Reproducing Kernel Functions Represented by Form of Polynomials

By re-defining the inner product of a reproducing kernel space, the reproducing kernel functions of that space can be represented by form of polynomials without changing any other conditions, and the higher order of the derivatives, the simpler of the reproducing kernel function expressions. Such expressions of reproducing kernel functions are the simplest from the computational point of view, resulting in speed and accuracy significant improvement in scientific and engineering applications. The performance of such reproducing kernel functions is shown to be very encouraging by experimental results.