Direct Estimates of the Weighted Simultaneous Approximation by the Baskakov Operator
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Results in Mathematics
Direct Estimates of the Weighted Simultaneous Approximation by the Baskakov Operator Borislav R. Draganov Abstract. We establish direct estimates of the rate of weighted simultaneous approximation by the Baskakov operator for smooth functions in the supremum norm on the non-negative semi-axis. We consider Jacobitype weights. The estimates are stated in terms of appropriate moduli of smoothness or K-functionals. Mathematics Subject Classification. 41A17, 41A25, 41A28, 41A35, 41A81. Keywords. Baskakov operator, Jackson inequality, direct estimate, simultaneous approximation, modulus of smoothness, K-functional.
1. Main Results The Baskakov operator for a function f defined on [0, ∞) and n ∈ N+ is given by (see [4, (6)]) ∞ k n+k−1 k Vn f (x) := f x (1 + x)−n−k , x ≥ 0. vn,k (x), vn,k (x) := k n k=0
Baskakov showed that if f is continuous on [0, ∞) and f (x) = f (R) for x > R with some R > 0, then (see [4, Theorem 1]) lim Vn f (x) = f (x),
n→∞
x ≥ 0,
(1.1)
This work was supported by Grant DN 02/14 of the Fund for Scientific Research of the Bulgarian Ministry of Education and Science. 0123456789().: V,-vol
156
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B. R. Draganov
Results Math
as the convergence is uniform on the compact subsets of [0, ∞). He also proved in [4, Corollary 3] that if f is continuous and bounded on [0, ∞) and has a second derivative at the point x0 , then 1 x(1 + x) f (x0 ) + o Vn f (x0 ) = f (x0 ) + . (1.2) 2n n Thus the differential operator (up to a multiplicative constant) that describes := ϕ2 g , where the approximation behaviour by the Baskakov operator is Dg ϕ(x) := x(1 + x). An estimate of the convergence in (1.1) in the uniform norm on [0, ∞) was proved by Totik [24, Theorem 1 and p. 175]. He showed that Vn f − f ≤ c ωϕ2 (f, n−1/2 )
(1.3)
for any continuous bounded function f on [0, ∞). Here and henceforward c denotes a constant (not necessarily the same at each occurrence), which is independent of f and n, ◦ stands for the supremum norm on [0, ∞), and ¯ 2hϕ f , ωϕ2 (f, t) := sup Δ 0 0 and n0 ∈ N such that for all f ∈ C[0, ∞) with f ∈ ACloc (0, ∞) and wf ∈ L∞ [0, ∞), and all n ∈ N+ , n ≥ n0 , there holds 1 ¯ 1+χ (f , n−1 )w . w(Vn f − f ) ≤ c ωϕ2 (f , n−1/2 )w + ω Direct point-wise estimates of the rate of simultaneous approximation in the unweighted case in terms of the modulus of continuity were obtained in [18,21,22,25]. A direct estimate in weighted spaces using again only a modulus of continuity, which is a modification of ω(f, t)w was established by L´opezMoreno [19, Theorem 4] with weights of the form w(0, −m), where m ∈ N0 . However, there the weight in the norm of the error is smaller than the one associated with the approximated function. Direct point-wise estimates for the simultaneous approximation by combinations of Baskakov operators were obtained in [26,27] for w = 1, and in [23] for w(a, −b), where 0 < a < 1 and b ≥ 0. In [23] weak converse estimates were also established. Recently, Aral and Tachev [2] proved general point-wise V
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