Browsing by Author "O'Neil, Richard"

Now showing 1 - 3 of 3
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    Metric function spaces and reflected spaces
    (1969) Gerber, Brian Paul; O'Neil, Richard
    In this paper we first define what is meant by the term metric function space. Basically, a metric function space consists of a set of functions F and a metric p on F which satisfies certain axioms. For example, the Lp spaces and the L(p, q) spaces are metric function spaces. For certain metric function spaces we can form what we will call the reflected space. Theorem 12 states that the reflected space to a metric function space is itself a metric function space. Theorem 13 shows that the reflected space to the reflected space of a metric function space is the original space. Theorem 14 gives a relation between a metric function space and its reflected space, namely, that a metric function space is absolutely continuous if and only if its reflected space has the truncation property.
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    Some compact operators on Orlicz spaces
    (1969) Chang, Shu-Ya; O'Neil, Richard
    Let f be a kernel in the Orlicz space LA. What is the necessary and sufficient condition on the Young's functions A, B, C so that the operator h(x) = S f(x-t)g(t) dt be compact from LB into LC? It is shown that the problem is impossible on the real line, or more generally, on a locally compact, commutative, connected but not compact group. If the group is compact, it is proved that the problem is possible, and the necessary and sufficient condition is that for every 0 > 0, there exists a number n > 0 such that for all x >/ 1: A-1 (x) B-1 (nx) /< onxc-1 (nx)
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    Some relations among Orlicz spaces
    (1965) Jodeit, Max August; O'Neil, Richard
    In this paper it is shown that for the study of Orlicz spaces the condition that a Young's function A be convex can be replaced by the more general (and more convenient) condition that A(x)/x be non-decreasing. Some properties of the lattice of Orlicz spaces ordered by inclusion are given. belong to Lc whenever f E LA, g E LB. The condition is also sufficient when the convolution is formed over the integers or (0,2pi]. It is proven here that the condition is also necessary; all triplets A, B, C of Y-functions for which the condition holds are determined.