On constructing new interpolation formulas using linear operators and an operator type of quadrature rules

Let T,UT,U be two linear operators mapped onto the function f   such that U(T(f))=fU(T(f))=f but T(U(f))≠fT(U(f))≠f. In this paper, interpolating the functions of type T(U(f))T(U(f)) is presented in a general case. As a special case, the linear operators T(f)=∫λxf(t)dt and U(f)=df(x)/dxU(f)=df(x)/dx are considered to interpolate the family of incomplete special functions. Three new examples of interpolation formulas together with their analytic error are also given as the special samples of the mentioned operator method. Finally, by using the foresaid method, a basic class of operator type quadrature rules is defined and its properties are investigated.