D'Alembert's and Wilson's functional equations for vector and 2×22\times 22×2 matrix valued functions (original) (raw)
Wilson's functional equation on C
Aequationes Mathematicae, 1997
We find the complete set of continuous solutions f, g of "Wilson's functional equation" E~ f (x + w'y) = Nf(x)g(y), x, y • C, given a primitive N th root w of unity. Disregarding the trivial solution f = 0 and g any complex function, it is known that g satisfies a version of d'Alembert's functional equation and so has the form g(z) = g~(z) = N-1 E,,JV=-o] E~,(w"z) for some p • C z. Here E~,~,,2)(x + iy) = exp(plx +#zY). For fixed g = g~ the space of solutions f of Wilson's functional equation can be decomposed into the N isotypic subspaces for the action of Z N on the continuous functions on C. We prove that the r th component, where r•{0, 1 ,. .. , N-l } , of any solution satisfies the signed functional equation N-1 E,= 0 f (x + w"y)w'r= Ng(x)f(y), x,y ~ C. We compute the solution spaces of each of these signed equations: They are l-dimensional and spanned by z ~ Effyd w"~E~,(w"z), except for g = 1 and r ¢ 0 where they are spanned by U and z N ~. Adding the components we get the solution of Wilson's equation. Analogous results are obtained with the action of Z N on C replaced by that of SO(2). The case of g = 0 in the signed equations is special and solved separately both for Z u and SO(2). (4) for each r e Z (Theorem IV.2 and in the exceptional case of g = 0 Theorem V.3), and use the result to show, again by a general decomposition theorem (Theorem II.2), that the solutions of Wilson's functional equation 1 ('2~ ~-I f(x + ei°y) dO =f(x)g(y), ZgJo x, yeC, (5) are superpositions of the solutions of the corresponding signed equations (4), when g # 0 (Theorem IV.3).
D'Alembert's functional equation on groups
Banach Center Publications, 2013
Given a (not necessarily unitary) character µ : G → (C \ {0}, •) of a group G we find the solutions g : G → C of the following version of d'Alembert's functional equation g(xy) + µ(y)g(xy −1) = 2g(x)g(y), x, y ∈ G.
On Wilson’s functional equations
Aequationes mathematicae, 2014
We find on a group G the solutions f, g : G → C of the functional equation f (xy) + f (y −1 x) = 2f (x)g(y), x, y ∈ G, in terms of characters, additive maps and matrix-elements of irreducible, 2-dimensional representations of G.
On a generalized functional equation of Abel
Publicationes Mathematicae, 2002
We present some results concerning the following generalization of a functional equation of Abel ψ (xf (y) + yg(x)) = ϕ(x) + ϕ(y). With f = g we get the original Abel's equation that was mentioned explixitly by D. Hilbert in the second part of his fifth problem. The present generalization implies many applications in the theory of functional equations, particularly those dealing with determination of parametrized subsemigroups. We solve the equation in the class of continuous real functions defined in an interval containing 0.
Functional equations and matrix-valued spherical functions
Aequationes mathematicae, 2005
We show that any solution of a certain functional equation, generalizing such equations as Wilson's, Levi-Cività's, Gajda's and the sine addition equation, can be expressed in terms of a matrix-valued spherical function. We compute this matrix-valued function for solutions of various functional equations, and use it to solve the equations. . 39B32 and 43A90.
Aequationes mathematicae
Let G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We find the solutions f,g,h \in C(G)f,g,h∈C(G)oftheLevi−Civitaequationf , g , h ∈ C ( G ) of the Levi-Civita equationf,g,h∈C(G)oftheLevi−Civitaequation\begin{aligned} f(xy) = f(x)h(y) + g(x)f(y), \ x,y \in G, \end{aligned}f(xy)=f(x)h(y)+g(x)f(y),x,y∈G,whichisanextensionofthesineadditionlaw.RepresentationsofGonf ( x y ) = f ( x ) h ( y ) + g ( x ) f ( y ) , x , y ∈ G , which is an extension of the sine addition law. Representations of G onf(xy)=f(x)h(y)+g(x)f(y),x,y∈G,whichisanextensionofthesineadditionlaw.RepresentationsofGon\mathbb {C}^2C2playanimportantrole.AsacorollarywegetthesolutionsC 2 play an important role. As a corollary we get the solutionsC2playanimportantrole.Asacorollarywegetthesolutionsf,g \in C(G)f,g∈C(G)ofthesinesubtractionlawf , g ∈ C ( G ) of the sine subtraction lawf,g∈C(G)ofthesinesubtractionlawf(xy^*) = f(x)g(y) - g(x)f(y)f(xy∗)=f(x)g(y)−g(x)f(y),f ( x y ∗ ) = f ( x ) g ( y ) - g ( x ) f ( y ) ,f(xy∗)=f(x)g(y)−g(x)f(y),x,y \in Gx,y∈G,inwhichx , y ∈ G , in whichx,y∈G,inwhichx \mapsto x^*x↦x∗isacontinuousinvolution,meaningthatx ↦ x ∗ is a continuous involution, meaning thatx↦x∗isacontinuousinvolution,meaningthat(xy)^* = y^*x^*(xy)∗=y∗x∗and( x y ) ∗ = y ∗ x ∗ and(xy)∗=y∗x∗andx^{**} = xx∗∗=xforallx ∗ ∗ = x for allx∗∗=xforallx,y \in G$$ x , y ∈ G .