Korovkin and Weierstrass Approximation via Lacunary Statistical Sequences

Abstract

In this study we shall extended Korovkin and Weierstrass approximation theorem to lacunary statistical convergent sequences. In addition, to these approximation theorems, we established also introduced lacunary statistically convergent of degree β and establish a corresponding Korovkin type theorem namely the following: If the sequence of positive linear operators P_n: C_M [a, b]→ B[a, b] satisfies the conditions: * ||P_n(1, x)-1||_β→0(S^β1_θ ) as r→ ∞, * ||P_n(t, x)-x||_B→0(S^β2_θ ) as r→ ∞ and * ||P_n(t², x)-x²||_B→0(S^β3_θ ) as r→ ∞, then for any function f ∈ C_M [a, b], we have ||P_n (f, x)- (x)||_B→0(S^β_θ ) as r→ ∞ and β = min{β₁, β₂, β₃}.