In this article we decide to define a modified homotopy perturbation for solving non-linear integral equations. Almost, all of the papers that was presented to solve non-linear problems by the homotopy method, they used from two non-linear and linear operators. But we convert a non-linear problem to two suitable non-linear operators also we use from appropriate bases functions such as Legendre ...

and B(v+1,n) being the beta function (see, e.g., [3]). It is easily verified that the operators Bn are linear positive operators. Also, Bn(1,x) = 1. It turns out that the order of approximation for the operators (1.1) is at best O(n−1) howsoever smooth the function may be. With the aim of improving the order of approximation, we have to slack the positive condition of these operators for which ...

Some inequalities for positive linear maps of continuous synchronous (asynchronous) functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved operators, are given. Applications for power function and logarithm are provided as well.

We present two general sequences of positive linear operators. The first is introduced by using a class of dependent random variables, and the second is a mixture between two linear operators of discrete type. Our goal is to study their statistical convergence to the approximated function. This type of convergence can replace classical results provided by Bohman-Korovkin theorem. A particular c...

Abstract. We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an rth order generalization of these operators and observe its approximation properties. Furthermore, we study ...

2 Linear Operators 6 2.1 Outer Products, Adjoints, and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Important Types of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Normal Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...