| Title:
|
Bernstein’s analyticity theorem for quantum differences (English) |
| Author:
|
Sjödin, Tord |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
57 |
| Issue:
|
1 |
| Year:
|
2007 |
| Pages:
|
67-73 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $I$. (English) |
| Keyword:
|
difference |
| Keyword:
|
quantum difference |
| Keyword:
|
quantum derivative |
| Keyword:
|
power series |
| MSC:
|
26A24 |
| MSC:
|
26A48 |
| MSC:
|
26E05 |
| idZBL:
|
Zbl 1174.26312 |
| idMR:
|
MR2309949 |
| . |
| Date available:
|
2009-09-24T11:43:53Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128155 |
| . |
| Reference:
|
[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) |
| Reference:
|
[2] S. G. Bernstein: Sur les fonctions absolument monotones.Acta Math. 52 (1928), 1–66. 10.1007/BF02592679 |
| Reference:
|
[3] G. Gasper, M. Rahman: Basic hypergeometric series.Encyclopaedia of Mathematics and its Applications 34, Cambridge University Press, Cambridge, 1990. MR 1052153 |
| Reference:
|
[4] J. H. B. Kemperman: On the regularity of generalized convex functions.Trans. Amer. Math. Soc. 135 (1969), 69–93. MR 0232900, 10.1090/S0002-9947-1969-0265531-3 |
| Reference:
|
[5] V. Kac, P. Cheung: Quantum Calculus.Springer-Verlag, New York, 2002. MR 1865777 |
| Reference:
|
[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. MR 1911224, 10.1112/S0024610702003198 |
| Reference:
|
[7] T. Sjödin: Bernstein’s analyticity theorem for binary differences.Math. Ann. 315 (1999), 251–261. MR 1721798, 10.1007/s002080050366 |
| Reference:
|
[8] T. Sjödin: On generalized differences and Bernstein’s analyticity theorem.Research report No 9, Department of Mathematics, University of Umeå, Umeå, 2003. |
| . |
