The Levi-Civita equation in function classes

Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A ), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G , then every element of V is an exponential polynomial. More precisely, every element o...

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Vydané v:	Aequationes mathematicae Ročník 94; číslo 4; s. 689 - 701
Hlavný autor:	Laczkovich, Miklós
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cham Springer International Publishing 01.08.2020 Springer Nature B.V
Predmet:	Analysis Combinatorics Continuity (mathematics) Functions (mathematics) Mathematical analysis Mathematics Mathematics and Statistics Polynomials Theorems Topology Primary 39B52 Exponential polynomials Translational invariant subspaces of function classes Secondary 43A45
ISSN:	0001-9054, 1420-8903
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Abstract	Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A ), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G , then every element of V is an exponential polynomial. More precisely, every element of V is of the form ∑ i = 1 n p i · m i , where m 1 , … , m n are exponentials belonging to V , and p 1 , … , p n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra A of complex valued functions such that whenever an exponential m belongs to A , then m - 1 ∈ A . As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G , or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary σ -algebra. We give two proofs of the result. The first is based on Theorem A . The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A .
AbstractList	Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G , then every element of V is an exponential polynomial. More precisely, every element of V is of the form $$\sum _{i=1}^np_i \cdot m_i$$ ∑ i = 1 n p i · m i , where $$m_1 ,\ldots ,m_n$$ m 1 , … , m n are exponentials belonging to V , and $$p_1 ,\ldots ,p_n$$ p 1 , … , p n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra $${{\mathcal {A}}}$$ A of complex valued functions such that whenever an exponential m belongs to $${{\mathcal {A}}}$$ A , then $$m^{-1}\in {{\mathcal {A}}}$$ m - 1 ∈ A . As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G , or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary $$\sigma $$ σ -algebra. We give two proofs of the result. The first is based on Theorem A. The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A. Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G, then every element of V is an exponential polynomial. More precisely, every element of V is of the form ∑i=1npi·mi, where m1,…,mn are exponentials belonging to V, and p1,…,pn are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra A of complex valued functions such that whenever an exponential m belongs to A, then m-1∈A. As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G, or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary σ-algebra. We give two proofs of the result. The first is based on Theorem A. The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A. Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A ), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G , then every element of V is an exponential polynomial. More precisely, every element of V is of the form ∑ i = 1 n p i · m i , where m 1 , … , m n are exponentials belonging to V , and p 1 , … , p n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra A of complex valued functions such that whenever an exponential m belongs to A , then m - 1 ∈ A . As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G , or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary σ -algebra. We give two proofs of the result. The first is based on Theorem A . The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A .
Author	Laczkovich, Miklós
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Cites_doi	10.1017/CBO9781139086578 10.2140/pjm.1982.103.583 10.1007/s00010-017-0489-4 10.2140/pjm.1979.80.503 10.2307/1969386 10.2307/2315197 10.4064/ap-22-2-189-198 10.1142/1406 10.5486/PMD.2013.5768 10.2140/pjm.1970.32.333 10.1007/s10474-013-0385-x 10.1007/BF01836418 10.1017/S0305004107000114 10.1090/S0002-9939-1964-0169048-7
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References	SchwartzLThéorie génerale des fonctions moyenne-périodiquesAnn. Math.19474848579292394810.2307/1969386 LaczkovichMLocal spectral synthesis on Abelian groupsActa Math. Hung.2014142313329323353510.1007/s10474-013-0385-x EngertMFinite dimensional translation invariant subspacesPac. J. Math.19703233334327466010.2140/pjm.1970.32.333 SzékelyhidiLNote on exponential polynomialsPac. J. Math.198210358358770525110.2140/pjm.1982.103.583 AczélJDhombresJFunctional Equations in Several Variables. Encyclopedia of Mathematics and its Applications, 311989CambridgeCambridge University Press10.1017/CBO9781139086578 LaczkovichMSzékelyhidiLSpectral synthesis on discrete Abelian groupsProc. Camb. Phil. Soc.2007143103120234097810.1017/S0305004107000114 LairdPGOn characterizations of exponential polynomialsPac. J. Math.197980250350753943110.2140/pjm.1979.80.503 Levi-CivitaTSulle funzioni che ammettono una formula d’addizione del tipo f(x+y)=∑i=1nXi(x)Yj(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x + y) = \sum _{i=1}^n X_i (x) Y_j (y)$$\end{document}Atti Accad. Nad. Lincei Rend.191322518118344.0502.03 Stone, J.J.: Exponential Polynomials on Commutative Semigroups, Appl. Math. and Stat. Lab. Technical Note No. 14, Stanford University (1960) SzékelyhidiLA characterization of exponential polynomialsPubl. Math. Debr.2013834757771315084110.5486/PMD.2013.5768 AnselonePMKorevaarJTranslation invariant subspaces of finite dimensionProc. Am. Math. Soc.19641574775216904810.1090/S0002-9939-1964-0169048-7 McKiernanMAEquations of the form H(x∘y)=∑ifi(x)gi(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(x\circ y) =\sum _i f_i (x)g_i (y)$$\end{document}Aequ. Math.197716515810.1007/BF01836418 SzékelyhidiLConvolution Type Functional Equations on Topological Abelian Groups1991SingaporeWorld Scientific10.1142/1406 LelandKOFinite dimensional translation invariant spacesAm. Math. Mon.19687575775823427110.2307/2315197 AlmiraJMShulmanEVOn certain generalizations of the Levi-Civita and Wilson functional equationsAequ. Math.2017915921931369717710.1007/s00010-017-0489-4 DjokovićDŽA representation theorem for (X1-1)(X2-1)…(Xn-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X_1 -1)(X_2 -1) \ldots (X_n -1)$$\end{document} and its applicationsAnn. Polon. Math.19692218919826579810.4064/ap-22-2-189-198 J Aczél (686_CR1) 1989 DŽ Djoković (686_CR4) 1969; 22 686_CR13 MA McKiernan (686_CR11) 1977; 16 L Székelyhidi (686_CR16) 2013; 83 PM Anselone (686_CR3) 1964; 15 L Schwartz (686_CR12) 1947; 48 PG Laird (686_CR8) 1979; 80 KO Leland (686_CR9) 1968; 75 M Engert (686_CR5) 1970; 32 M Laczkovich (686_CR7) 2007; 143 M Laczkovich (686_CR6) 2014; 142 T Levi-Civita (686_CR10) 1913; 22 L Székelyhidi (686_CR14) 1982; 103 JM Almira (686_CR2) 2017; 91 L Székelyhidi (686_CR15) 1991
References_xml	– reference: AnselonePMKorevaarJTranslation invariant subspaces of finite dimensionProc. Am. Math. Soc.19641574775216904810.1090/S0002-9939-1964-0169048-7 – reference: SzékelyhidiLConvolution Type Functional Equations on Topological Abelian Groups1991SingaporeWorld Scientific10.1142/1406 – reference: DjokovićDŽA representation theorem for (X1-1)(X2-1)…(Xn-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X_1 -1)(X_2 -1) \ldots (X_n -1)$$\end{document} and its applicationsAnn. Polon. Math.19692218919826579810.4064/ap-22-2-189-198 – reference: Stone, J.J.: Exponential Polynomials on Commutative Semigroups, Appl. Math. and Stat. Lab. Technical Note No. 14, Stanford University (1960) – reference: Levi-CivitaTSulle funzioni che ammettono una formula d’addizione del tipo f(x+y)=∑i=1nXi(x)Yj(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x + y) = \sum _{i=1}^n X_i (x) Y_j (y)$$\end{document}Atti Accad. Nad. Lincei Rend.191322518118344.0502.03 – reference: EngertMFinite dimensional translation invariant subspacesPac. J. Math.19703233334327466010.2140/pjm.1970.32.333 – reference: SchwartzLThéorie génerale des fonctions moyenne-périodiquesAnn. Math.19474848579292394810.2307/1969386 – reference: SzékelyhidiLNote on exponential polynomialsPac. J. Math.198210358358770525110.2140/pjm.1982.103.583 – reference: LelandKOFinite dimensional translation invariant spacesAm. Math. Mon.19687575775823427110.2307/2315197 – reference: LaczkovichMLocal spectral synthesis on Abelian groupsActa Math. Hung.2014142313329323353510.1007/s10474-013-0385-x – reference: AczélJDhombresJFunctional Equations in Several Variables. Encyclopedia of Mathematics and its Applications, 311989CambridgeCambridge University Press10.1017/CBO9781139086578 – reference: AlmiraJMShulmanEVOn certain generalizations of the Levi-Civita and Wilson functional equationsAequ. Math.2017915921931369717710.1007/s00010-017-0489-4 – reference: McKiernanMAEquations of the form H(x∘y)=∑ifi(x)gi(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(x\circ y) =\sum _i f_i (x)g_i (y)$$\end{document}Aequ. Math.197716515810.1007/BF01836418 – reference: LaczkovichMSzékelyhidiLSpectral synthesis on discrete Abelian groupsProc. Camb. Phil. Soc.2007143103120234097810.1017/S0305004107000114 – reference: LairdPGOn characterizations of exponential polynomialsPac. J. Math.197980250350753943110.2140/pjm.1979.80.503 – reference: SzékelyhidiLA characterization of exponential polynomialsPubl. Math. Debr.2013834757771315084110.5486/PMD.2013.5768 – ident: 686_CR13 – volume-title: Functional Equations in Several Variables. Encyclopedia of Mathematics and its Applications, 31 year: 1989 ident: 686_CR1 doi: 10.1017/CBO9781139086578 – volume: 103 start-page: 583 year: 1982 ident: 686_CR14 publication-title: Pac. J. Math. doi: 10.2140/pjm.1982.103.583 – volume: 91 start-page: 921 issue: 5 year: 2017 ident: 686_CR2 publication-title: Aequ. Math. doi: 10.1007/s00010-017-0489-4 – volume: 80 start-page: 503 issue: 2 year: 1979 ident: 686_CR8 publication-title: Pac. J. Math. doi: 10.2140/pjm.1979.80.503 – volume: 48 start-page: 857 issue: 4 year: 1947 ident: 686_CR12 publication-title: Ann. Math. doi: 10.2307/1969386 – volume: 75 start-page: 757 year: 1968 ident: 686_CR9 publication-title: Am. Math. Mon. doi: 10.2307/2315197 – volume: 22 start-page: 181 issue: 5 year: 1913 ident: 686_CR10 publication-title: Atti Accad. Nad. Lincei Rend. – volume: 22 start-page: 189 year: 1969 ident: 686_CR4 publication-title: Ann. Polon. Math. doi: 10.4064/ap-22-2-189-198 – volume-title: Convolution Type Functional Equations on Topological Abelian Groups year: 1991 ident: 686_CR15 doi: 10.1142/1406 – volume: 83 start-page: 757 issue: 4 year: 2013 ident: 686_CR16 publication-title: Publ. Math. Debr. doi: 10.5486/PMD.2013.5768 – volume: 32 start-page: 333 year: 1970 ident: 686_CR5 publication-title: Pac. J. Math. doi: 10.2140/pjm.1970.32.333 – volume: 142 start-page: 313 year: 2014 ident: 686_CR6 publication-title: Acta Math. Hung. doi: 10.1007/s10474-013-0385-x – volume: 16 start-page: 51 year: 1977 ident: 686_CR11 publication-title: Aequ. Math. doi: 10.1007/BF01836418 – volume: 143 start-page: 103 year: 2007 ident: 686_CR7 publication-title: Proc. Camb. Phil. Soc. doi: 10.1017/S0305004107000114 – volume: 15 start-page: 747 year: 1964 ident: 686_CR3 publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-1964-0169048-7
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Snippet	Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A ), if V is a finite dimensional translation invariant linear... Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space... Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space...
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SubjectTerms	Analysis Combinatorics Continuity (mathematics) Functions (mathematics) Mathematical analysis Mathematics Mathematics and Statistics Polynomials Theorems Topology
Title	The Levi-Civita equation in function classes
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