Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T06:41:49.706Z Has data issue: false hasContentIssue false
  • English
  • Français

On extensions of inequalities of Kolmogoroff and others and some applications to almost periodic functions

Published online by Cambridge University Press:  18 May 2009

C. J. F. Upton
C. J. F. Upton
Affiliation:
University Of Melbourne, Victoria, Australia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let f(x) be a complex function of a real variable, defined over the whole real line, which possesses n derivatives (the nth at least almost everywhere) and is such that . Then, if k is any integer for which 0< k < n, Kolmogoroff's inequality may be written as

,

or, by putting ,

The constant K=K (k, n) known explicitly and is the best possible, i.e., there is a (real) function for which equality holds (see Bang [1]).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 13 , Issue 1 , March 1972 , pp. 1 - 16
Copyright
Copyright © Glasgow Mathematical Journal Trust 1972

References

REFERENCES

1.Bang, T., Une inégalité de Kolmogoroff et les fonctions presque-périodiques, Danske Videnskabernes Selskab Mathematisk-Fysiske Meddelelser XIX, 4, Copenhagen, 1941.Google Scholar
2.Besicovitch, A. S., Almost periodic functions (Cambridge, 1932).Google Scholar
3.Bochner, S., Properties of Fourier series of almost periodic functions, Proc. London Math. Soc. (2) 26 (1927), 433–452.CrossRefGoogle Scholar
4.Bohr, H. and Folner, E., On some types of functional spaces, Acta Mathematica 76 (1944), 31–155.CrossRefGoogle Scholar
5.Burkill, H., Sums of trigonometric series, Proc. London Math. Soc. (3) 12 (1962), 690–706.CrossRefGoogle Scholar
6.Civin, P., Inequalities for trigonometric integrals, Duke Math. J. 8 (1941), 656–665.CrossRefGoogle Scholar
7.Hardy, G. H., Littlewood, J. E. and Landau, E., Some inequalities satisfied by the integrals or derivatives of real or analytic functions, Math. Zeit. 39 (1935), 677–695.CrossRefGoogle Scholar
8.Hardy, G. H., Littlewood, J. E. and Polya, G., Inequalities (Cambridge, 1952).Google Scholar
9.Love, E. R., More-than-uniform almost periodicity, J. London Math. Soc. 26 (1951), 14–25.CrossRefGoogle Scholar
10.Ogiewetski, I. I., Generalization of the inequality of P. Civin for the fractional derivative of a trigonometrical polynomial to L space, Acta Math. Hung. Tom. IX (1958), 133–135.CrossRefGoogle Scholar
11.Upton, C. J. F., Riesz almost periodicity, J. London Math. Soc. 31 (1956), 407–426.CrossRefGoogle Scholar