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Several authors studied the so called exponential polynomials, characterized as solutions to the equation f (x + y) = ∑ k = 1 n u i (x) v i (y). In the present paper we deal with a more general equation ∑ 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 are assumed to be unknown scalar functions on R d , while P j are polynomials. We prove that f j are ratios of exponential polynomials and polynomials, or sums of exponential functions multiplied by rational functions. [ABSTRACT FROM AUTHOR]

Applying medial limits we describe bounded solutions φ : S → R of the functional equation φ (x) = ∫ Ω g (ω) φ (f (x , ω)) d μ (ω) + G (x) , where (Ω , A , μ) is a measure space, S ⊂ R , f : S × Ω → S , g : Ω → R is integrable and G : S → R is bounded. The main purpose of this paper is to extend results obtained in Morawiec (Results Math 75(3):102, 2020) to the above general functional equation in wider classes of functions and under weaker assumptions. [ABSTRACT FROM AUTHOR]

Assume that (Ω , A , P) is a probability space, f : [ 0 , 1 ] × Ω → [ 0 , 1 ] is a function such that f (0 , ω) = 0 , f (1 , ω) = 1 for every ω ∈ Ω , g : [ 0 , 1 ] → R is a bounded function such that g (0) = g (1) = 0 , and a , b ∈ R . Applying medial limits we describe bounded solutions φ : [ 0 , 1 ] → R of the equation φ (x) = ∫ Ω φ (f (x , ω)) d P (ω) + g (x) satisfying the boundary conditions φ (0) = a and φ (1) = b . [ABSTRACT FROM AUTHOR]

A few misprints are corrected. [ABSTRACT FROM AUTHOR]