Dedicated to Professor Gabriel V. Orman

${}^{\ast }$First published in “Mathematics and its Applications” (Proc. 17th Scientific Session; ed. by G.V. Orman), 1–28, Braşov, Editura Universităţii “Transilvania”, 2003.

${}^{\ast \ast }$University of Duisburg-Essen, Faculty of Mathematics, Bismarckstr. 90, D-47057 Duisburg, Germany, e-mail: heiner.gonska@uni-due.de.

1 Introductory remarks and notations

Products of parametric extensions of $d$ univariate operators are appropriate tools to approximate functions defined on the product of $d$ compact metric spaces. We will deal in this paper with twodimensional tensor products over compact real intervals. These are used in Computer Aided Geometric Design (CAGD) to construct approximating or interpolating free form surfaces (see [ 21 ] , [ 42 ] , [ 7 ] and the references therein). According to Schumakers classification (see [ 66 ] ) the tensor product of two univariate operators can be viewed as a simple two-stage process. For the case of $d$ dimensions see, e.g., [ 29 ] , [ 33 ] where, however, no assertions concerning simultaneous approximation were made. This paper is a continuation of our recent paper [ 3 ] and focusses on applications. A more detailed presentation including full proofs was given in our technical report [ 2 ] .

The tensor product method is a restrictive one, especially because of the requirement that the information must be given in tensor product form, e.g., in the case of discretely defined operators on a Cartesian product grid. We also have to recall that some directions are preferred on a tensor product surface, and if these directions are not properly chosen, then the obtained surface might not be “good enough”.

Despite of these limitations one should use the tensor product method as often as it makes sense; as de Boor mentions in his book [ 16 ] , this method is extremely efficient when it comes to implement it in comparison with other surface approximation techniques. Furthermore, two recent preprints by Höllig, Reif and Wipper [ 43 ] , [ 44 ] show in an impressive way that the tensor product method is not dead at all. The method also remains useful in fitting functions given on the sphere. This was demonstrated in a talk given by Sbibih [ 64 ] at the 2001 International Conference on Numerical Algorithms (Marrakesh, Morocco).

The method of using the product of parametric extensions of univariate operators is quite a classic one. It was first used to a larger extent in the context of multivariate polynomial interpolation. Much historical information and many early references on the subject can be found in an article by Gasca and Sauer [ 23 ] . Here we mention in particular the 1926 paper by Neder [ 58 ] . To our knowledge the tensor product of the (noninterpolatory) Bernstein polynomial operators was first considered in 1933 by Hildebrandt and Schoenberg [ 41 ] . An early use of the tensor product method in combination with spline interpolation was made in de Boors 1962 article [ 14 ] on bicubic splines and–almost simultaneously–by Ferguson [ 22 ] . Both authors were guided by the work of Birkhoff and Garabedian [ 6 ] .

The tensor product of two linear operators was considered for example by Stancu [ 74 ] in 1964. For further references and results concerning the method (and other multivariate ones) see, e.g., a paper by Coatmelec [ 11 ] , the dissertation of Haussmann [ 39 ] , an article of Haussmann and Pottinger [ 40 ] , and the ones by Mastroianni [ 54 ] and Lancaster [ 49 ] . Many other notes treated the method as well. Our bibliography lists several of these, although our paper does not refer to them explicitely; there is no claim for completeness at all.

A recent deep result concerning $d$-variate tensor product Bernstein operators was given by Xinlong Zhou who proved a strong converse inequality (see [ 87 ] ). Similar results for $d$-variate tensor product Jackson operators were proved by Knoop and Zhou [ 48 ] .

For the bivariate case the method of parametric extensions can be described as follows (see [ 37 ] , [ 17 ] , [ 19 ] ); de Boor calls this the naive approach. We emphasize that this approach is the only one which is of interest in this note.

Let $I$ and $J$ be nontrivial compact intervals of the real axis $\mathbb{R}$. For $i=1,2$ let $G_i$ be linear subspaces of ${\mathbb{R}}^I$ $=\{g:I\ni x\to g(x)\in \mathbb{R}\}$, and $H_i$ be such of ${\mathbb{R}}^J=\{h:J\ni y\to h(y)\in \mathbb{R}\}$. We consider linear operators

For $f:I$ $\to \mathbb{R}$we define the partial functions $f_x:J\to \mathbb{R}$ and $f_y:I\to \mathbb{R}$ by

Furthermore, putting

the parametric extension of $L$ is given by ${}_{x}L:F\to {\mathbb{R}}^{I\times J}$,

Symmetrically, the parametric extension of $M$ has the form ${}_{y}M:F\to {\mathbb{R}}^{I\times J}$,

We assume, furthermore, that ${}_{x}L:F\to F$ and

so that the linear operators

are both defined, mapping $F$ into itself. The product of these parametric extensions is called the tensor product of $L$ and $M$ ($M$and $L$, respectively).

An answer to the question under which conditions the parametric extensions commute is known in certain cases. For example, Gordon and Cheney (see Theorem 9 in [ 37 ] ) and, independently Potapov and Jimenez Pozo (see Theorem 1 in [ 61 ] ) observed that this is so if $G_1=G_2=C(X),H_1=H_2=C(Y),F=C(X\times Y)$, where $X$ and $Y$ are compact metric spaces, and if $L:C(X)\to C(X),M:C(Y)\to C(Y)$ are continuous linear operators with respect to the sup norms. Other conditions are discussed by de Boor [ 17 , pp. 50–52 ] , and by Lancaster [ 49 , Th. 3 ] , for example.

The situation relevant to us is described as follows: For $k,l\in {\mathbb{N}}_0:=\mathbb{N}\cup \{0\}$, let the symbol $D^{(k,l)}$ denote the partial differential operator ${\partial }^{k+l}/\partial x^k\partial y^l;$ occasionally we will write $f^{(k,l)}$ instead of $D^{(k,l)}f$ . We define

The notation $(k,l)\le (p,q)$ indicates that $k\le p$ and $l\le q$. The corresponding symbols used for the univariate case will be $C^p\left(I\right),D^k,d^k/d{x}^k$ and $f^{\left(k\right)}$. For $p=q=0$ we obtain the spaces $C(I),C(J)$ and $C(I\times J)$ of real-valued functions which are continuous on $I,J$ and $I\times J$, respectively. Similarly, $D^0$ and $D^{(0,0)}$ mean the one and two dimensional identity operators.

Although we will not restrict our attention to these, many of our univariate building blocks will be discretely defined, a notion to be explained next.

Let$I=[a,b]$ and ${\mathrm{△}}_n;a\le x_0<x_1<\dots <x_n\le b$ be a partition of $[a,b]$. With this sequence of points we associate an incidence matrix

where $p$ is a nonnegative integer. Such matrices have as entries $|E|\ge n+1$ ones and $(n+1)(p+1)-|E|$ zeros and are such that in each row there is at least one entry equal to one. We also assume that the last column contains at least one entry equal to one.

Now suppose that $L_{{\mathrm{△}}_n,E}=L:C^p\left(I\right)\to \mathbb{R}$ is of the form

where $A_{i,j}\in {\mathbb{R}}^I$ are the fundamental functions of $L$. If, analogously, $M_{{\mathrm{△}}_m,F}=M:C^q\left(J\right)\to {\mathbb{R}}^J$ is given by

then for $f\in C^{p,q}(I\times J)$ one has

In our applications the univariate building blocks will be mostly of the form of $L$ and $M$, respectively, so that in these cases it will be justified to speak about the tensor product of $\{L,M$}.

For discretely defined operators on $C^{p,q}(I\times J)$ there also hold the following commutativity properties:

We note here that the functionals $g\to g^{(j)}(x_i),$ $e_{i,j}=1,$ $g\in C^p(I)$, figuring in the representation of $L_{\mathrm{△}}{,}_E$ and occuring, for example, in Birkhoff interpolation problems, can be replaced by more general linear functionals. This is, for instance, the case when dealing with the tensor product of two linear interpolation problems. See, e.g., [ 15 ] , [ 16 ] , [ 17 ] or [ 49 ] for details.

The space $C^p\left(I\right)$ will be equipped with the norm

here ${\Vert \cdot \Vert }_{\mathrm{\infty }}$ denotes the Čebyčëv norm on $I$. Clearly, ${\Vert \cdot \Vert }_{C\left(I\right)}={\Vert \cdot \Vert }_{\mathrm{\infty }}$. If $L$ is a continuous linear operator mapping $(C^p\left(I\right),{\Vert \cdot \Vert }_{C^p(I)})$ into $(C^q\left(J\right),{\Vert \cdot \Vert }_{C^q\left(J\right)})$, its operator norm will be denoted by ${\Vert \cdot \Vert }_{C^p\left(I\right)\to C^q\left(J\right)}$. This norm is defined as follows:

For $p=q=0$ we simply write $\Vert L\Vert$.

The main results of this paper and some applications will be given in terms of so-called partial moduli of smoothness of order $r$, given for the compact intervals $I,J\subset \mathbb{R},$ for $f\in C(I\times J),$ $r\in {\mathbb{N}}_0$ and $\delta \in {\mathbb{R}}_{+}$ by

and symmetrically by

Some other applications will be formulated in terms of total moduli of smoothness of order $r$, defined by

for the compact intervals $I\times J\subset \mathbb{R},$for $f\in C(I\times J),r\in {\mathbb{N}}_0$ and ${\delta }_1,{\delta }_2\in {\mathbb{R}}_{+}.$

The third type of moduli figuring in this note will be the mixed moduli of smoothness, given for $r,s\in {\mathbb{N}}_0$ by

Several properties of these moduli can be found in L.L. Schumakers book [ 67 ] and in [ 29 ] . Further notations will be introduced below when needed.

In this note we will first summarize how quantitative properties of certain univariate operators are inherited by the product of their parametric extensions, and this in a form as simple as possible. Details can be found in [ 2 ] and [ 3 ] . In order to illustrate what is meant by this we cite the following special case (for $n=1$) of Theorem 4.1 in the article [ 31 ] on generalized $n$-th order blending as

The quantities $p^{\prime }$ and $q^{\prime }$ were introduced mainly for two reasons. On the one hand we wanted a theorem applicable to a cubic spline operator defined on $C^4(I)$, say. In this case $p=4$, $p^{\prime }=2$ are the proper choices. On the other hand, we also want to be able to cover such methods which map $C^p(I)$ into itself, but where good inequalities are only available for $0\le k\le p^{\prime }<p$.

As was indicated before, generalized $n$-th order blending is one possible generalization of the tensor product method. For further extensions and modifications which have been of some interest in the past see our concluding remark. We emphasize here the fact that this is not intended to be a paper on scattered data interpolation (Where would the univariate building blocks be?).

Tensor product operators totally depend upon the quality of their building blocks, that is, they live on good or the best possible univariate results available by inheriting many of them, sometimes in a modified fashion. This is why in our sections on applications we will mention in detail some rather recent and very good results from the theory of functions of one variable which-due to the permanence principles below - more or less immediately carry over to tensor product operators.

The organization of the remainder of this report is as follows: In section 2 we give a summary of recent theoretical results and conclude the section with several remarks on the differences between the theorems presented. ?? contain applications of ??, respectively. In this way we arrive at new results for tensor products of several univariate operators which have been attracting the interest of many researchers for many years. We have chosen to add a rather extensive bibliography, also in order to draw the reader’s attention to some less-known articles dealing with the tensor product construct.

2 Degree of simultaneous approximation by products of parametric extensions. Recent theoretical results.

In this section let$I,J,I^{\prime }{J}^{\prime }$ be non-trivial compact intervals of the real axis $\mathbb{R}$, such that $I^{\prime }\subseteq I$ and $J^{\prime }\subseteq J.$ For $(p,q)\ge (0,0)$ let linear operators

be given. $\mathrm{\Gamma }$ and $\mathrm{\Lambda }$ are non-negative, bounded functions and $R,S\subset {\mathbb{N}}_0$ are finite, nonempty sets (preferably with few elements).

Can be found in [ 2 ] , [ 3 ] . In the case of discretely defined operators we are in a much more convenient situation: it suffices to only consider one fixed partial of order $(p,q)$ and its univariate predecessors. This is due to the fact that the equalities 1 are trivially satisfied for such operators.

Observe that in the upper bound of theorem 3–as compared to theorem 2–there is no $sup$ any more.

The following result is similar to theorem 2, but makes different assumptions for the univariate building blocks. Before discussing these further we state it as

Can be found in [ 2 ] .

3 Applications of Theorem 2

3.1 Brudnyı̆-Gopengauz operators Q${}_n$

The univariate operators $Q_n$ we use here are the result of many years of research which culminated in papers by Gonska and Hinnemann [ 34 ] and by Dahlhaus [ 13 ] . Guided by early work (in chronological order) of Timan, Dzjadyk, Trigub, Brudnyı̆, Teljakovskiı̆, Gopengauz and DeVore on the subject, the authors mentioned investigated linear polynomial operators satisfying certain interpolation conditions at the endpoints of $I=[-1,1]$. We decided to call the $Q_n$ below Brudnyı̆-Gopengauz operators because–according to our knowledge–Brudnyı̆ was the first to observe the possibility of using moduli of smoothness of arbitrary order in pointwise Jackson-type estimates for algebraic polynomial approximation. Furthermore, Gopengauz first used higher order moduli for estimates on simultaneous approximation and observed (correctly!) that in inequalities with the first modulus the quantity $\frac{\sqrt{1-x^2}}{n}$ can be used in simultaneous estimates. For the sake of completeness we mention here one further paper by Gonska et al. [ 35 ] in which the problem of pointwise interpolatory inequalities was treated in combination with questions concerning shape preservation.

We describe the work of Gonska, Hinnemann and Dahlhaus briefly in order to make the subsequent statements comprehensible. The first named two authors constructed in [ 34 ] a new linear operator $Q_n$ (not necessarily being discretely defined), namely

For details regarding this definition see [ 34 , p. 247 ] . The linearity of $Q_n$ follows straightforward from the linearity of its building blocks $R_n$ and $L_n$.

Extending the work in [ 34 ] , Dahlhaus showed in [ 13 ] that for $p,r\in {\mathbb{N}}_0$ fixed, $n\ge max\{4(p+1),p+r\},0\le k\le min\{p-r+2,p\},f\in C^p\left(I\right)$ and $x\in I$ there holds the inequality

Furthermore, Dahlhaus proved that his assertion is best possible in a most meaningful sense. The above inequality is of type 2 with bounded ${\mathrm{\Gamma }}_{r,k,Q_n}\left(x\right)=c_{p,r}\cdot {\left(\frac{\sqrt{1-x^2}}{n}\right)}^{p-k},{\mathrm{\Lambda }}_{r,Q_n}\left(x\right)=\frac{\sqrt{1-x^2}}{n}$ and $p^{\prime }:=min\{p-r+2,p\}\le p$.

Choosing a second copy $Q_m$, say, mapping $C^q\left(I\right)$ into ${\mathrm{\Pi }}_m\left(I\right)$ and satisfying

for $0\le l\le q^{\prime }:=min\{q-s+2,q\}$, all the assumptions of Theorem 2 are fulfilled. Thus we have, for example, for $(x,y)\in I^2,f\in C^{p,q}\left(I^2\right)$ and for $(0,0)\le (k,l)\le (p^{\prime },q^{\prime })$ the inequality

It follows from (5) that there is a constant $d_{p,r}$ which does not depend on $n$ such that

Hence the above upper bound of

simplifies accordingly.

3.2 Cubic interpolatory splines

As a second application of theorem 2 we deal with two types of cubic interpolatory splines. Part of the results presented here can also be found in [ 3 ] .

Let ${\mathrm{△}}_n$ be a partition of the real interval $I=[a,b]$. By $S_{{\mathrm{△}}_n}$ we denote the Type I cubic spline operator which attaches to each function $f\in C^{1}I$ the corresponding clamped cubic spline interpolant $s_f\in C^2\left(I\right)$. Further, let $T_{{\mathrm{△}}_n}$ be the cubic spline operator which attaches to each $f\in C\left(I\right)$ the corresponding natural cubic spline interpolant $t_f\in C^2\left(I\right)$. For more detailed definitions see, e.g., [ 32 ] .

In order to give two-dimensional estimates we use one-dimensional ones. These are, in the case of the Type I cubic spline operator, Theorem 3.4 in [ 32 ] , and Theorem 1 in [ 38 ] for the natural cubic spline operator. We recall them briefly.

In both theorems $\delta ={\delta }_n$ is the mesh gauge of ${\mathrm{△}}_n$.

Let now be $S_{{\mathrm{△}}_n}$ and $S_{{\mathrm{△}}_m}$ two Type I cubic spline operators. By theorem 5 we have

for $p=1,2,3$, or $4,f\in C^p\left(I\right),0\le k\le min\{p,2\},{\mathrm{\Gamma }}_{4-p,k,S_{{\mathrm{△}}_n}}:=c(p,k)\cdot {\delta }_1^{p-k},{\mathrm{\Lambda }}_{4-p,S_{{\mathrm{△}}_n}}:={\delta }_1$ (the mesh gauge of ${\mathrm{△}}_n$), and

for $q=1,2,3$, or $4,f\in C^q\left(J\right),0\le l\le min\{q,2\},{\mathrm{\Gamma }}_{4-q,l,S_{{\mathrm{△}}_m}}:=c(q,l)\cdot {\delta }_2^{q-1},{\mathrm{\Lambda }}_{4-q,S_{{\mathrm{△}}_m}}:{\delta }_2$ (the mesh gauge of ${\mathrm{△}}_m$). Applying now theorem 2 we obtain the following

In order to arrive at the latter inequality it is only necessary to observe that

Further let $T_{{\mathrm{△}}_n}$ and $T_{{\mathrm{△}}_m}$ be two natural cubic spline operators. According to theorem 6 we have the one-dimensional estimates

for $f\in C^2\left(I\right),$ $k\in \{0,1\},$ ${\mathrm{\Gamma }}_{0,k,T_{{\mathrm{△}}_n}}:=c\cdot {\delta }_1^{2-k},$ ${\mathrm{\Lambda }}_{0,T_{{\mathrm{△}}_n}}:={\delta }_1$ (the mesh gauge of ${\mathrm{△}}_n$), and

for $f\in C^2\left(J\right),l\in \{0,1\},{\mathrm{\Gamma }}_{0,l,T_{{\mathrm{△}}_m}}:=c\cdot {\delta }_2^{2-l},{\mathrm{\Lambda }}_{0,T_{{\mathrm{△}}_m}}:={\delta }_2$ (the mesh gauge of ${\mathrm{△}}_m$). Applying again theorem 2 there follows

3.3 Applications of theorem 3

We now turn to three classes of discretely defined operators and thus to applications of theorem 3.

Bivariate operators of binomial type

A sequence ${\left(p_n\right)}_{n\in \mathbb{N}}$ where for all n the polynomial $p_n$ is exactly of degree $n$ is called a polynomial sequence. A polynomial sequence is said to be of binomial type, iff for all $x,y$ and all $n\in \mathbb{N}$ it satisfies the identity

For a survey on this issue see [ 53 ] . Three further recent contributions on the subject are [ 78 ] , [ 80 ] and [ 79 ] .

We consider now a linear approximation operator of degree $n$ associated with the sequence ${\left(p_n\right)}_{n\in \mathbb{N}}$ of binomial type by

for all $f\in C[0,1]$, provided that $p_n\left(1\right)\ne 0$ for all $n\in {\mathbb{N}}_0$. These operators are called by D.D. Stancu and A.D. Vernescu in [ 81 ] Popoviciu type operators (see [ 60 ] ) and by P. Sablonnière in [ 63 ] Bernstein-Sheffer operators. See these and the further paper [ 78 ] for additional references. The well-known Bernstein-Stancu operator $S_n^{\alpha }$ (introduced in [ 75 ] ) corresponds to the sequence of factorial powers

and the classical Bernstein operator $B_n$ to the sequence of monomials $p_n\left(x\right)=x^n$.

Theorem 2 (ii) in [ 63 ] states that $T_n{e}_i=e_i$ for $i=0,1$, and $T_n{e}_2=e_2+\frac{b_n}{n}(e_1-e_2)$, where

For the specification of the sequence ${\left(r_n\left(x\right)\right)}_{n\in {\mathbb{N}}_0}$ see [ 63 ] . If for all $n\in {\mathbb{N}}_0$ and for$x\in [0,1]$ one has $p_n\left(x\right)\ge 0$, then the operator $T_n$ is positive and we can apply Păltănea’s Corollary 1 in [ 59 ] . With $p=1,$ $b=0,$ $\lambda =0$ and $h=\sqrt{x(1-x)\frac{b_n}{n}}$ it yields

for all $f\in C[0,1]$. If $\frac{r_{n-2}\left(1\right)}{p_n\left(1\right)}=o\left(1\right)$, then $\frac{b_n}{n}\to 0,$ and the latter inequality implies uniform convergence of $T_{n}f$ towards $f$for all $f\in C[0,1]$, at the same time constituing a significant refinement of theorem 3 by Sablonnière [ 63 ] (see also [ 53 , Th. 2.11 ] ).

Further we consider a second operator of binomial type $T_m$ with the corresponding estimate, namely

for all $f\in C[0,1{]}^2$. Applying theorem 3 with $I=I^{\prime }=J=J^{\prime }=[0,1],p=q=0,$ $R=S=\{2\}$, ${\mathrm{\Gamma }}_{2,0,T_n}\left(x\right)={\mathrm{\Gamma }}_{2,0,T_m}\left(y\right)=\frac{3}{2},$ ${\mathrm{\Lambda }}_{2,0,T_n}\left(x\right)=\sqrt{x(1-x)\frac{b_n}{n}},$ ${\mathrm{\Lambda }}_{2,0,T_m}\left(y\right)=\sqrt{y(1-y)\frac{b_m}{m}}$it follows that

for all $f\in C[0,1{]}^2$. This improves the estimate of the error given in [ 81 ] in the sense that now a second order modulus instead of a first order one is used.

For the Bernstein-Stancu operators $S_n^{\alpha }$ we have $b_n=\frac{1+n\alpha }{1+\alpha }$and thus

This inequality can be viewed as a pointwise version of ?? in Stancu’s paper [ 76 ] .

The best-known examples of tensor product Bernstein-Stancu operators are the tensor product Bernstein operators themselves. As mentioned earlier, these were investigated as early as 1933 by Hildebrandt and Schoenberg in order to prove the WeierstraSS approximation theorem in $C[0,1{]}^2$. Subsequently their research was supplemented by a number of authors who, among other results, proved quantitative assertions of different kinds. See [ 29 , sect. 6.2.1 ] for references and a brief discussion of those published until the mid 80’s.

If we put $\alpha =\beta =0$ in (8) we arrive at

which improves Corollary 9.2 in [ 29 ] for the case $d=2$. The latter estimate holds for any $f\in C[0,1{]}^2$.

3.4 Tensor Product Schoenberg Splines

As a second application of theorem 3 we summarize several results on the degree of simultaneous approximation by the tensor product of variation-diminishing Schoenberg spline operators. For details the reader is refered to [ 4 ] . We briefly recall some of the basic definitions and of the fundamental univariate results.

Consider the knot sequence ${△}_n=\{x_i{\}}_{-k}^{n+k}$ $(n>0,k>0)$ with

For a function $f\in {\mathbb{R}}^{[0,1]}$, the variation-diminishing spline of degree $k$ w.r.t. ${△}_n$ is given

with the nodes (Greville abscissas) ${\xi }_{j,k}:=\frac{x_{j+1}+\dots +x_{j+k}}{k},$ $-k\le j\le n-1$, and the normalized $B$-splines as fundamental functions

If the knots are equidistant, i.e., $x_j=\frac{j}{n},0\le j\le n,$ the $k$-th degree Schoenberg spline operator will be denoted simply by $S_{n,k}$.

The following inequalities will be given predominantly in terms of the so-called mesh gauge $\Vert {△}_n\Vert :=\underset{j}{max}(x_{j+1}-x_j)$.

For splines with $x_j=\frac{j}{n}$, $0\le j\le n$, and second order derivatives one has uniform convergence on compact subsets of $(0,1)$ only. In this case we have, for example,

For $m,l\ge 1$, we now consider a second operator $S_{{△}_{m,l}}$ defined for functions on $[0,1]$ and satisfying inequalities analogous to those in proposition 1–proposition 3. Hence all the univariate inequalities have the appropriate form to apply theorem 3.

For the tensor product of two Schoenberg spline operators we first state

This is the case $p=q=0,$ $R=S=\{2\}$. With ${\mathrm{\Gamma }}_{2,0,S_{{△}_{n,k}}}\left(x\right)=1+\frac{k+1}{24},$ ${\mathrm{\Lambda }}_{2,0,S_{{△}_{n,k}}}\left(x\right)=\Vert {△}_n\Vert$ and analogous choices with respect to the variable $y$ we arrive at the above upper bound, also observing that
${\Vert D^0\circ S_{{△}_{n,k}}\Vert }^{\ast }=1$.

For the partial derivatives up to order $(1,1)$ we obtain

It is mainly a consequence of properties of the partial moduli of smoothness; for details see [ 4 ] .

For the remaining partials up to order $2$ again we consider only the case of equidistant knots and the smaller intervals $[\frac{k-1}{n},1-\frac{k-1}{n}]\times [\frac{l-1}{m},1-\frac{l-1}{m}]$. We now have

Proof. See [ 4 ] for details.

3.5 Discretely defined $\mathcal{H}$-operators

We finish this section with certain operators which are of particular interest in the theory of positive linear operators. All three theorems from above are applicable in this case.

In order to solve the strong form of Butzer’s Problem, Gavrea, Gonska and Kacsó introduced in [ 25 ] a new class of discrete, linear and positive operators of the following form

For details regarding the quantities appearing in the above definition see [ 25 ] , [ 26 ] , [ 24 ] .

Theorem 7 in [ 26 ] states that the operater ${\mathcal{H}}_{n+s+2}^{\ast }:C[0,1]\to {\mathrm{\Pi }}_{n+s+2}$ satisfies the DeVore-Gopengauz inequality

In particular, $\Vert {\mathcal{H}}_{n+s+2}^{\ast }\Vert =1$. We consider a second operator ${\mathcal{H}}_{m+t+2}^{\ast }$ which satisfies an analogous inequality in the variable $y$. Proceeding as in the case of binomial-type operators we get for $f\in C{[0,1]}^2$ the following:

4 Applications of Theorem 4

4.1 Simultaneous Approximation by Bernstein Operators

We return to the classical Bernstein operators which we already considered in Section 4.1 as special operators of binomial type. Here we supplement the results from there by deriving assertions for the approximation of derivatives by derivatives.

Uniform convergence of certain mixed partial derivatives $[f-$
${(}_x{B}_n\circ {}_y{B}_m)f{]}^{(k,l)}$ to $0$ was considered in the past by many authors. As a more advanced paper we mention one by I. Badea and C. Badea [ 1 ] in which the following was shown.

The article by the two Badeas contains numerous references to earlier papers dealing with simultaneous approximation by tensor product Bernstein operators. In order to modify the above inequality we first cite a result of Kacsó, namely Theorem 5 in [ 46 ] :

The above holds for $p\in {\mathbb{N}}_0,n\ge max\{p+2,p(p+1)\},f\in C^p[0,1],x\in [0,1]$ and $h>0$.

Taking $h=\frac{1}{\sqrt{n}}$ leads to

for $p\ge 0$. This shows that theorem 4 is applicable if we consider $B_n$ as an operator from $C^p[0,1]$ into itself and make an analogous assumption for $B_m$. For simplicity we only discuss the derivative of fixed order $(p,q$) and derive

Due to the representation

we obtain ${\Vert D^p{B}_{n}f\Vert }_{\mathrm{\infty }}\le {\Vert f^{\left(p\right)}\Vert }_{\mathrm{\infty }}$, so that ${\Vert D^p\circ B_n\Vert }_{C^p[0,1]\to C[0,1]}\le 1$. Thus

Clearly the latter inequality shows that

Moreover, assuming that $f\in C^{p+2,q+2}{[0,1]}^2$ the inequality implies

This order cannot be derived from theorem 12.

4.2 Simultaneous approximation by Bernstein–Durrmeyer operators with Jacobi weights

In this subsection we consider certain operators which are not discretely defined so that, according to our above presentation, only theorem 4 will be applicable.

The univariate building blocks were recently investigated again by Kacsó [ 47 ] and we briefly recall here the basic definition and the result which we will use. For details, historical remarks and useful references the reader should consult [ 47 ] .

Consider the positive linear operators

where $p_{n,k}\left(x\right)=(\genfrac{}{}{0ex}{}{n}{k})x^k{(1-x)}^{n-k},{\omega }^{(\alpha ,\beta )}\left(x\right)=x^{\beta }{(1-x)}^{\alpha }$ and $\alpha ,\beta >-1$. These are the Bernstein-Durrmeyer operators w.r.t. the Jacobi weight ${\omega }^{(\alpha ,\beta )}$. Kacsó [ 47 , Th. 2.2 ] showed that

for $p\in \mathbb{N}\cup \{0\},$ any $f\in C^p[0,1],x\in [0,1]$ and $n$ sufficiently large.

We consider a second copy of the operators in question, namely $M_m^{⟨\overline{\alpha },\overline{\beta }⟩}$ acting on functions in $C^q[0,1]$, and the variable is $y\in [0,1]$.

For brevity again we only consider the derivative of order $(p,q)$. theorem 4 implies in this case

Recall that $n$ and $m$ have to be sufficiently large in the above.

In order to turn the latter inequality into a more compact and instructive assertion, we consider the quantity

The representation we need here can also be found in [ 47 ] .

where the Pochhammer symbol ${\left(a\right)}_b$ is defined as

From this representation it follows that

hence

Again we have

and for $f\in C^{p+2,q+2}{[0,1]}^2$ the inequality implies

5 Concluding remark

$\left(i\right)$ Tensor product operators (products of parametric extensions) as considered in this note were modified in the past in several ways in order to also handle less regularly spaced data. We mention here in chronological order the work on interpolation by Biermann [ 5 ] , Steffensen [ 83 ] , Stancu [ 70 ] , [ 74 ] , Delvos and Posdorf [ 18 ] , and that of Coman et al. [ 12 ] , among others. For example, Stancu considers bivariate Lagrange interpolation at an array of points $\{(x_{ij},y_{ij}):0\le i\le m,0\le j\le n\}$, thus generalizing Steffensen interpolation.

$\left(ii\right)$ Furthermore, also generalizing the notion of a tensor product, Bos et al. [ 8 ] introduced pseudo-tensor products of univariate schemes. In this context “pseudo” refers to the fact that ${}_{x}L(f;x,y)=L(f_y;x)$ may change with $y$ and that ${}_{y}M(f;x,y)=M(f_x;y)$ may change with $x$. This method was also investigated in the diploma thesis of Dyllong [ 20 ] to some extent.

For neither of the above methods significant quantitative inheritance statements seem to have been stated according to the authors knowledge until [ 2 ] and [ 3 ] were written.