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

AbstractLet gˆ(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 gˆh+=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 gˆ 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