Skip to main content
SpringerLink
Log in

Extension of Functional Equations

  • Chapter
The Mathematics of Paul Erdös II

Part of the book series: Algorithms and Combinatorics ((AC,volume 14))

  • 1022 Accesses

Abstract

Extension theorems are common in various areas of mathematics. In topology continuous extensions of continuous functions are studied. In functional analysis one is interested mainly in linear extensions of linear operators preserving continuity or some other properties like bounds or norm. In algebra extensions of homomorphisms and isomorphisms are investigated. The latter can be considered as extensions of functional equations.

Research supported by the Natural Sciences and Engineering Research Council of Canada grants nr. OGP002972 and CPG0164211

1991 Mathematics Subject Classification: 39 B 22, 39 B 52

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York-London, 1966 [Mathematics in Science and Engineering, Vol. 19].

    MATH  Google Scholar 

  2. J. Aczél, Some good and bad characters I have known and where they led. (Harmonic analysis and functional equations), In: 1980 Seminar on Harmonic Analysis. [Canad. Math. Soc. Conf. Proc., Vol. 1]. Amer Math. Soc., Providence, RI, 1981, pp. 177–187.

    Google Scholar 

  3. J. Aczél, Diamonds are not the Cauchy extensionist’s best friend, C. R. Math. Rep. Acad. Sci. Canada 5 (1983), 259–264.

    MATH  Google Scholar 

  4. J. Aczél, 28. Remark, Report of Meeting. The Twenty-second International Symposium on Functional Equations (December 16-December 22, 1984, Oberwolfach, Germany). Aequationes Math. 29 (1985), p. 101.

    Google Scholar 

  5. J. Aczél, J. A. Baker, D. Ž. Djoković, PL Kannappan and F. Radó, Extensions of certain homomorphisms of sub semigroups to homomorphisms of groups, Aequationes Math. 6 (1971), 263–271.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Aczél and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge-New York-New Rochelle-Melbourne-Sydney, 1989.

    MATH  Google Scholar 

  7. J. Aczél and P. Erdős, The non-existence of a Hamel-basis and the general solution of Cauchy’s functional equation for nonnegative numbers, Publ. Math. Debrecen 12 (1965), 259–265.

    MathSciNet  MATH  Google Scholar 

  8. J. Aczél, Pl. Kannappan, C. T. Ng and C. Wagner, Functional equations and inequalities in ‘rational group decision making’, In: General Inequalities 3 (Proc. Third Internat. Conf. on General Inequalities, Oberwolfach, 1981). Birkhäuser, Basel-Boston-Stuttgart, 1983, pp. 239–243.

    Google Scholar 

  9. J. Aczél, C. T. Ng and C. Wagner, Aggregation theorems for allocation problems, SIAM J. Alg. Disc. Meth. 5 (1984), 1–8.

    Article  MATH  Google Scholar 

  10. J. Aczél and C. Wagner, Rational group decision making generalized: the case of several unknown functions, C. R. Math. Rep. Acad. Sci. Canada 3 (1981), 139–142.

    MATH  Google Scholar 

  11. N. G. de Bruijn, On almost additive functions, Colloquium Math. 15 (1966), 59–63.

    MATH  Google Scholar 

  12. B. Crstici, I. Muntean and N. Vornicescu, General solution of the arctangent functional equation, Anal. Numér. Théor. Approx. 12 (1983), 113–123.

    MathSciNet  MATH  Google Scholar 

  13. Z. Daróczy and L. Losonczi, Über die Erweiterung der auf einer Punktmenge additiven Funktionen, Publ. Math. Debrecen 14 (1967), 239–245.

    MathSciNet  MATH  Google Scholar 

  14. J. Dhombres and R. Ger, Conditional Cauchy equations, Glas. Mat. Ser. III. 13 (33) (1978), 39–62.

    MathSciNet  Google Scholar 

  15. P. Erdős, P 310, Colloquium Math. 7 (1960), 311.

    Google Scholar 

  16. R. Ger, On almost polynomial functions, Colloquium Math. 24 (1971), 95–101.

    MathSciNet  MATH  Google Scholar 

  17. R. Ger, On some functional equations with a restricted domain, Fundamenta Math. 89 (1975), 95–101.

    MathSciNet  Google Scholar 

  18. S. Hartman, A remark about Cauchy’s equation, Colloquium Math. 8 (1961), 77–79.

    MathSciNet  MATH  Google Scholar 

  19. W. B. Jurkat, On Cauchy’s functional equation, Proc. Amer. Math. Soc. 16 (1965), 683–686.

    MathSciNet  MATH  Google Scholar 

  20. H. Kiesewetter, Über die arc tan-Funktionalgleichung, ihre mehr deutigen stetigen Lösungen und eine nichtstetige Gruppe, Wiss. Z. Friedrich-Schiller-Univ. Jena Math.-Natur. 14 (1965), 417–421.

    MathSciNet  Google Scholar 

  21. M. Kuczma, Almost convex functions, Colloquium Math. 21 (1970), 279–284.

    MathSciNet  MATH  Google Scholar 

  22. M. Kuczma, Functional equations on restricted domains, Aequationes Math. 18 (1978), 1–35.

    Article  MathSciNet  MATH  Google Scholar 

  23. K. Lajkó, Applications of extensions of additive functions, Aequationes Math. 11 (1974), 68–76.

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Losonczi, An extension theorem, Aequationes Math. 28 (1985), 293–299.

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Losonczi, Remark 32: The general solution of the arc tan equation, In: Proc. Twenty-third Internat. Symp. on Functional Equations (Gargnano, Italy, June 2–11, 1985). Univ. of Waterloo, Centre for Information Theory, Waterloo, Ont., 1985, pp.74–76.

    Google Scholar 

  26. L. Losonczi, Local solutions of functional equations, Glasnik Mat. 25 (45) (1990), 57–67.

    MathSciNet  Google Scholar 

  27. L. Losonczi, An extension theorem for the Levi-Cività functional equation and its applications, Grazer Math. Ber. 315 (1991), 51–68.

    MathSciNet  Google Scholar 

  28. S. C. Martin, Extensions and decompositions of homomorphisms of semigroups, Manuscript, University of Waterloo, Ont., 1977.

    Google Scholar 

  29. I. Muntean and N. Vornicescu, On the arctangent functional equation, (Roumanian), Seminarul “Theodor Angheluta”, Cluj-Napoca, 1983, pp. 241–246.

    Google Scholar 

  30. C. T. Ng, Representation for measures of information with the branching property, Inform, and Control 25 (1974), 45–56.

    Article  MATH  Google Scholar 

  31. K. E. Osondu, Extensions of homomorphisms of a subsemigroup of a group, Semigroup Forum 15 (1978), 311–318.

    Article  MathSciNet  MATH  Google Scholar 

  32. L. Paganoni and J. Rätz, Conditional functional equations and orthogonal additivity, Aequationes Math. 50 (1995), 134–141.

    Article  Google Scholar 

  33. F. Radó and J. A. Baker, Pexider’s equation and aggregation of allocations, Aequationes Math. 32 (1987), 227–239.

    Article  MathSciNet  Google Scholar 

  34. J. Rimán, On an extension of Pexider’s equation, Zbornik Radova Mat. Inst. Beograd N. S. 1(9) (1976), 65–72.

    Google Scholar 

  35. L. Székelyhidi, The general representation of an additive function on an open point set, (Hungarian), Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 21 (1972), 503–509.

    Google Scholar 

  36. L. Székelyhidi, An extension theorem for a functional equation, Publ. Math. Debrecen 28 (1981), 275–279.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    J. Aczél

  2. Department of Mathematics, Kuwait University, P.O.Box 5969, Safat, 13060, Kuwait

    L. Losonczi

Authors
  1. J. Aczél
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. Losonczi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. AT&T Bell Laboratories, 600 Mountain Avenue, 07974, Murray Hill, NJ, USA

    Ronald L. Graham

  2. Department of Applied Mathematics, Charles University, Malostranske nam. 25, 11800, Praha, Czech Republic

    Jaroslav Nešetřil

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Aczél, J., Losonczi, L. (1997). Extension of Functional Equations. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös II. Algorithms and Combinatorics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60406-5_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-60406-5_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64393-4

  • Online ISBN: 978-3-642-60406-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation