TY - JOUR
T1 - Exponential polynomials and the sine addition law on magmas
AU - Stetkær, Henrik
N1 - Publisher Copyright:
© 2023, The Author(s).
PY - 2023/12
Y1 - 2023/12
N2 - For any set X we let F(X) denote the complex vector space of functions f: X→ C . Let X= S be a magma, and let V be a subspace of F(S) , which is invariant under left or right translations. It is known for an abelian group S that if p1χ1, ⋯ , pnχn∈ F(S) are nonzero exponential polynomials with distinct exponentials χ1, ⋯ , χn then p1χ1+ ⋯ + pnχn∈ V⇒ p1χ1, ⋯ , pnχn∈ V and χ1, ⋯ , χn∈ V . We extend this to magmas. Our results imply that any exponential polynomial solution f∈ F(S) of f(xy) = f(x) χ(y) + χ(x) f(y) where χ∈ F(S) is an exponential, has the form f= aχ where a∈ F(S) is additive, even when χ has zeros.
AB - For any set X we let F(X) denote the complex vector space of functions f: X→ C . Let X= S be a magma, and let V be a subspace of F(S) , which is invariant under left or right translations. It is known for an abelian group S that if p1χ1, ⋯ , pnχn∈ F(S) are nonzero exponential polynomials with distinct exponentials χ1, ⋯ , χn then p1χ1+ ⋯ + pnχn∈ V⇒ p1χ1, ⋯ , pnχn∈ V and χ1, ⋯ , χn∈ V . We extend this to magmas. Our results imply that any exponential polynomial solution f∈ F(S) of f(xy) = f(x) χ(y) + χ(x) f(y) where χ∈ F(S) is an exponential, has the form f= aχ where a∈ F(S) is additive, even when χ has zeros.
KW - Exponential
KW - Exponential polynomial
KW - Levi–Civita
KW - Magma
KW - Sine addition law
UR - http://www.scopus.com/inward/record.url?scp=85161722424&partnerID=8YFLogxK
U2 - 10.1007/s00010-023-00965-y
DO - 10.1007/s00010-023-00965-y
M3 - Journal article
AN - SCOPUS:85161722424
SN - 0001-9054
VL - 97
SP - 963
EP - 979
JO - Aequationes Mathematicae
JF - Aequationes Mathematicae
IS - 5-6
ER -