On constructing new expansions of functions using linear operators

Let T , U be two linear operators mapped onto the function f such that U ( T ( f ) ) = f , but T ( U ( f ) ) ≠ f . In this paper, we first obtain the expansion of functions T ( U ( f ) ) and 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 T and U . In the third example, we apply the basic expansion 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 ) ) and U ( T ( f ) ) .