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N - 1, and satisf3' the following three conditions: 1. e. A P h j ( t , y) = 0 for all tj < t < tj+l and y in I~, f o r j = 1. . . N - 1. 2. The functions hj(t, y) belong to the class C2p-2(Stripj). Tlle bar over Stripj means smoothness, including the boundary of Stripj. 3. 1) 58 Multivariate polysplines and for all k = 0, 1 . . . 2p - 2, and j = l . . . N - 2. i(t, y) atpoints t ~ (tj, tj+l) and t ~ tj+l. We will say that the fimction h(t, y), which is equal to hi(t, y) on the strip Strip/ for j = 1.

1. The function h(t, v) given in the strip Strip(tl, tN) = {(t, y) E ~2 "tl < t < tN, and x' ~ R} is polyspline of order p = l if and only if Au(t,y)=0 for alltl < t < t N a n d t ¢ t j f o r j = 2 ..... N-I. and it satisfies the global smoothness condition h(t. 3") E C(Strip(ti. tN)). e. j+l. 3') for all ~, in IR, and j = l, 2 . . . N - 2. 5), p. 22. Consequently, the "smoothness condition" for harmonic polysplines simply means their continuity in the strip Strip(tl, t N). 4), p. 5), p. 22).

6, p. 47, forms a Chebyshev system. 20, p. 196, front the Appendix on Chebyshev systems. According to this theorem, the fimctions u k (t) and vk (t) are uniquely determined. 9), p. 48. Without going too deep into the theory of function spaces, we also have, roughly speaking, the reverse result: the boundary values provide us with a couple f o = (h(a, y), h(b, y)), fl (Oh(a, y) Oh(b, y ) ) = ~ ' Ot " Thus the couples fo, f l "parametrize" the space of periodic biharmonic functions in the strip. 2 The polyharmonic case Having considered the most important special case, which is that of the biharmonic polysplines, we proceed to the general o n e - for arbitrary p > 1.

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The apeiron of Anaximander; a study in the origin and function of metaphysical ideas. by Paul Seligman

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