Abstract
AbstractSeveral 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.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Mathematics (miscellaneous)