(PHD, 1966)

Abstract:

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 L_{2} 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 L _{2} can be rearranged so that it converges a.e. The main result of this thesis is a similar estimate of the L_{q} 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 L

]]>

(PHD, 1965)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

Let […] be a measure space, and T a positive contraction of […]. Let […] be a sequence of non-negative numbers whose sum is one, and […] a sequence defined by inductions as follows […]. Now let […], then we prove in this work that […] exists almost everywhere in the set […]. When […] we get that all […]. In this case (*) yields the abelian analog of the well-known ergodic theorem of Chacon-Ornstein dealing with the convergence of averages of the form […] whose proof we have generalized and adapted to show the convergence of […]. We have also considered the generalization of (**) to weighted averages […] whose convergence in […] was recently proved by G. E. Baxter. We have given a considerably simpler proof for this fact.
]]>

(PHD, 1964)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

Let {T[subscript n]} be a sequence of continuous linear transformations on Lsuperscript p for finite measure space X and 1<=p<=2. Assume further that lim T[subscript n]f(x) exists a.e. for all f(x) in Lsuperscript p. Then, under the added assumptions that X is a compact group or homogeneous space and that each operator T[subscript n] commutes with translations on X, E.M.Stein was able to prove the existence of a constant […] such that […] for all f(x) in Lsuperscript p and A > 0. The first result of this paper is to prove (1) from convergence under the weaker assumption that the sequence {T[subscript n]} commutes with each member of a family of measure-preserving transformations on X, a family which is large enough to have only trivial fixed sets. This result contains Stein’s theorem, concludes maximal ergodic theorems from individual ergodic theorems, and applies in situations arising in probability theory.

The conditions above are then weakened so that the domain of {T[subscript n]} becomes an F-space of functions satisfying a certain concordance condition on its topology, and the operators {T[subscript n]} become continuous in measure with range in the space of measurable functions on X. Then, under the assumption that {T[subscript n]} commutes with enough measure-preserving transformations as above, a slightly weaker version of (1) is concluded.

Now, assume that {T[subscript n]} is a sequence of continuous-in-measure linear transformations of an abstract F-space […] into measurable functions on finite measure space X, and that […] for every f in a dense subset of E. A decomposition of the measure space X is then obtained, such that […] on one of the sets for all f in E, and such that for all f in the complement of a set of the first category in E, […] a.e. on the other set of the decomposition. A, theorem of Banach then applies on the first set to give a result which can be viewed as similar to (1). The decomposition is then applied to the preceeding results to prove new theorems
]]>