Uniform approximation through partitioning

Uniform Approximation Through Partitioning

In this paper, the problem of best uniform polynomial approximation to a continuous function on a compact set X is approached through the partitioning of X and the definition of norms corresponding to the partition and each of the standard Lp norms 1 g p < oo. For computational convenience, a pseudo norm is defined corresponding to each partition. When the partition is chosen appropriately, the...

Partitioning 3-uniform hypergraphs

Bollobás and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges can be partitioned into r sets so that each set meets at least rm/(2r − 1) edges. For r = 3, Bollobás, Reed and Thomason proved the lower bound (1− 1/e)m/3 ≈ 0.21m, which was improved to 5m/9 by Bollobás and Scott (while the conjectured bound is 3m/5). In this paper, we show that this Bollobás-Thomason ...

Non-Uniform Graph Partitioning

We consider the problem of NON-UNIFORM GRAPH PARTITIONING, where the input is an edge-weighted undirected graph G = (V,E) and k capacities n1, . . . , nk, and the goal is to find a partition {S1, S2, . . . , Sk} of V satisfying |Sj | ≤ nj for all 1 ≤ j ≤ k, that minimizes the total weight of edges crossing between different parts. This natural graph partitioning problem arises in practical scen...

On Constrained Uniform Approximation

Weierstrass and Uniform Approximation

We present the theorem of uniform approximation of continuous functions by polynomials in the Weierstrass framework for the construction of analysis based on the representation of functions by sums of power series and analytic functions, and his effort to bring new standards of rigor to analysis that accompanied his differences with Riemann. We also describe the most original proofs of the appr...