A variant of d'Alembert's and Wilson's functional equations for matrix valued functions
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Given $M$ a monoid with a neutral element $e$. We show that the solutions of d'Alembert's functional equation for $n\times n$ matrices \begin{equation*} \Phi(pr,qs)+\Phi(sp,rq)=2\Phi(r,s)\Phi(p,q),\quad p,q,r,s\in M \end{equation*} are abelian. Furthermore, we prove under additional assumption that the solutions of the n-dimensional mixed vector-matrix Wilson's functional equation \begin{equation*} \left\lbrace\begin{array}{ll} f(pr,qs)+f(sp,rq)=2\Phi(r,s)f(p,q),\\ \Phi(p,q)=\Phi(q,p),\quad p,q,r,s\in M \end{array}\right. \end{equation*} are abelian. As an application we solve the first functional equation on groups for the particular case of $n=3$.
Abstract
Given $M$ a monoid with a neutral element $e$. We show that the solutions of d'Alembert's functional equation for $n\times n$ matrices \begin{equation*} \Phi(pr,qs)+\Phi(sp,rq)=2\Phi(r,s)\Phi(p,q),\quad p,q,r,s\in M \end{equation*} are abelian. Furthermore, we prove under additional assumption that the solutions of the n-dimensional mixed vector-matrix Wilson's functional equation \begin{equation*} \left\lbrace\begin{array}{ll} f(pr,qs)+f(sp,rq)=2\Phi(r,s)f(p,q),\\ \Phi(p,q)=\Phi(q,p),\quad p,q,r,s\in M \end{array}\right. \end{equation*} are abelian. As an application we solve the first functional equation on groups for the particular case of $n=3$.
참고문헌 (17)
- J. A. Baker, "Cosine, exponential and quadratic functions", Glas. Mat. Ser. III, 16(36), pp.269~274 (1981)
- B. Bouikhalene, "Inequality Theory and Applications", pp.6~ (2010), Novapublishers
- A. Chahbi, "A variant of d’Alembert’s and Wilson’s functional equations for 2 × 2 matrix valued functions", J. Math. Anal. Appl, 501(2), pp.18~ (2021)[DOI]
- W. Chojnacki, "Fonctions cosinus hilbertiennes bornées dans les groupes commutatifs localement compacts", Compos. Mat, 57(1), pp.15~60 (1986)
- W. Chojnacki, "Group representations of bounded cosine functions", J. Reine Angew. Math, 478, pp.61~84 (1996)[DOI]
- J. K. Chung, "Measures of distance between probability distributions", J. Math. Anal. Appl, 138(1), pp.280~292 (1989)[DOI]
- H. H. Elfen, "A variant of the quadratic functional equation on groups and an application", Bull. Korean Math. Soc, 54(6), pp.2165~2182 (2017)[DOI]
- H. O. Fattorini, "Uniformly bounded cosine functions in Hilbert space", Indiana Univ. Math. J, 20, pp.411~425 (1970)[DOI]
- J. Kisyński, "On operator-valued solutions of d’Alembert’s functional equation. I", Colloq. Math, 23, pp.107~114 (1971)[DOI]
- J. Kisyński, "On operator-valued solutions of d’Alembert’s functional equation. II", Studia Math, 42, pp.43~66 (1972)[DOI]
- S. Kurepa, "A cosine functional equation in Banach algebras", Acta Sci. Math. (Szeged), 23, pp.255~267 (1962)
- P. Sinopoulos, "Wilson’s functional equation for vector and matrix functions", Proc. Amer. Math. Soc, 125(4), pp.1089~1094 (1997)[DOI]
- P. Sinopoulos, "Wilson’s functional equation in dimension 3", Aequationes Math, 66(1-2), pp.164~179 (2003)[DOI]
- H. Stetkær, "Functional equations on abelian groups with involution", Aequationes Math, 54, pp.144~172 (1997)[DOI]
- H. Stetkær, "D’Alembert’s and Wilson’s functional equations for vector and 2×2 matrix valued functions", Math. Scand, 87(1), pp.115~132 (2000)[DOI]
- H. Stetkær, "A variant of d’Alembert’s functional equation", Aequationes Math, 89(3), pp.657~662 (2015)[DOI]
- L. Székelyhidi, "Functional equations on abelian groups", Acta Math. Acad. Sci. Hungar, 37(1-3), pp.235~243 (1981)[DOI]