Exponential polynomials and the sine addition law on magmas

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 -