Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Interpolation of Probability Measures on Graphs Read chapter Read first chapter

2017 | OriginalPaper | Chapter

The Chain Rule Operator Equation for Polynomials and Entire Functions

Authors : Hermann König, Vitali Milman

Published in: Convexity and Concentration

Publisher: Springer New York

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

After a short survey on operator functional equations we study the solutions of the chain rule operator equation
$$\displaystyle{T(f \circ g) = (Tf) \circ g \cdot Tg}$$
for maps T on spaces of polynomials and entire functions. In comparison with operators on C k -spaces with \(k \in \mathbb{N} \cup \{\infty \}\), which were studied before, there are entirely different solutions. However, these yield discontinuous maps T. Under a mild continuity assumption on T we determine all solutions of the chain rule equation on spaces of real or complex polynomials or analytic functions. The normalization condition T(−2 Id) = −2 yields Tf = f′ as the only solution.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Counting Integer Points in Higher-Dimensional Polytopes
Literature
[A]
J. Aczél; Lectures on functional equations and their applications, Academic Press, 1966.
[AAFM]
S. Alesker, S. Artstein-Avidan, D. Faifman, V. Milman; A characterization of product preserving maps with applications to a characterization of the Fourier transform, Illinois J. Math. 54 (2010), 1115–1132.
[AFM]
S. Artstein-Avidan, D. Faifman, V. Milman; On multiplicative maps of continuous and smooth functions, Geometric aspects of functional analysis, Lecture Notes in Math. 2050, 35–59, Springer, Heidelberg, 2012.
[AKM]
S. Artstein-Avidan, H. König, V. Milman; The chain rule as a functional equation, J. Funct. Anal. 259 (2010), 2999–3024.
[AO]
A. Alexiewicz, W. Orlicz; Remarque sur l’équation fonctionelle f(x + y) = f(x) + f(y), Fund. Math. 33 (1945), 314–315.
[B]
S. Banach; Sur l’équation fonctionelle f(x + y) = f(x) + f(y), Fund. Math. 1 (1920), 123–124.
[GS]
H. Goldmann, P. Šemrl; Multiplicative derivations on C(X), Monatsh. Math. 121 (1996), 189–197.
[KM]
H. König, V. Milman; Characterizing the derivative and the entropy function by the Leibniz rule, with an appendix by D. Faifman, J. Funct. Anal. 261 (2011), 1325–1344.
[M]
A.N. Milgram; Multiplicative semigroups of continuous functions, Duke Math. J. 16 (1940), 377–383.
[Mr]
J. Mrčun; On isomorphisms of algebras of smooth functions, Proc. Amer. Math. Soc. 133 (2005), 3109–3113.
[MS]
J. Mrčun, P. Šemrl; Multplicative bijections between algebras of differentiable functions, Ann. Acad. Sci. Fenn. Math. 32 (2007), 471–480.
[S]
W. Sierpinski; Sur l’équation fonctionelle f(x + y) = f(x) + f(y), Fund. Math. 1 (1920), 116–122.
Metadata
Title
The Chain Rule Operator Equation for Polynomials and Entire Functions
Authors
Hermann König
Vitali Milman
Copyright Year
2017
Publisher
Springer New York
Book
Convexity and Concentration

Print ISBN: 978-1-4939-7004-9

Electronic ISBN: 978-1-4939-7005-6

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-1-4939-7005-6_21

Premium Partner

    Image Credits