Weighted approximation of functions in L p -norm by Baskakov-Kantorovich operator
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WEIGHTED APPROXIMATION OF FUNCTIONS IN Lp -NORM BY BASKAKOV–KANTOROVICH OPERATOR P. E. PARVANOV Department of Mathematics and Informatics, University of Sofia, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria e-mail: [email protected] (Received November 21, 2019; revised April 21, 2020; accepted April 22, 2020)
Abstract. We investigate the weighted approximation of functions in Lp -norm by Kantorovich modifications of the classical Baskakov operator, with weights of type (1 + x)α , α ∈ R. By defining an appropriate K-functional we prove direct inequalities for them.
1. Introduction The classical Baskakov operator [1] is defined for every bounded function f (x) on the interval [0, ∞) by the formula (1)
Vn f (x) = Vn (f, x) =
∞ k vn,k (x), f n k=0
where (2)
vn,k (x) =
n+k−1 k x (1 + x)−n−k . k
But it is not suited for approximation of functions in the Lp -norm because it is not bounded in Lp . Some kind of modification is needed. Analogously to the Bernstein–Kantorovich operator, Ditzian and Totik [7] defined two Kantorovich modifications of Vn . For 0 ≤ x < ∞ they introduced k+1 ∞ n ∗ ∗ ˜ ˜ vn,k (x) n f (u) du Vn f (x) = Vn (f, x) = k=0
k n
Key words and phrases: Baskakov operator, K-functional, weighted approximation, Baskakov–Kantorovich operator. Mathematics Subject Classification: 41A36, 41A25, 41A27, 41A17. c 2020 Akad´ 0133-3852 emiai Kiad´ o, Budapest
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P. E. PARVANOV
and V˜n f (x) = V˜n (f, x) =
(3)
∞
vn,k (x) (n − 1)
k=0
k+1 n−1 k n−1
f (u) du.
The reason for introducing the second one is that the first one is not a contraction and because of that it is not very suitable for approximating of functions in Lp norm for p < ∞. There are many results for weighted approximation of functions by Baskakov operators in uniform norm, see, e.g. [20], [13], [15]. Regarding the weighted approximation of functions by Kantorovich type operators, the best results, to our knowledge, is the next theorem proved in [7, Theorem 10.1.3] (about the unweighted approximation of functions by Kantorovich type operators see [2], [3], [8], [4], [19], [16], [18] and [9]). We cite the theorem only for the Baskakov–Kantorovich operator V˜n∗ . Theorem 1.1. Let w∗ (x) = xγ(0) (1 + x)γ(∞) where γ(∞) is arbitrary and −1/p < γ(0) < 1 − 1/p for 1 ≤ p ≤ ∞, w∗ f ∈ Lp [0, ∞) and either 1 ≤ p ≤ ∞ and α < 1, or 1 < p < ∞ and α ≤ 1. Then for V˜n∗ the following equivalence is true: ∗ ∗ w (V˜n f − f ) = O(n−α) ⇔ w∗ Δ2 √ f = O(h2α). h ψ p Lp [2h2 ,∞) Here · p and · Lp (J) stand for the usual Lp -norm respectively on [0, ∞) and the interval J , ψ(x) = x(1 + x) and
(f, x) = f x − h ψ(x) − 2f (x) + f x + h ψ(x) . Δ2 √ h
ψ(x)
The same equivalency can be proved for V˜n with very little modifications in the proof. Our goal in this paper is to investigate the weighted approximation of functions in the Lp -norm by the Baskakov–Kantorovich operator V˜n and, by defining an appropriate K-functional, to prove a strong direct inequality for the error of approximation. Before stating our main result, l
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