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Regular Polytopes and Harmonic Polynomials

Published online by Cambridge University Press:  20 November 2018

Leopold Flatto  and
Sister Margaret M. Wiener
Leopold Flatto
Affiliation:
Belfer Graduate School of Science, Yeshiva University, New York, New York
Sister Margaret M. Wiener
Affiliation:
Marymount Manhattan College, New York, New York
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In this paper we study the following problem originally proposed by Walsh (8). To determine the class of functions f(x) continuous in a given n-dimensional region R and having the property that the value of f(x) be equal to the average of f(x) over the vertices of all sufficiently small regular polytopes similar to a given one, which are centred at x. This problem has been studied by several mathematicians (1; 6; 8) and has been completely solved except for the four-dimensional regular polytopes {3, 4, 3}, {3, 3, 5}, {5, 3, 3} (see 3, p. 129, for the meaning of these symbols) and the n-dimensional cube. In each case, the class of functions is identical with a class of harmonic polynomials which can be specified. In § 2, we solve the problem for the four-dimensional figures, thus leaving the problem open only for the n-dimensional cube.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 7 - 21
Copyright
Copyright © Canadian Mathematical Society 1970

References

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