Some problems of best approximation with constraints

AbstractWe discuss problems of best approximation with constraints in (a) an abstract Hilbert space setting and (b) a concrete form involving polynomial approximation. One problem is to compute the Hilbert space distance from a fixed vector h to the set of vectors Ad such that ∥Bd∥ ⩽ M, where A, B are given linear operators and M is a positive constant. A related concrete problem is to find the L2(μ)-distance from a fixed function h to the set of polynomials p that satisfy ∝ ¦p¦2 dv ⩽ M2, where μ, v are nonnegative, finite Borel measures on the unit circle and M is a positive constant. In particular, the dependence of this distance on the singular components of μ and v is investigated

Approximation by polynomials with restricted zeros

AbstractThis paper discusses convergence properties of polynomials whose zeros lie on the real axis or in the upper half-plane. A result of Levin shows that uniform convergence of such polynomials to a non-zero limit on a complex sequence converging not too East to a limit in the lower half-plane implies locally uniform convergence in C. We give a relatively simple proof of this result and present several extensions and examples which show that the criterion in Levin′s theorem is almost sharp