approximation
Let r > 0 be an integer. Let k and ℓ be integers with 0 ≤ k ≤ ℓ ≤ r. In this section, we consider a real function f on R such that f (r) (x) is continuous in R and we let a j (x), j = k, k + 1, ..., ℓ be bounded on R. Now, we define the linear differential operator (cf. where ω f (p) ; t is the modulus of continuity. In this section, we will obtain a similar result with exponential-type weighted L ∞ -norm as the above result. Our main theorem is as follows.