${}^{\mathrm{\S }}$Faculty of Mathematics and Natural Sciences, University of Wuppertal, GauSSstr. 20, Wuppertal, Germany, e-mail: heilmann@math.uni-wuppertal.de.

1 Introduction and definition of the operators

In 1957 Baskakov [ 5 ] introduced a general method to construct a class of positive linear operators depending on a real parameter $c$ including the classical Bernstein, Szász-Mirakjan and Baskakov operators as special cases. The so-called Bernstein-Durrmeyer operators were introduced by Durrmeyer in [ 17 ] and independently developed by Lupaş [ 31 ] . Afterwards this construction was carried over to many other classical operators; for instance see [ 32 , 35 ] for the Szász-Mirakjan and Baskakov operators, [ 20 , 22 ] in the general setting for so-called Baskakov-Durrmeyer type operators, [ 33 , 10 , 11 ] for the Jacobi weighted Bernstein-Durrmeyer operators, [ 14 , 15 ] for the non-weighted and Jacobi weighted multivariate Bernstein-Durrmeyer operators defined on a simplex. These operators have a lot of nice properties; they commute, they commute with certain differential operators, they are self-adjoint but they only reproduce constants. Let us mention also [ 2 , 24 ] for general Durrmeyer-type modifications of Meyer-König and Zeller operators and [ 3 ] for Durrmeyer variants of the Bleimann, Butzer and Hahn operators. The Durrmeyer modification of the Bleimann, Butzer and Hahn operators are closely connected to the Bernstein-Durrmeyer operators and the Meyer-König and Zeller operators to the Baskakov-Durrmeyer operators. Due to this relation we can carry over several results which will be discussed in a separate section at the end of this paper.

The consideration of so-called genuine Baskakov-Durrmeyer type operators leads to a class of operators reproducing linear functions and interpolating at (finite) endpoints of the corresponding interval. These operators are related to the Baskakov-Durrmeyer type operators in the same way as the Baskakov type operators to their corresponding Kantorovich variants, i. e., $D^1\circ B_n\circ I_1=B_n^{(1)}$ with the notation below.

In what follows for $c\in \mathbb{R}$ we use the notations

which can be considered as a generalization of rising and falling factorials. Note that $a^{-c,\bar{j}}=a^{c,\underset{―}{j}}$ and $a^{c,\bar{j}}=a^{-c,\underset{―}{j}}$. This notation enables us to state the results for the different operators in a unified form.

In the following definitions of the operators we omit the parameter $c$ in the notations in order to reduce the necessary sub- and superscripts.

Let $c\in \mathbb{R}$, $n\in \mathbb{R}$, $n>c$ for $c\ge 0$ and $-n/c\in \mathbb{N}$ for $c<0$. Furthermore let $j\in {\mathbb{N}}_0$, $x\in I_c$ with $I_c=[0,\mathrm{\infty })$ for $c\ge 0$ and $I_c=[0,-1/c]$ for $c<0$. Then the basis functions are given by

Note that $p_{n,j}(x)\equiv 0$ for $j>-n/c$ if $c<0$ and

with the convention $p_{n,l}(x)=0$, if $l<0$.

For $c<0$ we consider the space $L_1(I_c)$ and denote by $L_1^0(I_c)$ the set of all functions $f\in L_1(I_c)$ with finite limits $f(0)=\underset{x\to 0^{+}}{lim}f(x)$ and $f(-1/c)=\underset{x\to -1/c^{-}}{lim}f(x)$ at the endpoints of the interval. For $c\ge 0$, $\alpha \ge 0$ we denote by $W_{\alpha }(I_c)$ the space of all locally integrable functions on $I_c$, satisfying for $t\ge 0$ the growth condition

for some positive constant $M$. $W_{\alpha }^0(I_c)$ consists of all functions $f\in W_{\alpha }(I_c)$ with finite limit $f(0)=\underset{x\to 0^{+}}{lim}f(x)$. Furthermore ${\mathcal{P}}_l$ denotes the set of all polynomials of degree at most $l$.

Now we can define the genuine Baskakov-Durrmeyer type operators.

Setting $c=-1$ leads to the genuine Bernstein-Durrmeyer operators first defined in [ 12 ] and independently in [ 18 ] , $c=0$ to the Phillips operators [ 34 ] , $c>0$ was investigated in [ 36 ] .

Similar as in [ 27 , 28 , 6 ] we also consider the $k$th order Kantorovich modification of the operators $B_n$, i.e.,

where $D^k$ denotes the $k$th order ordinary differential operator and

For $k=0$ we omit the superscript $(k)$ as indicated by the definition above.

This general definition contains many known operators as special cases. For $k=1$ we get the Baskakov-Durrmeyer type operators $B_n^{(1)}$ (see [ 17 ] for $c=-1$, [ 32 ] for $c=0$ and [ 22 , (1.3) ] for $c\ge 0$, named $M_{n+c}$ there) and for $k\ge 2$ the auxiliary operators $B_n^{(k)}$ considered in [ 23 , (3.5) ] (named $M_{n+c,k-1}$ there).

For $k\in \mathbb{N}$, $f\in L_1(I_c)$ for $c<0$ and $f\in W_{\alpha }(I_c)$ for $c\ge 0$, we have the explicit representation [ 23 , (3.5) ]

where the upper limit of the sum is $-\frac{n}{c}-k$ in case $c<0$, as $p_{n+ck,j}(x)\equiv 0$ for $j>-\frac{n}{c}-k$ .

In this paper we summarize known results, give an overview of different methods for the proofs and establish general results for the $k$th order Kantorovich modification concerning the commutativity properties and results for the eigenfunctions of the operators and appropriate differential operators. The proofs are mainly based on the fact that for a suitable function $g$, $s\in {\mathbb{N}}_0$, $l\in \mathbb{N}$

where

Furthermore we need that for each $k\in {\mathbb{N}}_0$

(see [ 27 , 28 , Theorem 1, Theorem 2 ] ).

2 Commutativity of the operators

First we summarize known results and give a survey over the different methods of proofs.

In 1981 Derriennic [ 13 , Th é or è me III.3 ] proved that the eigenfunctions of the Bernstein-Durrmeyer operators $B_n^{(1)}$, i. e. , $c=-1$, $k=1$, are the Legendre polynomials

with corresponding eigenvalues

and deduced the representation of the operators in terms of these eigenfunctions, i. e.,

Ditzian and Ivanov [ 16 ] remarked that from this result it follows immediately that the operators commute:

So, the proof of the commutativity is quite elegant in case $c=-1$. The general case $c<0$, $k=1$ can be proved in the same way by using the corresponding eigenfunctions and eigenvalues given in Theorem 9.

For $c=0$ we have the eigenfunction $e_0=1$, for $c>0$ certain polynomial eigenfunctions (see [ 25 , Remark 2.2, Corollary 2.5 ] ). So, the method for $c=-1$ is not applicable to the non-compact interval $[0,\mathrm{\infty })$ in case $c\ge 0$.

In [ 20 , 21 ] the author proved the commutativity for $c\ge 0$, $k=1$, $f\in L_p[0,\mathrm{\infty })$, $1\le p\le \mathrm{\infty }$ with a completely different method. Here we give an outline of the main steps of the proof. Note that the proof is also valid for $f\in W_{\alpha }(I_c)$.

• First the integral representations

  $$\begin{array}{rcl}(B_n^{(1)}{B}_m^{(1)}f)(x)& =& {\int }_0^{\mathrm{\infty }}f(y)G_{n,m}(x,y)dy,\\ (B_m^{(1)}{B}_n^{(1)}f)(x)& =& {\int }_0^{\mathrm{\infty }}f(y)G_{m,n}(x,y)dy\end{array}$$

  for all $x\in [0,\mathrm{\infty })$ were derived with the kernel functions

  $$\begin{array}{rcl}{G}_{n,m}(x,y)& =& \sum _{j=0}^{\mathrm{\infty }}\sum _{l=0}^{\mathrm{\infty }}{p}_{n+c,j}(x)p_{m+c,l}(y)(\genfrac{}{}{0ex}{}{j+l}{j})\frac{n^{c,\overline{j+1}}{m}^{c,\overline{l+1}}}{(n+m+c{)}^{c,\overline{j+l+1}}},\\ G_{m,n}(x,y)& =& \sum _{j=0}^{\mathrm{\infty }}\sum _{l=0}^{\mathrm{\infty }}{p}_{n+c,j}(y)p_{m+c,l}(x)(\genfrac{}{}{0ex}{}{j+l}{j})\frac{n^{c,\overline{j+1}}{m}^{c,\overline{l+1}}}{(n+m+c{)}^{c,\overline{j+l+1}}}.\end{array}$$

• Next, the kernel functions were considered as functions of two complex variables and it was shown that they are holomorphic in a certain region.

• The equality of the kernel functions was proved in an open neighborhood of $(0,0)$ by considering the Taylor series at $(0,0)$.

• Finally, by using the identity theorem for analytic functions, the equality of the kernel functions was established for all $x,y\in [0,\mathrm{\infty })$.

In 2005 Abel and Ivan [ 4 ] presented a nice alternative proof for the commutativity in case $c=0$. They proved that for every $f\in W_{\alpha }(I_c)$, $n,m>\alpha$ with $\frac{nm}{n+m}>\alpha$

from which the commutativity follows as a corollary.

In 2011 Tachev and the author [ 29 ] proved an analogue for the case $c=0$, $k=0$, i. e., for every $f\in W_{\alpha }^0(I_c)$, $n,m>\alpha$ with $\frac{nm}{n+m}>\alpha$

Now we generalize (5) and (6), respectively, to $k\ge 2$.

Using the definition of $B_n^{(k)}$ and applying (3) for $g=B_m^{(1)}{I}_{k-1}f$ we derive

As $B_n^{(1)}{q}_{k-2}\in {\mathcal{P}}_{k-2}$ by (4) the last term on the right hand side vanishes. Together with (5) this leads to

From Theorem 2 together with (5) and (6) we now get the commutativity of the operators $B_n^{(k)}$ for each $k\in {\mathbb{N}}_0$ in case $c=0$.

Now we consider $c\ne 0$. Since identities as given in (5), (6) and (7), respectively, are not true for $c\ne 0$, the method by Abel and Ivan is not applicable in this case. For $k=0$ we need the following result.

We only prove the case $c<0$ as the case $c>0$ is completely analogue. Using integration by parts and (1) we have

Thus, again using (1), we derive

As

we get by applying $I_1$ on both sides of (8)

With Lemma 3 and the interpolation property of the genuine operators, i. e., $(B_{n}f)(0)=(B_{m}f)(0)=f(0)$, we get

Next we consider the case $k\ge 2$.

With similar arguments as in the proof of Theorem 4 we get

3 Adapted differential operators

The operators $B_n^{(k)}$ are strongly connected to appropriate differential operators. This was used for example for the construction of quasi-interpolants (see, e.g., [ 9 , 1 , 30 , 36 ] ).

In the following we use the notation $\phi (x)=\sqrt{x(1+cx)}$.

The following recursion formula for the differential operators was proved for the special cases $c=-1$, $k=1$ also in the multivariate setting in [ 8 , (4.4) ] , for $c\ge 0$, $k=1$ in [ 7 , Lemma 4 ] , for $c=-1$, $k=0$ in [ 30 , Lemma 3 ] and for $c\ge 0$, $k=0$ in [ 36 , Lemma 2.3 ] . In view of the already known results we only have to consider $k\ge 2$. We distinguish between the cases $2\le k\le r+1$ and $k\ge r+2$.
$2\le k\le r+1$:

By using Leibniz’ formula we derive

Thus,

Furthermore,

The proposition now follows from (9) and (10).
$k\ge r+2$: By using (3) for $l=s=k-r-1$ with $g=D^{r+1}{\phi }^2{I}_{k-2}$ we derive

Again by Leibniz’ formula we get

Thus,

Furthermore

The proposition now follows from (11) and (12).

From Theorem 7 the following product formula can be easily established by induction (see [ 8 , (4.5) ] for $k=1$ also in the multivariate setting, [ 30 , Lemma 4 ] for $c=-1$, $k=0$ and [ 36 , Lemma 2.4 ] for $c\in \mathbb{R}$, $k=0$).

For the special case $c=0$ this means

The commutativity of the differential operators now follows as a corollary.

4 Spectral properties

Next we generalize results concerning the spectral properties of the operators $B_n^{(k)}$ and the differential operators. For $B_n^{(k)}$ the special case $k=1,c=-1$ was considered in [ 13 , Th é or è me III.3 ] , for $c=-1,k=0$ see [ 19 , Theorem 4 ] , for $k=1,c=1$ [ 24 , Corollary 2.5 ] and for $k=0,c\ne 0$ [ 36 , Lemma 1.16 ] . References concerning the differential operators are [ 8 , Theorem 4 ] , [ 9 , (2.1), (2.2) ] for $c=-1$, $k=1$ (also in the Jacobi weighted multivariate setting) and [ 36 , Lemmas 2.2, 2.3, 2.4 ] .

First we consider $B_n^{(k)}$. We use the known results for $k=0$. For $k=1,l=0$ we have

as $B_n^{(1)}$ preserves constants.

Now let $k\in \mathbb{N}$, $l\in {\mathbb{N}}_0$ with $l+k\ge 2$. Then, again using (3) and (4),

Next we treat the differential operators. With

we derive

5 Commutativity of the operators and appropriate differential operators

In [ 23 , Lemma 3.1 ] the author proved that the operators $B_n^{(1)}$ and the differential operators ${\tilde{D}}^{2r,(1)}$ commute for sufficiently smooth functions. The corresponding result for $c=0$, $k=0$ was proved in [ 29 , Theorem 3.2, Remark 3.1 ] and was generalized for $c\in \mathbb{R}$, $k=0$ in [ 36 , Satz 2.8 ] .

With regard to the above mentioned known results we only have to treat the case $k\ge 2$ and prove our proposition by induction.
$r=1$: Using (3) with $l=k-2$ and $g=B_n^{(1)}{I}_{k-1}f$ if $k\ge 3$ we get

The conclusion $r\Rightarrow r+1$ follows easily from (13).

6 Related Durrmeyer type operators

In this section we consider $c\ne 0$. Let

The consideration of

leads to $k$th order Kantorovich modifications of Durrmeyer type variants of Bleimann, Butzer and Hahn operators (BBH-D operators) for $c<0$ and Meyer-König and Zeller operators (MKZ-D operators) for $c>0$.
With the notation

they are explicitly given by the following formulas.

For $c<0$, $n\in {\mathbb{R}}^{+}$, $-n/c\in \mathbb{N}$, $(1-c\cdot {)}^{-2}f(\cdot )\in L_1[0,\mathrm{\infty })$ with finite limits $f(0)=\underset{x\to 0^{+}}{lim}f(x)$ and $f_{\mathrm{\infty }}=\underset{x\to \mathrm{\infty }}{lim}f(x)$

$x\in [0,\mathrm{\infty })$, we have a genuine variant of BBH-D operators.
For $c>0$, $\alpha \ge 0$, $n\in {\mathbb{R}}^{+}$, $n>\alpha -c$, $f$ locally integrable on $[0,\frac{1}{c})$ satisfying $|f(t)|\le M(1-ct{)}^{-\frac{\alpha }{c}}$, $t\in [0,\frac{1}{c})$, and possessing a finite limit $f(0)=\underset{x\to 0^{+}}{lim}f(x)$

$x\in [0,\frac{1}{c})$, defines a genuine variant of MKZ-D operators.
For the $k$th order Kantorovich modification we derive for $f$ as above without the conditions for the limits with $c\ne 0$, $k\in \mathbb{N}$:

where the upper limit of the sum is $-\frac{n}{c}-k$ for $c<0$.

From the results in Section 2 we deduce that the operators ${\bar{B}}_n^{(k)}$ are commutative. For the special case $k=1,c=-1$ see [ 3 , Theorem 2.1 ] and for $k=1,c=1$ [ 26 , Theorem 1 ] . Furthermore they commute with the differential operators

where, with $\bar{\phi }(x)=\frac{\sqrt{x}}{1-cx}$,

and

From Section 4 we get the eigenfunctions

for the operators ${\bar{B}}_n^{(k)}$ and the differential operators ${\bar{D}}^{2r,(k)}$.