Abstract
In the theory of approximation, linear operators play an important role. The exponential-type operators were introduced four decades ago, since then no new exponential-type operator was introduced by researchers, although several generalizations of existing exponential-type operators were proposed and studied. Very recently, the concept of semi-exponential operators was introduced and few semi-exponential operators were captured from the exponential-type operators. It is more difficult to obtain semi-exponential operators than the corresponding exponential-type operators. In this paper, we extend the studies and define semi-exponential Bernstein, semi-exponential Baskakov operators, semi-exponential Ismail–May operators related to \(2x^{3/2}\) or \(x^{3}\). Furthermore, we present a new derivation for the semi-exponential Post–Widder operators. In some examples, open problems are indicated.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The exponential-type operators are important in the field of approximation theory. They were firstly considered by Ismail and May [4] in 1978. The exponential-type operators preserve the linear functions. Many generalizations of exponential-type operators are available in the literature. Tyliba and Wachnicki [7] extended the definition of Ismail and May [4] by proposing a more general family of operators. For a non-negative real number \(\beta \), they introduced the operators \(L_{\lambda }^{\beta }\). For \(\beta >0\), they are not of exponential type but similar to exponential-type operators. Recently, Herzog [3] further extended the studies and termed such operators as semi-exponential type operators. Actually, an operator of the form
is called a semi-exponential operator if its kernel \(W_{\beta }^{L}\left( \lambda ,x,t\right) \) satisfies the differential equation
In particular, for \(\beta >0\), one has \(L_{\lambda }^{\beta }e_{1}\ne e_{1}\), where \(e_{r}\left( t\right) =t^{r}\) \(\left( r=0,1,2,\ldots \right) \). In the case \(\beta =0\), the operator \(L_{\lambda }^{\beta =0}\) is simply the exponential-type operator studied by Ismail and May [4]. A collection of such operators may be found in the recent book [2, Ch. 1].
Choosing different functions \(p\left( x\right) \) several exponential-type operators were captured in Ismail and May [4]. It is difficult to construct new exponential-type operators or the corresponding semi-exponential operators by just taking different functions \(p\left( x\right) \). The essential obstacle is to fulfill the normalization condition
which means that \(L_{\lambda }^{\beta }\) preserves constant functions. Tyliba and Wachnicki [7] captured the semi-exponential operators of Weierstrass and Szász–Mirakyan operators, Herzog [3] got success to define the semi-exponential Post–Widder operators. We represent below the tabular form of known semi-exponential type operators available till date:
No. | Exponential operator | \(p\left( x\right) \) |
---|---|---|
(1) | Gauss–Weierstrass operators \((W_{n}f)\left( x\right) \) | 1 |
– | \((W_{n}f)\left( x\right) =\sqrt{\frac{n}{2\pi }}\int _{-\infty }^{\infty }\exp \left( \frac{-n(t-x)^{2}}{2}\right) f\left( t\right) dt\) | Exponential |
– | \((W_{n}^{\beta }f)\left( x\right) =\sqrt{\frac{n}{2\pi }}\int _{-\infty }^{\infty }\exp \left( \frac{-n(t-x-\beta /n)^{2}}{2}\right) f\left( t\right) dt\) | Semi-exponential |
(2) | Post–Widder operators \((P_{n}f)\left( x\right) \) | \(x^{2}\) |
– | \((P_{n}f)\left( x\right) =\frac{n^{n}}{\varGamma (n)x^{n}}\int _{0}^{\infty }e^{-nt/x}t^{n-1}f\left( t\right) dt\) | Exponential |
– | \((P_{n}^{\beta }f)\left( x\right) =\frac{n^{n}}{x^{n}e^{\beta x}} \sum _{k=0}^{\infty }\frac{\left( n\beta \right) ^{k}}{k!\varGamma \left( n+k\right) }\int _{0}^{\infty }e^{-nt/x}t^{n+k-1}f\left( t\right) dt\) | Semi-exponential |
(3) | Szász–Mirakyan operators \((S_{n}f)\left( x\right) \) | x |
– | \((S_{n}f)\left( x\right) =\sum _{k=0}^{\infty }e^{-nx}\frac{\left( nx\right) ^{k}}{k!}f\left( \frac{k}{n}\right) \) | Exponential |
– | \((S_{n}^{\beta }f)\left( x\right) =\sum _{k=0}^{\infty }e^{-(n+\beta )x} \frac{\left( \left( n+\beta \right) x\right) ^{k}}{k!}f\left( \frac{k}{n} \right) \) | Semi-exponential |
As pointed out earlier, one can obtain the exponential-type operator as the special \(\beta =0\) from semi-exponential operators, but the converse is not analogous. Here we capture some more semi-exponential operators viz. semi-exponential Bernstein polynomials, semi-exponential Baskakov operators, etc.
2 New semi-exponential operators
In this section, we establish some new exponential-type operators. In all listed cases it is possible to solve the differential equation (1) in the form \(W_{\beta }^{L}\left( n,x,t\right) =A_{L}\left( n,t,\beta \right) y\), but it is difficult to find the normalization, i.e., the factor \(A_{L}\left( n,t,\beta \right) \) of the solution y such that
or, in the discrete case,
respectively. Below we list some instances of \(p\left( x\right) \), which were considered for well-known exponential-type operators.
2.1 Semi-exponential Bernstein operators
If we take \(p\left( x\right) =x\left( 1-x\right) \), then for a kernel \( W_{\beta }^{B}\left( n,x,k/n\right) =A_{B}\left( n,k,\beta \right) y\), we have
where the derivative of y is with respect to the variable x. We conclude that
implying
In order to have normalization
we evaluate \(A_{B}\left( n,k,\beta \right) \) from the equation
For \(0\le x<1\), put \(z=x/\left( 1-x\right) \). Then \(x=z/\left( 1+z\right) \), and for any positive integer n, the generating function of the sequence \(\left( A_{B}\left( n,k,\beta \right) \right) _{k=0}^{\infty }\)
is analytic, for \(\left| z\right| <1\), with an essential singularity at \(z=-1\). Hence, it can be developed as a power series in the disk \(\left| z\right| <1\). The series
is convergent for all complex z different from \(-1\). It follows that, for \(\left| z\right| <1\),
where the binomial coefficient is to be read as \({\left( {\begin{array}{c}n-j\\ 0\end{array}}\right) =1}\) and \(\left( {\begin{array}{c}n-j\\ \ell \end{array}}\right) =\left( {\ell }! \right) ^{-1} \prod _{\nu =0}^{\ell -1}\left( n-j-\nu \right) \), for \( \ell \in {\mathbb {N}}\). We have
where
Thus the semi-exponential Bernstein polynomials \(B_{n}^{\beta }\) map a function f on \(\left[ 0,+\infty \right) \) to a function \(B_{n}^{\beta }f\) defined on \(\left[ 0,1/2\right) \), whenever the sum is convergent. It can be shown that the operators \(B_{n}^{\beta }\) apply to all polynomials. In the special case \(\beta =0\) we have \(j=0\) and \(\ell =k\) such that \( A_{B}\left( n,k,\beta \right) ={\left( {\begin{array}{c}n\\ k\end{array}}\right) }\). Hence, the sum defining \( B_{n}^{\beta =0}f\) is finite, and we get the Bernstein polynomials.
The operators \(B_{n}^{\beta }\) can be rewritten in the alternative form
\(\left( 0\le x<\frac{1}{2}\right) \). The latter representation immediately reveals the special case \(\beta =0\).
2.2 Semi-exponential Baskakov operators
If we take \(p\left( x\right) =x\left( 1+x\right) \), then for a kernel \( W_{\beta }^{V}\left( n,x,k/n\right) =A_{V}\left( n,k,\beta \right) y\), we have
where the derivative of y is with respect to the variable x. We conclude that
implying
In order to have normalization
Put, for \(x\ge 0\), \(z=x/\left( 1+x\right) \). Then \(x=z/\left( 1-z\right) \). We obtain
Thus, the semi-exponential Baskakov operators can be defined by
where
In special case \(\beta =0\) we have \(j=0\) and \(\ell =k\) such that we get the Baskakov operators.
2.3 Semi-exponential Ismail–May operators related to \(2x^{3/2}\)
If we take \(p\left( x\right) =2x^{3/2}\), then for a kernel \(W_{\beta }^{U}\left( n,x,t\right) =A_{U}\left( n,t,\beta \right) y\), we have
where the derivative of y is with respect to the variable x. We conclude that
implying
Our target is to obtain \(A_{U}\left( n,t,\beta \right) \) in order to have normalization
If we put, for abbreviation, \(s=n/\sqrt{x}\), the normalization condition takes the form
Since
we obtain
where \(\delta \left( t\right) \) denotes Dirac’s delta function. Hence, the operators are defined by
with
Thus, the semi-exponential operator, related to \(2x^{3/2}\), takes the form
In the special case \(\beta =0\), the definition reduces to the Ismail–May operator of exponential type
where \(I_{1}\left( x\right) \) is modified Bessel function of the first kind. Further results on the operators \(U_{n}^{\beta =0}\) can be found in [1].
2.4 Semi-exponential Post–Widder operators
Although the semi-exponential Post–Widder operators were captured in [3, Eq. (10)], using Laplace transform, we provide an alternative approach that is shorter. We proceed as follows.
If we take \(p(x)=x^{2}\), then for a kernel \(W_{\beta }^{P}\left( n,x,t\right) =A_{P}\left( n,t,\beta \right) y\), we have
For normalization, we look for a function \(A_{P}\left( n,t,\beta \right) \) such that
Putting
we have to choose coefficients \(a_{k}\) such that
This is equivalent to
It follows that \(\alpha =n-1\) and
Hence, \(A_{P}\left( n,t,\beta \right) \) is given by
Thus, semi-exponential Post–Widder operators take the form
Observing that \(A_{P}\left( n,t,\beta \right) =n\left( nt/\beta \right) ^{\left( n-1\right) /2}I_{n-1}\left( 2\sqrt{n\beta t}\right) \), where \(I_{n}\) denotes the modified Bessel function of the first kind, we obtain the alternative representation
2.5 Semi-exponential Ismail–May operators related to \(x\left( 1+x\right) ^{2}\)
If we take \(p\left( x\right) =x\left( 1+x\right) ^{2}\), then for a kernel \( W_{\beta }^{R}\left( n,x,k/n\right) =A_{R}\left( n,k,\beta \right) y\), we have
implying
If we put \(y=x/\left( 1+x\right) \) the normalization condition reads
Now we put \(z=ye^{1-y}\), so we have the inverse \(y=-W\left( -z/e\right) \), where W denotes the Lambert W function, i.e., the inverse of \(z\mapsto ze^{z}\). Hence, \(A_{R}\left( n,k,\beta \right) \) are the coefficients of the power series
which is convergent in a neighborhood of \(z=0\). Following Ismail and May [4, Eq. (3.13)] we take advantage of the identity [6, p. 348]
which is an easy consequence of the Lagrange expansion theorem. With \( w=-W\left( -z/e\right) \) we obtain
It follows
i.e., \(A_{R}\left( n,k,\beta \right) \) is the coefficient of \(z^{k}\) in the latter power series expansion. The semi-exponential operators related to \(x(1+x)^{2}\), take the form
In the special case \(\beta =0\) we have
and the operators reduce to
[4, Eq. (3.14)]. As Ismail and May remarked, the substitution \( y=x/\left( 1+x\right) \) leads to the operators
It may be considered as an open problem to find a closed form of the coefficients \(A_{R}\left( n,k,\beta \right) \).
2.6 Semi-exponential Ismail–May operators related to \(x^{3}\)
If we take \(p\left( x\right) =x^{3}\), then for a kernel \(W_{\beta }^{Q}\left( n,x,t\right) =A_{Q}\left( n,t,\beta \right) y\), we have
Thus
If we put \(s=n/\left( 2x^{2}\right) \) the normalization condition reads
such that \(A_{Q}\left( n,t,\beta \right) \) is the inverse Laplace transform \( {\mathcal {L}}^{-1}\) of \(\exp \left( \beta \sqrt{n/\left( 2s\right) }-\sqrt{2ns} \right) \). We have
which implies
where \(\delta \left( t\right) \) denotes Dirac’s delta function. It is well known that
We will take advantage of the convolution formula
where
Thus, semi-exponential operators related to \(p\left( x\right) =x^{3}\) take the form
where
In the special case \(\beta =0\) we have
and the operators reduce to
[4, Eq. (3.11)].
2.7 Semi-exponential Ismail–May operators related to \(1+x^{2}\)
If we take \(p\left( x\right) =1+x^{2}\), then for a kernel \(W_{\beta }^{T}\left( n,x,t\right) =A_{T}\left( n,t,\beta \right) y\), we have
implying
The operators related to \(1+x^{2}\) take the form
To have the normalization, we need
If we put \(s=n\arctan x,\) this is equivalent to
Using the identity [5, Section 9, p. 46] (see [4, Lemma 3.3])
Ismail and May [4, Eq. (3.10)] obtained in the special case \(\beta =0 \),
The main target to find a closed expression for \(A_{T}\left( n,t,\beta \right) \), for general \(\beta >0\), may be considered as an open problem.
References
Abel, U., Gupta, V.: Rate of convergence of exponential type operators related to \(p(x) =2x^{3/2}\) for functions of bounded variation. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114, Art. 188 (2020). https://doi.org/10.1007/s13398-020-00919-y
Gupta, V., Rassias, M.T.: Computation and Approximation. Ser. Springer Briefs in Mathematics. Springer Nature Switzerland AG, New York (2021)
Herzog, M.: Semi-exponential operators. Symmetry 13, 637 (2021). https://doi.org/10.3390/sym13040637
Ismail, M., May, C.P.: On a family of approximation operators. J. Math. Anal. Appl. 63, 446–462 (1978)
Oberhettinger, F.: Tabellen zur Fourier Transformation. Springer, Berlin (1957)
Polya, G., Szegö, G.: Problems and Theorems in Analysis, vol. 1 (English translation). Springer, New York (1972)
Tyliba, A., Wachnicki, E.: On some class of exponential type operators. Ann. Soc. Math. Pol. Ser. I Comment. Math. 45, 59–73 (2005)
Acknowledgements
The authors thank both anonymous reviewers for valuable suggestions that led to a better exposition of the paper.
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Meer Sisodia: Research Intern at NSUT.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Abel, U., Gupta, V. & Sisodia, M. Some new semi-exponential operators. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, 87 (2022). https://doi.org/10.1007/s13398-022-01228-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-022-01228-2
Keywords
- Semi-exponential Bernstein polynomials
- Semi-exponential Baskakov operators
- Semi-exponential Ismail–May operators
- Semi-exponential Post–Widder operators
- Approximation by operators