After an introduction to Hilbert spaces and convex analysis, the proximal average is studied and two smooth operators are provided. The first is a new version of an operator previously supplied by Goebel, while the second one is new and uses the proximal average of a function and a quadratic to find a smooth approximation of the function. Then, the kernel average of two functions is studied and a reformulation of the proximal average is used to extend the definition of the kernel average to

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... w for any number of functions. The Fenchel conjugate of this new kernel average is then examined by calculating the conjugate for two specific kernel functions that represent two of the simplest cases that could be considered. A closed form solution was found for the conjugate of the first kernel function and it was rewritten in three equivalent forms. A solution was also found for the conjugate of the second kernel function, but the two solutions do not have the same form which suggests that a general solution for the conjugate of any kernel function will not be found.