MSC. 41A36, 41A25, 41A63, 41A05.

Keywords. quantitative Korovkin-type theorems, compact metric spaces, bounded linear operators, positive linear operators, K-functional, modulus of continuity, least concave majorant of a modulus, optimal constants.

Introduction

The present paper discusses error estimates for the approximation of continuous functions defined on compact metric spaces. It is a contribution to quantitative approximation theory, and deals with pointwise error estimates for the approximation by bounded linear operators of continuous functions defined on compact metric spaces \((X,d)\). Most estimates will be given in terms of moduli of continuity or their least concave majorants.

While we conducted our preliminary research on this topic, it turned out that, although such problems are now considered as being classical, and in spite of the research of T. Nishishiraho and M.A. Jiménez Pozo, there was still a large discrepancy between the general theory and estimates for continuous functions defined on, say, a compact interval \([a,b]\). This observation was one of the main motivations for writing the author’s “Habilitationsschrift” from which the material presented here is taken. However, the thesis mentioned was never made available in the regular literature for no good reason, a fact which we would like to make up for (at least in part) by writing this note.

The main goals of this article are twofold. We present a new technique for deriving error estimates on approximation of functions defined on an arbitrary compact metric space (section 3) and we give a more careful analysis of the so-called direct technique for special metric spaces (section 4). According to the goals stated above, the paper is divided into the following parts.

Chapter 1 is of an auxiliary nature. In it we shall discuss several properties of (least concave majorants of) moduli of continuity and K-functionals which will be used throughout the remainder of this work. Among others, this part contains the refinement of a lemma due to M.A. Jiménez Pozo and the generalization of a lemma of N.P. Korneichuk to the case of continuous functions defined on a space with a finite coefficient of convex deformation.

Chapter 2 has its historical roots in a paper by G. Freud  [ 13 ] and was mostly motivated by the work of M.A. Jiménez Pozo and T. Nishishiraho. In their work they gave error estimates for functions defined on compact metric spaces satisfying certain additional conditions. The aim of Section 3 is to show that these constraints can be completely removed, if least concave majorants of moduli of continuity are used rather than the corresponding moduli themselves. It is also shown that our approach may be even better than theirs, provided \((X,d)\) satisfies the additional assumptions they imposed. Our new approach to give error estimates for the case of continuous functions defined on \((X,d)\) has some unexpected consequences even in the case where \(X = [a,b]\) is a compact interval of the real axis. Various further aspects are also discussed, such as a change of the test functions and the fact that the estimates of the main results of Section 3 are best possible in a certain sense.

Since the results of Section 3 are achieved via the intermediate use of a certain K-functional, we chose to call this the indirect approach. In Section 4 we discuss several consequences of the results of Section 3, and also present a careful analysis and a refinement of the direct technique. As is the case in Section 3, a general result for approximation by bounded linear operators is given. Subsequently the instance where \(L\) is also positive is discussed in depth. It will be shown in particular that the refined estimates of Section 4 are now best possible in a certain sense. As a consequence, we achieve a significant improvement over Nishishiraho’s technique inasmuch as best possible constants can now be obtained by evaluating the upper bounds of our inequalities.

The critical function \(L(d(\cdot ,x);x)\) and several upper bounds of it are discussed in detail. However, more explicit upper bounds for \(L(d(\cdot ,x);x)\) and similar expressions occurring in particular in Sections 3 and 4 were discussed in some detail by Nishishiraho. Since the use of neither of these bounds improves an order of an estimate or a constant, the reader is referred to Nishishiraho’s work for further information. It can be said in general that the construction of these upper bounds very closely follows the pattern suggested by classical Korovkin theory in order to introduce a finite number of test functions.

At certain central points of Sections 3 and 4 we also give a partial survey of related results which had been obtained earlier. Many of the examples discuss a univariate setting. This is the most convenient way to check the quality of both the order of approximation and of the constants.

The present article solely covers the general theory. For many applications the reader is referred to the author’s “Habilitationsschrift”. Among them are applications of the general results to certain operators of Vaida, Badea, and to the well-known Shepard operators. Part of the material given here was presented at the Third International Conference on Functional Analysis and Approximation Theory (Acquafredda di Maratea/Italy, September 1996).

Chapter I: Some Properties of Moduli of Smoothness and K-functionals

Most of the error estimates in this paper will be given in terms of moduli of smoothness (continuity) of various kinds and orders. Frequently so-called \(K\)-functionals will be used as intermediate tools for deriving these estimates. The aim of this section is to compile some information on both measures of smoothness, and to establish some new relationships between them and related quantities which will be useful in the sequel.

Section 1 lists several properties of moduli of continuity (smoothness) for real-valued and continuous functions of a single variable defined on a compact metric space \((X,d)\). Section 2 contains a brief discussion of certain \(K\)-functionals in \(C(X)\), where \(X\) is again a compact metric space.

As is clear from this summary, we do not attempt to present a comprehensive survey on moduli of smoothness and \(K\)-functionals in this chapter. It is solely intended to list those properties which will be needed in Chapter 2.

1 Moduli of Continuity of Functions Defined
on Compact Metric Spaces

In this section we shall collect some information on the metric modulus of continuity of a function \(f\in C(X)=C_{\mathbb {R}}((X,d))\). Here \(C_{\mathbb {R}}((X,d))\) denotes the space of all real-valued and continuous functions defined on the compact metric space \((X,d)\) having a diameter \(d(X){\gt}0\).

Throughout this paper we shall need the following

If the metric space has the property that for \(x,y\in X\) and \(d(x,y)=a+b\) with \(a,b{\gt}0\), there exists always an element \(z\in X\) such that \(d(x,z)=a\) and \(d(z,y)=b\), then the space is called metrically convex. This is a notion which was introduced by K. Menger  [ 37 ] . For such spaces the modulus of continuity satisfies the properties given in the following

There is no need to give a proof of lemma 1.2 at this point because lemma 1.6 covers a more general situation.

Examples of compact metric spaces being metrically convex are for instance given by compact and convex subsets of \(\mathbb {R}^m\), \(m\ge 1\), equipped with the Euclidean distance function, or the unit circle \(S^1\) with metric \(d\) given by \(d(e^{it}, e^{ix}) = \min \{ 2\pi -|t-x|, |t-x|\} , \; 0 \le t,x{\lt}2\pi \).

There is a generalization of the notion of a metrically convex space. To our knowledge, it was introduced by M.A. Jiménez Pozo  [ 23 ] and uses the concept of the so-called coefficient of convex deformation. A brief description follows.

The following remarks are due to M.A. Jiménez Pozo.

Jiménez Pozo’s motivation for introducing the coefficient of convex deformation \(\varrho \) was to make sure that an inequality of type

holds for the modulus of continuity as defined above. Here for \(a \in \mathbb {R}\), \([a]\) denotes the largest integer \(\le a\). One of the observations of the following lemma is that a somewhat stronger inequality holds. It constitutes a generalization of lemma 1.2.

(1) and (2) are immediate consequences of the definition of \({\omega }_d\).
(3) is obtained as follows. For \(r=0\) the inequality obviously holds. So let \(r{\gt}0\); because \(\varrho \ge 1\) this implies \(\varrho \cdot r{\gt}0\).

If \(\varrho \cdot r\notin \mathbb {N}\), then \(1+]\varrho \cdot r[ = [1 + \varrho \cdot r]\), where \([a] := \max \{ z\in \mathbb {Z}\; :\; z \le a\} \). For this case it was shown by M.A. Jiménez Pozo  [ 23 , Theor è me 2 ] that the above inequality holds.

Now let \(\varrho \cdot r\in \mathbb {N}\). Since \((X,d)\) is compact, it is also finitely compact, and thus, by a theorem of Hilbert (see W. Rinow  [ 51 , Statement 7, p. 141 ] ), assumption (1) of definition 1.4 implies that for each two points \(x,y\in X\), \(x\neq y\), there is a shortest rectifiable arc \(\Gamma _0\) (in German: “Kürzeste”) connecting \(x\) and \(y\) such that

Because of (2) of definition 1.4, we have in particular that

Using a slight modification of Jiménez Pozo’s approach for the case of \(\varrho \cdot r\notin \mathbb {N}\), we proceed further as follows.

Let \(x\) and \(y\) be such that \(0{\lt}d(x,y)\le r\delta \). Choosing \(\varrho \cdot r+1\) points \(t_i=i/(\varrho \cdot r)\), \(0\le i\le \varrho \cdot r\), and the parametrization \(\psi \) such that (see M.A. Jiménez Pozo  [ 23 , Lemme 1 ] )

where \(\Gamma _0[x,y]\) is chosen as above and in particular such that

one obtains (with \(z_i:=\psi (t_i)\) and \(n:=\varrho r\)) the inequalities

This implies

the latter equality being true by our assumption \(\varrho r\in \mathbb {N}.\)

(4) If \(\delta _1=0\) or \(\delta _2=0\), because \(\varrho \ge 1\) the statement is trivial.
Let \(\delta _1, \delta _2{\gt}0\) be given and fixed, and let \(x,y\in X\) be such that \(d(x,y) \le \delta _1+\delta _2\).
Choosing \(t_0=0\), \(t_1 = \delta _1/(\delta _1+\delta _2)\), and \(t_2=1\), the parametrization \(\psi \) and the arc \(\Gamma _0\) from (3) satisfy for \(i=1,2\)

Hence

showing that

This immediately implies

and the second inequality is a consequence of (3).

(5) is an immediate corollary of the definition.

Hence metric spaces \(X\) having a coefficient of convex deformation \(\varrho \) constitute an example of metric spaces such that for any \(f\in C(X)\) one has

where \(\eta {\gt}0\) is a fixed constant. For further properties of moduli of continuity given for \(f\in C(X)\) and satisfying an inequality of this type see Section 2.

2 K-functionals

In this section we shall compile several properties of certain classical and modified \(K\)-functionals, including their relationship to certain moduli of continuity and their least concave upper bounds.

2.1 Definition and elementary properties

A further means for measuring the smoothness of functions is Peetre’s \(K\)-functional. This measures the distance of an element \(f\) of a vector space \(E\) from a subspace \(U\), at the same time reflecting certain properties of the approximating elements \(g\in U\). The definition is as follows.

The following lemma collects some of the properties of \(K\).

2.2 K-functionals and least concave majorants of moduli of continuity

In the following we shall treat the approximation of continuous functions, and thus shall use only special cases of the \(K\)-functionals introduced in Section 2.1. In this section we consider again a compact metric space \((X,d)\) with diameter \(d(X){\gt}0\). If further properties of \(X\) are needed, these will be stated explicitly.

For \(0{\lt}r\le 1\) let \({\rm Lip~ }r\) denote the set of all functions \(g\) in \(C(X)\) such that

Then \({\rm Lip~ }r\) is a dense subspace of \(C(X)\), and \(|\cdot |_{{\rm Lip~ }r}\) is a seminorm on \({\rm Lip~ }r\). Thus it makes sense to use the \(K\)-functional with respect to \(({\rm Lip~ }r, |\cdot |_{{\rm Lip~ }r})\) in order to prove quantitative assertions. As indicated above, this functional is given by

where \(f\in C(X)\) and \(t\ge 0\).

Another tool for our proofs will be the following lemma of Yu.A. Brudnyı̌ relating the above \(K\)-functional for \(r=1\) to the least concave majorant of the metric modulus of continuity as introduced in Section 1. For a sketch of proof see B.S. Mitjagin and E.M. Semenov  [ 38 ] .

The definition of \(\widetilde{\omega }_d(f,\cdot )\) shows that \({\omega }_d(f,\cdot ) \le \widetilde{\omega }_d(f,\cdot )\). However, an estimate from above by some multiple of \({\omega }_d(f,\cdot )\) is not possible for all metric spaces \((X,d)\), as will be seen below. For some further properties of \(\widetilde{\omega }_d(f,\cdot )\) see e.g., V.K. Dzjadyk  [ 12 , p. 153ff ] .

If \((X,d)\) is such that the modulus of continuity \({\omega }_d(f,\cdot )\) of some function \(f\in C(X)\) is continuous, nondecreasing and subadditive on \([0,\infty )\), then it was shown by N.P. Korneičuk  [ 30 , p. 670 ] that for any \(\varepsilon \ge 0\) and \(\xi {\gt}0\) the functions \({\omega }_d(f,\cdot )\) and its least concave majorant \(\widetilde{\omega }_d(f,\cdot )\) are related by the inequality

and that this inequality cannot be improved for each \(\varepsilon {\gt}0\) and \(\xi =1,2,\ldots \) In lemma 2.4 a generalization of this lemma to the setting of certain metric spaces including those with a finite coefficient of convex deformation will be accomplished.

The generalization just mentioned will be for metric spaces \((X,d)\) of the following type.

There exists a fixed constant \(\eta {\gt}0\) such that for all \(\xi ,\varepsilon {\gt}0\) and all \(f\in C(X),\) the inequality

holds.

Examples of such spaces are e.g., compact metric spaces being convex in the sense of K. Menger  [ 37 ] , or compact convex subsets \((X,d)\) of a metric linear space \((Y,d)\) with translation invariant metric and starshaped \(d(\cdot ,0)\) (see T. Nishishiraho  [ 43 ] ). In both cases the above inequality holds for \(\eta =1\). Other examples are given by spaces \((X,d)\) having a coefficient of convex deformation \(\varrho =\varrho (X)\). In this case the above inequality holds for \(\eta =\varrho \) (see corollary 1.7). Provided an inequality of the above type holds, \(\widetilde{\omega }_d(f,\cdot )\) and \({\omega }_d(f,\cdot )\) are related as described in the following lemma; for the case \(\eta =1\) it was obtained by N.P. Korneičuk  [ 30 , p. 670 ] .

Obviously (1) is true if at least one of the numbers \(\xi \) and \(\varepsilon \) is equal to zero. So let \(\xi , \varepsilon {\gt}0\) and such that \(\xi \cdot \varepsilon \le d(X)\). As mentioned earlier, for \(0\le \xi \cdot \varepsilon \le d(X)\) one has

Putting for instance \(x=\xi \cdot \varepsilon \) shows that \({\omega }_d(f,\xi \cdot \varepsilon ) \le \widetilde{\omega }_d(f,\xi \cdot \varepsilon )\).
For the proof of the second inequality in (1) let \(0\le x \le \xi \cdot \varepsilon \le y \le d(X),\; x\neq y,\) be arbitrarily given, and write \({\omega }(\cdot )={\omega }_d(f,\cdot )\) for the sake of brevity. Then

This shows that inequality (1) holds for \(\xi \varepsilon \le d(X)\). For \(\xi \varepsilon {\gt}d(X)\) we have

Thus the lemma is proved for all possible choices of \(\xi \varepsilon .\)

However, it is not always possible to estimate \(\widetilde{\omega }_d(f,\cdot )\) from above by some multiple of \({\omega }_d(f,\cdot )\). This can be seen from

Chapter II: Approximation in C(X)

This chapter deals with quantitative Korovkin-type theorems for approximation by bounded linear operators defined on \(C(X)\), and in particular by positive ones. Here \(C(X)=C_{\mathbb {R}}((X,d))\) denotes the Banach lattice of real-valued continuous functions defined on the compact metric space \((X,d)\) equipped with the canonical ordering, and with norm given by \(\| f\| =\| f\| _X = \max \{ |f(x)|:x\in X\} \). We also assume that \(X\) has diameter \(d(X){\gt}0\).

The first such theorem for general positive linear operators and \(X=[a,b]\) equipped with the euclidian distance is apparently due to R.G. Mamedov  [ 34 ] . For spaces \((X,d)\) being metrically convex in the sense of K. Menger  [ 37 ] , D.J. Newman and H.S. Shapiro  [ 41 ] proved the following theorem similar to that of Mamedov (see also H.S. Shapiro  [ 55 , Chapter 8.8.2 ] , and papers by M.W. Müller and H. Walk  [ 40 , p. 225 ] and G. Mastroianni  [ 36 ] ).

This direct approach (i.e., one which avoids the intermediate use of \(K\)-functionals) was further developed in papers of T. Nishishiraho  [ 42 , 43 , 44 ] , where additional references can be found. For compact spaces \((X,d)\) having a coefficient of convex deformation \(\varrho {\lt}\infty \) (see definition 1.4), M.A. Jiménez Pozo  [ 23 ] published a generalization of the result of Newman and Shapiro involving mainly the modulus of continuity of \(f\). See Jiménez Pozo  [ 26 ] and Jiménez Pozo–Baile Baldet  [ 27 ] for a discussion of several earlier results.

Furthermore, for arbitrary compact \((X,d)\), M.A. Jiménez Pozo  [ 21 , 26 ] proved a certain generalization of the following

A disadvantage of this latter type of estimate is the fact that, for operators satisfying \(L(1_X)=1_X\), the upper bound is not given solely in terms of a modulus of continuity. This observation is the main motivation for our investigations in Section 3, where we shall prove several inequalites in terms of the least concave majorant of the modulus of continuity (see Section 2.2), thus avoiding the problems mentioned.

As will be discussed in detail in Section 4, there are basically two ways to obtain estimates in terms of the metric modulus of continuity from definition 1.1. One way is to use the results from Section 3 and the relationship between \(\widetilde{\omega }\) and \({\omega }\) as given in lemma 2.4. The other method is the direct approach mentioned above. Section 4 contains a refined version of this approach which is more sensitive than the technique employed by T. Nishishiraho. See the introductory remarks to Section 4 for further details.

3 Estimates on Approximation by Bounded Linear Operators in terms of Least Concave Majorants of Moduli of Continuity

This section contains several results most of which were presented in the author’s papers [ 16 ] and [ 17 ] . The main point is to show that the assumption of metric convexity or of existence of a finite coefficient of convex deformation can be completely dropped when using the least concave majorant of \({\omega }(f,\cdot )\) instead of \({\omega }(f,\cdot )\) itself. As will be seen below, our general results imply estimates similar to (and sometimes even better than) those of T. Nishishiraho and M.A. Jiménez Pozo provided \((X,d)\) satisfies the additional assumptions they imposed.

It was observed by H. Berens and G.G. Lorentz  [ 5 ] among others that the results on approximation of lattice homomorphisms \(A:C(X)\to E\), \(E\) a Banach lattice, by positive linear operators are similar to those for approximation of the injection \(i:C(X)\to B(X)\). Here \(B(X)\) is the space of bounded real-valued functions on \(X\). This is our motivation for proving estimates on approximation of mappings \(A:C(X)\to B(Y)\), \(Y\neq \emptyset \), given by \(A(f,y)=\psi _A(y)\cdot f(g_A(y))\); here \(\psi _A\) is a bounded real-valued function on \(Y\) and \(g_A\) maps \(Y\) into \(X\). For the relationship between such mappings and lattice homomorphisms see e.g., M. Wolf  [ 62 ] .

It is possible to generalize the operators to be approximated. Some results in this direction are due to M.A. Jiménez Pozo  [ 22 , 25 ] and to the author  [ 15 ] . However, this problem will not be discussed here.

One of the key results of this section is the following theorem 3.1. Its simple proof reveals the smoothing technique in spaces of real-valued continuous functions defined on compact metric spaces mentioned above, and uses some of the results of Section 2.

The inequality is obviously correct for \(\Delta =0\); so let \(\Delta \neq 0\). Let \(g\) be arbitrarily given in \({\rm Lip~ }1\). For any \(f\in C(X)\) we have

Passing to the inf in \({\rm Lip~ }1\) implies for each \(\varepsilon {\gt}0\) (see lemma 2.2 (3)):

Brudnyı̌’s lemma 2.3 implies

The next theorem deals with approximation of operators \(A\) given for \(f\in C(X)\) and \(y\in Y\) by \(A(f,y) = \psi _A(y) \cdot f(g_A(y))\) where \(\psi _A\in B(Y)\) and \(g_A\) maps \(Y\) into \(X\). It contains a general inequality for the approximation of operators \(A\) by bounded linear operators, and was announced in the author’s paper  [ 17 , Theorem 2.2 ] . If \(A\) is given as above, then

Thus for all \(f\in C(X)\) and all \(y\in Y\)

Defining \(\widetilde A(f,y) :=L(1_X,y)\cdot f(g_A(y))\) we have to consider \(|(L-\widetilde A)(f,y)|\). First observe that \(L-\widetilde A\) is a bounded operator. Moreover, if \(g\in {\rm Lip~ }1\) is arbitrarily given, then

In order to estimate this quantity we introduce the functions

Note that \(h_1\le g\le h_2\). Now

and

As mentioned above, both \(g-h_1\) and \(h_2-g\) are nonnegative functions. We prove lower estimates for the quantities \(L(g-h_1;y)\) and \(L(h_2-g;y)\).

For \(y\) fixed, the continuous linear functional

can be identified with some real measure \(\mu \) on the compact space \(X\). Hence \(L(f,y) = \int _X f d\mu \) for \(f\in C_{\mathbb {R}}(X)\) (see, e.g., J. Dieudonné  [ 11 , Ch. 13 ] ). It is also known that \(\mu \) can be represented as \(\mu =\mu ^+ - \mu ^-\) with both \(\mu ^+\) and \(\mu ^-\) being positive measures. Because of

for every nonnegative function \(h\in C(X)\) we have

We estimate \(\int _X h \, d\mu ^-\) from above to obtain

(For the definition and existence of \(\sup (-\mu ,0),\; |\mu |\), and \(\| \mu \| \) see J. Dieudonné  [ 11 , Chapts. 13.3, 13.9, 13.15, and 13.20 ] ). Moreover,

Thus

or

This implies

or

and

These inequalities yield

Now

and

Thus

This settles the estimate for Lipschitz functions \(g\). Applying theorem 3.1 to \(\Delta =L-\widetilde A\) gives

Together with the decomposition for \((L-A)(f;y)\) from the beginning of the proof, we arrive at the estimate of theorem 3.2. \(\Box \)

For positive linear operators we have the simpler estimate of

Let \(0{\lt}\varepsilon \). Since \(L\) is a positive linear operator, \(\varepsilon _y\circ L = L(\cdot ,y)\) given by \(L(\cdot ,y):C(X) \ni f\mapsto L(f,y)\in \mathbb {R}\) is a positive linear functional such that \(\| \varepsilon _y\circ L\| = \| L(\cdot ,y)\| = |L(1_X,y)|=L(1_X,y)\). Moreover, \(\| L\| = \| L(1_X)\| _Y\), so that the estimate from theorem 3.2 recudes to the one in theorem 3.4.

However, for positive linear operators a simpler proof of theorem 3.4 is available which is given below.

Second proof of theorem 3.4. First recall that \(\widetilde A(f,y)=L(1_X,y) \cdot f(g_A(y))\). The estimate of \(|(L-\widetilde A)(g,y)|\) for functions \(g\in {\rm Lip~ }1\) may now be obtained in the following way.

Applying theorem 3.1 to the bounded operator \(\Delta =L-\widetilde A\) implies for any \(f\in C(X)\), and any \(\varepsilon {\gt}0\)

and the remaining part of the proof may be carried out as that of theorem 3.2. \(\Box \)

In the example to follow we illustrate theorem 3.4 by discussing two univariate cases.

The right hand side in the estimate of theorem 3.4 contains two test functions, namely

and

It is possible to replace both of them by other test functions as will be seen below. We do this first for \(1_X\). The estimate of the next theorem resembles one obtained by T. Nishishiraho  [ 42 , Lemma 4 ] .

If \(A\) is given as above, and if \(h\in C(X)\) is such that for the fixed \(y\) the expression \((f/h)(g_A(y))\) is defined, then

The first term of the last sum can be estimated as follows:

Here \(\widetilde A\) is the operator from the proofs of theorem 3.2, theorem 3.4, for which

Using this estimate for both \(|(L-\widetilde A)(f,y)|\) and \(|(\widetilde A -L)(h,y)|\) gives

which yields the claim of theorem 3.6.

Obviously, for \(h=1_X\) our last estimate is again the one from theorem 3.4. It is also possible to replace \(1_X\) in a different way: As in the proof of theorem 3.6, we first have

The second term is already part of the right hand side in the above estimate. For the first term we have

Now let \(g\in {\rm Lip~ }1\) be arbitrarily given. Then the last quantity is equal to

Thus

This concludes the proof.

Next we show how the test functions

can be replaced. To this end let \(\Phi \) be a function on \(X^2\) such that \(\Phi (\cdot ,y)\in C(X)\) for all \(y\in X\), and that for some fixed \(q{\gt}0\) the condition

holds. For possible choices of \(\Phi \) see the examples of T. Nishishiraho  [ 43 , p. 445f ] . The following theorem is an extension of Theorem 3.2 in our paper  [ 17 ] .

(1) is obtained by using theorem 3.6, estimate (2) with the aid of theorem 3.7. In both cases it is only necessary to estimate \(L(d(\cdot ,g_A(y));y)\) in order to obtain the claim of theorem 3.9.

For \(y\) fixed, the functional \(L(*,y)\) is a positive linear form on \(C(X)\). Thus if \(p\ge 1\), Hölder’s inequality implies

If \(q{\gt}0\) is fixed and such that

we have

Hence

Combining this with the inequalties from theorem 3.6, theorem 3.7, respectively, shows the validity of our claim.

Our next example continues the discussion of example 3.5.

Our next theorem generalizes theorem 3.4 in another direction. It shows that this theorem implies a variety of estimates including uniform ones and estimates in \(L_1\) spaces.

If \(g\in {\rm Lip~ }1\) is arbitrarily given, then, as in the proof of theorem 3.2 and in the second proof of theorem 3.4, the following holds:

Applying the positive functional \(\mu \in M\) to this inequality means

If \(L=0\), then it reads

implying

Hence in this case our claim is true.

If \(L\neq 0\), then the right hand side of \((*)\) may be estimated from above by

Thus in this case we arrive at

For \(\varepsilon {\gt}0\) the second term on the right hand side may be written as

and thus we have

Passing to the supremum in \(M\) now gives the estimate of theorem 3.12.

Note that if in theorem 3.12 one has \(M=\{ \varepsilon _y\} \) for some point evaluation functional \(\varepsilon _y\), \(y\in Y\), then the estimate given reduces to that given in theorem 3.4. The following corollary 3.13 establishes a similar generalization of theorem 3.9 (1) (for the case \(h=1_X\)).

The assertion in corollary 3.13 results from an estimate of
\(p_M[L(d(\cdot , g_A(*));*)]\) in theorem 3.12. For any \(p\ge 1\) we have

Applying Hölder’s inequality again, we conclude as in the proof of theorem 3.12 that

Passing to the inf over all \(p\ge 1\) implies

This yields the claim of corollary 3.13. \(\Box \)

As a further consequence of theorem 3.4 we mention

Use theorem 3.9 (1) with \(Y=X,\, A(f,x)=f(x),\, h=1_X,\, \Phi (x,y)=d(x,y)\) and \(p=2\).

Under the assumptions of theorem 3.15 it is of course also possible to give estimates similar to those in theorem 3.12 or corollary 3.13.

4 Estimates on Bounded and Positive Linear Operator Approximation in terms of Moduli of Continuity

In this section we shall prove inequalities in terms of the modulus of continuity. In Section 4.1 we shall give some consequences of theorem 3.12, thus employing the relationship between \(\widetilde{\omega }\) and \({\omega }\) as given in lemma 2.4. As will be seen in remark 4.3, in many important cases our technique yields results as good as those obtained by T. Nishishiraho. Thus even for the special compact spaces \((X,d)\) considered by Nishishiraho (who used the direct approach), our general approach via least concave majorants may be more powerful.

Another main point of Section 4 is motivated by an example in Nishishiraho’s paper  [ 43 , p. 453f. ] dealing with Bernstein operators. The estimates given there are consequences of one of his more general theorems. As will be discussed in Section 4.2, however, his version of the direct technique is not sensitive enough to imply a certain best possible result obtained by P.C. Sikkema [ 58 ] . Thus the second purpose of this section is to refine the direct technique in a way such that best possible constants can be obtained when evaluating the general upper bound.

Finally, in Section 4.3 it will be shown that the indirect and the direct approach are of equal value in the sense that each of them may give better results than the other one.

4.1 Estimates in terms of \({\omega }\; \)via Least Concave Majorants

We first discuss some of the consequences of theorem 3.12.

The inequality of (1) is an immediate consequence of \(\widetilde{\omega }(f,\varepsilon ) = {\omega }(f,\varepsilon )\). That of (2) is obtained by using the estimate of theorem 3.12 (with \(h\) instead of \(\varepsilon \)) and then observing that because of lemma 2.4 (1) one has

where \(\varepsilon {\gt}0\) again is arbitrarily given (and independent of \(h\)).  \(\Box \)

4.2 Refined estimates in terms of \({\omega }\; \)using the direct technique

All estimates of Section 4.1 in terms of \({\omega }\) were consequences of a single estimate being true for arbitrary metric spaces. The assumption \({\omega }(f,\xi \cdot \varepsilon )\le (1+\eta \xi )\cdot {\omega }(f,\varepsilon )\) was made in order to enable us to apply lemma 2.4 and thus to achieve inequalities in terms of \({\omega }\) instead of \(\widetilde{\omega }\).

As a first result of this section we shall show how this assumption can be used together with the so-called direct technique for the case of certain bounded operators. The result given below in theorem 4.4 resembles that of theorem 3.2; the technique of proof is due to M.A. Jiménez Pozo  [ 22 ] .

Similar to the proof of theorem 3.2, we first construct two auxiliary functions \(h_1\) and \(h_2\). Their definition is based upon the following observations. If \(f\in C(X)\) and \(x\in X\), then for all \(t\in X\)

For \(t\in X\) (and \(y\in Y\) fixed) we now define

Due to the above observations for the differences \(f(t)-f(g_A(y))\) it is clear that the continuous functions \(h_i\), \(i=1,2\), satisfy

Furthermore,

Also,

Hence,

where \(d(X)\) is the diameter of \(X\).
The assumption that \(L(1_X,y)\neq 0\) (so that \(\| \varepsilon _y \circ L\| \neq 0\), too) allows us to introduce the auxiliary functional \(T\) given by

For fixed \(y\in Y\) this is a continuous functional on \(C_{\mathbb {R}}(X)\). Using the same technique as in the proof of theorem 3.2 (representing measures), it may be seen that for each \(f\in C_{\mathbb {R}}(X)\), \(f\ge 0\), the inequality

holds; here

Applying the latter inequality to \((f-h_1)\) and \((h_2-f)\) shows that

Consequently,

Similarly, from

it follows that

Thus

Recalling the definition of \(T\), the final estimate is now obtained by observing first that

and then estimating the first term of the upper bound as follows:

This yields the inequality of theorem 4.4.

In the remainder of this section we shall assume again that \(L\) is a positive linear operator. We give a development which parallels that of Section 3. The slightly stronger assumption mentioned in remark 4.6 follows.

In the sequel we shall assume the existence of a fixed number \(\varrho {\gt}0\) such that we have

here \(]\varrho \xi [ = \max \{ z\in \mathbb {Z}\; :\; z {\lt} \varrho \cdot \xi \} \). As shown in lemma 1.6 (3), this type of inequality holds at least for spaces \((X,d)\) having a coefficient of convex deformation \(\varrho \ge 1\). In particular, for space \((X,d)\) being metrically convex in the sense of Menger (and thus \(\varrho =1\)), we have the better inequality

For approximation by positive linear operators this stronger inequality (as opposed to the weaker one \({\omega }(f,\xi \cdot \varepsilon ) \le (1+\xi )\cdot {\omega }(f,\varepsilon )\)) is of great importance when determining optimal constants (see e.g., P.C. Sikkema  [ 58 ] ). However, the use of the weaker form frequently has the advantage of leading to simpler estimates.

corollary 4.5 (2) already contains an inequality concerning the approximation by positive linear operators which was obtained via the direct technique. The following theorem is a first step to refine this method in connection with approximation by such operators. It constitutes an analogy of theorem 3.4.

An immediate consequence of theorem 4.7 is

Proof of theorem 4.7. As in the proof of theorem 3.2 first observe that

Hence it remains to estimate the first term on the right hand side of this inequality. Next observe that for \(t\in X\)

Thus

Application of \(L\) yields

This gives inequality (1).

Inequality (2) is obtained by observing that for \(\varrho =1\) and all \(t\in X\) one has

Finally, the estimates of the corollary are achieved by observing that

As was the case for the estimate of theorem 3.4, the direct approach also allows a partial replacement of the function \(1_X\) by suitable functions \(h\). This will be shown in the following two theorems. The first one constitutes an analogy to theorem 3.6.

As in the proof of theorem 3.6 the first observation is that

Here \(\widetilde A\) is again given by \(\widetilde A(f,y)=L(1_X,y)\cdot f(g_A(y))\). We now estimate \(|(L-\widetilde A)(f,y)|\) and \(|(L-\widetilde A)(h,y)|\) as in the proof of theorem 4.7 (1) to arrive at

Adding this quantity to the term \(|(f/h)(g_A(y))|\cdot |(L-A)(h,y)|\) yields the estimate of theorem 4.10.

As will be seen in discussion 4.13 (1) and discussion 4.13 (9), the estimate of theorem 4.10 contains all the main results of Nishishiraho’s papers cited in this work. Before showing this we are going to partially replace \(1_X\) by the second method already employed in theorem 3.7.

As in ?? it suffices to estimate

Also, as in the proof of theorem 4.7 it may be observed that for \(t\in X\) one has

which implies

Combining this with the remaining term \(|(f/h)(g_A(y))|\cdot |(L-A)(h,y)|\) gives the estimate of theorem 4.11.

The analogy of corollary 3.8 is the following

The following discussion will show how the bounded functions

may be replaced in the estimates of Theorems 4.7, 4.10, and 4.11. This discussion will complete the information available from corollary 4.8 and will yield results analogous to the ones from theorem 3.9, among others. It is also intended to relate the main results of this section to earlier work. At the same time it will give us an opportunity to briefly review univariate results of several types, and will thus parallel the discussion conducted in ??.

It is possible to obtain analogies of theorem 3.12 and corollary 3.13 using the direct approach. For instance, the following is a consequence of corollary 4.8 (1).

The proof of theorem 4.15 is a simple consequence of corollary 4.8 and is similar to that of theorem 3.12.

It is easily seen that it is also possible to find slightly improved estimates by using theorem 4.7 or such involving certain upper bounds of \(d(\cdot ,g_A(*))\) as was for instance done in corollary 3.13.

As was the case for theorem 3.12 and corollary 3.13, the approach of theorem 4.15 implies, for instance, uniform estimates if \(M\) is a set of point evaluation functionals, or estimates in \(L_1\)-spaces if the functional \(\mu \) is an integral operator.

The next remark will discuss the optimality of the estimates given in this section. Its importance is also evident in view of discussion 4.13 (see discussion 4.13 (1) in particular.)

As was the case in Section 3 (theorem 3.15), there is also a particular consequence of the direct approach for the pre-Hilbert space setting. For the sake of brevity we restrict ourselves to only giving the following corollary of discussion 4.13; two of Nishishiraho’s papers  [ 43 , Theorem 4 ] and  [ 44 , Theorem 5 ] contain similar estimates. However, it has to be noted that the use of refined estimates of discussion 4.13 leads to slightly improved inequalities sometimes yielding better results in applications. We decline to cite these at this point since their proofs are straightforward.

If \(X\) is a compact and convex set of a real pre-Hilbert space \(X\), then \(X\) equipped with metric \(d(x,y)=\sqrt{\langle x-y, x-y\rangle }\) is metrically convex. Thus the inequality of discussion 4.13 (10) is applicable to this case, so that the error estimate above is an immediate consequence of this observation.

4.3 Comparison of the results of 4.1 and 4.2

In order to conclude this section we show that both approaches discussed so far in order to obtain estimates in terms of \(\widetilde{\omega }\) or \({\omega }\) are useful, i.e., none of them yields a better estimate than the other one in general. Thus the following discussion is of particular importance. To demonstrate the fact mentioned, we consider the special situation \(Y=X\), \(\varrho =1\), \(A= {\rm Id}\), \(L(1_X,x)=1\) for some \(x\in X\).

5 Concluding Remarks

A) After submission of the author’s “Habilitationschrift” in 1985 and its subsequent 1986 publication in “Schriftenreihe des Fachbereichs Mathematik” of the University of Duisburg, several further papers dealing with related questions appeared. For obvious reasons, none of them contained estimates better than the ones given in the above. As a general source for related material we mention here the book by F. Altomare and M. Campiti  [ 1 ] . Recent contributions influenced by the Romanian school are due to D. Andrica, C. Badea, I. Raşa, and several of their collaborators (see, e.g.,  [ 2 ] , [ 3 ] , [ 4 ] , [ 47 ] ). As articles from China papers by Tian-ping Chen and Wen-ge Zhu ( [ 63 ] , [ 8 ] ) have to be mentioned. Related work was also carried out in the work of Ch. Richter and I. Stephani from the University of Jena (cf.  [ 48 ] , [ 49 ] , [ 50 ] , [ 60 ] , [ 61 ] ). An excellent survey on Shepard’s method, probably the most significant application of the general theory presented here, was recently finished by H. Knauf  [ 28 ] . In regard to the latter technique, there is also the interesting work of the group around Gh. Coman from Cluj-Napoca (see, e.g.,  [ 9 ] and the references given there).

B) In our recent paper “The second order modulus again: some still (?) open problems” (see  [ 19 ] ) we asked the following question: Is there a generalization of \({\omega }_2\) to \(C(X,d)\), where \((X,d)\) is a compact metric space with at least some suitable geometric structure? Here we would like to suggest the following definition.

Suppose \((X,d)\) is metrically convex in the sense of Menger. Then, for given \(x,y\in X\) there is always a point \(z\in X\) such that

Now consider all differences of the form

Then put

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