History of Functional Equations (HFE)

* Historical papers

The following are seminal papers in Functional Equations:

J. Aczel, J. D'Hombres, Functional equations in several variables, Encyclopedia of Maths. and its Appl. 31, Cambridge University Press, 1989.

A. Aksoy, J. M. Almira, On Montel and Montel-Popoviciu Theorems in several variables, Aequationes Mathematicae, 89 (5) (2015) 1335–1357

J. M. Almira, Montel's theorem and subspaces of distributions which are $\Delta^m$-invariant, Numer. Functional Anal. Optimiz. \textbf{35} (4) (2014) 389–403.

J. M. Almira, A Montel-type theorem for mixed differences, Rendiconti Sem. Mat. Univ. Pol. Torino Vol. 74, 1 (2017), 5–10.

\bibitem{A_popo} {\sc J. M. Almira, } On Popoviciu-Ionescu functional equation, Annales Mathematicae Silesianae, \textbf{30} (2016) 5–15.

\bibitem{A_JJA} {\sc J. M. Almira, } On Loewner's characterization of polynomials, Ja\'{e}n Journal on Approximation, \textbf{8} (2) (2016) 175–181.

\bibitem{AK_CJM} {\sc J. M. Almira, K. F. Abu-Helaiel, } On Montel's theorem in several variables, Carpathian Journal of Mathematics, \textbf{31} (2015), 1–10.

\bibitem{AK_gaceta} {\sc J. M. Almira, K. F. Abu-Helaiel, } Los teoremas de Fréchet, Montel y Popoviciu y los grafos de los polinomios discontinuos, La Gaceta de la R.S.M.E. 18 (2) (2015) 239-268

\bibitem{AK_ATPSFE} {\sc J. M. Almira, K. F. Abu-Helaiel, } A note on invariant subspaces and the solution of some classical functional equations, Annals of the Tiberiu Popoviciu Seminar of Functional Equations, Approximation and Convexity \textbf{11} (2013) 3-17.

\bibitem{almira_antonio} {\sc J. M. Almira, A. J. L\'{o}pez-Moreno, } On solutions of the Fr\'{e}chet functional equation, \emph{J. Math. Anal. Appl. } \textbf{332} (2007), 1119–1133.

\bibitem{AS_LCFE} {\sc J. M. Almira, E. V. Shulman, } On certain generalizations of the Levi-Civita and Wilson functional equations, Aequationes Mathematicae, to appear, Published First Online with DOI: 10.1007/s00010-017-0489-4, 2017.

\bibitem{AS_AM} {\sc J. M. Almira, L. Sz\'{e}kelyhidi, } Local polynomials and the Montel theorem, Aequationes Mathematicae, \textbf{89} (2015) 329-338.

\bibitem{AS_DM} {\sc J. M. Almira, L. Sz\'{e}kelyhidi, } Montel–type theorems for exponential polynomials, Demonstratio Mathematica, \textbf{49} (2) (2016) 197-212.

\bibitem{AK_TP} {\sc J. M. Almira, K. F. Abu-Helaiel, } A note on invariant subspaces and the solution of certain classical functional equations, \emph{Annals of the Tiberiu Popoviciu Seminar on Functional Equations, Approximation and Convexity,} \textbf{11} (2013) 3-17.

\bibitem{AK_D} {\sc J. M. Almira, K. F. Abu-Helaiel, } A qualitative description of graphs of discontinuous polynomials, to appear in \emph{Annals of Functional Analysis,} 2015; available at arXiv:1401.3273.

\bibitem{A_L_montel} {\sc J. M. Almira, L. Sz\'{e}kelyhidi,} Local polynomials and the Montel Theorem, manuscript, submitted, 2014.

\bibitem{AK_D} {\sc J. M. Almira, K. F. Abu-Helaiel, } A qualitative description of graphs of discontinuous polynomials, to appear in \emph{Annals of Functional Analysis,} 2015; available at arXiv:1401.3273.

\bibitem{anselone} {\sc P. M. Anselone, J. Korevaar, } Translation invariant subspaces of finite dimension, \emph{Proc. Amer. Math. Soc. } \textbf{15} (1964), 747-752.

\bibitem{baker}{\sc J. A. Baker} Functional equations, tempered distributions and Fourier transforms, \emph{Trans. Amer. Math. Soc.} \textbf{315} (1989), no.1, 57-68.

%%% \bibitem{banach} {\sc S. Banach, } Sur l'equation fontionnelle $f(x+y)=f(x)+f(y)$, \emph{Fundamenta Mathematicae} \textbf{1} (1920) 123-124.

\bibitem{B_J} {\sc K. Baron, W. Jarczyk, } Recent results on functional equations in a single variable, perspectives and open problems, \emph{Aequationes Mathematicae} \textbf{61} (2001) 1-48.

\bibitem{ciesielski} {\sc Z. Ciesielski, } Some properties of convex functions of higher order, \emph{Ann. Polon. Math.} \textbf{7} (1959) 1-7.

\bibitem{Nilp} {\sc A. E. Clement, S. Majewicz, M. Zyman, } \emph{The Theory of Nilpotent Groups}, Birkh\“{a}user, 2017.

\bibitem{Cra} {\sc H. Cram\'{e}r, } \emph{Random variables and probability distributions, } Cambridge Tracts in Maths. and Math. Physics \textbf{36} Cambridge University Press, London, 1937.\bibitem{darboux} {\sc G. Darboux, } Memoire sur les fonctions discontinues, \emph{Ann. Sci. \'{E}cole Norm. Sup.} \textbf{4} (1875) 57-112.

\bibitem{Dj} {\sc D. Z. Djokovi\'{c}, } A representation theorem for $(X_1-1)(X_2-1)\cdots(X_n-1)$ and its applications, \emph{Ann. Polon. Math.} \textbf{22} (1969/1970) 189-198.

\bibitem{donoghue} {\sc W. F. Donoghue, Jr., } \emph{Distributions and Fourier Transforms, } Academic Press, New York and London, 1969.

\bibitem{E} {\sc M. Engert, } Finite dimensional translation invariant subspaces, Pacific J. Maths. \textbf{32} (2) (1970) 333-343.

\bibitem{Feld} {\sc G. Feldman, } \emph{Functional equations and characterization problems on locally compact Abelian groups, } EMS, 2008.

\bibitem{fe} {\sc I. Feny\”{o}, } Sur les \'{e}quations distributionnelles, in \emph{Functional Equations and Inequalities,} Ed. B. Forte, C.I.M.E. summer schools, \textbf{54}, Springer, 1971 (Reprinted in 2010), 45-109.

\bibitem{frechet} {\sc M. Fr\'{e}chet, } Une definition fonctionelle des polynomes, \emph{Nouv. Ann.} \textbf{9} (1909), 145-162. \bibitem{Gu_O} {\sc S. G. Ghurye, I. Olkin, } A characterization of the multivariate normal distribution, Ann. Math. Statist. \textbf{33} (2) (1962) 533–541.

\bibitem{ger1} {\sc R. Ger}, On some properties of polynomial functions, \emph{Ann. Pol. Math.} \textbf{25} (1971) 195-203.

\bibitem{ger} {\sc R. Ger, } On extensions of polynomial functions, \emph{Results in Mathematics} \textbf{26} (1994), 281-289.

\bibitem{hamel} {\sc G. Hamel}, Eine Basis aller Zahlen und die unstetigen L\“{o}sungen der Funktionalgleichung $f(x+y)=f(x)+f(y)$, \emph{Math. Ann.} \textbf{60} (1905) 459-462.

\bibitem{HW} {\sc G. H. Hardy, E. M. Wright, } \emph{An Introduction to the Theory of Numbers. Fifth edition.} The Clarendon Press, Oxford University Press, New York, 1979.

\bibitem{h1} {\sc S. Haruki, } On the mean value property of harmonic and complex polynomials, Proc. Japan Acad. Ser. A, \textbf{57} (1981) 216-218.

\bibitem{h2} {\sc S. Haruki, } On the theorem of S. Kakutani-M. Nagumo and J.L. Walsh for the mean value property of harmonic and complex polynomials, Pacific J. Math. \textbf{94} (1) (1981) 113-123.

\bibitem{h3} {\sc S. Haruki, } On two functional equations connected with a mean-value property of polynomials, Aequationes Math. \textbf{6} (1971) 275-277.

\bibitem{Hu} {\sc Y.Q. Hu,} Polynomial maps and polynomial sequences in groups, Journal of Group Theory, \textbf{27} (4) (2024) 739–787.

\bibitem{HIR} {\sc D. H. Hyers, G. Isac, T. M. Rassias, } \emph{Stability of functional equations in several variables, } Birkh\”{a}user, 1998.

\bibitem{I_K} {\sc E. Isaacson, H. B. Keller,} \emph{Analysis of Numerical Methods,} Wiley, New York, 1966.

\bibitem{Jacobi} {\sc C. G. J. Jacobi, } De usu theoriae integralium ellipticorum et integralium abelianorum in analysi diophantea, Werke \textbf{2} (1834) 53-55.

\bibitem{jarai}{\sc A. J\'{a}rai, } \emph{Regularity properties of functional equations in several variables, } Springer Verlag, 2005.

\bibitem{J_Laszlo} {\sc A. J\'{a}rai, L. Sz\'{e}kelyhidi, } Regularization and General Methods in the Theory of Functional Equations, \emph{Aequationes Mathematicae} \textbf{52} (1996) 10-29.

\bibitem{JS} {\sc G. A. Jones, D. Singerman}, \emph{Complex functions. An algebraic and geometric viewpoint}, Cambridge Univ. Press, 1987.

\bibitem{KLR} {\sc A.M. Kagan, Yu. V. Linnik, C. R. Rao, } \emph{Characterization problems in Mathematical Statistics, } John Wiley \& Sons, 1973.

\bibitem{kn} {\sc S. Kakutani, M. Nagumo, } About the functional equation $\sum_{v=0}^{n-1}f(z+e^{(\frac{2 v \pi}{2})i})=nf(z)$, Zenkoku Shij\^{o} Danwakai, \textbf{66} (1935) 10-12 (in Japanese).

\bibitem{kormes} {\sc M. Kormes, } On the functional equation $f(x+y)=f(x)+f(y)$, \emph{Bulletin of the Amer. Math. Soc. } \textbf{32} (1926) 689-693.

\bibitem{kuczma} {\sc M. Kuczma}, \emph{An introduction to the theory of functional equations and inequalities, } (Second Edition, Edited by A. Gil\'{a}nyi), Birkh\“{a}user, 2009.

\bibitem{kuczma0} {\sc M. Kuczma, } On some analogies between measure and category and their applications in the theory of additive functions, \emph{ Ann. Math. Sil.}No.\textbf{13} (1985), 155-162.

\bibitem{kuczma1} {\sc M. Kuczma}, On measurable functions with vanishing differences, \emph{Ann. Math. Sil.} \textbf{6} (1992) 42-60.

\bibitem{kurepa} {\sc S. Kurepa, } A property of a set of positive measure and its application, \emph{J. Math. Soc. Japan} \textbf{13} (1) (1961) 13-19.

\bibitem{L} {\sc M. Laczkovich, } Polynomial mappings on Abelian groups, \emph{Aequationes Mathematicae} \textbf{68} (3) (2004) 177-199.

\bibitem{Leib0} {\sc A. Leibman,} Polynomial sequences in groups, J. Algebra 201 (1998), no. 1, 189–206.

\bibitem{Leib1} {\sc A. Leibman,} Polynomial mappings of groups, Israel J. Math. 129 (2002) 29–60.

\bibitem{leland} {\sc K. O. Leland, } Finite dimensional translation invariant spaces, \emph{Amer. Math. Monthly} \textbf{75} (1968) 757-758.

\bibitem{LC} {\sc T. Levi-Civit\`a, } Sulle funzioni che ammettono una formula d'addizione del tipo $f(x+y)=\sum_{i=1}^n X_i(x)Y_i(y)$, Atti Accad. Naz. Lincei Rend. \textbf{22} (5) (1913) 181-183.

\bibitem{Lo} {\sc C. Loewner, } On some transformation semigroups invariant under Euclidean and non-Euclidean isometries, J. Math. Mech. 8 (1959) 393-409.

\bibitem{MaPe} {\sc A. M. Mathai, G. Pederzoli, } \emph{Characterizations of the Normal Probability Law, } Wiley Eastern Limited, 1977.

\bibitem{mckiernan} {\sc M. A. Mckiernan, } On vanishing n-th ordered differences and Hamel bases, \emph{Ann. Pol. Math.} \textbf{19} (1967) 331-336.

\bibitem{montel_1935} {\sc P. Montel, } Sur un th\'{e}oreme du Jacobi, \emph{Comptes Rend. Acad. Sci. Par\'{\i}s,} \textbf{201} (1935) 586.

\bibitem{montel} {\sc P. Montel, } Sur quelques extensions d'un th\'{e}or\`{e}me de Jacobi, \emph{Prace Matematyczno-Fizyczne} \textbf{44} (1) (1937) 315-329.

\bibitem{pales} {\sc Zs. P\'{a}les, } Problems in the regularity theory of functional equations, \emph{Aequationes Mathematicae} \textbf{63} (2002) 1-17.

\bibitem{popoviciu_tesis} {\sc T. Popoviciu, } Sur quelques propiétés des fonctions d'une ou de deux variables réelles, Thèse, Paris, 12 June 1933. Published in \emph{Mathematica} vol. VIII, 1934, pp. 1-85.

\bibitem{popoviciu} {\sc T. Popoviciu, } Remarques sur la définition fonctionnelle d'un polynôme d'une variable réelle, \emph{Mathematica (Cluj)} \textbf{12} (1936) 5-12.

\bibitem{R_B} {\sc Th. Rassias, J. Brzdek, } \emph{Functional Equations in Mathematical Analysis, } Springer Verlag , 2011.

\bibitem{rudin} {\sc W. Rudin, } \emph{Functional Analysis} (Second Edition), McGraw-Hill, 1991.

\bibitem{sanjuan} {\sc R. San Juan, } Una aplicaci\'{o}n de las aproximaciones diof\'{a}nticas a la ecuaci\'{o}n funcional $f(x_1+x_2)=f(x_1)+f(x_2)$, \emph{Publicaciones del Inst. Matem\'{a}tico de la Universidad Nacional del Litoral} \textbf{6} (1946) 221-224. .

\bibitem{S1} {\sc E. Shulman, } Subadditive set-functions on semigroups, applications to group representations and functional equations, \emph{J. Funct. Anal.} \textbf{263} (2012) 1468-1484.

\bibitem{S} {\sc E. Shulman, } Decomposable functions and representations of topological semigroups, \emph{Aequat. Math.} \textbf{79} (2010) 13-21.

\bibitem{S_PhD} {\sc E. Shulman, } \emph{Functional equations of homological type,} PhD Thesis, Moscow State Pedagogical University (1994).

\bibitem{sierpinsky} {\sc W. Sierpinsky, } Sur l'equation fontionnelle $f(x+y)=f(x)+f(y)$, \emph{Fundamenta Mathematicae} \textbf{1} (1920) 116-122.

\bibitem{steinhaus} {\sc H. Steinhaus, } Sur les distances des points dans les ensembles de mesure positive, \emph{Fundamenta Mathematicae} \textbf{1} (1920) 93-104

\bibitem{laszlo1} {\sc L. Sz\'{e}kelyhidi, } \emph{Convolution type functional equations on topological abelian groups, } World Scientific, 1991.

\bibitem{laszlo_discrete} {\sc L. Sz\'{e}kelyhidi, } \emph{Discrete spectral synthesis and its applications, } Springer, 2006.

\bibitem{laszlo_harm} {\sc L. Sz\'{e}kelyhidi, } \emph{Harmonic and Spectral Analysis, } World Scientific, 2014.

\bibitem{laszlo_mona} {\sc L. Sz\'{e}kelyhidi, } On Fréchet’s functional equation, to appear in \emph{Monatshefte f\”{u}r Mathematik,} 2014. \bibitem{V} {\sc V. S. Vladimirov, } \emph{Generalized Functions in Mathematical Physics,} 1979.

\bibitem{W} {\sc M. Waldschmidt, } \emph{Topologie des Points Rationnels, } Cours de Troisi\`{e}me Cycle 1994/95 Universit\'{e} P. et M. Curie (Paris VI), %1995.

\bibitem{wal} {\sc J. L. Walsh, } A mean value theorem for polynomials and harmonic polynomials, Bull. Amer. Math. Soc. \textbf{42} (1936) 923-930.

% %\bibitem{A_D} {\sc J. Aczel, J. D'Hombres, } \emph{Functional equations in several variables, } Encyclopedia of Maths. and its Appl. \textbf{31}, Cambridge University Press, 1989.

%\bibitem{I_K} {\sc E. Isaacson, H. B. Keller,} \emph{Analysis of Numerical Methods,} Wiley, New York, 1966.

\bibitem{tesis_correct_interp} {\sc V. Vitrih, } \emph{Correct interpolation problems in multivariate interpolation spaces, } Ph. Thesis, Department of Mathematics, University of Ljubljana, 2010.

\bibitem{vla} {\sc V. S. Vladimirov, } \emph{Generalized functions in Mathematical Analysis, } Mir Publishers, Moscow, 1979.

\bibitem{W} {\sc M. Waldschmidt, } \emph{Topologie des Points Rationnels, } Cours de Troisi\`{e}me Cycle 1994/95 Universit\'{e} P. et M. Curie (Paris VI), 1995.

\end{thebibliography}

* Articles on History of Functional Equations and obituaries

* Functional equations People