Yet another look at positive linear operators, -monotonicity and applications

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文摘

For each q∈N₀

q \in N_{0}

, we construct positive linear polynomial approximation operators M_n

M_{n}

that simultaneously preserve k

k

-monotonicity for all 0≤k≤q

0 \leq k \leq q

and yield the estimate

| f (x) - M_{n} (f, x) | \leq c ω_{2}^{φ^{λ}} (f, n^{- 1} φ^{1 - λ / 2} (x) {(φ (x) + 1 / n)}^{- λ / 2}),

for x∈[0,1]

x \in [0, 1]

and λ∈[0,2)

λ \in [0, 2)

, where

φ (x) : = \sqrt{x (1 - x)}

and

ω_{2}^{ψ}

is the second Ditzian–Totik modulus of smoothness corresponding to the “step-weight function” ψ

ψ

. In particular, this implies that the rate of best uniform q

q

-monotone polynomial approximation can be estimated in terms of

ω_{2}^{φ} (f, 1 / n)

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