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By constructing a special set of A-orthonormal functions, it is shown that, under'certain smoothness assumptions, the variational and Fourier series representations for the solution of first initial boundary value problems for the simple parabolic differential equation coincide. This result is then extended in order to construct a variational representation for the solution of a very general first initial boundary value problem for this equation.doi:10.1017/s0004972700047250 fatcat:k54jqyv2hnhurid2z7ccqhh25y