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Using some results of the theory of functional equations we deduce some properties of the Jacobian sn z function which seem to be new. Some functional equations have also been found which are fulfilled by the sn z function which the author did not find in the literature.
Si consideri l’equazione e se ne ricerchi una soluzione nella classe delle distribuzioni (secondo L. Schwarz), che verifichi le condizioni , dove e sono due distribuzioni. I valori locali e si intendono nel senso di S. Łojasievicz. Il problema posto ha soluzioni se e soltanto se le distribuzioni , sono dispari. Soddisfatta questa condizione le soluzioni le più generali sono date dalla (19) e (20), essendo una distribuzione pari qualunque.
Let X be an arbitrary Abelian group and E a Banach space. We consider the difference-operators ∆n defined by induction:
(∆f)(x;y) = f(x+y) - f(x), (∆nf)(x;y1,...,yn) = (∆n-1(∆f)(.;y1)) (x;y2,...,yn)
(n = 2,3,4,..., ∆1=∆, x,yi belonging to X, i = 1,2,...,n; f: X --> E).
...
In this note we solve the inhomogeneous Cauchy functional equation f(x+y) - f(x) - f(y) = d(x,y), x,y belonging to R, in the case where d is bounded.
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