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Regular Polytopes and Harmonic Polynomials

  • Leopold Flatto (a1) and Sister Margaret M. Wiener (a2)

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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.

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References

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1. Beckenbach, E. F. and Reade, M., Regular solids and harmonic polynomials, Duke Math. J. 12 (1945), 629644.
2. Chevalley, C., Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778782.
3. Coxeter, H. S. M., Regular polytopes, 2nd ed. (Macmillan, New York, 1963).
4. Coxeter, H. S. M., The product of the generators of a finite group generated by reflections, Duke Math. J. 18 (1951), 765782.
5. Coxeter, H. S. M., Introduction to geometry (Wiley, New York, 1961).
6. Flatto, L., Functions with a mean value property. II, Amer. J. Math. 85 (1963), 248270.
7. Flatto, L. and Sister M. M., Wiener, Invariants of finite reflection groups and mean value problems, Amer. J. Math, (to appear).
8. Walsh, J. L., A mean value theorem for polynomials and harmonic polynomials, Bull. Amer. Math. Soc. 42 (1936), 923930.
9. Wiener, Sister M. M., Invariants of finite reflection groups, Thesis, Yeshiva University, New York, 1968.
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Regular Polytopes and Harmonic Polynomials

  • Leopold Flatto (a1) and Sister Margaret M. Wiener (a2)

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