This work is concerned with estimating the upper envelopes S* of the absolute values of the partial sums of rearranged trigonometric sums.A.M. Garsia [Annals of Math. 79 (1964), 634-9] gave an estimate for the L2 norms of the S*, averaged over all rearrangements of the original (finite) sum.This estimate enabled him to prove that the Fourier series of any function in L2 can be rearranged so that it converges a.e.The main result of this thesis is a similar estimate of the Lq norms of the S*, for all even integers q.This holds for finite linear combinations of functions which satisfy a condition which is a generalization of orthonormality in the L2 case.This estimate for finite sums is extended to Fourier series of Lq functions; it is shown that there are functions to which the Men’shov-Paley Theorem does not apply, but whose Fourier series can nevertheless be rearranged so that the S* of the rearranged series is in Lq.
