Approximation By Bernstein Operator And Its Generalizations

Since its simplicity in construction, and its good properties to keep the mono-tonicity and convexity of the objective functions, Bernstein operator has always been a most important part of theory of approximation by operators. Bernstein operator has been applied widely in Functional Analysis, Computational Mathe-matics and Learning Theory. The main purpose of the present thesis is to consider some approximation properties of Bernstein operator and one of its importan-t generalizations-Bernstein-Stancu operator. The main results can be read as follows:Chapter â… . We give a brief survey on the research of Bernstein operator and its generalizations.Chapter â…¡. Let Cw:={f(x) ∈ C(0,1):limxâ†'1(wf)(x)=limxâ†'0(wf)(x)=0} with w(x)=xα(1-x)b, a,b>0, that is, CW is consist of functions with singularities at endpoints. By using the weighted K-functional and the weighted modulus of smoothness, we obtain a type of Stechkin-Marchaud inequalities of approximation by a kind of modified Bernstein operators Bn*(f,x) in Cw.In Chapter â…¢. We further give an equivalent characteristic description be-tween the derivatives of the above mentioned operator Bn*(f, x) and the smoothness of the approximated function in Cw spaces. Our results generalize the related one of Wang ([35]) from C[0,1] to Cw, and also remove the restriction a, b<1 on the parameters a, b of Jacobi weights w(x)=xa(1-x)b。Chapter â…£. We obtain both direct and converse results of approximation by a type of weighted Bernstein-Durrmeyer operators in weighted Lwp spaces. In our results, there is no restrictions on the upper bounds of the parameters a, b of Jacobi weight function w(x)=xα(1-x)b, and thus our results essentially generalize the related results of Zhang ([43]).In Chapter â…¤. we consider a new type of Bernstein-Stancu Sn,α,β(f,x) intro-duced by Gadjiev et al ([17]). In [17], Gadjiev et al showed that Sn,α,β(f, x) can be applied to approximate the continuous functions defined on An (a subset of [0,1]). We dicover that Sn,α,β(f, x) actually can approximate the functions in C[0,1] well. Furthermore, we establish both direct and converse theorems for approximation by Sn,α,β(f,x), which contains both global and ponitwise results. We also gener-alize a result of Tasdelen et al ([28]) on approximation by the operator Sn,α,β[Ï„](f,x) for functions in C[0,1]Ï„.Chapter VI. we give both direct and converse results on approximation by a kind of Kantorovich-Bernstein-Stancu operators Sn,α,β*(f,x) in L[0,1]p spaces, which generalize the vrelated results of Igoz([19]) from C[0,1] to L[0,1]p, spaces.