A Step Forward to Formalize Tailored to Problem Specificity Mathematical Transforms

Abstract

Linear functional analysis historically founded by Fourier and Legendre played a significant role to provide a unified vision of mathematical transformations between vector spaces. The possibility of extending this approach is explored when basis of vector spaces is built Tailored to the Problem Specificity (TPS) and not from the convenience or effectiveness of mathematical calculations. Standardized mathematical transformations, such as Fourier or polynomial transforms, could be extended toward TPS methods, on a basis, which properly encodes specific knowledge about a problem. Transition between methods is illustrated by comparing what happens in conventional Fourier transform with what happened during the development of Jewett Transform, reported in previous articles. The proper use of computational intelligence tools to perform Jewett Transform allowed complexity algorithm optimization, which encourages the search for a general TPS methodology.