Abstract
The present article deals with the approximation of certain exponential type operators defined by Ismail and May. We estimate the rate of convergence of these operators for functions of bounded variation.
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1 Introduction
Ismail and May [9] studied exponential type operators \(L_{n}\). These operators have the form
with kernels \(k_{n}\left( x,t\right) \) satisfying the partial differential equation
The special case \(p\left( x\right) =2x^{3/2}\) leads to the kernel
where \(\delta \left( \cdot \right) \) denotes the Dirac delta function and \( I_{1}\) the modified Bessel function of first kind given by
The corresponding operator [9, (3.16)] has the explicit representation
where \(x\in \left( 0,\infty \right) \).
Note that the operators (3) can alternatively be written in the form
where \(x\in \left( 0,\infty \right) \) and
We observe that the operators \(T_{n}\) are closely related to the well-known Phillips operators [10] (see [11, Eq. (1.1), case c=0]) given by
If we substitute n in the Phillips operators by the real parameter \(n/ \sqrt{x}\), we immediately get the operators (3) by Ismail and May in the form (4). More precisely, we have \(\left( P_{n/ \sqrt{x}}f\right) \left( x\right) =\left( T_{n}f\right) \left( x\right) \). However, unlike the operators \(T_{n}\), the Phillips operators \(P_{n}\) are not exponential type operators. Many other integral type operators (see, e.g., [1, 2, 6, 7]) including the Phillips operators are not of exponential type.
Recently Gupta [5] gave some direct results including an estimate in terms of the second order modulus of continuity and a Voronovskaja-type result for these operators. The recent paper [8] contains a quantitative asymptotic formula in terms of the modulus of continuity with exponential growth, a Korovkin-type result for exponential functions and also a Voronovskaja-type asymptotic formula in the simultaneous approximation.
The present paper deals with the rate of convergence of the operators \(T_{n}\) for functions of bounded variation.
2 Some Lemmas
Let \(e_{r}\) \(\left( r=0,1,2,\ldots \right) \) denote the monomials \( e_{r}\left( x\right) =x^{r}\). For real A, put \(h_{A}\left( t\right) =e^{At} \). The moments \(T_{n}e_{r}\) of the operators (3) are encoded in the moment generating function (m.g.f.) defined by \(\left( T_{n}h_{A}\right) \left( x\right) \). Gupta [5, Remark 1] derived the following formula.
Lemma 1
The moment generating function of the operators (3) is given by
Expanding in powers of A, we obtain, for \(\left| A\right| <n/\sqrt{ x}\),
Obviously, the coefficients of \(A^{r}/r!\) provide the r-th order moments \( \left( T_{n}e_{r}\right) \left( x\right) \). Moreover, we have
as \(n\rightarrow \infty \).
For \(x\in \mathbb {R}\), define \(\psi _{x}\left( t\right) =t-x\). The central moments \(\mu _{n,r}\left( x\right) :=\left( T_{n}\psi _{x}^{r}\right) \left( x\right) \) are crucial for the approximation properties of the operators (3).
Remark 1
The central moments \(\mu _{n,r}\left( x\right) =\left( T_{n}\psi _{x}^{r}\right) \left( x\right) \) of the operators (3) are the coefficients of \(A^{r}/r!\) in the expansion
In particular, we have \(\mu _{n,0}\left( x\right) =1\), \(\mu _{n,1}\left( x\right) =0\), \(\mu _{n,2}\left( x\right) =2x^{3/2}/n\).
Lemma 2
Let \(x\in (0,\infty )\) and let the kernel \(k_{n}(x,t)\) be as defined in (2). Then we have
and
Proof
For \(0<y<x\) in view of Remark 1, we have
The other estimate follows analogously. \(\square \)
For the proof of the main result we need an estimation of \(\left( T_{n}\text { {sgn }}\psi _{x}\right) \left( x\right) \). It is easy to see that
where \(\chi _{\left( 0,x\right) }\) denotes the characteristic function of the interval \(\left( 0,x\right) \). Therefore, we consider
The next result describes the asymptotic behaviour of the latter expression.
Lemma 3
For each \(x\in (0,\infty )\), we have
Proof
From the definition (3) we obtain
The change of variable replacing t with \(\left( \sqrt{x}-t\right) ^{2}\) leads to the representation
Taking advantage of the integral representation [3, (9.6.19)]
we obtain
where
Note that \(h\left( 0,0\right) =0\). For a sufficiently small positive real \( \delta \), there exists \(\varepsilon \in \left( 0,\sqrt{x}\right) \), such that \(h\left( t,\vartheta \right) >\delta \) if \(t\in \left[ \varepsilon , \sqrt{x}\right] \) and \(\vartheta \in \left[ 0,\pi \right] \). This implies that, for arbitrary small \(a>0\), there is a constant \(c>0\) such that
We study
This equation can be rewritten in the form
where
A further change of variable replacing t with \(\sqrt{x/n}t\) and replacing \( \vartheta \) with \(\vartheta /\sqrt{n}\) yields
Noting that \(-2\left( 1-\cos \vartheta \right) +\vartheta ^{2}=:\vartheta ^{4}F\left( \vartheta \right) \) and \(2\left( 1-\cos \vartheta \right) =:\vartheta ^{2}G\left( \vartheta \right) \), where F and G are entire functions, we see that
Therefore, we have
as \(n\rightarrow \infty \). After some calculations we arrive at
as \(n\rightarrow \infty \). \(\square \)
3 Rate of convergence
Theorem 1
Let f be a function of bounded variation on each finite subinterval of \(\left( 0,\infty \right) \). Suppose that f satisfies the growth condition \(\left| f\left( t\right) \right| \le C e^{\gamma t}\) , for \(t>0\). Then, it exists a sequence \(\left( \varepsilon _{n}\left( x\right) \right) \) with \(\varepsilon _{n}\left( x\right) \rightarrow 0\) as \( n\rightarrow \infty \), such that, for \(n>2\gamma \sqrt{x}\),
where
Proof
Let \(x\in (0,\infty )\). Our starting point is the inequality
Firstly, by Lemma 3, we obtain
as \(n\rightarrow \infty \). Next we estimate \(\left( T_{n}f_{x}\right) \left( x\right) \) as follows:
Integrating the first term by parts we obtain
Since \(\left| f_{x}\left( y\right) \right| \le V_{y}^{x}\left( f_{x}\right) \), we have
Applying Lemma 2, and in the next step integrating by parts, we get
Next for \(t\in \left[ x-x/\sqrt{n},x+x/\sqrt{n}\right] \) and by the fact \( \int _{x-x/\sqrt{n}}^{x+x/\sqrt{n}}d_{t}(\eta _{n}\left( x,t\right) )\le 1\), we conclude that
Finally,
Arguing analogously as in estimate of \(E_{1}\), we have
By assumption, we have \(\left| f_{x}\left( t\right) \right| \le \left| f\left( x+\right) \right| +\left| f\left( t\right) \right| \le C\left( e^{\gamma x}+e^{\gamma t}\right) \), such that
Since \(\left| t-x\right| /x\ge 1\), for \(t\ge 2x\), we conclude that
Collecting the estimates of \(E_{1},E_{2},E_{3}\), we get the desired result. \(\square \)
Remark 2
One may extend the present result to generalized bounded variations of integral type (see [4]) and for \(\bigwedge BV^{(p)}\) and \(\varphi \bigwedge BV\) discussed in [12]. This may be treated as further research in this direction.
Change history
24 May 2021
The original version of this article was revised due to a retrospective Open Access order
29 May 2021
A Correction to this paper has been published: https://doi.org/10.1007/s13398-021-01069-5
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The authors are extremely grateful to the three anonymous reviewers for their valuable advice leading to several improvements in the manuscript.
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Abel, U., Gupta, V. Rate of convergence of exponential type operators related to \(p\left( x\right) =2x^{3/2}\) for functions of bounded variation. RACSAM 114, 188 (2020). https://doi.org/10.1007/s13398-020-00919-y
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DOI: https://doi.org/10.1007/s13398-020-00919-y