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On Linear Functional Equations with Nonpolynomial C∞ Solutions

Published online by Cambridge University Press:  20 November 2018

Halina Światak
Halina Światak*
Affiliation:
Jagello University, Cracow, Poland, McGill University, Montreal, Quebec
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It is known (cf. M. A. McKiernan [6]) that the only measurably bounded solutions ƒ of the equations

1

where x ∊ Rn, t ∊ R, αi(i= 1, …, m) span the space , and Σi ∊ 1 μi ≠0 for any I ⊂ {1, …, m}, are polynomials. The degree of these polynomials and the dimension of the solution space can be estimated by numbers depending on m and n. (For estimates and other details concerning equations (1) see see [1], [2], [3], [4], [5], [6].)

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 14 , Issue 2 , June 1971 , pp. 239 - 244
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Flatto, L., Functions with a mean value property, J. Math. Mech. 10 (1961), 11-18.Google Scholar
2. Flatto, L., Functions with a mean value property, Amer. J. Math. 85 (1963), 248-270.Google Scholar
3. Flatto, L., On polynomials characterized by a certain mean value property, Proc. Amer. Math. Soc. 17(1966), 598-601.Google Scholar
4. Friedman, A. and Littman, W., Functions satisfying the mean value property, Trans. Amer. Math. Soc. 102 (1962), 167-180.Google Scholar
5. Garsia, A. M., Note on the mean value property, Trans. Amer. Math. Soc. 102 (1962), 181-186.Google Scholar
6. McKiernan, M. A., Boundedness on a set of positive measure and the mean value property characterizes polynomials on a space Vn, Aequationes Math, (to appear).Google Scholar
7. Światak, H., A generalization of the Haruki functional equation, Ann. Polon. Math. 22 (1970), 371–376.Google Scholar
8. Światak, H., Criteria for the regularity of continuous and locally integral)le solutions of a class of linear functional equations, Aequationes Math, (to appear).Google Scholar