This paper seeks to present the fundamental features of the category of conditional exponential convex functions (CECFs). Additionally, the study of continuous CECFs contributes to the characterization of convolution semigroups. In this context, we expand our focus to include a much broader class of Gaussian processes, where we define the generalized Fourier transform in a more straightforward manner. This approach is closely connected to the method by which we derived the Gaussian process, utilizing the framework of a Gelfand triple and the theorem of Bochner–Minlos. A part of this work involves constructing the reproducing kernel Hilbert spaces (RKHS) directly from CECFs.