Application of finite integral transforms to analog simulations

Integral transformations are defined by the expression $(p) = \int^{b}_{a} {f(x)k(p, x)dx} \eqno {\hbox{(1)}}$ where K(p, x) is termed the transformation kernel. Theoretically a wide variety of functions could serve as kernels, and an infinite number of transformations could be generated. However, only a limited number of kernels possess those properties which make their transforms useful for the solution of practical problems.