Commutativity and Spectral Properties of Genuine Baskakov-Durrmeyer Type Operators and their $k$th order Kantorovich Modification$^\ast $

Margareta Heilmann$^\S $

January 20, 2016.

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

In this paper we present an overview of commutativity results and different methods for the proofs for Baskakov-Durrmeyer type operators and associated differential operators. We discuss the spectral properties and generalize all results to $k$th order Kantorovich modifications and corresponding Durrmeyer type variants of Bleimann, Butzer and Hahn operators and Meyer-König and Zeller operators.

MSC. 41A36, 41A28

Keywords. Positive linear operators, Durrmeyer type operators, Kantorovich type modification, commutativity, differential operators, spectral properties.

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

\[ a^{c,\overline{j}} := \prod _{l=0}^{j-1} (a+cl) , \; a^{c,\underline{j}} := \prod _{l=0}^{j-1} (a-cl) , \; j \in \mathbb {N}; \quad a^{c,\overline{0}}= a^{c,\underline{0}} :=1 \]

which can be considered as a generalization of rising and falling factorials. Note that $ a^{-c,\overline{j}} = a^{c,\underline{j}}$ and $ a^{c,\overline{j}} = a^{-c,\underline{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 $c \in \mathbb {R}$, $n \in \mathbb {R}$, $n {\gt} c$ for $c\geq 0$ and $-n/c \in \mathbb {N}$ for $c{\lt}0$. Furthermore let $j \in \mathbb {N}_0$, $x \in I_c$ with $I_c = [0,\infty )$ for $c\geq 0$ and $I_c=[0,-1/c]$ for $c {\lt} 0$. Then the basis functions are given by

\[ p_{n,j}(x) = \left\{ \begin{array}{cl} \displaystyle \frac{n^j}{j!} x^j e^{-nx} & , \, c = 0 ,\\ \displaystyle \frac{n^{c,\overline{j}}}{j!} x^j (1+cx)^{-\left( \frac{n}{c}+j\right)} & , \, c \not= 0 . \end{array} \right. \]

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

\begin{eqnarray} \label{eq0.7} p_{n,j}’ (x) & = & n [ p_{n+c,j-1} (x) - p_{n+c,j} (x)] \end{eqnarray}

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

For $c{\lt}0$ 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 $f(0) = \lim _{x \to 0^+} f(x)$ and $f(-1/c) = \lim _{x \to -1/c^-} f(x)$ at the endpoints of the interval. For $c \geq 0$, $ \alpha \geq 0$ we denote by $W_\alpha (I_c)$ the space of all locally integrable functions on $I_c$, satisfying for $ t \geq 0$ the growth condition

\[ |f(t)| \leq M e^{\alpha t} \mbox{ if } c=0 \mbox{ and } |f(t)| \leq M (1+ct)^{\frac{\alpha }{c} } \mbox{ if } c{\gt}0 \]

for some positive constant $M$. $W_\alpha ^0 (I_c)$ consists of all functions $ f \in W_\alpha (I_c)$ with finite limit $f(0) = \lim _{x \to 0^+} f(x)$. 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.

Definition 1

For $c{\lt}0$, $n \in \mathbb {R}^+$, $ -n/c \in \mathbb {N}$, $f \in L_1^0 (I_c)$ define

\begin{eqnarray*} (B_{n} f)(x) & = & f(0) p_{n,0}(x) + f\left( -\tfrac {1}{c} \right) p_{n,-\frac{n}{c}} (x) \\ & & + (n+c) \sum _{j=1}^{-\frac{n}{c}-1} p_{n,j}(x) \int _0^{-\frac{1}{c}} p_{n+2c,j-1} (t) f (t) dt , \, x \in \left[ 0, -\tfrac {1}{c} \right]. \end{eqnarray*}

For $c\geq 0$, $ \alpha \geq 0$, $n \in \mathbb {R}^+$, $n {\gt} \alpha -c $, $f \in W_\alpha ^0 (I_c) $ define

\begin{eqnarray*} (B_{n} f)(x) & = & f(0) p_{n,0}(x) \\ & & +(n+c) \sum _{j=1}^{\infty } p_{n,j}(x) \int _0^{\infty } p_{n+2c,j-1} (t) f (t) dt , \, x \in [0,\infty ). \end{eqnarray*}

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 ] , $c{\gt}0$ 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.,

\begin{equation} \label{eq0.1} B_{n}^{(k)}:=D^k \circ B_{n}\circ I_k, \, k \in \mathbb {N}_0, \end{equation}

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

\[ I_k f = f, \mbox{ if } k=0, \mbox{ and } (I_k f)(x) = \int _0^x \frac{(x-t)^{k-1}}{(k-1)!} f(t) dt, \mbox{ if } k \in \mathbb {N}. \]

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 $c \geq 0$, named $M_{n+c}$ there) and for $k \geq 2$ 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 $c {\lt} 0$ and $f \in W_\alpha (I_c)$ for $c \geq 0$, we have the explicit representation [ 23 , (3.5) ]

\[ (B_{n}^{(k)} f)(x) = \frac{n^{c,\overline{k}}}{n^{c,\underline{k-1}}} \sum _{j=0}^{\infty } p_{n+ck,j} (x) \int _{I_c} p_{n-c(k-2),j+k-1} (t) f(t) dt , \]

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

\begin{equation} \label{eq1.2} I_{l} D^{s} g = \left\{ \begin{array}{ccc} D^{s-l}g - q_{l-1} & , & s \geq l, \\ I_{l-s} g -q_{l-1} & , & s \leq l, \end{array} \right. \end{equation}

where

\[ q_{l-1} (x) = \sum _{i=\max \{ 0,l-s\} }^{l-1} \frac{g^{(i+s-l)}(0)}{i!}x^{i} \in \mathcal{P}_{l-1}. \]

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

\begin{equation} \label{eq1.3} p \in \mathcal{P}_{l} \quad \Longrightarrow \quad B_n^{(k)} p \in \mathcal{P}_{l} \end{equation}

(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

\[ Q_0(x) =1, \, Q_l (x) = \tfrac {\sqrt{2l+1}}{l!} D^l \left[ x^l(1-x)^l \right], \, l \in \mathbb {N}, \]

with corresponding eigenvalues

\[ \lambda _{n,l} = \left\{ \begin{array}{ccc} \displaystyle \tfrac {n!(n-1)!}{(n-l-1)!(n+l)!} & , & l \leq n-1,\\ 0 & , & l\geq n , \end{array} \right. \]

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

\[ (B_n^{(1)} f) (x) = \sum _{l=0}^{n-1} \lambda _{n,l} Q_l (x) \int _0^1 Q_l(t) f(t) dt, \, f \in L_1[0,1]. \]

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

\[ B_m^{(1)} B_n^{(1)} f = B_n^{(1)} B_m^{(1)} f = \sum _{l=0}^{\min {\{ m-1,n-1\} }} \lambda _{m,l} \lambda _{n,l} Q_l \int _0^1 Q_l(t) f(t) dt . \]

So, the proof of the commutativity is quite elegant in case $c=-1$. The general case $c{\lt}0$, $ 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 $c{\gt}0$ 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 $[0, \infty )$ in case $c\geq 0$.

In [ 20 , 21 ] the author proved the commutativity for $c \geq 0$, $k=1$, $ f \in L_p[0,\infty )$, $1 \leq p \leq \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{eqnarray*} (B_n^{(1)} B_m^{(1)} f ) (x) & = & \int _0^\infty f(y) G_{n,m} (x,y) dy, \\ (B_m^{(1)} B_n^{(1)} f ) (x) & = & \int _0^\infty f(y) G_{m,n} (x,y) dy \end{eqnarray*}

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

\begin{eqnarray*} G_{n,m} (x,y) & = & \sum _{j=0}^\infty \sum _{l=0}^\infty p_{n+c,j} (x) p_{m+c,l} (y) \textstyle {j+l \choose 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}^\infty \sum _{l=0}^\infty p_{n+c,j} (y) p_{m+c,l} (x) \textstyle {j+l \choose j} \frac{n^{c,\overline{j+1}}m^{c,\overline{l+1}}}{(n+m+c)^{c,\overline{j+l+1}}} . \end{eqnarray*}
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 $x,y \in [0,\infty )$.

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 {\gt} \alpha $ with $\frac{nm}{n+m} {\gt} \alpha $

\begin{equation} \label{eq1.1} B_{n}^{(1)} B_m^{(1)} f = B_{\frac{nm}{n+m}}^{(1)} f \end{equation}

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 {\gt} \alpha $ with $\frac{nm}{n+m} {\gt} \alpha $

\begin{equation} \label{eq1.1a} B_{n} B_m f = B_{\frac{nm}{n+m}} f . \end{equation}

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

Theorem 2

Let $c=0$, $ k \geq 2$, $f \in W_\alpha (I_c)$, $\alpha \geq 0$, $n,m {\gt} \alpha $ with $\frac{nm}{n+m} {\gt} \alpha $. Then

\begin{equation} \label{eq1.4} B_{n}^{(k)} B_m^{(k)} f = B_{\frac{nm}{n+m}}^{(k)} f \end{equation}

Proof â–¼

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

\begin{eqnarray*} B_n^{(k)} B_m^{(k)} f & = & D^{k-1} B_n^{(1)} I_{k-1} D^{k-1} B_m^{(1)} I_{k-1} f \\ & = & D^{k-1} B_n^{(1)} B_m^{(1)} I_{k-1} f - D^{k-1} B_n^{(1)} q_{k-2} . \end{eqnarray*}

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

\[ B_n^{(k)} B_m^{(k)} f = D^{k-1} B_{\frac{nm}{n+m}}^{(1)} I_{k-1} f = B_{\frac{nm}{n+m}}^{(k)} f . \]

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

Lemma 3

For $c{\lt}0$ let $ n \in \mathbb {R}^+ $, $ -n/c \in \mathbb {N}$, $f \in L_1^0(I_c)$ such that $D^1 f \in L_1 (I_c)$. For $c{\gt}0$, $\alpha \geq 0$ let $n \in \mathbb {R}^+$, $n {\gt} \alpha -c$, $f \in W_\alpha ^0 (I_c)$ such that $D^1f \in W_\alpha (I_c)$. Then

\[ B_nf = f(0) +I_1 B_n^{(1)}D^1 f . \]

Proof â–¼

We only prove the case $c{\lt}0$ as the case $c {\gt} 0$ is completely analogue. Using integration by parts and (1) we have

\begin{eqnarray*} {\int _0^{-1/c} p_{n+c,j}(t) f’(t) dt} & = & - (n+c) \int _0^{-1/c} [p_{n+2c,j-1} (t) - p_{n+2c,j} (t) ] f(t) dt \\[3mm]& & \qquad + \left\{ \begin{array}{ccl} f\left( -\frac{1}{c} \right) & , & j= -\frac{n}{c} -1, \\[3mm] -f(0) & , & j=0, \\[3mm] 0 & , & 1 \leq j \leq -\frac{n}{c} -2. \end{array} \right. \end{eqnarray*}

Thus, again using (1), we derive

\begin{eqnarray} \label{eq1.5} \lefteqn{(B_n^{(1)} D^1 f)(x)} \\ \nonumber & = & n \left[ f\left( -\tfrac {1}{c} \right) p_{n+c,-\frac{n}{c}-1}(x)-f(0) p_{n+c,0}(x) \right] \\ \nonumber & & - n \sum _{j=0}^{-\frac{n}{c}-1} p_{n+c,j} (x) (n+c) \int _0^{-1/c} (p_{n+2c,j-1} (t) - p_{n+2c,j} (t) ) f(t) dt \\ \nonumber & = & n \left[ f\left( -\frac{1}{c} \right) p_{n+c,-\frac{n}{c}-1}(x)-f(0) p_{n+c,0}(x) \right] \\ \nonumber & & + (n+c) \sum _{j=1}^{-\frac{n}{c}-1} p’_{n,j} (x) \int _0^{-1/c} p_{n+2c,j-1} (t) f(t) dt . \end{eqnarray}

\begin{eqnarray*} \int _0^x p_{n+c,-\tfrac {n}{c}-1}(u) du & = & \tfrac {1}{n} (-cx)^{-\frac{n}{c}} = \tfrac {1}{n} p_{n,-\frac{n}{c}} (x), \\ \int _0^x p_{n+c,0}(u) du & = & \tfrac {1}{n} \left[1 - (1+cx)^{-\frac{n}{c}} \right] = \tfrac {1}{n} (1-p_{n,0}(x)), \end{eqnarray*}

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

\begin{eqnarray*} (I_1 B_n^{(1)} D^1 f)(x) & = & f\left( -\tfrac {1}{c} \right) p_{n,-\frac{n}{c}} (x) -f(0) (1-p_{n,0}(x)) \\ & & + (n+c) \sum _{j=1}^{-\frac{n}{c}-1} p_{n,j} (x) \int _0^{-1/c} p_{n+2c,j-1} (t) f(t) dt \\ & = & -f(0) + (B_n f)(x). \end{eqnarray*}

Theorem 4

With the same assumptions as in Lemma 3 we have

\[ B_n B_m f = B_m B_n f . \]

Proof â–¼

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

\begin{eqnarray*} B_n B_m f & = & f(0) + I_1 B_n^{(1)} D^1 I_1 B_m^{(1)} D^1 f \\ & = & f(0) + I_1 B_n^{(1)} B_m^{(1)} D^1 f \\ & = & f(0) + I_1 B_m^{(1)} B_n^{(1)} D^1 f \\ & = & B_m B_n f . \end{eqnarray*}

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

Theorem 5

Let $ k \in \mathbb {N}$, $k \geq 2$. For $c{\lt}0$ let $ n \in \mathbb {R}^+ $, $ -n/c \in \mathbb {N}$, $f \in L_1(I_c)$. For $c{\gt}0$, $\alpha \geq 0$ let $n \in \mathbb {R}^+$, $n {\gt} \alpha -c$, $f \in W_\alpha (I_c)$. Then

\[ B_n^{(k)} B_m^{(k)} f = B_m^{(k)} B_n^{(k)} f . \]

Proof â–¼

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

\begin{eqnarray*} B_n^{(k)} B_m^{(k)} f & = & D^{k-1} B_n^{(1)} I_{k-1} D^{k-1} B_m^{(1)} I_{k-1} f \\ & = & D^{k-1} B_n^{(1)} B_m^{(1)} I_{k-1} f - D^{k-1} B_n^{(1)} q_{k-2} \\ & = & D^{k-1} B_m^{(1)} B_n^{(1)} I_{k-1} f \\ & = & B_m^{(k)} B_n^{(k)} f . \end{eqnarray*}

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 $\varphi (x) = \sqrt{x(1+cx)}$.

Definition 6

For $ r \in \mathbb {N}$ we define

\[ \widetilde{D}^{2r,(k)} = \left\{ \begin{array}{lcc} \displaystyle D^{r-1+k} \varphi ^{2r} D^{r+1-k} & , & k \leq r+1, \\ \displaystyle D^{r-1+k} \varphi ^{2r} I_{k-r-1} & , & k \geq r+1. \end{array} \right. . \]

Formally we denote $ \widetilde{D}^{0,(k)} = Id $.

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 $c \geq 0$, $k=1$ in [ 7 , Lemma 4 ] , for $c=-1$, $k=0$ in [ 30 , Lemma 3 ] and for $ c \geq 0$, $k=0$ in [ 36 , Lemma 2.3 ] .

Theorem 7

For $ r \in \mathbb {N}_0$ we have

\[ \widetilde{D}^{2r+2,(k)} = \widetilde{D}^{2r,(k)} \left[ \widetilde{D}^{2,(k)} -cr(r+1) Id \right] . \]

Proof â–¼

In view of the already known results we only have to consider $ k \geq 2$. We distinguish between the cases $ 2 \leq k \leq r+1$ and $ k \geq r+2$.
$ 2 \leq k \leq r+1$:

\begin{eqnarray*} \widetilde{D}^{2r,(k)} \widetilde{D}^{2,(k)} & = & D^{r+k-1} \varphi ^{2r} D^{r-k+1} D^k \varphi ^2 I_{k-2} \\ & = & D^{r+k-1} \varphi ^{2r} D^{r+1} \varphi ^2 I_{k-2} . \end{eqnarray*}

By using Leibniz’ formula we derive

\begin{eqnarray*} & & D^{r+1} \varphi ^2 I_{k-2} \\ & = & \sum _{l=0}^{r+1} \textstyle {r+1 \choose l} \left( D^l \varphi ^2 \right) \left( D^{r+1-l} I_{k-2} \right) \\ & = & \varphi ^2 D^{r+1} I_{k-2} + (r+1) \left( D \varphi ^2 \right) \left( D^{r} I_{k-2} \right) + cr(r+1) \left( D^{r-1} I_{k-2} \right) \\ & = & \varphi ^2 D^{r+3-k}+(r+1) \left( D \varphi ^2 \right) D^{r+2-k} + cr(r+1) D^{r+1-k} . \end{eqnarray*}

Thus,

\begin{eqnarray} \label{eq3.1} \lefteqn{\widetilde{D}^{2r,(k)} \widetilde{D}^{2,(k)}} \\ \nonumber & = & D^{r+k-1}\varphi ^{2r+2} D^{r+3-k} +(r+1) D^{r+k-1} \varphi ^{2r} \left( D \varphi ^2 \right) D^{r+2-k} \\ \nonumber & & + cr(r+1) D^{r+k-1} \varphi ^{2r} D^{r+1-k} . \end{eqnarray}

Furthermore,

\begin{eqnarray} \label{eq3.2} \widetilde{D}^{2r+2,(k)} & = & D^{r+k-1} D \varphi ^{2r+2} D^{r+2-k} \\ \nonumber & = & D^{r+k-1} \left[ (r+1) \varphi ^{2r} (D \varphi ^2 ) D^{r+2-k} + \varphi ^{2r+2} D^{r+3-k} \right] . \end{eqnarray}

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

\begin{eqnarray*} \widetilde{D}^{2r,(k)} \widetilde{D}^{2,(k)} & = & D^{r+k-1} \varphi ^{2r} I_{k-r-1} D^{k-r-1} D^{r+1} \varphi ^2 I_{k-2} \\ & = & D^{r+k-1} \varphi ^{2r} D^{r+1} \varphi ^2 I_{k-2} . \end{eqnarray*}

Again by Leibniz’ formula we get

\begin{eqnarray*} & & D^{r+1} \varphi ^2 I_{k-2}= \\ & = & \varphi ^2 D^{r+1} I_{k-2} + (r+1) \left( D \varphi ^2 \right) \left( D^{r} I_{k-2} \right) + cr(r+1) \left( D^{r-1} I_{k-2} \right) \\ & = & \varphi ^2 D I_{k-2-r}+(r+1) \left( D \varphi ^2 \right) I_{k-2-r} + cr(r+1) I_{k-r-1} . \end{eqnarray*}

Thus,

\begin{eqnarray} \label{eq3.3} \lefteqn{\widetilde{D}^{2r,(k)} \widetilde{D}^{2,(k)}=} \\ \nonumber & = & D^{r+k-1}\varphi ^{2r+2} D I_{k-2-r} +(r+1) D^{r+k-1} \varphi ^{2r} \left( D \varphi ^2 \right) I_{k-2-r} \\ \nonumber & & + cr(r+1) D^{r+k-1} \varphi ^{2r} I_{k-r-1} . \end{eqnarray}

Furthermore

\begin{eqnarray} \label{eq3.4} \widetilde{D}^{2r+2,(k)} & = & D^{r+k-1} D \varphi ^{2r+2} I_{k-r-2} \\ \nonumber & = & D^{r+k-1} \left[ (r+1) \varphi ^{2r} (D \varphi ^2 ) I_{k-r-2} + \varphi ^{2r+2} D I_{k-r-2} \right] . \end{eqnarray}

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 $c \in \mathbb {R}$, $k=0$).

\begin{eqnarray} \label{nr-neu} \widetilde{D}^{2r,(k)} & = & \prod _{j=0}^{r-1} \left[ \widetilde{D}^{2,(k)} -j(j+1)c Id \right] \\ \nonumber & = & \widetilde{D}^{2,(k)} \circ \left( \widetilde{D}^{2,(k)} -2c Id \right) \circ \dots \circ \left( \widetilde{D}^{2,(k)} -(r-1)rc Id \right). \end{eqnarray}

For the special case $c=0$ this means

\[ \widetilde{D}^{2r,(k)} = \left( \widetilde{D}^{2,(k)} \right)^r . \]

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

Corollary 8

Let $r,l \in \mathbb {N}$ , $ k \in \mathbb {N}_0$. Then

\[ \widetilde{D}^{2r,(k)} \widetilde{D}^{2l,(k)} = \widetilde{D}^{2l(k)} \widetilde{D}^{2r,(k)} . \]

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 $k=0, c\not= 0$ [ 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 ] .

Theorem 9

For $c \not= 0$, $l \in \mathbb {N}_0$ and $ n {\gt} c(l+k-1)$ in case $c{\gt}0$ it holds

\[ B_n^{(k)} g_{l,k} = \lambda _{n,l,k} g_{l,k} \mbox{ and } \widetilde{D}^{2r,(k)} g_{l,k} = \gamma _{n,l,k} g_{l,k}, \]

where

\[ g_{0,0} (x) = 1 , \; g_{1,0} (x) = x, \; g_{l,k} (x) = D^{l+2(k-1)} \varphi ^{2(l+k-1)}, \; l+2(k-1) \geq 0 \]

and

\[ \lambda _{n,l,k} = \frac{n^{c,\overline{l+k}}}{n^{c,\underline{l+k}}} , \quad \gamma _{r,l,k} = \left\{ \begin{array}{ccc} \displaystyle c^r \frac{(l+k+r-1)!}{(l+k-r-1)!} & , & l+k-1 \geq r, \\ 0 & , & l+k-1 {\lt} r. \end{array} \right. \]

Proof â–¼

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

\[ g_{0,1}=1 \mbox{ and } B_n^{(1)} g_{0,1} = g_{0,1} \]

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

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

\begin{eqnarray*} B_n^{(k)} g_{l,k} & = & D^k B_n I_k D^{l+2(k-1)} \varphi ^{2(l+k-1)} \\ & = & D^k B_n D^{l+k-2} \varphi ^{2(l+k-1)} \\ & = & D^k B_n g_{l+k,0} \\ & = & \frac{n^{c,\overline{l+k}}}{n^{c,\underline{l+k}}} g_{l,k}. \end{eqnarray*}

Next we treat the differential operators. With

\[ \gamma _{r,l,k} = \gamma _{r,l+k,0} \mbox{ and } g_{l,k} = D^k g_{l+k,0} \]

we derive

\begin{eqnarray*} \gamma _{r,l,k} g_{l,k} & = & D^k \gamma _{r,l+k,0} g_{l+k,0} \\ & = & D^k \widetilde{D}^{2r,(0)} g_{l+k,0} \\ & = & D^{k+r-1} \varphi ^{2r} D^{r+1} g_{l+k,0} \\ & = & \left\{ \begin{array}{cr} D^{k+r-1} \varphi ^{2r} D^{r+1-k} D^k g_{l+k,0} , & k \leq r+1 \\ D^{k+r-1} \varphi ^{2r} I_{k-r-1} D^k g_{l+k,0} , & k \geq r+1 \end{array} \right. \\ & = & \widetilde{D}^{2r,k} g_{l,k}. \end{eqnarray*}

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 $ \widetilde{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 ] .

Theorem 10

For $k \geq 2$ we have

\[ \widetilde{D}^{2r,(k)} B_n^{(k)} = B_n^{(k)} \widetilde{D}^{2r,(k)} . \]

Proof â–¼

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

\begin{eqnarray*} \widetilde{D}^{2,(k)} B_n^{(k)} f & = & D^{k-1} D \varphi ^2 D B_n^{(1)} I_{k-1} f \\ & = & D^{k-1} B_n^{(1)} D \varphi ^2 D I_{k-1} f \\ & = & D^{k-1} B_n^{(1)} D \varphi ^2 I_{k-2} f \\ & = & D^{k-1} B_n^{(1)} I_{k-1} D^k \varphi ^2 I_{k-2} f \\ & = & B_n^{(k)} \widetilde{D}^{2,(k)} . \end{eqnarray*}

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

6 Related Durrmeyer type operators

In this section we consider $c \not=0$. Let

\begin{eqnarray*} \sigma : I_c \longrightarrow I_{-c} & \qquad & \sigma (x) = \frac{x}{1+cx} \\ \psi : I_c \longrightarrow I_{-c} & \qquad & \psi (x) = \frac{x}{1-cx} . \end{eqnarray*}

The consideration of

\[ \left(\overline{B}_n^{(k)} f (t) \right)(x) := \left(B_n^{(k)} f(\sigma (t)) \right)(\psi (x)) \]

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

\[ \overline{p}_{n,j}(x) := p_{n,j} \left( \psi (x) \right) = \frac{n^{c,\overline{j}}}{j!} x^j (1-cx)^{\frac{n}{c}} \]

they are explicitly given by the following formulas.

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

\begin{eqnarray*} (\overline{B}_n f)(x) & = & f(0) \overline{p}_{n,0}(x) + f_{\infty } \overline{p}_{n,-\frac{n}{c}} (x) \\ & & +(n+c) \sum _{j=1}^{-\frac{n}{c}-1} \overline{p}_{n,j} (x) \int _{I_{-c}} \overline{p}_{n+2c,j-1} (t) f(t) (1-ct)^{-2} dt, \end{eqnarray*}

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

\begin{eqnarray*} (\overline{B}_n f)(x) & = & f(0) \overline{p}_{n,0} \\ & & +(n+c) \sum _{j=1}^{\infty } \overline{p}_{n,j} (x) \int _{I_{-c}} \overline{p}_{n+2c,j-1} (t) f(t) (1-ct)^{-2} dt, \end{eqnarray*}

$ 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 $c\not=0$, $k \in \mathbb {N}$:

\begin{eqnarray*} (\overline{B}_n^{(k)} f)(x) & = & \frac{n^{c,\overline{k}}}{n^{c,\underline{k-1}}} \sum _{j=0}^{\infty } \overline{p}_{n+ck,j} (x) \int _{I_{-c}} \overline{p}_{n-c(k-2),j+k-1} (t) f(t) (1-ct)^{-2} dt \end{eqnarray*}

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

From the results in Section 2 we deduce that the operators $\overline{B}_n^{(k)}$ 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

\[ \overline{D}^{2r,(k)} = \left\{ \begin{array}{lcc} \displaystyle \widehat{D}^{r-1+k} \overline{\varphi }^{2r} \widehat{D}^{r+1-k} & , & k \leq r+1, \\ \displaystyle \widehat{D}^{r-1+k} \overline{\varphi }^{2r} \widehat{I}_{k-r-1} & , & k \geq r+1, \end{array} \right. \]

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

\[ (\widehat{D} f)(x) = (1-cx)^{-2} f'(x) , \, \widehat{D}^m f = \widehat{D}^{m-1} (\widehat{D} f) \]

and

\[ \widehat{(I} f)(x) = \int _0^x \frac{f(t)}{(1-ct)^2} dt, \, \widehat{I}_m f = \widehat{I}_{m-1} (\widehat{I} f) . \]

From Section 4 we get the eigenfunctions

\[ \overline{g}_{0,0} (x) = 1 , \; \overline{g}_{1,0} (x) = \frac{x}{1-cx} , \; \overline{g}_{l,k} (x) = \widehat{D}^{l+2(k-1)} \overline{\varphi }^{2(l+k-1)}, \; l+2(k-1) \geq 0 \]

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

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This work is licensed under a Creative Commons Attribution 4.0 International License.

Commutativity and Spectral Properties of Genuine Baskakov-Durrmeyer Type Operators and their \(k\)th order Kantorovich Modification\(^\ast \)

1 Introduction and definition of the operators

2 Commutativity of the operators

3 Adapted differential operators

4 Spectral properties

5 Commutativity of the operators and appropriate differential operators

6 Related Durrmeyer type operators

Bibliography