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

Let ˆg(ξ) = aeiαξ + b + ce−iβξ with α,β ∈ ]0, 1[ such that α +β <1, αβ−1 /∈ Q and a, b, c ∈ C \ {0}. In this paper the existence of almost-periodic polynomial (APP) solutions to the equation ˆ gh+ = El+ +l− (with h+ ∈ H+∞ ∩ EH−∞ 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 αZ + βZ + Z. Due to the difficulty in dealing with the problem in that generality, solutions are sought with spectrum in the group αZ + βZ. Several interesting and totally new results are obtained. It is shown that, if 1 /∈ αZ + βZ, no polynomial solutions exist, i.e., almost periodic polynomial solutions exist only if αZ+βZ = αZ+βZ+Z. Keeping to this setting, it is shown that APP solutions exist if and only if the function ˆ 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. © 2006 Elsevier Inc. All rights reserved.