Abstract
Continuing previous investigations concerning Bernstein polynomials, the purpose of this paper is to establish the weak-type inequality ( f∈ L p(0,1), n∈ℕ) ω ϕ ( n − 1 / 2 , f ) ≤ Μ p n − 1 ∑ k = 1 n || K K f - f || p in terms of the Kantorovitch polynomial K kƒ and the modulus of continuity ( ϕ 2( x): = x(1 − x)) ω ϕ ( t , f ) : = sup 0 < h ≤ t | | Δ h ϕ 2 f | | p + sup 0 < h ≤ t 2 | | Δ h 2 f | | p . Such estimates which immediately imply well-known inverse results are also obtained for the Kantorovitch version of the Szász-Mirakjan and Baskakov operators, respectively.
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