2016/02/22 by Belaid Bouikhalene, Bouikhalene Belaid, Belaid, Bouikhalene +3
Mathematics · #39B32 #39B52 #39B82 #Advanced Operator Algebra Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Mathematical and Theoretical Analysis #advanced mathematical theories #math.CA #math.FA #msc:39B32 #msc:39B52 #msc:39B82
paper · pdf · doi:10.48550/arxiv.1603.02064
19 pages
arxiv created 2016/02/22 · openalex publication_date 2016/02/22 · arxiv updated 2016/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a locally compact group, and let K be a compact subgroup of G. Let μ: G\longrightarrowℂ\backslash\0\ be a character of G. In this paper, we deal with the integral equations Wμ(K): ∫Kf(xkyk-1)dk+μ(y)∫Kf(xky-1k-1)dk=2f(x)g(y), and Dμ(K): ∫Kf(xkyk-1)dk+μ(y)∫Kf(xky-1k-1)dk=2f(x)f(y) for all x, y∈ G where f, g: G\longrightarrow ℂ, to be determined, are complex continuous functions on G. When K⊂ Z(G), the center of G, Dμ(K) reduces to the new version of d'Almbert's functional equation f(xy)+μ(y)f(xy-1)=2f(x)f(y), recently studied by Davison [18] and Stetkær [35]. We derive the following link between the solutions of Wμ(K) and Dμ(K) in the following way : If (f,g) is a solution of equation Wμ(K) such that CKf=∫Kf(kxk-1)dωK(k)≠ 0 then g is a solution of Dμ(K). This result is used to establish the superstability problem of Wμ(K). In the case where (G,K) is a central pair, we show that the solutions are expressed by means of K-spherical functions and related functions. Also we give explicit formulas of solutions of Dμ(K) in terms of irreducible representations of G. These formulas generalize Euler's formula cos(x)=\fraceix+e-ix2 on G=ℝ.