PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 63 (77), 1998, 152–162

ASYMPTOTIC FORMULAE FOR RECURSIVELY DEFINED BASKAKOV-TYPE OPERATORS

O. Agratini

Communicated by Stevan Pilipović

Abstract. We deal with Baskakov-type operators which are defined by replacing the binomial coefficients with general ones satisfying a recursively relation. We establish a Voronovskaja-type formula and also we give a global approximation property using some weighted norms.

Introduction. In [7] Campiti and Metafune introduced and studied a generalization of the classical Bernstein operators consisting in replacing the binomial coefficients with more general ones satisfying suitable recursive relations. Their work was motivated from the development of the study of connections between approximation processes and evolution problems through semigroup theory. During the last years, F. Altomare and his Bari school considered the study of ellipticparabolic equations by means of positive operators. Among the former papers we quote [2], [3] and for a unified treatment of this subject we mention the monograph [5]. The class of evolution equations whose solutions can be approximated by constructive approximation processes has been enlarged and this fact has led to the new types of operators, see [4].

In the same manner, we considered [1] a generalization of the Baskakov operators which are related to functions defined on an unbounded interval. We obtained a decomposition of the Baskakov operator as a sum of elementary operators and our operators are expressed through a linear combination of these last ones. Also, we proved that these sequences of linear operators converge towards an operator multiplied by an analytic function.

In this paper we continue to study these Baskakov-type operators previously introduced pointing out new regularity properties. The main result is a Voronovskaja-type formula where the second order derivative is perturbed by a term depending on two fixed sequences. Also we establish global approximation properties of these operators in some spaces of functions of polynomial growth.

AMS Subject Classification (1991): Primary 41A35; Secondary: 26A15 Key words and phrases: Baskakov-type operators, degree of approximation, Voronovskajatype formula, weighted norms.

Background. Let two fixed sequences of real numbers be: a = (a_{n})_{n≥1}, b = (b_{k})_{k≥0} with a_{1} = b_{0}. We consider the numbers c_{n,k} (n ≥ 1, k ≥ 0) which satisfy the following recursive formulae:

(1)  \begin{matrix}c_{n,0} = a_{n}, n ≥ 1; c_{1,k} = b_{k}, k ≥ 0; \\ c_{n+1,k} = c_{n,k} + c_{n+1,k−1}, n ≥ 1 and k ≥ 1. \end{matrix}

It is obvious that the sequences a and b determine uniquely these coefficients. Actually, the coefficients from (1) follow the same rule as the binomial ones. Now, we can define the linear operators L_{n}^{⟨a,b⟩} as follows:

(2)  (L_{n}^{⟨a,b⟩}f)(x) = ∑_{k=0}^{∞} c_{n,k} \frac{x^{k}}{(1 + x)^{n+k}} f(\frac{k}{n}),

where f ∈ C_{B}[0, ∞), the space of real functions continuous on [0, ∞) such that the above series converges uniformly and \frac{f(x)}{1+x^{2}} is convergent as x → ∞. The space is endowed with the norm ∥f∥_{∗} = sup_{x≥0} \frac{|f(x)|}{1+x^{2}}. If a_{n} = λ, n ≥ 1, and b_{k} = λ, k ≥ 0, then

c_{n,k} = λ\binom{n+k−1}{k} and L_{n}^{⟨λ,λ⟩} = λV_{n}, where V_{n} represent the Baskakov operators.

We would like to mention some results from [1] which will be useful for the purposes of this paper. For the sake of simplification we set:

(3)  r_{k,n}(x) = \frac{x^{k}}{(1 + x)^{n+k}} , x ≥ 0.

Theorem A. For f ∈ C_{B}[0, ∞) and n ≥ 1 the following identity

(4)  (L_{n}^{⟨a,b⟩}f)(x) = ∑_{m=1}^{∞} a_{m}(A_{m,n}f)(x) + ∑_{m=0}^{∞} b_{m}(B_{m,n}f)(x)

holds, where:

(A_{1,n}f)(x) = (B_{0,n}f)(x) = \frac{δ_{1,n}}{2(1 + x)}f(0);

for any m ≥ 2,

(5)  (A_{m,n}f)(x) = \{ \begin{matrix}0, & m ≥ n + 1 \\ ∑_{k=0}^{∞} r_{k,n}(x)f(k/n), & m = n \\ ∑_{k=1}^{∞} \binom{n−m+k−1}{k−1}r_{k,n}(x)f(k/n), & m ≤ n − 1 \end{matrix}

for any m ≥ 1, (6)

(B_{m,n}f)(x) = r_{m,n}(x)f(m/n) + (1 − δ_{1,n}) ∑_{k=m+1}^{∞} \binom{n + k − m − 2}{k − m}r_{k,n}(x)f(k/n).

Here δ_{n,m} represents Kronecker’s symbol. Also we shall put e_{j}(x) = x^{j}, j ∈ N.

Theorem B. For n ≥ 3, the following identity holds:

(7)  (L_{n}^{⟨a,b⟩}e_{0})(x) = ∑_{m≥1} b_{m}r_{m,1}(x) + ∑_{m=2}^{n−1} a_{m}r_{1,m−1}(x) + a_{n}/(1 + x)^{n−1}

Theorem C. Assume that f belongs to C_{B}[0, ∞) and the sequences a, b are bounded. Let the function φ be defined on (0, ∞) as follows:

(8) φ = σ + τ where σ(x) = ∑_{m=2}^{∞} a_{m}r_{1,m−1}(x), τ(x) = ∑_{m=1}^{∞} b_{m}r_{m,1}(x).

Then lim_{n→∞} L_{n}^{⟨a,b⟩}f = φf, uniformly on any compact K ⊂ (0, ∞).

Results. The first comment concerns our notation (3) which implies:

∑_{k=0}^{∞} r_{k,0}(x) = 1 + x, ∑_{k=0}^{∞} kr_{k,0}(x) = x(1 + x), (9)

∑_{k=0}^{∞} k^{2}r_{k,0}(x) = x(1 + x)(2x + 1) and r_{k,n}(x) = (1 + x)^{−n}r_{k,0}(x).

Also, we recall some useful relations which are fulfilled by V_{n}:

(10)  (V_{n}e_{0})(x) = 1, (V_{n}e_{1})(x) = x, (V_{n}e_{2})(x) = x^{2} + \frac{x(1 + x)}{n}.

In what follows, we consider n ≥ 2 and we calculate the expression of the elementary operators A_{m,n}, B_{m,n} defined in Theorem A for the test function e_{1}. It is evident (A_{1,n}e_{1})(x) = (B_{0,n}e_{1})(x) = 0 and (A_{m,n}e_{1})(x) = 0 too, for m ≥ n + 1. In accordance with (9) we can write:

(A_{n,n}e_{1})(x) = \frac{1}{n(1 + x)^{n}} ∑_{k=0}^{∞} kr_{n,0}(x) = \frac{x}{n(1 + x)^{n−1}} .

Asymptotic formulae for recursively defined Baskakov-type operators 155

For m ≤ n − 1, by using (10), we have: (A_{m,n}e_{1})(x)

= \frac{x}{n(1 + x)^{m}} \{∑_{i=0}^{∞} \binom{n − m + i}{i}r_{i,n−m+1}(x)i + ∑_{i=0}^{∞} \binom{n − m + i}{i}r_{i,n−m+1}(x)\}

= \frac{x}{n(1 + x)^{m}} ((n − m + 1)x + 1).

Similarly, for any m ≥ 1, we can write:

(B_{m,n}e_{1})(x) − \frac{m}{n}r_{m,n}(x) = \frac{x^{m}}{(1 + x)^{m+1}} ∑_{i=1}^{∞} \binom{n − 2 + i}{i}r_{i,n−1}(x)\frac{m + i}{n}

= \frac{m}{n} \frac{x^{m}}{(1 + x)^{m+1}} ((V_{n−1}e_{0})(x) − \frac{1}{(1 + x)^{n−1}} ) + \frac{n − 1}{n} \frac{x^{m}}{(1 + x)^{m+1}} (V_{n−1}e_{1})(x),

which implies (B_{m,n}e_{1})(x) = n^{−1}r_{m,1}(x)((n − 1)x + m).

The above relations and Theorem A lead us to the following result: Lemma 1. If the operators L_{n}^{⟨a,b⟩} are defined by (2) then, for n ≥ 2, we have:

n(L_{n}^{⟨a,b⟩}e_{1})(x)

= ∑_{m=2}^{n−1} a_{m}r_{1,m−1}(x)((n−m+1)x+1)+ \frac{a_{n}x}{(1 + x)^{n−1}} + ∑_{m≥1} b_{m}r_{m,1}(x)((n−1)x+m).

This lemma allows us to state: Theorem 1. If the sequences a, b are bounded then the operators L_{n}^{⟨a,b⟩} sat-

isfy:

lim_{n→∞} n(e_{1}(x)(L_{n}^{⟨a,b⟩}e_{0})(x) − (L_{n}^{⟨a,b⟩}e_{1})(x)) = −x(1 + x)φ′(x),

where x > 0 and φ is defined by (8).

Proof. Let n ≥ 3. Taking lemma 1 and identity (7) into account we can write:

(11) n(x(L_{n}^{⟨a,b⟩}e_{0})(x) − (L_{n}^{⟨a,b⟩}e_{1})(x)) = \frac{(n − 1)a_{n}x}{(1 + x)^{n−1}}

+ ∑_{m≥1} b_{m}r_{m,1}(x)(x − m) + ∑_{m=2}^{n−1} a_{m}r_{1,m−1}(x)((m − 1)x − 1).

Because of boundedness of a, the term \frac{(n − 1)a_{n}x}{(1 + x)^{n−1}} converges to zero as n → ∞.

On the other hand, from (8) we obtain:

σ′(x) = ∑_{m=2}^{∞} a_{m} \frac{1 − (m − 1)x}{(1 + x)^{m+1}}, τ′(x) = ∑_{m≥1} b_{m} \frac{x^{m−1}(m − x)}{(1 + x)^{m+2}},

consequently:

∑_{m≥2} a_{m}(m − 1)xr_{1,m−1}(x) = σ(x) − x(1 + x)σ′(x)

and

∑_{m≥1} b_{m}(x − m)r_{m,1}(x) = −x(1 + x)τ′(x).

Using these relations in (11), for n → ∞ we get to the desired result. Now, we need the expression of operators A_{m,n}, B_{m,n} (n ≥ 2) for the test

function e_{2}. From (9) we deduce:

(A_{n,n}e_{2})(x) = \frac{1}{n^{2}(1 + x)^{n}} ∑_{k=0}^{∞} k^{2}r_{k,0}(x) = \frac{x(2x + 1)}{n^{2}(1 + x)^{n−1}} .

For 2 ≤ m ≤ n − 1, we take p = n − m + 1 and in view of (10) we can write successively:

(A_{m,n}e_{2})(x) = \frac{x}{n^{2}(1 + x)^{m}} ∑_{k=0}^{∞} \binom{p + k − 1}{k}r_{k,p}(x)(k + 1)^{2}

= \frac{xp^{2}}{n^{2}(1 + x)^{m}} ((V_{p}e_{2})(x) + \frac{2}{p}(V_{p}e_{1})(x) + \frac{1}{p^{2}} (V_{p}e_{0})(x))

= \frac{x}{n^{2}(1 + x)^{m}} (p(p + 1)x^{2} + 3px + 1).

For m ≥ 1, we have:

(B_{m,n}e_{2})(x) − \frac{m^{2}}{n^{2}} r_{m,n}(x) = \frac{x^{m}}{n^{2}(1 + x)^{m+1}} ∑_{i=1}^{∞} \binom{n + i − 2}{i}r_{i,n−1}(x)(m + i)^{2}

= \frac{x^{m}}{n^{2}(1 + x)^{m+1}} \{m^{2}((V_{n−1}e_{0})(x) − \frac{1}{(1 + x)^{n−1}} )
+2m(n − 1)(V_{n−1}e_{1})(x) + (n − 1)^{2}(V_{n−1}e_{2})(x)}.

Thus,

(B_{m,n}e_{2})(x) = \frac{x^{m}}{n^{2}(1 + x)^{m+1}} (n(n − 1)x^{2} + (n − 1)(2m + 1)x + m^{2}).

Asymptotic formulae for recursively defined Baskakov-type operators 157

We can summarize these relations below. Lemma 2. If the operators L_{n}^{⟨a,b⟩} are defined by (2) then, for n ≥ 2, we have:

n(L_{n}^{⟨a,b⟩}e_{2})(x) = ∑_{m=2}^{n−1} \frac{a_{m}}{n} r_{1,m−1}(x)(p(p + 1)x^{2} + 3px + 1) + \frac{a_{n}x(2x + 1)}{n(1 + x)^{n−1}}

+ ∑_{m≥1} \frac{b_{m}}{n} r_{m,1}(x)((n − 1)nx^{2} + (n − 1)(2m + 1)x + m^{2}),

where p = n − m + 1.

Set ϕ_{x,r}(t) := (t − x)^{r} (t ≥ 0, x ≥ 0). Theorem 2. If the sequences a, b are bounded then the operators L_{n}^{⟨a,b⟩} sat-

isfy:

lim_{n→∞} n(L_{n}^{⟨a,b⟩}ϕ_{x,2})(x) = x(1 + x)φ(x),

where x > 0 and φ is defined by (8).

Proof. Because L_{n}^{⟨a,b⟩} are linear operators, it is clear that:

(L_{n}^{⟨a,b⟩}ϕ_{x,2})(x) = (L_{n}^{⟨a,b⟩}e_{2})(x) − 2x(L_{n}^{⟨a,b⟩}e_{1})(x) + x^{2}(L_{n}^{⟨a,b⟩}e_{0})(x). Using (7), Lemma 1 and Lemma 2 after a short calculation, we get:

n(L_{n}^{⟨a,b⟩}ϕ_{x,2})(x) = (x^{2} + x) ∑_{m=2}^{n−1} a_{m}r_{1,m−1}(x) + (x^{2} + x)τ(x) + \frac{1}{n}R(n; x),

where

R(n; x) = ∑_{m=2}^{n−1} α(m, x)a_{m} \frac{x}{(1 + x)^{m}} + ∑_{m≥1} β(m, x)b_{m} \frac{x^{m}}{(1 + x)^{m+1}} + γ(n, x)

and

α(m, x) = (m^{2} − 3m + 2)x^{2} − 3(m − 1)x + 1,

β(m, x) = −(2m + 1)x + m^{2},

γ(n, x) = a_{n} \frac{x}{(1 + x)^{n−1}} ((n^{2} − 2n + 2)x + 1).

But R(n; x)/n tends to 0 for n → ∞ and according to (8) this completes the proof. Now, we can state and prove the announced Voronovskaja-type formula.

Theorem 3. Let f ∈ C_{B}[0, ∞) be twice differentiable at some point x > 0 and assume that f(t) = O(t^{2}) as t → ∞. If the operators L_{n}^{⟨a,b⟩} are defined by (2) and a, b are bounded then (12)

lim_{n→∞} n((L_{n}^{⟨a,b⟩}f)(x) − f(x)(L_{n}^{⟨a,b⟩}e_{0})(x)) = \frac{x(1 + x)}{2}(φ(x)f′′(x) + 2φ′(x)f′(x)).

Proof. In order to prove this identity we use Taylor’s formula:

f(\frac{k}{n}) − f(x) = (\frac{k}{n} − x)f′(x) + (\frac{k}{n} − x)^{2}(\frac{1}{2}f′′(x) + ε(\frac{k}{n} − x)),

where ε is bounded and lim_{t→0} ε(t) = 0. Then

(13) (L_{n}^{⟨a,b⟩}f)(x) − f(x)(L_{n}^{⟨a,b⟩}e_{0})(x) = ∑_{k=0}^{∞} c_{n,k} \frac{x^{k}}{(1 + x)^{n+k}} (f(\frac{k}{n}) − f(x))

= f′(x)((L_{n}^{⟨a,b⟩}e_{1})(x) − x(L_{n}^{⟨a,b⟩}e_{0})(x)) + \frac{1}{2}f′′(x)(L_{n}^{⟨a,b⟩}ϕ_{x,2})(x)

+ ∑_{k=0}^{∞} c_{n,k} \frac{x^{k}}{(1 + x)^{n+k}} (\frac{k}{n} − x)^{2}ε(\frac{k}{n} − x).

If λ is an upper bound for the sequences |a|, |b|, then (2) implies |c_{n,k}| ≤ λ\binom{n+k−1}{k}. It follows:

lim_{n→∞} n| ∑_{k=0}^{∞} c_{n,k}r_{k,n}(x)(\frac{k}{n} − x)^{2}ε(\frac{k}{n} − x)| ≤ λ lim_{n→∞} n(V_{n}ψ_{x,2})(x) = 0,

where ψ_{x}(t) = ϕ_{x,2}(t)ε(t − x). We also used the Voronovskaja’s formula for Baskakov operators [8], that is lim_{n→∞} n((V_{n}f)(x) − f(x)) = (x^{2} + x)f′′(x)/2. Therefore, from (13), Theorem 1 and Theorem 2, we obtain the desired result.

In what follows we assume φ(x) ≠ 0 for every x > 0.

Remark. The right-hand side of (12) can be arranged under the form (2φ(x))^{−1}x(1 + x)\frac{d}{dx}(φ^{2}(x)f′(x)). Therefore, theorem 3 shows that (L_{n}^{⟨a,b⟩}f)(x) − f(x)(L_{n}^{⟨a,b⟩}e_{0})(x) is of order not better than 1/n if \frac{d}{dx}(φ^{2}(x)f′(x)) ≠ 0.

In order to prove a global approximation theorem for L_{n}^{⟨a,b⟩} in polynomial weight spaces, we define the weights w_{N} as follows w_{0}(x) = 1, w_{N}(x) = (1 + x^{N})^{−1}, x ≥ 0, N ≥ 1. Also, we introduce the spaces C_{N} = {f ∈ C[0, ∞); w_{N}f uniformly continuous and bounded on [0, ∞)} endowed with the norm ∥f∥_{N} = sup_{x≥0} w_{N}(x)|f(x)|. Finally, we recall the notations

Δ_{h}^{2}f(x) = f(x + 2h) − 2f(x + h) + f(x), ω_{N}^{2}(f, δ) = sup_{0<h≤δ} ∥Δ_{h}^{2}f∥_{N}.

In the sequel, we consider that a, b are non negative bounded sequences and we shall denote by K_{N} a constant which may be different at each occurrence, it depends only on a, b and N.

Introducing μ_{n,r}, m_{n,r} the r-th order moment (r ∈ N) of V_{n} respectively of L_{n}^{⟨a,b⟩}, we need the following result due to Becker.

Lemma 3 [6, p. 131]. Let r ≥ 2 and δ_{r} = \{ \begin{matrix}1, & r odd \\ 0, & r even. \end{matrix}

Then, the r-th moment of the Baskakov operators is

μ_{n,r}(x) = ∑_{j=1}^{[r/2]} b_{n,r,j}(\frac{x(1 + x)}{n})^{j}(\frac{1 + 2x}{n})^{δ_{r}},

where the b_{n,r,j} coefficients are positive and bounded with respect to n. Moreover, μ_{n,r}(x) is a polynomial of degree r.

Now, we can state the following result

Lemma 4. Let r be integer, 0 ≤ r ≤ N. The quantities m_{n,r}x^{N−r}(1 + x^{N})^{−1} are bounded with respect to x (x ≥ 0) and to n (n ∈ N).

Proof. If the sequences a, b are non-negative and λ = max { sup_{n≥1} a_{n}, sup_{k≥0} b_{k}}, then L_{n}^{⟨a,b⟩}|g| ≤ L_{n}^{⟨λ,λ⟩}|g| = λV_{n}|g|. We easily deduce

(14)  m_{n,2i} ≤ λμ_{n,2i}, i ≥ 0.

If r is odd, according to Cauchy’s inequality, we get

(15)  |m_{n,2i+1}| ≤ m_{n,2}^{1/2}m_{n,4i}^{1/2} ≤ λμ_{n,2}^{1/2}μ_{n,4i}^{1/2} = O(x^{2i+1}), x → ∞.

Also, the boundedness with respect to n follows from the coefficients of μ_{n,i} being bounded in n, see Lemma 3.

Lemma 5. For f ∈ C_{N}[0, ∞) the following inequality

(16)  ∥L_{n}^{⟨a,b⟩}f∥_{N} ≤ K_{N}∥f∥_{N}

holds.

Proof. We can write

w_{N}(x)L_{n}^{⟨a,b⟩}(1/w_{N}; x) = w_{N}(x) ∑_{k=0}^{∞} c_{n,k}r_{k,n}(x)(1 + ((\frac{k}{n} − x) + x)^{N})

≤ λ(V_{n}e_{0})(x)(1 + x^{N})^{−1} + ∑_{j=0}^{N} \binom{N}{j}m_{n,j}(x)x^{N−j}(1 + x^{N})^{−1} ≤ K_{N}

160 Agratini

since the sum is bounded in x and in n - see lemma 4. For f ∈ C_{N}, we have

w_{N}(x)|(L_{n}^{⟨a,b⟩}f)(x)| ≤ w_{N}(x) ∑_{k=0}^{∞} w_{N}(\frac{k}{n})|f(\frac{k}{n})|(w_{N}(\frac{k}{n}))^{−1}c_{n,k}r_{k,n}(x)

≤ ∥f∥_{N}w_{N}(x)L_{n}^{⟨a,b⟩}(1/w_{N}; x) ≤ K_{N}∥f∥_{N}

and this implies (16).

Lemma 6. For g ∈ C_{N}^{2} = {g : g, g′, g′′ ∈ C_{N}} the inequality

(17) w_{N}(x)|(L_{n}^{⟨a,b⟩}g)(x) − g(x)(L_{n}^{⟨a,b⟩}e_{0})(x)|

≤ K_{N}(\sqrt{\frac{x(x + 1)}{n}}∥g′∥_{N} + \frac{x(x + 1)}{n}∥g′′∥_{N}), n ≥ 1, x ≥ 0,

holds.

Proof. We use the Taylor expansion

g(t) − g(x) = (t − x)g′(x) + ∫_{x}^{t}(t − u)g′′(u)du

where

|∫_{x}^{t}(t − u)g′′(u)du| ≤ ∥g′′∥_{N} ∫_{x}^{t} |t − u|\frac{1}{w_{N}(u)}du

≤ \frac{1}{2}∥g′′| N(t − x)^{2}(\frac{1}{w_{N}(t)} + \frac{1}{w_{N}(x)}).

This implies

(18) w_{N}(x)|(L_{n}^{⟨a,b⟩}g)(x) − g(x)(L_{n}^{⟨a,b⟩}e_{0})(x)|

≤ (L_{n}^{⟨a,b⟩}|ϕ_{x,1}|)(x)∥g′∥_{N} + \frac{1}{2}∥g′′∥_{N}(m_{n,2}(x) + w_{N}(x)L_{n}^{⟨a,b⟩}(\frac{ϕ_{x,2}}{w_{N}}; x)).

Taking into account (15), (14), (10) and Lemma 3 we can infer

(L_{n}^{⟨a,b⟩}|ϕ_{x,1}|)(x) ≤ λμ_{n,2}^{1/2} = λ\sqrt{\frac{x(x + 1)}{n}},

and

(L_{n}^{⟨a,b⟩} \frac{ϕ_{x,2}}{w_{N}} )(x) = m_{n,2}(x) + ∑_{j=0}^{N} \binom{N}{j}m_{n,j+2}x^{N−j}

≤ \frac{x(x + 1)}{n} + P_{N+2,n}(x) = P_{N,n}(x)\frac{x(x + 1)}{n},

P_{k,n} being a polynomial of degree k with all coefficients bounded in n.

Inserting the above relations into (18) we obtain (17).

Theorem 4. Let f belong to C_{N}[0, ∞). If the sequences a, b are non-negative and bounded then

(19) w_{N}(x)|(L_{n}^{⟨a,b⟩}f)(x) − f(x)(L_{n}^{⟨a,b⟩}e_{0})(x)| ≤ K_{N}ω_{N}^{2}(f, \sqrt{\frac{x(x + 1)}{n}}).

Proof. For any h > 0 we introduce the modified Steklov means by

f_{h}(x) = (2/h)^{2} ∫_{0}^{h/2} ∫_{0}^{h/2}(2f(x + s + t) − f(x + 2s + 2t))ds dt.

From above, we get

f(t) − f_{h}(x) = (2/h)^{2} ∫_{0}^{h/2} ∫_{0}^{h/2} Δ_{s+t}^{2}f(x)ds dt,
f′_{h}(x) = (2/h)(Δ_{t_{0}}^{2}f(x) − Δ_{t_{0}+h/2}^{2}f(x)), 0 < t_{0} ≤ h/2,
f′′_{h}(x) = h^{−2}(8Δ_{h/2}^{2}f(x) − Δ_{h}^{2}f(x))

and hence

∥f − f_{h}∥_{N} ≤ ω_{N}^{2}(f, h), (20)

∥f′_{h}∥_{N} ≤ 4h^{−1}ω_{N}^{2}(f, h), ∥f′′_{h}∥_{N} ≤ 9h^{−2}ω_{N}^{2}(f, h). (21)

Clearly, we have:

w_{N}(x)|(L_{n}^{⟨a,b⟩}f)(x) − f(x)(L_{n}^{⟨a,b⟩}e_{0})(x)| ≤ w_{N}(x)|L_{n}^{⟨a,b⟩}(f − f_{h}; x)| + w_{N}(x)|(L_{n}f_{h})(x) − f_{h}(x)(L_{n}e_{0})(x)| + w_{N}(x)|f_{h}(x) − f(x)||(L_{n}e_{0})(x)|.

Below we estimate the terms on the right side separately. Using (16), (20) and (14) we can write

w_{N}(x)|L_{n}^{⟨a,b⟩}(f − f_{h}; x)| ≤ K_{N}ω_{N}^{2}(f, h)

and

w_{N}(x)|f_{h}(x) − f(x)||(L_{n}^{⟨a,b⟩}e_{0})(x)| ≤ λω_{N}^{2}(f, h)

where λ is an upper bound for a and b.

Using (17) and (21) we get

w_{N}(x)|(L_{n}^{⟨a,b⟩}f_{h})(x) − f_{h}(x)(L_{n}^{⟨a,b⟩}e_{0})(x)|

≤ K_{N}(\sqrt{\frac{x(x + 1)}{n}} \frac{4}{h} + \frac{x(x + 1)}{n} \frac{9}{h^{2}} )ω_{N}^{2}(f, h).

Choosing h = \sqrt{\frac{x(x + 1)}{n}} and collecting the above inequalities we obtain (19).

Remark. In particular, if 0 < α ≤ 2 and f ∈ Lip_{N}^{2}α = {f ∈ C_{N}[0, ∞) : ω_{N}^{2}(f, δ) = O(δ^{α}), δ → 0^{+}} then the conclusion of the theorem becomes

w_{N}(x)|(L_{n}^{⟨a,b⟩}f)(x) − f(x)(L_{n}^{⟨a,b⟩}e_{0})(x)| ≤ K_{N}(\frac{x(x + 1)}{n})^{α/2}.

References

[1] O. Agratini, Properties of a new class of recursively defined Baskakov-type operators, Archivum Mathematicum 33 (1997), to appear

[2] F. Altomare, Limit semigroups of Bernstein-Schnabl operators associated with positive projections, Ann. Sc. Norm. Sup. Pisa, Serie IV, (16)2 (1989), 259–279.

[3] F. Altomare, Positive projections approximation processes and degenerate diffusion equations, Conf. Sem. Mat. Univ. Bari, 241 (1991), 43–68.

[4] F. Altomare, On some sequences of positive linear operators on unbounded intervals, in: Approximation and Optimization, Proceedings of ICAOR: International Conference on Approximation Theory and Optimization, Cluj-Napoca, July 29 – August 1, 1996, D.D. Stancu et al. Eds., Transilvania Press Cluj-Napoca, 1997, pp. 1–16.

[5] F. Altomare, M. Campiti, Korovkin-type Approximation Theory and its Applications, De Gruyter Studies in Mathematics, Vol. 17, De Gruyter, Berlin, New York, 1994.

[6] M.Becker, Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J. 27 (1978), 127–142.

[7] M Campiti, G. Metafune, Approximation properties of recursively defined Bernstein-type operators, J. Approx. Theory 87 (1996), 243–269.

[8] P.C. Sikkema, On some linear positive operators, Indagationes Mathematicae 32 (1970), 327– 337.

Faculty of Mathematics and Informatics (Received 09 04 1997) Babeş–Bolyai University (Revised 20 08 1997 and 28 04 1998) Kogălniceanu 1 RO–3400 Cluj-Napoca Romania

agratini@math.ubbcluj.ro