We use the algebra of unbounded differentiation t operators acting on the ring of differentiable functions. The analytical representation of the fractional degree of the t operator is used to construct the resolvents of three boundary problems for the Fourier equation. Periodic solutions of limiting Fourier problems in the algebra of differentiation operators coincide with classical solutions. The t+2 extension is a continuous spectrum of the Fourier transform and allows us to obtain exact solutions of three limit problems for a domain of any dimension d > 1.