GEORGIAN MATHEMATICAL JOURNAL: Vol. 1, No. 5, 191994, 523-536

ON THE DURRMEYER-TYPE MODIFICATION OF SOME DISCRETE APPROXIMATION OPERATORS

PAULINA PYCH-TABERSKA

Abstract. In [10], for continuous functions f from the domain of certain discrete operators L_{n} the inequalities are proved concerning the modulus of continuity of L_{n}f. Here we present analogues of the results obtained for the Durrmeyer-type modification _{L_{n}} of L_{n}. Moreover, we give the estimates of the rate of convergence of _{L_{n}}f in Hölder-type norms

1. Introduction and Notation

Let I be a finite or infinite interval. Consider a sequence (J_{k})_{1}^{∞} of some index sets contained in Z := {0, ±1, ±2, … }, choose real numbers ξ_{j,k} ∈ I and fix non-negative functions p_{j,k} continuous on I. Write, formally,

L_{k}f(x) := ∑_{j=J_{k}} f(ξ_{j,k})p_{j,k}(x) (x ∈ I, k ∈ N := {1, 2, … })  (1)

for univariate (complex-valued) functions f defined on I. If for f_{0}(x) ≡ 1 on I the values L_{k}f_{0}(x) (x ∈ I, k ∈ N) are finite, then L_{k}f are well-defined for every function f bounded on I. Under appropriate additional assumptions, operators (1) are meaningful also for some locally bounded functions f on infinite intervals I. The fundamental approximation properties of operators (1) in the space C(I) of all continuous functions on I can be deduced, for example, via the general Bohman–Korovkin theorems ([5], Sect. 2.2).

Recently, several authors have investigated relations between the smoothness properties of the functions f and L_{k}f ([1], [10], [15]). For example, taking an arbitrary function f ∈ C(I) ∩ Dom(L_{n}), n ∈ N, Kratz and Stadtmüller [10] obtained the following result. Let

∑_{j∈J_{k}} p_{j,k}(x) ≤ c_{1} for all x ∈ I, k ∈ N,  (2)

523 and let the sum of the above series be independent of x; if, moreover,

1991 Mathematics Subject Classification. 41A17, 41A25, 26A15.

1072-947X/1994/0900-0523$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

p′_{j,k} ∈ C(I^{∘}), ∑_{j∈J_{k}} |(ξ_{j.k} − x)p′_{j,k}(x)| ≤ c′_{1} for all x ∈I^{∘}, k ∈ N,

where c_{1}, c′_{1} are positive constants and I^{∘} denotes the interior of I, then the ordinary moduli of continuity of f and L_{n}f satisfy the inequality

ω(L_{n}f; δ) ≤ 2(c_{1} + c′_{1})ω(f; δ) (δ ≥ 0).

They proved an analogous inequality for the suitable weighted moduli of continuity of f and L_{n}f when I is an infinite interval and f has the modulus |f| of polynomial growth at infinity. In [12] their result is extended to functions f having |f| of a stronger growth than the polynomial one. [12] also presents some applications of the above-mentioned inequalities in problems of approximation of continuous functions f by L_{n}f in some Hölder-type norms.

Suppose that for every j ∈ J_{k} and every k ∈ N the integral ∫_{I} p_{j,k}(t)dt coincides with a positive number, say, 1/q_{j,k}. Denote by L̃_{k} the operators given by

L̃_{k}f(x) ≡ L̃_{k}(f)(x) := ∑_{j∈J_{k}} q_{j,k}p_{j,k}(X) ∫_{I} f(t)p_{j,k}(t)dt (x ∈ I, k ∈ N)(3)

for these measurable (complex-valued) functions f for which the right-hand side of (3) is meaningful. This modification of the classical Bernstein polynomials was first introduced by J.I. Durrmeyer (see [4]). The approximation properties of these polynomials were investigated, for example, in [4], [7], [2]. Some results on the approximation of functions by the Durrmeyertype modification of the Szász–Mirakyan operators, Baskakov operators or Meyer–König and Zeller operators can be found, for example, in [8], [9], [13], [14], [16].

In this paper we derive Kratz and Stadtmüller type inequalities involving ordinary or weighted moduli of continuity of the functions f and L̃_{n}f on I. Using these inequalities, we obtain estimates of the degree of approximation of f by L̃_{n}f in some Hölder-type norms. Theorems 1–3 show that the smoothness properties of L̃_{n}f are slightly different from those of L_{n}f.

We adopt the following notation. Given any non-negative function w defined on I and any x, y ∈ I, we write w̌(x, y) := min{w(x), w(y)}.

For an arbitrary function f defined on I we introduce the quantities

∥f∥_{w} := sup{|f(x)|w(x) : x ∈ I},

Ω_{w}(f; δ) := sup{|f(x) − f(y)||w̌(x, y) : x, y ∈ I, |x − y| ≤ δ} (δ ≥ 0).

If f is continuous on I and ∥f∥_{w} < ∞, we say that f ∈ C_{w}(I). The quantity Ω_{w}(f; δ) is called the weighted modulus of continuity of f on I. In case w(x) = 1 for all x ∈ I, Ω_{w}(f; δ) becomes ω(f; δ) and the symbol ∥f∥ is used instead of ∥f∥_{w}. If the weight w is nondecreasing [nonincreasing] on I, then

Ω_{w}(f; δ) := sup{|f(x) − f(y)|w(x)} [Ω_{w}(f; δ) := sup{|f(x) − f(y)|w(y)}],

where the supremum is taken over all x, y ∈ I such that 0 < y − x ≤ δ.

We denote by W the set of all continuous functions w on I with values not greater than 1, which are positive in the interior of I and satisfy the inequality w̌(x, y) ≤ w(t) for any three points x, t, y ∈ I such that x ≤ t ≤ y (obviously, this inequality holds if, for example, w is nondecreasing, nonincreasing or concave on I). When I is an infinite interval, we indtroduce, in addition, the set Λ of all positive functions η belonging to W such that η(x) → 0 as |x| → 0.

Given two weights w, η ∈ W , we define a more general modulus of continuity of f on I by

Ω_{w,η}(f; δ) := sup{|f(x) − f(y)|w̌(x, y)η̌(x, y) : x, y ∈ I, |x − y| ≤ δ}.

It reduces to Ω_{w}(f; δ) if η ≡ 1 on I, and to Ω_{η}(f; δ) if w ≡ 1 on I. Taking into account that the positive function φ is nondecreasing on the interval (0, 1] and has values not greater that 1, we put

∥f∥w,η^{(φ)} := ∥f∥_{wη} +

+ sup \{\frac{|f(x) − f(y)|w̌(x, y)η̌(x, y)}{φ(|x − y|)} : x, y ∈ I, < |x − y| ≤ 1\}.

If this quantity is finite, we call it the Hölder-type norm of f on I. Under the assumption f ∈ C_{η}(I), ∥f∥w,η^{(φ)} < ∞ if and only if there exists a positive constant K such that Ω_{w,η}(f; δ) ≤ Kφ(δ) for every δ ∈ (0, 1]. We write

∥f∥w^{(φ)} for ∥f∥w,η^{(φ)} if η ≡ 1 on I, and ∥f∥η^{(φ)} if w ≡ 1 on I.

Throughout this paper the symbols c_{ν} (ν = 1, 2, …) will mean some positive constants depending only on a given sequence (L_{k})_{1}^{∞} and eventually on the considered weights w, η, ρ. The integer part of the real number will be denoted by [a].

2. Smoothness Properties

Let L̃_{k}, k ∈ N, be the operators defined by (3) such that L̃_{k}f_{0}(x) are finite at every x ∈ I. Put

r_{k}(x) := ∑_{j∈J_{k}} p_{j,k}(x) − 1 (x ∈ I, k ∈ N)

and make the standing assumption that all functions p_{j,k} (j ∈ J_{k}, k ∈ N) are absolutely continuous on every compact interval contained in I. Consider measurable functions f locally bounded on I and belonging to Dom(L̃_{n}) for

some n ∈ N. Write, as in Section 1, I^{∘}= Int I.

Theorem 1. Suppose that condition (2) is satisfied and

∑_{j∈J_{k}} q_{j,k}|p′_{j,k}(x)| ∫_{I} |t − x|p_{j,k}(t)dt ≤ \frac{c_{2}}{w(x)}  (4)

for x ∈I^{∘} and all k ∈ N, w being a function of the class W. Then

Ω_{w}(L̃_{n}f; δ) ≤ c_{3}ω(f; δ) + ∥f∥_{w}ω(r_{n}; δ) (δ ≥ 0),  (5)

where c_{3} = 2(c_{1}∥w∥ + c_{2}).

Proof. Let x, y ∈ I 0 < y − x ≤ δ and let x_{0} := (x + y)/2. Clearly,

L̃_{n}f(x)−L̃_{n}f(y)= ∑_{j∈J_{n}}q_{j,n}(p_{j,n}(x)−p_{j,n}(y))∫_{I} (f(t)−f(x_{0}))p_{j,n}(t)dt+

+f(x_{0})(r_{n}(x) − r_{n}(y)).  (6)

Taking into account (2) and the well-known inequality |f(t) − f(x_{0})| ≤ (1 + [|t − x_{0}|δ^{−1}])ω(f; δ), we obtain |L̃_{n}f(x) − L̃_{n}f(y)| ≤ (2c_{1} + A_{n}(x, y))× ω(f; δ) + |f(x_{0})|ω(r_{n}; δ), where

A_{n}(x, y) := ∑_{j∈J_{n}} q_{j,n}|p_{j,n}(x) − p_{j,n}(y)|δ^{−1} ∫_{I\I_{δ}} |t − x_{0}|p_{j,n}(t)dt ≤

≤ δ^{−1} ∫_{x}^{y} ( ∑_{j∈J_{n}} q_{j,n}|p′_{j,n}(s)| ∫_{I\I_{δ}} |t − x_{0}|p_{j,n}(t)dt)ds

and I_{δ} := I ∩ (x_{0} − δ, x_{0} + δ). If x < s < y and |t − x_{0}| ≥ y − x, then |t − x_{0}| ≤ 2|t − s|. Hence, applying (4), we get

A_{n}(x, y):=2δ^{−1}∫_{x}^{y}( ∑_{j∈J_{n}} q_{j,n}|p′_{j,n}(s)|∫_{I} |t−s|p_{j,n}(t)dt)ds≤2c_{2}δ^{−1}∫_{x}^{y} \frac{1}{w(s)}ds,

and inequality (5) follows.

The result of Theorem 1 is interesting if ω(f; δ) < ∞. This holds, for example, for functions f ∈ C(I) on the compact interval I. If I is an infinite interval, the assumption ω(f; δ) < ∞ implies the restriction f(x) = O(|x|) as |x| → ∞. So, in this case, it is convenient to use the weighted modulus of continuity Ω_{η}(f; δ) with some η ∈ Λ. If f ∈ C_{η}(I), then this modulus is a nondecreasing function of δ on the interval [0, ∞). It is easy to verify that, for every δ > 0 and for all x, y ∈ I there holds the inequality

|f(x) − f(y)|η̌(X, y) ≤ (1 + [δ^{−1}|x − y|])Ω_{η}(f; δ).  (7)

Moreover, in case ρ ∈ Λ and ρ(x)/η(x) → 0 as |x| → ∞ we have Ω_{ρ}(f; δ) → 0 as δ → 0+, whenever f ∈ C_{η}(I) is uniformly continuous on each finite interval contained in I.

Note that under the assumptions η ∈ Λ, f ∈ C_{η}(I) and L̃_{k}(1/η)(x) < ∞ we have |L_{k}f(x)| < ∞. If, moreover, ρ ∈ Λ and

L̃_{k}(\frac{1}{η} )(x) ≤ \frac{c_{4}}{ρ(x)} for all x ∈ I and k ∈ N  (8)

then ∥L̃_{k}f∥_{ρ} < ∞.

In the next two theorems it is assumed that I is an infinite interval. ■

Theorem 2. Let condition (2) be satisfied. Suppose, moreover, that there exist functions w ∈ W, ρ, η ∈ Λ, ρ ≤ η such that (4), (8) and

∑_{j∈J_{k}} q_{j,k}|p′_{j,k}(x)| ∫_{I} \frac{|t − x|}{η(t)}p_{j,k}(t)dt ≤

≤ \frac{c_{5}}{w(x)ρ(x)} for a.e. x ∈I^{∘} and k ∈ N  (9)

hold. Then

Ω_{w,ρ}(L̃_{n}f; δ) ≤ c_{6}Ω_{η}(f; δ) + ∥f∥_{wρ}ω(r_{n}; δ) (δ ≥ 0),  (10)

where c_{6} = 2((c_{1} + c_{4})∥w∥ + c_{2} + c_{5}).

Proof. Consider x, y ∈ I such that 0 < y − x ≤ δ. Retain the symbol x_{0} used in the proof of Theorem 1 and start with identity (6). In view of (7), |L̃_{n}f(x) − L̃_{n}f(y)| ≤ B_{n}(x, y)Ω_{η}(f; δ) + |f(x_{0})||r_{n}(x) − r_{n}(y)|, where

B_{n}(x, y) := ∑_{j∈J_{n}} q_{j,n}|p_{j,n}(x) − p_{j,n}(y)| ∫_{I} (1 + [δ^{−1}|t − x_{0}|])\frac{1}{η̌(t, x_{0})}p_{j,n}(t)dt.

Observing that for every t ∈ I

\frac{ρ̌(x, y)}{η̌(t, x_{0})} ≤ 1 + \frac{ρ̌(x, y)}{η(t)}  (11)

and applying (2), we obtain

B_{n}(x, y)ρ̌(x, y) ≤ 2c_{1} + ∑_{j∈J_{n}} q_{j,n}|p_{j,n}(x) − p_{j,n}(y)| ∫_{I} \frac{ρ̌(x, y)}{η(t)}p_{j,n}(t)dt +

+δ^{−1} ∑_{j∈J_{n}} q_{j,n} ∫_{x}^{y} |p′_{j,n}(s)|ds ∫_{I−I_{δ}} (1 + \frac{ρ̌(x, y)}{η(t)})|t − x_{0}|p_{j,n}(t)dt.

Further, the inequality |t − x_{0}| ≤ 2|t − s| (t ∈ I\I_{δ}, x < s < y) and assumptions (4), (8), (9) lead to

B_{n}(x, y)ρ̌(x, y) ≤ 2(c_{1} + c_{4}) + 2δ^{−1} ∫_{x}^{y} \frac{c_{2} + c_{5}}{w(s)} ds.

The desired estimate is now evident.

For functions f for which |f| is of the polynomial growth at infinity our result can be stated as follows. ■

Theorem 3. Let conditions (2), (4) be satisfied and let η(x) = (1+|x|)^{−σ} x ∈ I σ > 0. Suppose that inequality (9) in which ρ = η holds. Then

Ω_{w,η}(L̃_{n}f; δ) ≤ c_{7}Ω_{η}(f; δ) + ∥f∥_{wη}ω(r_{n}; δ) (δ ≥ 0),

where c_{7} = 2(c_{1} + 2 · 3^{σ}c_{1} + c_{2} + 2c_{5}).

Proof. To see this it is enough to make a slight modification in the evaluation of the term B_{n}(x, y) occurring in the proof of Theorem 2. Namely, let us divide the interval I into two sets I_{n} and I\I_{h}, where I_{h} := I∩(x_{0}−h, x_{0}+h), h = y − x. If t ∈ I_{h}, then [δ^{−1}|t − x_{0}|] = 0 and

\frac{η̌(x, y)}{η(t)} ≤ 3^{σ}η̌(x, y)(\frac{1}{η(x)} + \frac{1}{η(y)}) ≤ 2 · 3^{σ}.

This inequality, (11) and (2) imply

B_{n}(x, y)η̌(x, y) ≤ 2(1 + 2 · 3^{σ})c_{1} +

+ ∑_{j∈J_{n}} q_{j,n}|p′_{j,n}(s)ds| ∫_{I\I_{h}} (\frac{|t − x_{0}|}{δ} + \frac{η̌(x, y)}{η(t)}(1 + \frac{|t − x_{0}|}{y − x}))p_{j,n}(t)dt.

Observing that |t − x_{0}| ≤ 2|t − s|, |t − x_{0}| ≤ y − x whenever t ∈ I\I_{h}, x < s < y, we obtain, on account of (4) and (9) (with ρ = η),

B_{n}(x, y)η̌(x, y) ≤ 2(1 + 2 · 3^{σ})c_{1} +

+\frac{2}{δ} ∫_{x}^{y} \frac{c_{2}}{w(s)}ds + 4\frac{η̌(x, y)}{y − x} ∫_{x}^{y} ( ∑_{j∈J_{n}} q_{j,n}|p′_{j,n}(s)| ∫_{I} \frac{|t − s|}{η(t)}p_{j,n}(t)dt)ds ≤

≤ 2(1 + 2 · 3^{σ})c_{1} + \frac{2}{y − x} ∫_{x}^{y} \frac{c_{2} + c_{5}}{w(s)} ds .

Thus

B_{n}(x, y)w̌(x, y)η̌(x, y) ≤ 2(1 + 2 · 3^{σ})c_{1}∥w∥ + 2c_{2} + 4c_{5}. ■

Remark 1. For many known operators the functions r_{k}(x) ≡ 0 on I, the quantities μ_{2,k}(x) := ∑_{j∈J_{k}}(ξ_{j,k} − x)^{2}p_{j,k}(x) are finite at every x ∈ I and

positive in I^{∘}; moreover,

p′_{j,k}(x)μ_{2,k}(x) = p_{j,k}(x)(ξ_{j,k} − x)  (12)

for every x ∈I^{∘} and every k ∈ N. In view of identity (12) and the Cauchy– Schwartz inequality the left-hand side of (4) can be estimated from above by (μ̃_{2,k}(x)/μ_{2,k}(x))^{1/2}, where μ̃_{2,k}(x) := ∑_{j∈J_{k}} q_{j,k}|p_{j,k}(x)| ∫_{I}(t−x)^{2}p_{j,k}(t)dt. Therefore, in this case, assumption (4) can be replaced by

\frac{μ̃_{2,k}(x)}{μ_{2,k}(x)} ≤ \frac{c_{2}^{2}}{w^{2}(x)} for all x ∈I^{∘}, k ∈ N.  (13)

Analogously, the left-hand side of (9) can be estimated by

\frac{1}{μ_{2,k}(x)}(μ̃_{2,k}(x) ∑_{j∈J_{k}} q_{j,k}(ξ_{j,k} − x)^{2}p_{j,k}(x) ∫_{I} \frac{p_{j,k}(t)}{η^{2}(t)}dt)^{1/2}.

Hence, if

\frac{1}{μ_{2,k}(x)} ∑_{j∈J_{k}} q_{j,k}p_{j,k}(x)(ξ_{j,k} − x)^{2} ∫_{I} \frac{p_{j,k}(t)}{η^{2}(t)}dt ≤ \frac{c_{8}^{2}}{ρ^{2}(x)}  (14)

for all x ∈I^{∘}, k ∈ N, then (9) holds with c_{5} = c_{2} · c_{8}.

Remark 2. Let w ∈ W, η ∈ Λ. Define the weighted modulus Φ_{w}(f; δ) and Φ_{w,η}(f; δ) as in Section 1, replacing w̌(x, y) by

\overline{w}(x, y) := \begin{cases}0 & if w(x) = 0 or w(y) = 0, \\ (\frac{1}{w(x)} + \frac{1}{w(y)})^{−1} & otherwise, \end{cases}

and η̌(x, y) by \overline{η}(x, y), respectively. Since \overline{w}(x, y) ≤ w̌(x, y) for every pair of points x, y ∈ I, Theorem 1 remains valid for Φ_{w}(L̃_{n}f; δ). Further, in this case, inequality (7) becomes |f(x) − f(y)|\overline{η}(x, y) ≤ 2(1 + [δ^{−1}|x − y|])Φ_{η}(f; δ). Consequently, under the assumptions of Theorem 2, the modulus Φ_{w,ρ}(L̃_{n}f; δ) and Φ_{η}(f; δ) satisfy inequality (10) with the constant 2c_{6} instead of c_{6}.

Note that, for the weight η(x) = (1 + |x|)^{−σ} with the parameter σ > 0, the modulus Φ_{η}(f; δ) is equivalent to the one introduced in [10], p. 331 (see also [12]).

3. Approximation Properties

Considering still the functions f as in Section 2 we first estimate the ordinary weighted norm of the difference L̃_{n}f − f.

Theorem 4. Let condition (2) be satisfied and let

ρ(x)L̃_{k}(\frac{1}{η^{2}} )(x) ≤ \frac{c_{9}}{η(x)} for all x ∈ I, k ∈ N,  (15)
ρ(x)μ̃_{2,k}(x) ≤ c_{10}η(x)δ_{k}^{2} for all x ∈ I, k ∈ N,  (16)

where (δ_{k})_{1}^{∞} is a sequence of positive numbers, η is a positive function on I and ρ is a non-negative one such that ρ ≤ η. Then

∥L̃_{n}f − f∥_{ρ} ≤ c_{11}Ω_{η}(f; δ_{n}) + ∥f∥_{ρ}∥r_{n}∥,  (17)

where c_{11} = c_{1} + (c_{1}c_{9})^{1/2} + (c_{9}c_{10})^{1/2} + c_{10}.

Proof. Start with the obvious identity

L̃_{n}f(x) − f(x) = ∑_{j∈J_{n}} q_{j,n}p_{j,n}(x) ∫_{I} (f(t) − f(x))p_{j,n}(t)dt + f(x)r_{n}(x)

and take a positive number δ. In view of (7) and the inequality (η̌(x, t))^{−1} ≤ (η(x))^{−1} + (η(t))^{−1} we have |L̃_{n}f(x) − f(x)| ≤ γ_{n}(x)Ω_{η}(f; δ) + |f(x)| · ∥r_{n}∥, where

γ_{n}(x) := ∑_{j∈J_{n}} q_{j,n}p_{j,n}(x) ∫_{I} (1 + [δ^{−1}|t − x|])(\frac{1}{η(x)} + \frac{1}{η(t)})p_{j,n}(t)dt.

Further, by (2), (15) and (16) and the Cauchy–Schwartz inequality we obtain

γ_{n}(x)ρ(x) ≤ c_{1} + L̃_{n}(\frac{1}{η} )(x)ρ(x) + δ^{−2} \frac{ρ(x)}{η(x)}μ̃_{2,n}(x) +

+ρ(x)δ^{−1} ∑_{j∈J_{n}} q_{j,n}p_{j,n}(x) ∫_{I} \frac{|t − x|}{η(t)}p_{j,n}(t)dt ≤

≤ c_{1} + (c_{1}L̃_{n}(\frac{1}{η^{2}} )(x))^{1/2}ρ(x) + c_{10}δ^{−2}δ_{n}^{2} +

+ρ(x)δ^{−1}(μ̃_{2,n}(x))^{1/2}(L̃_{n}(\frac{1}{η^{2}} )(x))^{1/2} ≤

≤ c_{1} + (c_{1}c_{9})^{1/2} + c_{10}δ^{−2}δ_{n}^{2} + (c_{9}c_{10})^{1/2}δ^{−1}δ_{n}.

Choosing δ = δ_{n}, we get (17) at once. ■

Remark 3. In the case when η(x) = 1 for all x ∈ I, the constant c_{11} in (17) is equal to c_{1} + c_{10}. If we use the modulus Φ_{η}(f; δ) (defined in Remark 2) instead of Ω_{η}(f; δ), the constant c_{11} should be multiplied by 2.

Passing to approximation in the Hölder-type norm we note that, for an arbitrary ν_{n} ∈ (0, 1],

∥L̃_{n}f − f∥_{w,η}^{(φ)} ≤ (1 + \frac{2}{φ(ν_{n})})∥L̃_{n}f − f∥_{wη} +

+ sup \{\frac{1}{φ(δ)}(Ω_{w,η}(L̃_{n}f; δ) + Ω_{w,η}(f; δ)) : 0 < δ ≤ ν_{n}\}  (18)

(see, for example, [11], [12]). This inequality, Theorem 4 and the estimates obtained in Section 2 allow us to state a few standard results. We will formulate only one of them. Namely, combining inequality (18) with Theorems 1 and 2 gives

Theorem 5. Let conditions (2), (4) be satisfied and let (δ_{k})_{1}^{∞} be a sequence of numbers from (0, 1] for which (16) holds with ρ = w and η ≡ 1 on I. Then

∥L̃_{n}f − f∥_{w}^{(φ)} ≤ c_{12} sup \{\frac{ω(f; δ)}{φ(δ)} : 0 < δ ≤ δ_{n}\} + ∥f∥_{w}Δ_{n}^{(φ)} ,

where c_{12} = 3c_{1} + 2c_{2} + 3c_{10} + (1 + 2c_{1})∥w∥ and

Δ_{n}^{(φ)} = 3∥r_{n}∥/φ(δ_{n}) + sup{ω(r_{n}; δ)/φ(δ) : 0 < δ ≤ δ_{n}}.

Remark 4. Clearly, if the assumptions of Theorems 1 − −5 hold for positive integers k belonging to a certain subset N_{1} of N, then the corresponding assertions remain valid only if n ∈ N_{1}.

4. Examples

1) The Bernstein polynomials B_{k}f ≡ L_{k}f are defined by (1) with ξ_{j,k} =

j/k, p_{j,k} = (\begin{matrix}k \\ j \end{matrix}) x^{j}(1 − x)^{k−j}, I = [0, 1], J_{k} = \{0, 1, 2, …, k\}. The corre-

sponding Bernstein–Durrmeyer polynomials L̃_{k}f ≡ L̃_{k}f are of the form (3) in which q_{j,k} = k+1 for all j ∈ J_{k}, k ∈ N. In this case r_{k}(x) = 0 for all x ∈ I, the constant c_{1} in (2) equals 1, μ_{2,k}(x) = x(1 − x)/k and equality (12) is true. Since μ̃_{2,k}(x) = \frac{2x(1−x)(k−3)+2}{(k+2)(k+3)} (x ∈ I, k ∈ N) (see [4]), we easily state that condition (13) is satisfied with c_{2} = 1, w(x) = (x(1 − x))^{1/2}. Hence, in view of Theorem 1 (and Remark 1), for every f ∈ C(I) and every n ∈ N, Ω_{w}(B̃_{n}f; δ) ≤ 3ω(f; δ) (δ ≥ 0) Further, μ̃_{2,k}(x) ≤ \frac{1}{2k} for all x ∈ I, k ∈ N (see [4], p. 327). Therefore (16) holds with ρ(x) = η(x) = 1 for all x ∈ I, δ_{k} = k^{−1/2} and c_{10} = 1/2. Thus Theorem 4 gives ∥B̃_{n}f −f∥ ≤ \frac{3}{2}ω(f; n^{−1/2}) for all n ∈ N (cf. [4], Theorem II.2). Also, Theorem 5 applies with w(x) = (x(1 − x))^{1/2}, δ_{n} = n^{−1/2}, c_{12} = 8 and Δ_{n}^{(φ)} = 0.

2) The Meier–König and Zeller operators M_{k} ≡ L_{k} are defined by ξ_{j,k} =

j/(j + k), p_{j,k}(x) = (\begin{matrix}k + j − 1 \\ j \end{matrix}) x^{j}(1 − x)^{k}, x ∈ I = [0, 1), j ∈ J_{n} = N_{0},

N_{0} := {0, 1, … }. Their Durrmeyer modification M̃ ≡ L̃_{k} are of the form (3) in which q_{j,k} = (k + j)(k + j + 1)/k. Condition (2) holds with c_{1} = 1. Since

p′_{j,k}(x)\frac{x(1 − x)^{2}}{k} = p_{j,k+1}(x)(\frac{j}{k + j} − x)^{2} (0 < x < 1),

the left-hand side of (4) can be estimated from above by

\frac{k}{x(1 − x)^{2}} (\{ ∑_{j=0}^{∞} (\frac{j}{k + j} − x)^{2}p_{j,k+1}(x)\} ×

×{ ∑_{j=0}^{∞} q_{j,k}p_{j,k+1}(x) ∫_{0}^{1}(t − x)^{2}p_{j,k}(t)dt})^{1/2}

for all x ∈ (0, 1), k ∈ N. If k ≥ 3, the expression in the first curly brackets is not greater than 2x(1 − x)^{2}/k (see [3]); straightforward calculation shows that the expression in the second ones does not exceed 7(1 − x)^{2}/k. Thus, for the functions f ∈ C(I) ∩ Dom(M̃_{n}) and M̃_{n}f (n ≥ 3), inequality (5) applies with c_{3} = 10, w(x) = x^{1/2} and r_{n}(x) = 0 for all x ∈ I.

3) The Baskakov–Durrmeyer operators Ũ_{k,c} ≡ L̃_{k} (with a parameter c ∈ N_{0}) are defined by (3) in which I = [0, ∞), J_{k} = N_{0}, p_{j,k}(x) = (−1)^{j}x^{j}ψ_{k,c}^{(j)}(x)/j!, ψ_{k,c}(x) = e^{−kx} if c = 0, and ψ_{k,c}(x) = (1 + cx)^{−k/c} if c ≥ 1, q_{j,k} = k − c for k > c (see [9]). Now r_{k}(x) = 0 for all x ∈ I, k ∈ N, c_{1} = 1, μ_{2,k}(x) = x(1 + cx)/k for all x ∈ I, k > c and condition (12) holds with ξ_{j,k} = j/k. Further,

μ̃_{2,k} = \frac{2x(1 + cx)(k + 3c) + 2}{(k − 2c)(k − 3c)} for x ∈ I, k > 3c.

Hence Theorem 1 (via Remarks 1, 4) applies for n > 3c, with w(x) = (x/(1 + x))^{1/2}, c_{3} = 2(1 + c_{2}), c_{2} = (2(1 + 3c)(1 + 6c)/(1 + c))^{1/2}.

4) The Szász–Mirakyan–Durrmeyer operators S̃_{k} are the special case of operators Ũ_{k,c} defined in 3), with c = 0. From 3) we know that, for these operators, conditions (2) and (13) hold with c_{1} = 1, c_{2} = 2^{1/2} and w(x) = (x/(1 + x))^{1/2}. Consider f ∈ C_{η}(I) with the weight η(x) = (1 + x)^{−σ} where σ ∈ N. It is easy to see that, for k ≥ 2σ,

∫_{0}^{∞}\frac{1}{η^{2}(t)}p_{j,k}(t)dt= \frac{k^{j}}{j!} ∫_{0}^{∞}(1+t)^{2σ}t^{j}e^{−kt}dt≤2^{2σ−1}(\frac{1}{k} + \frac{k^{j}}{j!} ∫_{0}^{∞}t^{2σ+j}e^{−kt}dt)=

= 2^{2σ−1} \frac{1}{k} (1+ \frac{(2σ + j)!}{j!}k^{−2σ})≤2^{2σ−1} \frac{1}{k} (1+(\frac{j}{k} +1)^{2σ}).

Consequently, the left-hand side of (14) is not greater than

\frac{2^{2σ−1}}{μ_{2,k}(x)} ∑_{j=0}^{∞} (\frac{j}{k} − x)^{2}p_{j,k}(x)(1 + 2^{2σ−1}((1 + x)^{2σ} + (\frac{j}{k} − x)^{2σ})) =

= 2^{2σ−1}(1 + 2^{2σ−1}(1 + x)^{2σ}) + \frac{4^{2σ−1}}{μ_{2,k}(x)} ∑_{j=0}^{∞} (\frac{j}{k} − x)^{2σ+2}p_{j,k}(x) ≤

≤ c_{13}(1 + x)^{2σ}

(see [10], p. 334). Applying Theorem 3 (together with Remarks 1, 4), we get the estimate

Ω_{w,η}(S̃_{n}f; δ) ≤ c_{14}Ω_{η}(f; δ) (δ ≥ 0, n ≥ 2σ).  (19)

Since μ̃_{2,k}(x) ≤ 2(1 + x)/k, conditions (15) and (16) are satisfied with ρ(x) = (1 + x)^{−σ−1} and δ_{k} = k^{−1/2}. Consequently, Theorem 4 gives

∥S̃_{n}f − f∥_{ρ} ≤ c_{15}Ω_{η}(f; n^{−1/2}) for all n ∈ N.

Combining this result and (19) with the general inequality (18), we easily verify that, for n ≥ 2σ,

∥S̃_{n}f − f∥_{w,ρ}^{(φ)} ≤ c_{16} sup \{\frac{1}{φ(δ)}Ω_{η}(f; δ) : 0 < δ ≤ n^{−1/2}\}.

5) The generalized Favard operators F_{k} ≡ L_{k} are deefined by (1) with ξ_{j,k} = j/k, J_{k} = Z, I = (−∞, ∞) and

p_{j,k}(x) ≡ p_{j,k}(γ; x) = (\sqrt{2π}kγ_{k})^{−1} exp ( − \frac{1}{2}γ_{k}^{−2}(\frac{j}{k} − x)^{2}),

γ = (γ_{k})_{1}^{∞} being a positive null sequence satisfying

k^{2}γ_{k}^{2} ≥ \frac{1}{2}π^{−2} log k for k ≥ 2, γ_{1}^{2} ≥ \frac{1}{2}π^{−2} log 2

(see [6]). Denote by F̃_{k} their Durrmeyer modification of form (3) in which q_{j,k} = k for all j ∈ Z and k ∈ N. As is known ([6], [12]), for all x ∈ I and k ∈ N,

|r_{k}(x)| ≡ |r_{k}(γ; x)| = | ∑_{j=−∞}^{∞} p_{j,k}(γ; x) − 1| ≤ 2 or |r_{k}(γ; x)| ≤ 7πγ_{k}.

μ_{2,k}(x) ≡ μ_{2,k}(γ; x) ≤ 51γ_{k}^{2}; moreover, ω(r_{k}(γ; x)| ≤ 16πδ for every δ ≥ 0 (see [10], p. 336). It is easy to see that

μ̃_{2,k}(x) ≡ μ̃_{2,k}(γ; x) = μ_{2,k}(γ; x) + γ_{k}^{2}(1 + r_{k}(γ; x)) ≤ 54γ_{k}^{2}.

Observing that

p′_{j,k}(γ; x) = γ_{k}^{−2}(\frac{j}{k} − x)p_{j,k}(γ; x)

and applying the Cauchy–Schwartz inequality, we estimate the left-hand side of (4) by

kγ_{k}^{−2} ∑_{j=−∞}^{∞} |\frac{j}{k} − x|p_{j,k}(γ; x) ∫_{−∞}^{∞} |t − x|p_{j,k}(γ; t)dt ≤

≤ γ_{k}^{−2}(μ_{2,k}(γ; x))^{1/2}(μ̃_{2,k}(γ; x))^{1/2},

i.e., w(x) = 1 for all real x and c_{2} = 52, 5. Thus Theorem 1 yields the estimate

ω(F̃_{n}f; δ) ≤ 111ω(f; δ) + 16πδ∥f∥ (δ ≥ 0)

for every n ∈ N and every f ∈ C(I). Clearly, this inequality is interesting if f ∈ C(I) is bounded on I.

Consider now f ∈ C_{η}(I) where η(x) = exp(−σx^{2}) σ > 0. If σγ_{k}^{2} ≥ 3/32, then

exp(σx^{2}) exp ( − \frac{1}{2}γ_{k}^{−2}(\frac{j}{k} − x)^{2}) exp ( − \frac{1}{2}γ_{k}^{−2}(\frac{j}{k} − t)^{2}) ≤

≤ exp(4σx^{2}) exp ( − \frac{1}{8}γ_{k}^{−2}(\frac{j}{k} − x)^{2}) exp ( − \frac{1}{8}γ_{k}^{−2}(\frac{j}{k} − t)^{2});

whence

F̃_{k}(1/η)(x) ≤ 2(1 + r_{k}(2γ; x)) exp(4σx^{2}).

Analogously, one can show that the left-hand side of (9) is not greater than

2γ_{k}^{−2}μ_{2,k}(γ; x))^{1/2}(μ̃_{2,k}(2γ; x))^{1/2} exp(4σx^{2})

provided that σγ_{k}^{2} ≤ 3/64. Further (see [12]),

r_{k}(2γ; x) ≤ 2/15, μ_{2,k}(2γ; x) ≤ 23γ_{k}^{2}

and

μ̃_{2,k}(2γ; x) = μ_{2,k}(2γ; x) + (2γ_{k})^{2}(1 + r_{k}(2γ; x)) ≤ \frac{413}{15}γ_{k}^{2}.

Thus Theorem 2 applies with w(x) ≡ 1, ρ(x) = exp(−4σx^{2}), c_{4} = 68/15, c_{5} = 75 (i.e. c_{6} = 271) and n such that σγ_{n}^{2} ≤ 3/64. In the same way one can show that Theorem 4 is true with ρ(x) = ρ_{1}(x) := exp(−7σx^{2}), δ_{n} = γ_{n}, σγ_{n}^{2} ≤ 3/64 and a positive absolute constant c_{11}. From these results the estimate of ∥F̃_{n}f − f∥_{ρ_{1}}^{(φ)} follows at once via inequality (18).

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(Received 05.05.1993)

Author’s Address: Institute of Mathematics Adam Mickiewicz University Matejki 48/49 60-769 Poznań, Poland