Discrepancy norm: Approximation and variations

Fuzzy Best Simultaneous Approximation of a Finite Numbers of Functions

Fuzzy best simultaneous approximation of a finite number of functions is considered. For this purpose, a fuzzy norm on $Cleft (X, Y right )$ and its fuzzy dual space and also the  set of subgradients of a fuzzy norm are introduced. Necessary case of a proved theorem about characterization of simultaneous approximation will be extended to the fuzzy case.

The Supremum Norm of the Discrepancy Function: Recent Results and Connections

A great challenge in the analysis of the discrepancy function DN is to obtain universal lower bounds on the L∞ norm of DN in dimensions d ≥ 3. It follows from the L2 bound of Klaus Roth that ‖DN‖∞ ≥ ‖DN‖2 & (logN)(d−1)/2. It is conjectured that the L∞ bound is significantly larger, but the only definitive result is that of Wolfgang Schmidt in dimension d = 2. Partial improvements of the Roth ex...

A new discrepancy principle

The aim of this note is to prove a new discrepancy principle. The advantage of the new discrepancy principle compared with the known one consists of solving a minimization problem (see problem (2) below) approximately, rather than exactly, and in the proof of a stability result. To explain this in more detail, let us recall the usual discrepancy principle, which can be stated as follows. Consid...

Approximating Hereditary Discrepancy via Small Width Ellipsoids

The Discrepancy of a hypergraph is the minimum attainable value, over twocolorings of its vertices, of the maximum absolute imbalance of any hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum discrepancy of a restriction of the hypergraph to a subset of its vertices, is a measure of its complexity. Lovász, Spencer and Vesztergombi (1986) related the natural extension ...

Chebyshev Centers and Approximation in Pre-Hilbert C*-Modules