The aim of this work is to establish in great detail the conformable Fourier analysis related to the cosine. We define and analyze the properties of the even conformable translation and the convolution. By adopting the approximation of the identity approach, we study the conformable cosine Fourier transform and its inverse formula. The second theme of this paper is an application of the Fourier analysis developed earlier. We extend the heat representation theory inaugurated by P.C. Rosenbloom and D.V. Widder to the conformable analysis. We construct the solution source and the heat polynomials and constitute with the associated functions a biorthogonal system. We conclude by solving the analytic Cauchy problem related to the conformable heat equation.
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