Abstract
In this note we construct the B€zier variant of summation integral type operators based on a non-negative real parameter. We present a direct approximation theorem by means of the first order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation. In the last section, we study the quantitative Voronovska ja type theorem
Highlights
In 1912 Bernstein introduced the most famous algebraic polynomials Bn(f ; x) in approximation theory in order to give a constructive proof of Weierstrass’s theorem which is given by Xn kBn(f ; x) = pn;k(x)f n ; x 2 [0; 1]; k=0 where pn;k(x) =n xk(1 k x)n k and he proved that if f 2 C[0; 1] Bn(f ; x) converges uniformly to f (x) in [0; 1]: The Bernstein operators have been used in many branches of mathematics and computer science
In this note we construct the Bézier variant of summation integral type operators based on a non-negative real parameter
We present a direct approximation theorem by means of the ...rst order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation
Summary
In 1912 Bernstein introduced the most famous algebraic polynomials Bn(f ; x) in approximation theory in order to give a constructive proof of Weierstrass’s theorem which is given by. N xk(1 k x)n k and he proved that if f 2 C[0; 1] Bn(f ; x) converges uniformly to f (x) in [0; 1]: The Bernstein operators have been used in many branches of mathematics and computer science. The pioneer works in this direction are due to [3, 5, 9, 13, 24, 26, 28, 29, 30] In these works, the direct approximation results were obtained and the rate of convergence for functions of bounded variation were established. The order of approximation of the summation-integral type operators for functions with derivatives of bounded variation is estimated in [1, 4, 6, 7, 14, 15, 16, 17, 18, 21, 20, 22, 25]. For x 2 [0; 1] we have k Dn( )(f ) k k f k :
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