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An Integral Transformation and its Applications to Harmonic Analysis on the Space of Solutions of the Heat Equation
We introduce an integral transformation T defined by (Tf)(x,t) = ∫ e^{−ixy−t|y|^2} f(y)dy,\quad f\in L^1(\boldsymbol R^n) in order to do harmonic analysis on the space of C^∞ solutions of the heat equation. First, the Paley–Wiener type theorem for the transformation T will be given for the C^∞ functions and distributions …