Previous |  Up |  Next


difference; quantum difference; quantum derivative; power series
We consider real valued functions $f$ defined on a subinterval $I$ of the positive real axis and prove that if all of $f$’s quantum differences are nonnegative then $f$ has a power series representation on $I$. Further, if the quantum differences have fixed sign on $I$ then $f$ is analytic on $I$.
[1] S. G. Bernstein: Leçons sur les propriété extrémales et la meilleure approximation des functions analytiques d’une variable réelle. Gautier-Villars, Paris, 1926. (French)
[2] S. G.  Bernstein: Sur les fonctions absolument monotones. Acta Math. 52 (1928), 1–66. DOI 10.1007/BF02592679
[3] G.  Gasper, M.  Rahman: Basic hypergeometric series. Encyclopaedia of Mathematics and its Applications 34, Cambridge University Press, Cambridge, 1990. MR 1052153
[4] J. H. B.  Kemperman: On the regularity of generalized convex functions. Trans. Amer. Math. Soc. 135 (1969), 69–93. DOI 10.1090/S0002-9947-1969-0265531-3 | MR 0232900
[5] V.  Kac, P.  Cheung: Quantum Calculus. Springer-Verlag, New York, 2002. MR 1865777
[6] J. M.  Ash, S. Catoiu, and R.  Rios-Collantes-de-Teran: On the $n$th quantum derivative. J.  London Math. Soc. 66 (2002), 114–130. DOI 10.1112/S0024610702003198 | MR 1911224
[7] T.  Sjödin: Bernstein’s analyticity theorem for binary differences. Math. Ann. 315 (1999), 251–261. DOI 10.1007/s002080050366 | MR 1721798
[8] T.  Sjödin: On generalized differences and Bernstein’s analyticity theorem. Research report No  9, Department of Mathematics, University of Umeå, Umeå, 2003.
Partner of
EuDML logo