Some Integral Transformations with Reproducing Properties

By means of an elementary consideration, families of integral transformations in certain spaces (e.g., in , where is the unit disk) are constructed. These transformations map elements of certain subspaces either to itself or to their derivatives, respectively. As a special case, we obtain a family of integral transformations generating a decomposition of into a direct sum. By introducing appropriate new scalar products, these direct sums become decompositions into orthogonal complements, and the corresponding integral transformations become self-adjoint operators of into itself positive with respect to the new scalar products. In further special cases, these integral transformations possess bounded and injective extensions mapping onto well-defined subspaces of . The latter property is a consequence of the connection of our transformations with the complex Hilbert transformation. Bibliography: 10 titles.