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

摘要

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 alpha Z + beta Z + Z (alpha, beta epsilon ]0, 1[, alpha + beta > 1, alpha/,beta Q). The factorization problem is solved by a method that is based on the calculation of one solution of the Riemann-Hilbert problem G phi(+) = phi_ in L-infinity (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 phi(+) = phi_ In particular it is shown that a solution exists with Fourier spectrum in the additive group alpha Z + beta 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. (c) 2005 Elsevier Inc. All rights reserved.