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Distributional analog of a functional equation. (English) Zbl 1049.39028

Let I:=(0,1), and let D(I) and D(I 2 ) denote the space of all infinitely differentiable functions with compact support on I and on I 2 , respectively. The symbol D ' (I) denotes the dual space of D(I).

Formulas

Q + [ϕ](x)= R ϕ(x-y,y)dy= I ϕ(x-y,y)dy,Q - [ϕ](x)= R ϕ(x+y,y)dy= I ϕ(x+y,y)dyandR[ϕ](x)= I ϕ(x·y,y)1 ydy

define linear operators Q + ,Q - and R from D(I 2 ) into D(I), whereas Q + * ,Q - * and R * denote their adjoint operators.

If f 1 ,f 2 and f 3 are locally integrable functions and every T i is the regular distribution corresponding to f i (i = 1, 2, 3) (this is written as T i =λ f i ) and

Q + * [T 1 ]+Q - * [T 2 ]+R * [T 3 ]=0,(1)

then

f 1 (x+y)+f 2 (x-y)+f 3 (xy)=0

almost everywhere on I 2 .

If T 1 ,T 2 ,T 3 D ' (I) satisfy equation (1), then they are of the form: T 1 =λ f 1 , T 2 =λ f 2 and T 3 =λ f 3 , where f 1 (x)=-γx 2 +α 2 , f 2 (x)=γx 2 +β 2 , f 3 =4γx+a for some real a,γ,α 2 and β 2 such that α 2 +β 2 +a=0.

MSC:
39B52Functional equations for functions with more general domains and/or ranges
46F10Operations with distributions (generalized functions)