Uniform approximation on manifolds

Approximation on Manifolds

Let (r); r = 1; 2; : : : be a positive decreasing sequence such that P 1 r=1 (r) k diverges. Using a powerful variance argument due to W. M. Schmidt, an asymptotic formula is obtained for the number of integer solutions q of the system of Diophantine inequalities maxfkqx i k: 1 i kg < (q) which holds for almost all points (x 1 ; : : : ; x k) on a smooth m-dimensional subman-ifold M of R k. The ...

On Constrained Uniform Approximation

On uniform approximation by splines

for 0 ≤ r ≤ k − 1. In particular, dist (f, S π) = O(|π| ) for f ∈ C(I), or, more generally, for f ∈ C(I), such, that f (k−1) satisfies a Lipschitz condition, a result proved earlier by different means [2]. These results are shown to be true even if I is permitted to become infinite and some of the knots are permitted to coalesce. The argument is based on a “local” interpolation scheme Pπ by spl...

Uniform Approximation on Riemann Surfaces

This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is “usually” not possible to approximate f uniformly by functions holomorphic on all of R. In Chapter 2, we show, however, that for every open Riemann surface...

Holomorphic Approximation on Compact Pseudoconvex Complex Manifolds

Let M be a smoothly bounded compact pseudoconvex complex manifold of finite type in the sense of D’Angelo such that the complex structure of M extends smoothly up to bM . Let m be an arbitrary nonnegative integer. Let f be a function in H(M) ∩Hm(M), where Hm(M) is the Sobolev space of order m. Then f can be approximated by holomorphic functions on M in the Sobolev space Hm(M). Also, we get a ho...