In this paper, we introduce the theory of a generalized Fourier
transform in order to solve differential equations with a generalized
fractional derivative, and we state its main properties. In particular,
we obtain the corresponding convolution, inverse and Plancherel
formulas, and Hausdorff-Young inequality. We show that this generalized
Fourier transform is useful in the study of fractional partial
differential equations, by solving the fractional heat equation on the
real line.