Polynomial almost periodic solutions for a class of Riemann-Hilbert problems with triangular symbols

Let over(g, ̂) (ξ) = a e^{i α ξ} + b + c e^{- i β ξ} with α, β ∈] 0, 1 [ such that α + β < 1, α β^-1 ∉ Q and a, b, c ∈ C {set minus} {0}. In this paper the existence of almost-periodic polynomial (APP) solutions to the equation over(g, ̂) h⁺ = E l⁺ + l^- (with h⁺ ∈ H_∞⁺ ∩ E H_∞^- and l^± ∈ H_∞^±) is studied. The natural space in which to seek a solution to the above problem is the space of almost periodic functions with spectrum in the group α Z + β Z + Z. Due to the difficulty in dealing with the problem in that generality, solutions are sought with spectrum in the group α Z + β Z. Several interesting and totally new results are obtained. It is shown that, if 1 ∉ α Z + β Z, no polynomial solutions exist, i.e., almost periodic polynomial solutions exist only if α Z + β Z = α Z + β Z + Z. Keeping to this setting, it is shown that APP solutions exist if and only if the function over(g, ̂) satisfies the simple spectral condition α + β > 1 / 2. The proof of this result is nontrivial and has a number-theoretic flavour. Explicit formulas for the solution to the above problem are given in the final section of the paper. The derivation of these formulas is to some extent a byproduct of the proof of the result on the existence of APP solutions.