P. L. BUTZER,and G. SU~OUCHI Math. Annalen 155, 316--330 (1964) Approximation Theorems for the Solution of Fourier's Problem and Dirichlet's Problem By P. L. BUTZER and G. StJNoucm* in Aachen § 1. Introduction One important and suggestive problem in the conduction of heat where the temperature depends only upon one coordinate and the time is Fourier's problem of the ring. Consider the distribution of temperature in a homogeneous isotropic ring of unit radius, when there is no radiation at the surface, in other words the normalized partial ditt~rential equation aU O~U a--7 - = Ox--- T (-zt <x< ~;0 <t < ¢¢) with the boundary conditions U(zt;t)=U(-~;t), U~(:t;t)=U~(-z~;t) and initial condition lim U(x; t) =/(x). t~o In case ] is a periodic function with period 2 g with complex Fourier coefficients given by i~F[l] = [(n), f(n) = ~ e-~ l(u) du -~ (n = O, ±1, ±2 .... ), y(x; t) = I ](n) e-'te''~, which may be rewritten in the form (1.1) U(x; t) = -~ z~a(x - u; t) l(u) du, * The contribution of the last named author was supported in part by the Deutseher Akademischer Austauschdienst while holding a Visiting Professorship in Aachen. 1) For a study of the problem and properties of the solution, one may see S. BOCHN~R [1], H. S. CARS~W [5, p. 211--217], E. HI~I¢ [8, p. 402] and S. MX~SHISU~I)ARA~ [10]. l(x)~ the solution 1) is given by