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3, p. 46, where we considered the solution of the Dirichlet problem for the biharmonic equation. 5 First, we assume that the biharmonic polyspline h(t, y) ~t'hich satisfies the interpolation and boundary conditions exists. The conditions oll the data f j. c. d which provide the existence of a "reasonably smooth" solution h (t. 3') will be thoroughly studied in Chapter 20, p. 409, in the terms of Sobolev and H61der spaces. 6), p. 46, we also assume that the pieces hj(t, y) of the polyspline h(t.

We likewise define the function gk(t ) :-- (Vl,k(t), V2,k(t) . . . ON-l,k(t)) to coincide with the function Vj,k(t) on the interval (tj, tj+l) for all j = 1, 2 . . . N - 1. 5), p. 6), p. 60, and compare the coefficients. We see that the newly defined functions Uk (t) and Vk (t) satisfy the following properties: 1. The following smoothness conditions are satisfied: Uk(t) is in C2(tl, tN), for k = 0, 1, 2 . . . Vk(t) is in C2(tl, tN), for k = 1,2, 3 . . . Uk(t) i u I, k u2, k . 1. j,k .

Let us note that the case k = 0 coincides with the one-dimensional Laplace equation Ah(t) = h"(t) = O. Now we are ready to understand the structure of the space of periodic harmonic functions in the strip. 2 "Parametrization" of the space of periodic harmonic functions in the strip: the Dirichlet problem Obviously, the boundary values of the periodic harmonic function h (t, y) on the edges of the strip, the sets {(t, y) 6 /K2 : t = a} and {(t, y) 6 ~2 : t = b}, also have to be 2zr-periodic. Now let us assume that a function, which is in fact a couple of functions, f ( y ) (fa(Y), fb(Y)), be given such that the two functions fa(Y) and fb(Y) are 2:r-periodic, and both are in L2(0, 2zr).

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