Generalized hyperbolic functions, circulant matrices and functional equations

There is a contrast between the two sets of functional equations f 0 ( x + y ) = f 0 ( x ) f 0 ( y ) + f 1 ( x ) f 1 ( y ) , f 1 ( x + y ) = f 1 ( x ) f 0 ( y ) + f 0 ( x ) f 1 ( y ) , and f 0 ( x - y ) = f 0 ( x ) f 0 ( y ) - f 1 ( x ) f 1 ( y ) , f 1 ( x - y ) = f 1 ( x ) f 0 ( y ) - f 0 ( x ) f 1...

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Veröffentlicht in:	Linear algebra and its applications Jg. 406; S. 272 - 284
1. Verfasser:	Muldoon, Martin E.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York, NY Elsevier Inc 01.09.2005 Elsevier Science
Schlagworte:	Algebra Circulant matrices Exact sciences and technology Finite Fourier transform Functional equations Generalized exponential functions Generalized hyperbolic functions Linear and multilinear algebra, matrix theory Mathematics Sciences and techniques of general use Primary 39B30 Secondary 15A57 Generalized hyperbolic functions Functional equations Generalized exponential functions Circulant matrices Finite Fourier transform Fourier transformation Generalized function Functional equation Secondary 15A57 Generalized exponential functions Exponential function Hyperbolic equation Circulant matrix Primary
ISSN:	0024-3795, 1873-1856
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Abstract	There is a contrast between the two sets of functional equations f 0 ( x + y ) = f 0 ( x ) f 0 ( y ) + f 1 ( x ) f 1 ( y ) , f 1 ( x + y ) = f 1 ( x ) f 0 ( y ) + f 0 ( x ) f 1 ( y ) , and f 0 ( x - y ) = f 0 ( x ) f 0 ( y ) - f 1 ( x ) f 1 ( y ) , f 1 ( x - y ) = f 1 ( x ) f 0 ( y ) - f 0 ( x ) f 1 ( y ) satisfied by the even and odd components of a solution of f( x + y) = f( x) f( y). Schwaiger and, later, Förg-Rob and Schwaiger considered the extension of these ideas to the case where f is sum of n components. Here we shorten and simplify the statements and proofs of some of these results by a more systematic use of matrix notation.
AbstractList	There is a contrast between the two sets of functional equations f 0 ( x + y ) = f 0 ( x ) f 0 ( y ) + f 1 ( x ) f 1 ( y ) , f 1 ( x + y ) = f 1 ( x ) f 0 ( y ) + f 0 ( x ) f 1 ( y ) , and f 0 ( x - y ) = f 0 ( x ) f 0 ( y ) - f 1 ( x ) f 1 ( y ) , f 1 ( x - y ) = f 1 ( x ) f 0 ( y ) - f 0 ( x ) f 1 ( y ) satisfied by the even and odd components of a solution of f( x + y) = f( x) f( y). Schwaiger and, later, Förg-Rob and Schwaiger considered the extension of these ideas to the case where f is sum of n components. Here we shorten and simplify the statements and proofs of some of these results by a more systematic use of matrix notation.
Author	Muldoon, Martin E.
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Keywords	Primary 39B30 Secondary 15A57 Generalized hyperbolic functions Functional equations Generalized exponential functions Circulant matrices Finite Fourier transform Fourier transformation Generalized function Functional equation Secondary 15A57 Generalized exponential functions Exponential function Hyperbolic equation Circulant matrix Primary
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References	Battioni (bib4) 1969; 10 Erdélyi, Magnus, Oberhettinger, Tricomi (bib8) 1955.; vol. 3 Aczél, Dhombres (bib1) 1989. J. Schwaiger, Remark 15, Report on 28th International Symposium on Functional Equations, Aequationes Math. 41 (1991) 289–290 Ger (bib12) 1990; 64 Davis (bib7) 1994. Baker (bib2) 1980; 80 Forti (bib11) 1995; 50 Vietoris (bib19) 1944; 186 Förg-Rob, Schwaiger (bib10) 1994; 26 Muldoon, Ungar (bib14) 1996; 69 Ben Cheikh (bib6) 1998; 99 Hyers, Isac, Rassias (bib13) 1998.; vol. 34 Baker, Lawrence, Zorzitto (bib3) 1979; 74 Ungar (bib18) 1984; 15 Zorzitto (bib21) 1994; 48 Ben Cheikh (bib5) 1997; 52 Wilson (bib20) 1919–1920; 26 Schwaiger (bib16) 1992; 43 Ungar (bib17) 1982; 89 Förg-Rob, Schwaiger (bib9) 1993; 45 Ben Cheikh (10.1016/j.laa.2005.04.011_bib5) 1997; 52 Ger (10.1016/j.laa.2005.04.011_bib12) 1990; 64 Aczél (10.1016/j.laa.2005.04.011_bib1) 1989 Zorzitto (10.1016/j.laa.2005.04.011_bib21) 1994; 48 Schwaiger (10.1016/j.laa.2005.04.011_bib16) 1992; 43 10.1016/j.laa.2005.04.011_bib15 Battioni (10.1016/j.laa.2005.04.011_bib4) 1969; 10 Davis (10.1016/j.laa.2005.04.011_bib7) 1994 Förg-Rob (10.1016/j.laa.2005.04.011_bib9) 1993; 45 Hyers (10.1016/j.laa.2005.04.011_bib13) 1998; vol. 34 Wilson (10.1016/j.laa.2005.04.011_bib20) 1919; 26 Baker (10.1016/j.laa.2005.04.011_bib2) 1980; 80 Ben Cheikh (10.1016/j.laa.2005.04.011_bib6) 1998; 99 Erdélyi (10.1016/j.laa.2005.04.011_bib8) 1955; vol. 3 Vietoris (10.1016/j.laa.2005.04.011_bib19) 1944; 186 Baker (10.1016/j.laa.2005.04.011_bib3) 1979; 74 Ungar (10.1016/j.laa.2005.04.011_bib17) 1982; 89 Muldoon (10.1016/j.laa.2005.04.011_bib14) 1996; 69 Ungar (10.1016/j.laa.2005.04.011_bib18) 1984; 15 Forti (10.1016/j.laa.2005.04.011_bib11) 1995; 50 Förg-Rob (10.1016/j.laa.2005.04.011_bib10) 1994; 26
References_xml	– volume: 52 start-page: 365 year: 1997 end-page: 378 ident: bib5 article-title: Decomposition of the Bessel functions with respect to the cyclic group of order publication-title: Le Matematiche – volume: 43 start-page: 198 year: 1992 end-page: 210 ident: bib16 article-title: On generalized hyperbolic functions and their characterization by functional equations publication-title: Aequationes Math. – volume: 26 start-page: 300 year: 1919–1920 end-page: 312 ident: bib20 article-title: On certain related functional equations publication-title: Bull. Amer. Math. Soc. – volume: 26 start-page: 274 year: 1994 end-page: 280 ident: bib10 article-title: On the stability of some functional equations for generalized hyperbolic functions and for the generalized cosine equation publication-title: Results Math. – volume: 64 start-page: 75 year: 1990 end-page: 84 ident: bib12 article-title: Stability of addition formulae for trigonometric mappings publication-title: Zeszyty Nauk. Politech. Ślask. mat.—Fiz. – volume: 15 start-page: 301 year: 1984 end-page: 304 ident: bib18 article-title: Higher order alpha-hyperbolic functions publication-title: Indian J. Pure Appl. Math. – volume: 186 start-page: 1 year: 1944 end-page: 15 ident: bib19 article-title: Zur Kennzeichnung des Sinus und verwandter Funktionen durch Funktionalgleichungen publication-title: J. Reine Angew. Math. – volume: 10 start-page: 39 year: 1969 end-page: 48 ident: bib4 article-title: Su una generalizzazione delle funzioni iperboliche e delle funzioni circolari publication-title: Riv. Mat. Univ. Parma (2) – volume: 74 start-page: 242 year: 1979 end-page: 246 ident: bib3 article-title: The stability of the equation publication-title: Proc. Amer. Math. Soc. – reference: J. Schwaiger, Remark 15, Report on 28th International Symposium on Functional Equations, Aequationes Math. 41 (1991) 289–290 – volume: 99 start-page: 55 year: 1998 end-page: 66 ident: bib6 article-title: Decomposition of Laguerre polynomials with respect to the cyclic group of order publication-title: J. Comput. Appl. Math. – volume: 45 start-page: 285 year: 1993 end-page: 296 ident: bib9 article-title: On the stability of a system of functional equations characterizing generalized hyperbolic and trigonometric functions publication-title: Aequationes Math. – volume: 69 start-page: 3 year: 1996 end-page: 14 ident: bib14 article-title: Beyond sin and cos publication-title: Math. Mag. – year: 1989. ident: bib1 article-title: Functional Equations in Several Variables – volume: vol. 3 year: 1955. ident: bib8 publication-title: Higher Transcendental Functions – volume: vol. 34 year: 1998. ident: bib13 article-title: Stability of functional equations in several variables publication-title: Progress in Nonlinear Differential Equations and their Applications – year: 1994. ident: bib7 article-title: Circulant Matrices – volume: 50 start-page: 143 year: 1995 end-page: 190 ident: bib11 article-title: Hyers–Ulam stability of functional equations in several variables publication-title: Aequationes Math. – volume: 48 start-page: 294 year: 1994 end-page: 305 ident: bib21 article-title: Groups acting on circulant matrices publication-title: Aequationes Math. – volume: 80 start-page: 411 year: 1980 end-page: 416 ident: bib2 article-title: The stability of the cosine equation publication-title: Proc. Amer. Math. Soc. – volume: 89 start-page: 688 year: 1982 end-page: 691 ident: bib17 article-title: Generalized hyperbolic functions publication-title: Amer. Math. Monthly – volume: 69 start-page: 3 year: 1996 ident: 10.1016/j.laa.2005.04.011_bib14 article-title: Beyond sin and cos publication-title: Math. Mag. doi: 10.1080/0025570X.1996.11996374 – volume: 45 start-page: 285 year: 1993 ident: 10.1016/j.laa.2005.04.011_bib9 article-title: On the stability of a system of functional equations characterizing generalized hyperbolic and trigonometric functions publication-title: Aequationes Math. doi: 10.1007/BF01855886 – volume: 52 start-page: 365 year: 1997 ident: 10.1016/j.laa.2005.04.011_bib5 article-title: Decomposition of the Bessel functions with respect to the cyclic group of order n publication-title: Le Matematiche – year: 1994 ident: 10.1016/j.laa.2005.04.011_bib7 – volume: vol. 34 year: 1998 ident: 10.1016/j.laa.2005.04.011_bib13 article-title: Stability of functional equations in several variables – volume: 15 start-page: 301 year: 1984 ident: 10.1016/j.laa.2005.04.011_bib18 article-title: Higher order alpha-hyperbolic functions publication-title: Indian J. Pure Appl. Math. – volume: 186 start-page: 1 year: 1944 ident: 10.1016/j.laa.2005.04.011_bib19 article-title: Zur Kennzeichnung des Sinus und verwandter Funktionen durch Funktionalgleichungen publication-title: J. Reine Angew. Math. – year: 1989 ident: 10.1016/j.laa.2005.04.011_bib1 – volume: 80 start-page: 411 year: 1980 ident: 10.1016/j.laa.2005.04.011_bib2 article-title: The stability of the cosine equation publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-1980-0580995-3 – volume: 74 start-page: 242 year: 1979 ident: 10.1016/j.laa.2005.04.011_bib3 article-title: The stability of the equation f(x+y)=f(x)+f(y) publication-title: Proc. Amer. Math. Soc. – volume: 50 start-page: 143 year: 1995 ident: 10.1016/j.laa.2005.04.011_bib11 article-title: Hyers–Ulam stability of functional equations in several variables publication-title: Aequationes Math. doi: 10.1007/BF01831117 – volume: 64 start-page: 75 year: 1990 ident: 10.1016/j.laa.2005.04.011_bib12 article-title: Stability of addition formulae for trigonometric mappings publication-title: Zeszyty Nauk. Politech. Ślask. mat.—Fiz. – volume: 48 start-page: 294 year: 1994 ident: 10.1016/j.laa.2005.04.011_bib21 article-title: Groups acting on circulant matrices publication-title: Aequationes Math. doi: 10.1007/BF01832991 – volume: 26 start-page: 274 year: 1994 ident: 10.1016/j.laa.2005.04.011_bib10 article-title: On the stability of some functional equations for generalized hyperbolic functions and for the generalized cosine equation publication-title: Results Math. doi: 10.1007/BF03323049 – volume: 43 start-page: 198 year: 1992 ident: 10.1016/j.laa.2005.04.011_bib16 article-title: On generalized hyperbolic functions and their characterization by functional equations publication-title: Aequationes Math. doi: 10.1007/BF01835702 – volume: 10 start-page: 39 year: 1969 ident: 10.1016/j.laa.2005.04.011_bib4 article-title: Su una generalizzazione delle funzioni iperboliche e delle funzioni circolari publication-title: Riv. Mat. Univ. Parma (2) – volume: vol. 3 year: 1955 ident: 10.1016/j.laa.2005.04.011_bib8 – ident: 10.1016/j.laa.2005.04.011_bib15 – volume: 99 start-page: 55 year: 1998 ident: 10.1016/j.laa.2005.04.011_bib6 article-title: Decomposition of Laguerre polynomials with respect to the cyclic group of order n publication-title: J. Comput. Appl. Math. doi: 10.1016/S0377-0427(98)00145-9 – volume: 89 start-page: 688 year: 1982 ident: 10.1016/j.laa.2005.04.011_bib17 article-title: Generalized hyperbolic functions publication-title: Amer. Math. Monthly doi: 10.1080/00029890.1982.11995514 – volume: 26 start-page: 300 year: 1919 ident: 10.1016/j.laa.2005.04.011_bib20 article-title: On certain related functional equations publication-title: Bull. Amer. Math. Soc. doi: 10.1090/S0002-9904-1920-03310-0
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SubjectTerms	Algebra Circulant matrices Exact sciences and technology Finite Fourier transform Functional equations Generalized exponential functions Generalized hyperbolic functions Linear and multilinear algebra, matrix theory Mathematics Sciences and techniques of general use
Title	Generalized hyperbolic functions, circulant matrices and functional equations
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