The confirmatory hypothesis is that subjects with resistant hypertension and OSA will show a greater degree of blood pressure reduction after 6 weeks of treatment with fixed CPAP in comparison to those treated with APAP. Explanatory and exploratory hypotheses are that a greater reduction in arterial stiffness will be noted when compared with the benefit in blood pressure, that reductions in arterial stiffness will be associated with benefits to serum measurements of inflammatory markers and glucose regulation, and that control of sleep apnea and the attendant improvement in sleep quality will be better with fixed CPAP than APAP.