Kronecker’s approximation theorem

.1 Kronecker's Approximation Theorem

Theorem 2 (Kronecker Approximation Theorem (1884)) ∃x ∈ Z s.t. Ax = b if and only if ∀y, y>b is an integer whenever y>A is an integral vector. Proof: To prove the forward implication, take an integral solution x∗. For any y, since y>Ax∗ = y>b, if y>A is integral then y>b must be an integer too. We next prove the converse. First, suppose that Ax = b does not have a solution (over the reals). The...

The Block Approximation Theorem

The block approximation theorem is an extensive generalization of both the well known weak approximation theorem from valuation theory and the density property of global fields in their henselizations. It guarantees the existence of rational points of smooth affine varieties that solve approximation problems of local-global type (see e.g. [HJP07]). The theorem holds for pseudo real closed field...

Analytic Approximation Theorem

Best Uniform Approximation 3 Theorem 1

We present a Chebyshev-type characterization for the best uniform approximations of periodic continuous functions by functions of the class M = ff(x) : f(x) = Z 2 0 K(x; y)h(y) dy; jh(y)j 1 a: e:; y 2 0; 2)g; where K(x; y) is a strictly cyclic variation diminishing kernel. x1 Introduction Let W r 1 a; b] be the Sobolev class of functions in C r?1 a; b] whose (r?1)st derivative is absolutely con...

The PCP Theorem and Hardness of Approximation

In this report, we give a brief overview of Probabilistically Chekable proofs, and present Irit Dinur’s proof of the celebrated PCP theorem. We also briefly deal with the importance of the PCP theorem in deciding hardness of approximation of various problems.