We establish the precise asymptotics of the quantization and entropy coding errors for fractional Brownian motion with respect to the supremum norm and L[0, 1]-norm distortions. We show that all moments in the quantization problem lead to the same asymptotics. Using a general principle, we conclude that entropy coding and quantization coincide asymptotically. Under supremum-norm distortion, our...

Step-asynchronous successive overrelaxation updates the values contained in a single vector using the usual Gauß–Seidel-like weighted rule, but arbitrarily mixing old and new values, the only constraint being temporal coherence— you cannot use a value before it has been computed. We show that given a nonnegative real matrix A, a σ ≥ ρ(A) and a vector w > 0 such that Aw ≤ σw, every iteration of ...

In 1924, Szegő showed that the zeros of the normalized partial sums, sn(nz), of ez tended to what is now called the Szegő curve S, where S := { z ∈ C : |ze1−z| = 1 and |z| ≤ 1 } . Using modern methods of weighted potential theory, these zero distribution results of Szegő can be essentially recovered, along with an asymptotic formula for the weighted partial sums {esn(nz)}n=0. We show that G := ...

This paper is concerned with the problem of approximating a homeomorphism by piecewise affine homeomorphisms. The main result is as follows: every homeomorphism from a planar domain with a polygonal boundary to R that is globally Hölder continuous of exponent α ∈ (0, 1], and whose inverse is also globally Hölder continuous of exponent α can be approximated in the Hölder norm of exponent β by pi...

We study the norm induced by the supremum metric on the space of fuzzy numbers. And then we propose a method for constructing a norm on the quotient space of fuzzy numbers. This norm is very natural and works well with the induced metric on the quotient space.

Following the development of weighted asymptotic approximation properties matrices, we introduce analogous uniform (that is, study improvability Dirichlet's Theorem). An added feature is use general norms, rather than supremum norm, to quantify approximation. In terms homogeneous dynamics, an $m \times n$ matrix are governed by a trajectory in $\mathrm{SL}_{m+n}({\mathbb R})/\mathrm{SL}_{m+n}({...

This paper considers the minimax filtering problem in which the supremum norm of weighted error sequence is minimized. It is shown that the minimax solution is also the optimal Set-Membership Filtering (SMF) solution. An adaptive algorithm is derived that is based on approximating the minimax cost function at each time instant using an optimal quadratic lower bound. The proposed recursions are ...

Dedicated to our friends Beresford and Velvel on the occasion of their sixtieth birthdays. ABSTRACT We show that a certain matrix norm ratio studied by Parlett has a supremum that is O(p n) when the chosen norm is the Frobenius norm, while it is O(log n) for the 2-norm. This ratio arises in Parlett's analysis of the Cholesky decomposition of an n by n matrix.