Richard B Darst

Best L1-approximation of bounded, approximately continuous functions on [0, 1] by nondecreasing functions

Abstract Let Ω denote the closed interval [0, 1] and let bA denote the set of all bounded, approximately continuous functions on Ω. Let f ϵ bA . It is shown that f has an (essentially) unique best L 1 -approximation f 1 by nondecreasing functions; f 1 is shown to be continuous. For 1 p -approximations f p are shown to be continuous, and they are shown to converge uniformly to f 1 as p → 1. A characterization of f 1 is given. It is also shown that if f n ϵ bA , 0 ≦ n n converges to f 0 in L 1 as n → ∞, then f 1 n → f 1 0 in L 1 as n → ∞.

Approximation of continuous and quasi-continuous functions by monotone functions

Let Q denote the Banach space (sup norm) of quasi-continuous functions defined on the interval [0, 1]. Let C denote the subspace comprised of continuous functions. Met M denote the closed convex cone in Q comprised of nondecreasing functions. For f ϵ Q and 1 < p < ∞, let fp denote the best Lp-approximation to f by elements of M. It is shown that fp converges uniformly as p → ∞ to a best L∞-approximation to f by elements of M. If f ϵ C, then each fp ϵ C; so f∞ ϵ C.

Hausdorff dimension of sets of non-differentiability points of Cantor functions

Each number a in the segment (0, ½) produces a Cantor set, C a , by putting b = 1 − 2 a and recursively removing segments of relative length b from the centres of the interval [0, 1] and the intervals which are subsequently generated. The distribution function of the uniform probability measure on C a is a Cantor function, f a . When a = 1/3 = b , C a is the standard Cantor set, C , and f a is the standard Cantor function, f . The upper derivative of f is infinite at each point of C and the lower derivative of f is infinite at most points of C in the following sense: the Hausdorff dimension of C is ln(2)/ln(3) and the Hausdorff dimension of S = { x ∈ C : the lower derivative of f is finite at x } is [ln(2)/ln(3)] 2 . The derivative of f is zero off C , the derivative of f is infinite on C — S , and S is the set of non-differentiability points of f . Similar results are established in this paper for all C a : the Hausdorff dimension of C a is ln (2)/ln (1/ a ) and the Hausdorff dimension of S a is [ln (2)/ln (1/ a )] 2 . Removing k segments of relative length b and leaving k + 1 intervals of relative length a produces a Cantor set of dimension ln( k + l)/ln(1/ a ); the dimension of the set of non-differentiability points of the corresponding Cantor function is [ln ( k + l)/ln (1/ a )] 2 .

FRACTAL TILINGS IN THE PLANE

Tilings have appeared in human activity since prehistoric times. They are used in the design of floor and wall coverings for cathedrals, commercial buildings, and personal dwellings. Mathematicians stucly the geometric structure of tilings. A checkerboard is an elementary example of a sinilarity tiling, one that is composed of smaller tiles (,rep tiles) of the same size, each having the same shape as the whole. Each rep tile in the checkerboard is the scaled and translated image of the entire board. For the checkerboard in FIGURE la, the lower left tile is the image of the checkerboard under

Convergence of best best _{∞}-approximations

We begin with a brief introduction in which we present notation and terminology, a related result [1] for 1 0 u(E) = essup(f, E) = inf{X; ,({x E E;f(x) > )}) = 0) and

Prediction in Orlicz spaces

In this paper, we investigate the prediction (or best approximation) operator from a uniformly convex real Orlicz space to a subset of σ-lattice measurable functions. In particular, a counterexample to the monotoncity property, which holds in Lp spaces, is given. Also, a sufficient condition for monotonicity to hold is given. Finally, nested σ-lattices, as occur in isotonic regression, are considered.