Abstract
In this note we provide a new way to capture operators involving Laguerre polynomials by composition of an integral operator and a discrete operator. The new operator so obtained is a discrete operator. We give three examples by considering composition of Szász-Durrmeyer operator, exponential-type operator related to \(2x^{3/2}\) and the Phillips operator, respectively, with Szász-Mirakyan operators. In all cases we obtain positive linear, discretely defined operators which are based on Laguerre polynomials and approximate functions on the positive real half-axis.
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1 Introduction
If we take the composition of some integral-type operator with a discrete operator, we get a discrete operator. More precisely, we take the composition \(U_{m,n}=V_{m}\circ W_{n}\) of two operators
where I is a certain real interval and \(k_{m}\left( x,t\right) \) is a kernel function, and
For a suited class of functions f, we obtain a discrete operator
In several concrete cases it is possible to give a closed expression for the coefficients
Throughout the paper we tacitly assume that the functions f have the property that interchanging the order of the limit processes integration and summation is justified.
In this note we give three examples of the composition of an integral operator with a discrete operator to obtain a discrete operator. It turns out that the resulting operators are based on Laguerre polynomials. To each of the discrete operators so obtained we find the moment generating function providing the moments which are essential for studying the approximation properties of the new operators. Without a proof we present some basic convergence results for the new operators including a Voronovskaja-type formula.
2 Preliminaries
In this section we present some auxiliary results, in particular on Laguerre polynomials which play an important role in the following.
For nonnegative integers n and arbitrary real \(\alpha \), the polynomial solutions of the differential equation \(xy^{\prime \prime }+\left( \alpha +1-x\right) y^{\prime }+ny=0\) are called generalized Laguerre polynomials \( L_{n}^{\left( \alpha \right) }\), or associated Laguerre polynomials. A concise representation is the Rodrigues’ formula
The explicit form is given by
The ordinary Laguerre polynomials \(L_{n}=L_{n}^{\left( 0\right) }\) are the special case \(\alpha =0\), where
We gather some representations and properties of Laguerre functions which will be useful in the following. Note that the definition of Laguerre functions can be extended to real values of n and \(\alpha \) by writing them in terms of confluent hypergeometric functions
[3, Eq. (13.6.9)], where \(\left( {\begin{array}{c}n+\alpha \\ n\end{array}}\right) =\Gamma \left( n+\alpha +1\right) /\left( \Gamma \left( n+1\right) \Gamma \left( n+\alpha +1\right) \right) \) and
is the confluent hypergeometric function (also denoted by \(_{1}F_{1}\)) and \( \left( a\right) _{k}=\Gamma \left( a+k\right) /\Gamma \left( a\right) \), \( k=0,1,2,\ldots \), is the Pochhammer symbol. When n is an integer the function reduces to a polynomial of degree n. Recall the Kummer’s transformation [3, Eq. (13.1.27)]
for the confluent hypergeometric function. Furthermore, we have the recursive formula [3, Eq. (22.7.31)]
and its special instance \(\alpha =0\), i.e.,
The following lemma will frequently be of use in the following.
Lemma 1
For \(\alpha ,n=0,1,2,\ldots \), it holds
Proof
Using the identity
and according to (4) we obtain
Application of the Kummer transformation (5) leads to
Now the desired formula follows by Eq. (3). \(\square \)
Throughout the paper we consider, for each complex A, the function \(\exp _{A}\) defined by \(\exp _{A}\left( x\right) =\exp \left( Ax\right) \). Furthermore, we denote by \(e_{r}\) \(\left( r=0,1,2,\ldots \right) \) the monomials \(e_{r}\left( x\right) =x^{r}\).
The moment generating function (m.g.f.) of a linear operator U on a space of functions, is defined by
provided that \(U\exp _{A}\) exists. It can be utilized to obtain the moments \(Ue_{r}\) of the operator U via
The m.g.f. of several approximation operators are gathered in [4]. In this note we determine the m.g.f. of the new composition operators.
3 Composition operators based on Laguerre polynomials
The Szász–Mirakyan operator \(S_{n}\) associates to each function \(f: \left[ 0,+\infty \right) \rightarrow {\mathbb {R}}\) of at most exponential growth the function
where \(s_{k}\left( x\right) =e^{-x}x^{k}/k!\) \(\left( k=0,1,2,\ldots \right) \). Direct computation [4, Eq. (8)] confirms that its m.g.f. is given by
Now we list three compositions leading to positive linear approximation operators based on Laguerre polynomials.
3.1 Example 1
For each nonnegative integer \(\alpha \), the generalized Durrmeyer variant of the Szász–Mirakyan operator is defined by
provided that all integrals exist. Note that \(\left| f\left( t\right) \right| =O\left( e^{\gamma t}\right) \) as \(t\rightarrow +\infty \) implies that \(\left( \overline{S}_{n}^{\left[ \alpha \right] }f\right) \left( x\right) \) is well-defined, for \(n>\gamma \). We take the composition of the generalized Szász–Durrmeyer operator \(\overline{S}_{m}^{\left[ \alpha \right] }\) with the Szász–Mirakyan operators \(S_{n}\), to obtain a new approximation operator \(A_{m,n}^{\left[ \alpha \right] }\) as follows:
It is possible to give a concise form of \(A_{m,n}^{\left[ \alpha \right] }\) in terms of generalized Laguerre polynomials (1).
Theorem 1
The operator \(A_{m,n}\) can be represented in the form
where
Remark 1
Since \(L_{k}^{\left( \alpha \right) }\left( x \right) \ge 0\), for \(x \le 0\), the operator \(A_{m,n}^{\left[ \alpha \right] }\) is positive.
Remark 2
In the special case \(m=n\) the operator takes the form
This operator was discussed in [6, Eq. (3.4)].
Proof of Theorem 1
We have
such that
By Lemma 1, we infer that
Noting that \(e^{-mx}\exp \left( \frac{m^{2}x}{m+n}\right) =\exp \left( - \frac{mn}{m+n}x\right) \) completes the proof. \(\square \)
Remark 3
It is easily verified that the generalized Szász–Mirakyan–Durrmeyer operator has the m.g.f.
Thus, by Eq. (8),
provided that \(m>\left| n\left( e^{A/n}-1\right) \right| \). In the special case \(m=n\), the m.g.f. takes the form
for all n satisfying \(\left| e^{A/n}\right| <2\).
Without giving the standard proofs we present some convergence results.
Theorem 2
Let f be a bounded function f on \(\left[ 0,+\infty \right) \). Then, for \(x\ge 0\), it holds
We state the following Voronovskaja-type result.
Theorem 3
Given \(x>0\), for each bounded function f on \(\left[ 0,+\infty \right) \) it holds
provided that the second derivative \(f^{\prime \prime }\left( x\right) \) exists.
3.2 Example 2
The exponential-type operator \(U_{n}\) associated with \(2x^{3/2}\) [5, Eq. (3.16)] (see also [1, 2]) is given by
The composition of the exponential-type operator \(U_{m}\) associated with \(2x^{3/2}\) and \(S_{n}\) yields
It is possible to give a concise form of \(B_{m,n}\) in terms of the classical Laguerre polynomials (2).
Theorem 4
The operator \(B_{m,n}\) can be represented in the form
where
and, for \(k\ge 1\),
Remark 4
In particular
where
and, for \(k\ge 1\),
Proof of Theorem 4
By definition, we have
When \(k=0\), then, we have
For \(k\ge 1\),
By Lemma 1, we infer that
Furthermore, note that \(e^{-m\sqrt{x}}\exp \left( \frac{m^{2}\sqrt{x}}{m+n \sqrt{x}}\right) =\exp \left( \frac{-mnx}{m+n\sqrt{x}}\right) \). Hence,
Now an application of Eq. (7) completes the proof of the desired formula. \(\square \)
Remark 5
The exponential-type operator related to \(2x^{3/2}\) has the m.g.f
Thus
Without giving the proofs we present some convergence results.
Theorem 5
Let f be a bounded function f on \(\left[ 0,+\infty \right) \). Then for \(x\ge 0\), we have
We state the following Voronovskaja-type result.
Theorem 6
Given \(x>0\), for each bounded function f on \(\left[ 0,+\infty \right) \) it holds
provided that the second derivative \(f^{\prime \prime }\left( x\right) \) exists.
3.3 Example 3
The Phillips operator is given by
The composition of Phillips operator \(\overline{P}_{m}\) and Szász–Mirakyan operator \(S_{n}\) yields
It is possible to give a concise form of \(C_{m,n}\) in terms of Laguerre polynomials (2).
Theorem 7
A concise form of \(C_{m,n}\) is given by
where
and for \(k\ge 1\)
Remark 6
In particular
where
and for \(k\ge 1\)
Proof of Theorem 7
By definition, we have
where
and, for \(k\ge 1\),
As in the proof of Theorem 4 it follows, by Lemma 1, that
We get
Now an application of Eq. (7) completes the proof of the desired formula. \(\square \)
Remark 7
For each complex number A, the moment generating function of the operator \(C_{m,n}\) is given by
This is a consequence of the combination of equation (8) and the m.g.f. of the Phillips operator
which implies that
Without giving the proofs we present some convergence results.
Theorem 8
Let f be a bounded function f on \(\left[ 0,+\infty \right) \). Then for \(x\ge 0\), we have
We state the following Voronovskaja-type result.
Theorem 9
Given \(x>0\), for each bounded function f on \(\left[ 0,+\infty \right) \) it holds
provided that the second derivative \(f^{\prime \prime }\left( x\right) \) exists.
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The authors are grateful to the anonymous reviewers for their advice, which improved the exposition of the paper, and for pointing out a typo.
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Abel, U., Gupta, V. Composition of integral-type operators and discrete operators involving Laguerre polynomials. Positivity 28, 39 (2024). https://doi.org/10.1007/s11117-024-01058-z
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DOI: https://doi.org/10.1007/s11117-024-01058-z