Averaged Integral Transforms
Dale Murray Rognlie, B. C. Carlson · SIAM Journal on Mathematical Analysis · 1973
An integral transform $\bar f(s) = \int f(t)\varphi (s,t)dt$ can be averaged over the convex hull of $\{ s_1 , \cdots ,s_k \} $ to produce an analogous function $\bar F\{ s_1 , \cdots ,s_k \} $ of several real or complex variables. The question arises whether it is legitimate to take the average under the integral sign, so that $\bar F\{ s_1 , \cdots ,s_k \} = \int f(t)\Phi (s_1 , \cdots ,s_k ;t)dt$, where $\Phi $ is the corresponding average of $\varphi $. Conditions for the validity of this equation are of interest in the theory of special functions because the kernel $\Phi $ may be a Bessel function, elliptic integral, or other hypergeometric function when $\varphi $ is the kernel of a Laplace, Fourier, or Stieltjes transformation. Sufficient conditions of validity are established in the case of these three transformations and the inverse Laplace, Fourier, and Mellin transformations. The averages have some but not all of the operational properties of the corresponding ordinary transforms. Some examples are given involving various special functions.