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We define a class of functions which have a known decay rate coupled with aperiodic fluctuation. We identify conditions on the kernel of a linearsummation convolution Volterra equation which give the equivalence of thekernel lying in this class of functions and the solution lying in this class offunctions. Some specific examples are examined. In particular this theory isused to provide a counter--example to a result regarding the rate of decay ofthe auto--covariance function of an ARCH($\infty$) process.

Necessary and sufficient conditions for periodic decaying resolvents in  linear discrete convolution Volterra equations and applications to  ARCH$(\infty)$ processes