On constructing new expansions of functions using linear operators

Let T,UT,U be two linear operators mapped onto the function ff such that U(T(f))=fU(T(f))=f, but T(U(f))≠fT(U(f))≠f. In this paper, we first obtain the expansion of functions T(U(f))T(U(f)) and U(T(f))U(T(f)) in a general case. Then, we introduce four special examples of the derived expansions. First example is a combination of the Fourier trigonometric expansion with the Taylor expansion and the second example is a mixed combination of orthogonal polynomial expansions with respect to the defined linear operators TT and UU. In the third example, we apply the basic expansion U(T(f))=f(x)U(T(f))=f(x) to explicitly compute some inverse integral transforms, particularly the inverse Laplace transform. And in the last example, a mixed combination of Taylor expansions is presented. A separate section is also allocated to discuss the convergence of the basic expansions T(U(f))T(U(f)) and U(T(f))U(T(f)).