1 Introduction

In the present paper we deal with pointwise estimates on approximation by bounded linear operators of real-valued continuous functions defined on the cartesian product of \(d\) compact intervals \(I_{\delta }\). This space will be denoted by \(C\big( \times _{\delta =1}^{d}I_{\delta }\big) \). The main purpose is to provide a unified theory to deal with pointwise estimates on approximation processes of the above type which are generated by the tensor product method. Thus it constitutes an extension and a refinement of papers of W. Haussmann and P. Pottinger [ 3 ] , [ 4 ] , [ 5 ] who treated the case of uniform estimates. Since all function to be approximated are defined on a rectangular domain in \(d\) dimensions, it is possible to take full advantage of refined estimates for the univariate case, many of which were obtained only recently. This is exemplified for the case of positive operators.

While 2 will deal with the case of products of arbitrary bounded linear operators, several more instructive pointwise inequalities on tensor product of positive linear operators will be given in 3.

See for instance W. Haussmann and P. Pottinger [ 5 ] for references concerning among other things existence and uniqueness theorems or non-quantitative assertions on convergence. Throughout the paper we write \(I_{\delta }=\left[ a_{\delta },b_{\delta }\right] ,1\leq \delta \leq d\in N\), where \(\left[ a_{\delta },b_{\delta }\right] \) are compact intevals with non-empty interior. The space of continuous functions on such an interval will be \(C\left( I_{\delta }\right) \). The definition of some further notation used in this paper may be found in Haussmann’s and Pottinger’s article.

2 Estimates on approximation by bounded linear operators

The following is a modification of a result due to W. Haussmann and P. Pottinger [ 5 , Proposition 1 ] .

If as a realization for the last product we also choose the linear hulls of the corresponding complex product, this yields

If we equip \(\otimes _{\delta =1}^{d}C\left( I_{\delta }\right) \) with the \(\varepsilon \)-norm, that is, if for \(g\in \otimes _{\delta =1}^{d}C\left( I_{\delta }\right) \), \(g=\sum _{i=1}^{n}g_{i,1}\cdot \ldots \cdot g_{i,d}\)

and if we do the same in \(\big(\otimes _{1\leq \delta \leq \delta _{0}-1}C\left( I_{\delta }\right)\big)\otimes R\otimes \big(\otimes _{\delta _{0}+|\leq \delta \leq d}C\left( I_{\delta }\right) \big) \), then \(\left( \varepsilon ,\varepsilon \right) \) are uniform crossnorms with respect to the pair
\(\Big\{ \otimes _{\delta =1}^{d}C\left(I_{\delta }\right),\big( \otimes _{1\leq \delta \leq \delta _{0}-1}C\left( I_{\delta }\right)\big) \otimes R\otimes \big( \otimes _{\delta _{0}+1\leq \delta \leq d}C\left( I_{\delta }\right) \big) \Big\} \) see W. Haussmann and P. Pottinger [ 5 , Theorem 2 ] . Thus the tensor product operator

is continuous.

We now consider the \(\delta _{0}\)-th partial mappings \(f_{\delta _{0}}^{\bar{\xi }}\) belonging to \(f,\delta _{0}\) and the fixed \(\left( d-1\right) \) tuples \(\bar{\xi }:=\left( x_{1},\ldots ,x_{\delta _{0}-1},x_{\delta _{0}+1},\ldots ,x_{d}\right). \) The mapping \(f_{\mu }L:\times _{\delta =1,\delta \neq \delta _{0}}^{d}I_{\delta }\rightarrow R\) (where \(\mu \) is the linear functional from above), given by \(F_{\mu }\big( \bar{\xi }\big) :=\mu \big( f_{\delta _{0}}^{\bar{\xi }}\big) \) is continuous, since for \(\bar{\xi }:=\left( x_{1},\ldots ,x_{\delta _{0}-1},x_{\delta _{0}+1},\ldots ,x_{d}\right) \) and \(\widehat{\bar{\xi }}~ =\hat{x}_{1},\ldots ,\hat{x}_{\delta _{0}-1},\hat{x}_{\delta _{0}+1},\ldots ,\hat{x}_{d})\in \times _{\delta =1,\delta \neq \delta _{0}}^{d}I_{\delta }\), one has

where \(\left\Vert \mu \right\Vert \) denotes the norm of \(\mu \) with respect to \(\Big( \left( C\left( I_{\delta }\right) ,\left\Vert \cdot \right\Vert _{\infty }\right) ,R\left\vert \cdot \right\vert \Big)\). This and the uniform continuity of \(f\) imply the continuity of \(f_{\mu }\).

We now define

The fact that \(H_{\mu }\) is continuous is a consequence of the following chain of (in)equalities showing that the operator norm \(\Vert H_{\mu }\Vert \) is bounded:

If \(g\in \Big\langle \prod \limits _{\delta =1}^{d}C(I_{\delta })\Big\rangle \), \(g=\sum \limits _{i=1}^{n}g_{i,1\cdot \dots \cdot g_{i,d}}\) for some \(n\in \mathbb {N}\), then

Hence the mappings \(H_{\mu }\) and \(\big(id^{1}\otimes \dots \otimes id^{\delta _{0}-1}\otimes \mu \otimes id^{\delta _{0}+1}\otimes \dots \otimes id^{d}\big)\) coincide on \(\otimes _{\delta =1}^{d}C(I_{\delta })=\Big\langle \prod \limits _{\delta =1}^{d}C(I_{\delta })\Big\rangle \). Since the Chebyshev norm \(\Vert \cdot \Vert _{\infty }\) on \(\otimes _{\delta =1,\delta \neq \delta _{0}}^{d}C(I_{\delta })\) induces the \(\varepsilon \)-norm, both norms also coincide on the completion \(\widehat{\otimes }_{1\leq \delta \leq d,\delta \neq \delta _{0}}^{\varepsilon }C(I_{\delta })\), which is thus isometrically isomorphic to \(C\big(\times _{\delta =1,\delta \neq \delta _{0}}^{d}I_{\delta }\big)\). From this and from the continuity of both mappings considered above, it follows for each \(h\in C(\times _{\delta =1}^{d}I_{\delta })\) that

Here \(\Vert \cdot \Vert _{\varepsilon }\) is the \(\varepsilon \)-norm on \(\widehat{\otimes }_{1\leq \delta \leq d,\delta \neq \delta _{0}}^{\varepsilon }C(I_{\delta })\), and \(\Vert \cdot \Vert _{\infty }\) denotes the Chebyshev norm on \(C(\times _{\delta =1,\delta \neq \delta _{0}}^{d}I_{\delta })\). Furthermore,

Since by Haussmann’s and Pottinger’s [ 5 , Theorem 2 ] \((\varepsilon ,\varepsilon )\) are uniform cross norms with respect to the couple \((\otimes _{\delta =1}^{d}X_{\delta },\otimes _{\delta =1}^{d}\mathbb {R}=\mathbb {R})\), the mappings

and

are continuous. This implies the continuity of

For the same reason we also have continuity of

Together with the observation made at the beginning of the proof this also yields equality of the extensions of the two mappings considered above, i.e.,

An analogous density argument shows the validity of

From this the claim of the lemma immediately follows.

If \(h\in \widehat{\otimes }_{1\leq \delta \leq d}^{\varepsilon }X_{\delta }\), then

The uniform cross norm property of \((\varepsilon ,\varepsilon )\) with respect to the couple

(see W. Haussmann and P. Pottinger [ 5 , Theorem 2 ] ) first implies

For density reasons the extension of \((id_{R}\circ \mu _{\delta 1}\circ id^{1})\otimes \ldots \otimes id_{R}\otimes \ldots \otimes id_{R}\circ \mu _{d}\circ id^{d})\) has the same norm; hence

Since \(\delta _{0} \in \{ 1, \dots , d \} \) was arbitrarily chosen, the claim of the lemma follows.

Here \(S_{d} \) is the symmetric group of all permutations of \(\{ 1, \dots , d\} \).

Let \(\sigma \in S_{d}\) be an arbitrary permutation. A decomposition of \(\widehat{\otimes }_{\delta =1}^{d}id^{\delta }-\widehat{\otimes }_{\delta =1}^{d}P^{\delta }\) analogous to the one employed by Haussmann and Pottinger together with a density argument yields the equality

Thus by lemma 2 for all \(h\in \widehat{\otimes }_{1\leq \delta \leq d}^{\varepsilon }X_{\delta }\), one obtains

From lemma 3 we conclude that the difference considered above may be estimated as follows:

In the above, we have used the convention that an empty product equals \(1\). Since this is true for all permutations \(\sigma \in S_{d}\), we may pass to the minimum over all \(\sigma \in S_{d}\) on the right-hand side of the last inequality. We shall show that for all \(h\in \otimes _{\delta =1}^{d}X_{\delta }\) and all \(\mu _{\delta }:X_{\delta }\rightarrow \mathbb {R}\) (\(\mu _{\delta }\) linear and continuous), the equality

holds; this will suffice to prove the theorem. Let \(h\in \otimes _{\delta =1}^{d}X_{\delta }\). Hence \(h=\sum \limits _{i=1}^{n}x_{i,1}\otimes \ldots \otimes x_{i,d}\) for some \(n\in \mathbb {N}\). Thus

and

For density reasons this equality also holds for all \(h\in \widehat{\otimes }_{\delta =1}^{d}X_{\delta }\) and for the extension \(\widehat{\otimes }_{\delta =1}^{d}\mu _{\delta }\) of \(\otimes _{\delta =1}^{d}\mu _{\delta }\). Thus theorem 4 is proved.

In the sequel we shall discuss the case where \((X_{\delta },\Vert \cdot \Vert _{\delta })=\big(C(I_{\delta }),\Vert \cdot \Vert _{\infty }\big)\).

Here \(h_{\sigma (d-\nu +1)}^{\bar{\xi }}\) is the partial mapping belonging to fixed \(\bar{\xi }\in \times _{\delta =1,\delta \neq \sigma (d-\nu +1)}^{d}I_{\delta }\).

Using theorem 1 the above may be replaced by

where \(\overline{\xi }\) is a point in \(X_{\delta =1,\delta \neq \sigma -(d-v+1)}^{d}I_{\delta }\). Plugging this upper bound into the estimate of theorem 4 gives our claim.

If we neglect to pass to the \(\min \) over \(\sigma \in S_{d}\) and use \(\sigma =\text{id}\) in the proof of theorem 5 we obtain the somewhat weaker

A particularly important consequence of corollary 6 is given in the following theorem. It shows how certain univariate inequalities may be directly used when striving for error estimates on approximation by the tensor product of \(d\) univariate operators. For the definition of the (higher order) modulus of continuity \(\omega _{r_{\delta }}(f;\cdot )\) and that of the partial moduli \(\omega _{rv}(h;0,\ldots ,0,\ldots ,0)\), see, e.g., the books by Timan [ 8 ] and Schumaker [ 7 ] .

Here \(A\) may be chosen as \(\max \left\{ 1,\Vert P^{\delta }\Vert :1\leq \delta \leq d\right\} \) .

(point evaluation functional) that

where \(A\geq 1\) is such that \(\big\Vert P^{\delta }\big\Vert \leq A\) for \(1\leq \delta \leq d\).

Note that the constant \(A\) indeed exists because the \(\Gamma _{\delta }\) are bounded and \(\omega _{r}(f,\delta )\leq 2^{r}\Vert f\Vert _{\infty }\). By the above assumption on \(P^{\delta }\), \(1\leq \delta \leq d\) , and the fact that for fixed \(\bar{\xi }=(x_{1},\ldots ,x_{v-1},x_{v+1},\ldots ,x_{d})\) the function \(h_{\nu }^{\overline{\xi }}\) is given by

it is seen that

Hence because \(A\geq 1\),

where \(\varepsilon \) figures in the \(v\)-th component of

3 Examples: Pointwise Inequalities for Products of Positive Linear Operators

In the above we mainly considered continuous linear mappings \(\mu _{\delta }:C(I_{\delta })\rightarrow \mathbb {R}\) and \(P^{\delta }:C(I_{\delta })\rightarrow C(I_{\delta })\). We shall assume throughout this section that \(\mu _{\delta }\) is a point evaluation functional and that \(P^{\delta }\) is positive. The assertions proved here will be based upon a special instance of a theorem by the author (see [ 1 , Theorem 4.6 ] ) and of an improvement of part of the same theorem due to Păltănea [ 6 ] . We summarize as follows.

These inequalities will now be combined with the results from section 2 The following theorem gives an estimate in terms of first order partial moduli of continuity.

For \(\bar{\xi }\) fixed, the expression

is a univariate difference which may be estimated from above using (1). Note that for each coordinate we may choose a separate couple \((h_{\nu },e_{\nu }){\gt}(0,0)\). Hence

(Here \(e_{1}\) simultaneously denotes the functions \(I_{\nu }\ni x_{\nu }\mapsto x_{\nu }\in \) \(R\), \(1\leq \nu \leq d\)). Thus

Since the function \(\alpha (P^{\nu };h_{\nu },\varepsilon _{\nu };x_{\nu })\) does not depend on \(\bar{\xi }\in \times _{\delta =1,\delta \neq \nu }^{d}I_{\delta }\), the latter sum may be rewritten as

which is the upper bound of theorem 9 in terms of a sum of first order partial moduli of continuity.

We also have

The univariate differences figuring in the sups may now be estimated using 2 from which we get

Note again that for each coordinate a separate \(h_{v}\) may be chosen. Since both \(\alpha \) and \(\beta \) do not depend on \(\overline{\xi }\) we may write

Making appropriate choices for \(h_{\nu }\), \(1\leq \nu \leq d\), gives the above inequality.

4 CONCLUDING REMARK

All fundamental estimates given in section 2 and section 3 are those concerning the differences \(\left\vert k(\xi )-\widehat{\otimes }_{\delta =1}^{d}P^{\delta }(k,\xi )\right\vert ,\) where \(k\in C(\times _{\delta =1}^{d},I_{\delta })\). It is also possible to modify the assumptions made in theorem 7 by assuming that similar inequalities hold in order to arrive at somewhat improved estimates for subspaces of smooth functions.

Furthermore, no assertions were made concerning the pointwise degree of simultaneous approximation of partial derivatives. While this is also possible, we decline to do so for the sake of brevity. Related material can be found in the author’s “Habilitationsschrift”  [ 2 ] .

1

H. Gonska, On approximation by linear operators: improved estimates, Anal. Numer. Theor. Approx., 14 (1985), pp. 7–32.

2

H. Gonska, Quantitative Approximation in \(C(X)\), Habilitationsschrift, University of Duisburg, 1985.

3

W. Haussmann and P. Pottinger, Zur Konvergenz mehrdimensionaler Interpolationsverfahren, Z. Angew. Math. Mech., 53 (1973), pp. T195–T197. https://doi.org/10.1002/zamm.197305312101

4

W. Haussmann and P. Pottinger, On multivariate approximation by continuous linear operators, in: Constructive Theory of Functions of Several Variables, Proc. Conf. Oberwolfach 1976 (W. Schempp and K. Zeller, eds.), Springer, Berlin, 1977, pp. 101–108.

5

W. Haussmann and P. Pottinger, On the construction and convergence of multivariate interpolation operators, J. Approx. Theory, 19 (1977), pp. 205–221.

6

R. Păltănea, Optimal estimates with moduli of continuity, Result. Math., 32 (1997), pp. 318–331. https://doi.org/10.1007/BF03322143

7

L.L. Schumaker, Spline Functions: Basic Theory, J. Wiley & Sons, New York, 1981.

8

A.F. Timan, Theory of Approximation of Functions of a Real Variable, Macmillan Co., New York, 1963.