Abstract
We consider a variant of the Bernstein–Chlodovsky polynomials approximating continuous functions on the entire real line and study its rate of convergence. The main result is a complete asymptotic expansion. As a special case we obtain a Voronovskaja-type formula previously derived by Karsli [11].
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1 Introduction
Let f be a real function on \(\mathbb {R}\) which is bounded on each finite interval. For \(a,b\in \mathbb {R}\) with \(a<b\), define the function \(f_{a,b}\) on \(\left[ 0,1\right] \) by \(f_{a,b}\left( t\right) =f\left( a+\left( b-a\right) t\right) \). Furthermore, put
Obviously, \(f_{0,1}\) is the restriction of f to \(\left[ 0,1\right] \) and we have \(\left\| f\right\| _{a,b}=\left\| f_{a,b}\right\| _{0,1}\).
The Bernstein–Chlodovsky operators applied to the function f described above are defined by
where \(B_{n}\) denote the Bernstein operators defined by
with Bernstein basis polynomials
In the special case \(\left[ a,b\right] =\left[ 0,1\right] \), we have \( C_{n,0,1}\equiv B_{n}\).
The symmetric version
with \(-a=b=c>0\) was recently introduced by Kilgore [12, Eq. (7.1)]. Using this Chlodovsky generalization of the Bernstein operators he [12, Theorems 1 and 2 ] gave a constructive proof for the Weierstrass approximation theorem in weighted spaces of continuous functions defined on \(\left[ 0,\infty \right) \) or on \(\left( -\infty ,\infty \right) \). See also [13].
In the following we suppose that the parameters a, b are coupled with n, i.e., \(a=a_{n}\) and \(b=b_{n}\). Because the difference between two nodes of \( C_{n,a,b}\) is at least \(\left( b-a\right) /n\) it is clear that the condition \(b_{n}-a_{n}=o\left( n\right) \) as \(n\rightarrow \infty \) is necessary for having convergence of \(\left( C_{n,a_{n},b_{n}}f\right) \left( x\right) \) to \(f\left( x\right) \).
In the special case \(a_{n}=0<b_{n}\), for \(n\in \mathbb {N}\), these polynomials were introduced by I. Chlodovsky [7] in 1937 in order to approximate functions on infinite intervals. He showed that under the condition (1.3), if a function f satisfies
for every \(\sigma >0\), then
at each point x of continuity of f. Moreover, he proved convergence in each continuity point for the large class of functions f satisfying the growth condition \(f\left( t\right) =O\left( \exp \left( t^{p}\right) \right) \) as \(t\rightarrow +\infty \), if the sequence \(\left( b_{n}\right) \) satisfies the condition
for an arbitrary small \(\eta >0\). For more results on Chlodovsky operators see the survey article [9] by Karsli.
Explicit expressions of the coefficients \(c_{k}^{\left[ b_{n}\right] }\left( f,x\right) \) in terms of Stirling numbers were given by Karsli [10]. He derived the asymptotic expansion if the function f satisfies condition (1.1) for every \(\sigma >0\).
Throughout the paper we assume that the sequences \(\left( a_{n}\right) \) and \(\left( b_{n}\right) \) satisfy
The purpose of this note is a pointwise complete asymptotic expansion for the sequence of Bernstein–Chlodovsky operators in the form:
as \(n\rightarrow \infty \), for sufficiently smooth functions f satisfying \( f\left( t\right) =O\left( \exp \left( \alpha t^{p}\right) \right) \) as \( t\rightarrow +\infty \), provided that the sequences \(\left( a_{n}\right) ,\) \( \left( b_{n}\right) \) satisfy \(\left( -a_{n}b_{n}\right) =o\left( n^{1/\left( p+1\right) }\right) \) as \(n\rightarrow \infty \). The coefficients \(c_{k}^{\left[ a_{n},b_{n}\right] }\left( f,x\right) \), which depend on f and \(a_{n},b_{n}\), are bounded with respect to n.
The latter formula means that, for each fixed \(x>0\) and for all positive integers q:
as \(n\rightarrow \infty \).
2 Main Result
For real constants \(\alpha \ge 0\) and \(p\ge 0\), let \(W_{\alpha ,p}\) denote the class of functions \(f\in C\left( \mathbb {R}\right) \) satisfying the growth condition:
Note that in the special instance \(p=0\) the class \(W_{\alpha ,0}\) consists of the bounded continuous functions on \(\mathbb {R}\). Since \(W_{0,p}\) and \( W_{\alpha ,0}\) coincide we consider only the case \(\alpha >0\).
Recall that the Stirling numbers \(s\left( n,k\right) \) and \(S\left( n,k\right) \) of first and second kind, respectively, are defined by the relations:
where \(z^{\underline{0}}=1\) and \(z^{\underline{n}}=z\left( z-1\right) \cdots \left( z-n+1\right) \), for \(n\in \mathbb {N}\), denote the falling factorials.
The following theorem is the main result.
Theorem 2.1
Let \(\alpha ,p\ge 0\). Suppose that the function \(f\in W_{\alpha ,p}\) is 2q times differentiable in the point \( x>0\). Let \(\left( -a_{n}\right) \) and \(\left( b_{n}\right) \) be sequences of reals tending to infinity and satisfying the growth condition:
Then, for any positive integer q, the Bernstein–Chlodovsky operators \( C_{n,a_{n},b_{n}}\) possess the asymptotic expansion:
as \(n\rightarrow \infty \), where
The coefficients \(c_{k}^{\left[ a,b\right] }\left( f,x\right) \) have the explicit representation:
with numbers
Remark 2.2
Note that the coefficients \(c_{k}^{\left[ a_{n},b_{n}\right] }\left( f,x\right) \) depend on n but are bounded with respect to n.
Remark 2.3
Our assumption (2.1) on the sequence \(\left( -a_{n}b_{n}\right) \) corresponds to Chlodovsky’s condition (1.2). Furthermore, it is related to the assumption in the case \(a_{n}=0\) (see, [3, Theorem 1, Eq. (4)]).
In the special case \(q=1\) Theorem 2.1 implies the following Voronovskaja-type result.
Corollary 2.4
Let \(\alpha ,p\ge 0\). Suppose that the function \(f\in W_{\alpha ,p}\) admits a second derivative at the point \(x>0\). Let \(\left( -a_{n}\right) \) and \(\left( b_{n}\right) \) be sequences of reals tending to infinity and satisfying the growth condition (2.1) . Then, the Bernstein–Chlodovsky operators \(C_{n,a_{n},b_{n}}\) satisfy the asymptotic relation:
Remark 2.5
The expansion in Theorem 2.1 is completely different to the (pointwise) complete asymptotic expansion:
for the classical Bernstein polynomials \(B_{n}\), which is valid for all bounded functions \(f:\left[ 0,1\right] \rightarrow \mathbb {R}\) being sufficiently smooth in \(x\in \left[ 0,1\right] \). The Voronovskaja formula states that
The same is true in the case of the classical Bernstein–Chlodovsky operators \(C_{n,0,b_{n}}\). Their Voronovskaja-type formula:
was derived in 1960 by Albrycht and Radecki [5]. For further history consult the survey article [9].
3 Auxiliary Results and Proof of the Main Theorem
Our starting-point is an explicit representation of the central moments of the Bernstein polynomials in terms of Stirling numbers of the first and second kind. In the following we write \(e_{m}\left( x\right) =x^{m}\), \(m\in \mathbb {N}_{0}\), for the m-th monomial and \(\psi _{x}\left( t\right) =t-x\) for \(x\in \mathbb {R}\).
Lemma 3.1
The central moments of the Bernstein polynomials possess the representation
\(\left( s=0,1,2,\ldots \right) \), where the coefficients are given by Eq. (2.3).
For a proof see, e.g., [2].
Lemma 3.2
The central moments of the Bernstein–Chlodovsky operators possess the representation:
where the coefficients \(A\left( k,s,j\right) \) are given by Eq. (2.3) .
Proof
We have
and the lemma follows by Lemma 3.1. \(\square \)
As we have seen in the proof of Lemma 3.2 the central moments of the Bernstein–Chlodovsky operators can be expressed in terms of Bernstein polynomials:
As a consequence from well-known properties of the Bernstein polynomials we obtain the following result:
Lemma 3.3
For \(a\le x\le b\), it holds
where \(d_{s,i}\) are certain real numbers.
Proof
Taking advantage of the well-known formulas for the central moments of the Bernstein polynomials (see [8, Chapt. 10, Theorem 1.1]):
where \(d_{s,i}\) are certain real numbers, we obtain
This can be rewritten in the form as stated in the lemma. \(\square \)
For the sake of brevity, in the following, we write
Note that, for \(a\le x\le b\), with \(a<0<b\), we have
This immediately implies the following estimate for the central moment of the Bernstein–Chlodovsky operators.
Lemma 3.4
Let \(\left( -a_{n}\right) \) and \(\left( b_{n}\right) \) be sequences of reals tending to infinity and satisfying the condition \(-a_{n}b_{n}=o\left( n\right) \) as \(n\rightarrow \infty \). Then, for \(s=0,1,2,\ldots \), the quantities \(Q_{k,s}\left( a_{n},b_{n},s;x\right) \) \(\left( \left\lfloor \frac{s+1}{2}\right\rfloor \le k\le s\right) \) are bounded with respect to n, and
A crucial tool is the following estimate due to Bernstein (see [14, Theorem 1.5.3, p. 18f]).
Lemma 3.5
(Bernstein)For \(0\le t\le 1\), the inequality
implies
The next lemma presents a form of Lemma 3.5 which is more useful for application to Chlodovsky operators on the real line. It follows the idea of Albrycht and Radecki [5] who proved a similar result for the classical Chlodovsky operators.
Lemma 3.6
Let \(a<x<b\). If \(0<\delta \le 3\left( x-a\right) \left( b-x\right) /\left( b-a\right) \) it holds
Proof of Lemma 3.6
Putting \(t=\left( x-a\right) /\left( b-a\right) \) in Lemma 3.5, we have
if \(0\le z\le \frac{3}{2}\sqrt{n\frac{x-a}{b-a}\frac{b-x}{b-a}}\). Choose \( \delta =2z\sqrt{n^{-1}\left( x-a\right) \left( b-x\right) }\). Then
if \(0\le \delta /\left( 2\sqrt{n^{-1}\left( x-a\right) \left( b-x\right) } \right) \le \frac{3}{2}\sqrt{n\frac{x-a}{b-a}\frac{b-x}{b-a}}\). The latter inequality is equivalent to
which is a condition of the lemma. \(\square \)
Lemma 3.7
Let \(a<x<b\) and \(0<\delta \le 3\left( x-a\right) \left( b-x\right) /\left( b-a\right) \). If a bounded function \(f:\left[ a,b\right] \rightarrow \mathbb { R}\) satisfies \(f\left( t\right) =0\), for all \(t\in \left( x-\delta ,x+\delta \right) \cap \left[ a,b\right] \), it follows the estimate
Proof
Because of \(f\left( a+\left( b-a\right) \frac{\nu }{n}\right) =0\) for all \( \nu \in \left\{ 0,\ldots ,n\right\} \) with \(\left| a+\left( b-a\right) \frac{\nu }{n}-x\right| <\delta \) we have
and the assertion follows by an application of Lemma 3.6. \(\square \)
A direct consequence is the following localization result for Bernstein–Chlodovsky polynomials which is interesting in itself.
Proposition 3.8
(Localization theorem). Let \(\alpha ,p\ge 0\) be fixed constants and propose that \(f\in W_{\alpha ,p} \) satisfies the estimate
Furthermore, fix the real number \(x\in \left( a,b\right) \) and let \(\delta >0 \). Then \(f\left( t\right) =0\), for all \(t\in \left( x-\delta ,x+\delta \right) \), implies
Suppose that \(\lim _{n\rightarrow \infty }\left( -a_{n}\right) =\lim _{n\rightarrow \infty }b_{n}=+\infty \). Then, for sufficiently large values of n, we can assume that \(a_{n}<-1<1<b_{n}\), such that \(\max \left\{ \left| a_{n}\right| ,\left| b_{n}\right| \right\} \le -a_{n}b_{n}\). Since \(\left( x-a_{n}\right) \left( b_{n}-x\right) =O\left( -a_{n}b_{n}\right) \) as \(n\rightarrow \infty \), there is as positive constant \(M\left( x\right) \) (independent of n), such that
for sufficiently large values of n. Hence, we have
We conclude that, for large n,
Proof of Theorem 2.1
Suppose that f is continuous on \(\mathbb {R}\) being 2q times differentiable at the point \(x\in \mathbb {R}\). Define the function \(h_{x}\) by
and \(h_{x}\left( x\right) =0\). It is a consequence of Taylor’s theorem that \( h_{x}\) is continuous at x. Hence, \(h_{x}\in C\left( \mathbb {R}\right) \). Applying the operator \(C_{n,a,b}\) to both sides of Eq. (3.3) we obtain
The first sum is equal to
Note that \(c_{0}^{\left[ a,b\right] }\left( f,x\right) =1\). Eq. (2.2) is a consequence of Lemma 3.4. We conclude that
as \(n\rightarrow \infty \). In order to complete the proof we have to show that the remainder can be estimated by
To this end let \(\left( \delta _{n}\right) \) be a sequence of positive numbers such that
Note that the conditions (1.3) and (2.1) imply that \(\delta _{n}=o\left( 1\right) \) as \( n\rightarrow \infty \). Define
Because \(h_{x}\) is continuous with \(h_{x}\left( x\right) =0\) we have \( \varepsilon _{n}=o\left( 1\right) \) as \(n\rightarrow \infty \). We split the remainder into two parts
say. Let us start with the estimate of the first sum:
as \(n\rightarrow \infty \), where we used Lemma 3.4. By the Taylor formula (3.3), the second sum can be rewritten as
and we obtain
where in the last step Lemma 3.6 was applied. Note that
Hence,
as \(n\rightarrow \infty \). In the case \(p=0\), i.e., f is bounded on \(\mathbb {R}\), we have
We can assume that \(\alpha >0\). Therefore, in the case \(p>0\), we have
Obviously, it is sufficient to estimate the latter relation. By Eq. (3.4), we infer that
Finally, we conclude that the remainder can be estimated by
which completes the proof of the theorem. \(\square \)
Proof of Corollary 2.4
In the special case \(q=1\), Theorem 2.1 states that
which can be rewritten in the form
We have
and the desired formula follows because
This completes the proof. \(\square \)
References
Abel, U.: A Voronovskaya-type result for simultaneous approximation by Bernstein–Chlodovsky polynomials, Results Math. 74 (2019): Paper No. 117, 12 p. https://doi.org/10.1007/s00025-019-1036-5
Abel, U., Ivan, M.: Asymptotic expansion of the multivariate Bernstein polynomials on a simplex. Approx. Theory Appl. 16, 85–93 (2000)
Abel, U., Karsli, H.: Complete Asymptotic Expansions for Bernstein–Chlodovsky Polynomials. In: CONSTRUCTIVE THEORY OF FUNCTIONS, Sozopol 2016 (K. Ivanov, G. Nikolov and R. Uluchev, Eds.), pp. 1–12, Prof. Marin Drinov Academic Publishing House, Sofia, (2017)
Abel, U., Karsli, H.: Asymptotic expansions for Bernstein–Durrmeyer–Chlodovsky polynomials. Results Math. 73 (2018): Paper No. 104, 12 p. https://doi.org/10.1007/s00025-018-0863-0
Albrycht, J., Radecki, J.: On a generalization of the theorem of Voronovskaya. Zeszyty Naukowe UAM, Zeszyt 2, Poznan, (1960), 1–7
Butzer, P.L., Karsli, H.: Voronovskaya-type theorems for derivatives of the Bernstein–Chlodovsky polynomials and the Szász-Mirakyan operator. Comment. Math. 49, 33–58 (2009)
Chlodovsky, I.: Sur le développement des fonctions définies dans un intervalle infini en séries de polynomes de M. S. Bernstein. Compositio Math. 4, 380–393 (1937)
DeVore, R. A., Lorentz, G. G.: Constructive approximation. Grundlehren der Mathematischen Wissenschaften 303, Springer, Berlin, Heidelberg (1993)
Karsli, H.: Recent results on Chlodovsky operators. Stud. Univ. Babeş-Bolyai Math. 56, 423–436 (2011)
Karsli, H.: Complete asymptotic expansions for the Chlodovsky polynomials. Numer. Funct. Anal. Optim. 34, 1206–1223 (2013)
Karsli, H.: A complete extension of the Bernstein-Weierstrass Theorem to the infinite interval\((-\infty ,+\infty )\)via Chlodovsky polynomials. Submitted
Kilgore, T.: On a constructive proof of the Weierstrass Theorem with a weight function on unbounded intervals. Mediterr. J. Math. 14, No. 6, Paper No. 217, 9 p. (2017)
Kilgore, T., Szabados, J.: On weighted uniform boundedness and convergence of the Bernstein-Chlodowsky operators. J. Math. Anal. Appl. 473, 1165–1173 (2019)
George, G.: Lorentz. University of Toronto Press, Bernstein Polynomials (1953)
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Abel, U., Karsli, H. A Complete Asymptotic Expansion for Bernstein–Chlodovsky Polynomials for Functions on \(\mathbb {R}\). Mediterr. J. Math. 17, 201 (2020). https://doi.org/10.1007/s00009-020-01632-1
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DOI: https://doi.org/10.1007/s00009-020-01632-1