Abstract
The functional equation
has the linear functions ƒ(x) = a + bx (a, b ∈ ℝ) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ƒ(x) = eλi x (i = 1, 2, …), where the λi∈ ℂ are the fixed points of the map z ↦ sinh z, and (ii) C∞-solutions ƒ for which the values in the interval [−1,1] can be prescribed arbitrarily, but with the provision that ƒ(j)(− 1) = ƒ(j)(0) = ƒ(j)(1) = 0 for all j = 0, 1, 2 …
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Literature
G. Pólya, Geometrisches über die Verteilung der Nullstellen gewisser ganzer transzendenter Funktionen, Sitz.-Ber. Bayer. Akad. Wiss. (1920) 285-290 (Collected Papers, Vol. II, p. 198-203, The MIT Press, Cambridge, Mass. 1974).
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Dedicated to Prof. Jáinos Aczél on the occasion of this 70th birthday.
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Walter, W. A Functional Equation Of J.A. Baker Involving Integral Means. Results. Math. 26, 399–402 (1994). https://doi.org/10.1007/BF03323066
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DOI: https://doi.org/10.1007/BF03323066