A new approach to factorization of a class of almost-periodic triangular symbols and related Riemann–Hilbert problems

Abstract

The factorization of almost-periodic triangular symbols, G, associated to finite-interval convolution operators is studied for two classes of operators whose Fourier symbols are almost periodic polynomials with spectrum in the group α Z + β Z + Z ( α , β ∈ ] 0 , 1 [ , α + β > 1 , α / β ∉ Q ). The factorization problem is solved by a method that is based on the calculation of one solution of the Riemann–Hilbert problem G Φ + = Φ − in L ∞ ( R ) and does not require solving the associated corona problems since a second linearly independent solution is obtained by means of an appropriate transformation on the space of solutions to the Riemann–Hilbert problem. Some unexpected, but interesting, results are obtained concerning the Fourier spectrum of the solutions of G Φ + = Φ − . In particular it is shown that a solution exists with Fourier spectrum in the additive group α Z + β Z whether this group contains Z or not. Possible application of the method to more general classes of symbols is considered in the last section of the paper.