Functional equations for exponential polynomials
Abstract: Abstract. The aim of the present paper is to describe some properties of functions with finite dimensional difference spaces by means of spectral analysis and spectral synthesis. We are going to apply these results to a version of a Levi-Civita functional equation, which has been recently studied by J. M. Almira and E. Shulman.Mathematics Subject Classification. 39B99, 43A62, 20N20.
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Several authors studied the so called exponential polynomials, characterized as solutions to the equation $$\begin{aligned} f(x+y) = \sum _{k=1}^nu_i(x)v_i(y). \end{aligned}$$ f ( x + y ) = ∑ k = 1 n u i ( x ) v i ( y ) . In the present paper we deal with a more general equation $$\begin{aligned} \sum _{j=1}^M P_j(x,y) f_j(a_jx + c_j y) = \sum _{k=1}^n u_k(x)v_k(y). \end{aligned}$$ ∑ j = 1 M P j ( x , y ) f j ( a j x + c j y ) = ∑ k = 1 n u k ( x ) v k ( y ) . Here all $$f_j, \ u_k, \ v_k$$ f j , u k , v k are assumed to be unknown scalar functions on $${\mathbb {R}}^d,$$ R d , while $$P_j$$ P j are polynomials. We prove that $$f_j$$ f j are ratios of exponential polynomials and polynomials, or sums of exponential functions multiplied by rational functions.
Several authors studied the so called exponential polynomials, characterized as solutions to the equation $$\begin{aligned} f(x+y) = \sum _{k=1}^nu_i(x)v_i(y). \end{aligned}$$ f ( x + y ) = ∑ k = 1 n u i ( x ) v i ( y ) . In the present paper we deal with a more general equation $$\begin{aligned} \sum _{j=1}^M P_j(x,y) f_j(a_jx + c_j y) = \sum _{k=1}^n u_k(x)v_k(y). \end{aligned}$$ ∑ j = 1 M P j ( x , y ) f j ( a j x + c j y ) = ∑ k = 1 n u k ( x ) v k ( y ) . Here all $$f_j, \ u_k, \ v_k$$ f j , u k , v k are assumed to be unknown scalar functions on $${\mathbb {R}}^d,$$ R d , while $$P_j$$ P j are polynomials. We prove that $$f_j$$ f j are ratios of exponential polynomials and polynomials, or sums of exponential functions multiplied by rational functions.
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