This paper contains some theorems related to the best approximation n (f ; E) to a function f in the uniform metric on a compact set E ⊂ C by rational functions of degree at most n. We obtain results characterizing the relationship between n (f ; K) and n (f ; E) in the case when complements of compact sets K and E are connected, K is a subset of the interior of E, and f is analytic in and continuous on E.

In this paper, we revisit the idea of locked inflation, which does not require a potential satisfying the normal slow-roll condition, but suffers from the problems associated with "saddle inflation". We propose a scenario based on locked inflation, however, with an alternative evolution mechanism of the "waterfall field" φ. Instead of rolling down along the potential, the φ field will tunnel to end the inflation stage like in old inflation, by which the saddle inflation could be avoided. Further, we study a cascade of old locked inflation, which can be motivated by the string landscape. Our model is based on the consideration of making locked inflation feasible so as to give a working model without slow roll; It also can be seen as an effort to embed the old inflation in string landscape.

Let E ⊂ (−1, 1) be a compact set, let µ be a positive Borel measure with support supp µ = E, and let H p (G), 1 ≤ p ≤ ∞, be the Hardy space of analytic functions on the open unit disk G with circumference = {z: |z| = 1}. Let n, p be the error in best approximation of the Markov functionin the space L p ( ) by meromorphic functions that can be represented in the form h = P/Q, where P ∈ H p (G), Q is a polynomial of degree at most n, Q ≡ 0. We investigate the rate of decrease of n, p , 1 ≤ p ≤ ∞, and its connection with n-widths. The convergence of the best meromorphic approximants and the limiting Date distribution of poles of the best approximants are described in the case when 1 < p ≤ ∞ and the measure µ with support E = [a, b] satisfies the Szegő condition b a log(dµ/dx)

A Hankel form on a Hilbert function space is a bounded, symmetric, bilinear form [., .] satisfying [fx, y] = [x, f y] for a class of multipliers f . We prove analogs of Weyl-Horn and Ky Fan inequalities for compact Hankel forms, and apply them to estimate the related eigenvalues, both for HardySmirnov and Bergman spaces norms associated to multiply connected planar domains. In the case of the unit disk, we investigate the asymptotic of some measures constructed by eigenfunctions of Hankel operators with certain Markov functions as symbols. Mathematics Subject Classification (2000). Primary 41A20; Secondary 30E10, 47B35.

Let E be a closed subset of the open unit disk G ¼ fz : jzjo1g; and let m be a positive Borel measure with support supp m ¼ E: Denote by A p the restriction on E of the closed unit ball of the Hardy space H p ðGÞ; 1pppN: In this paper we investigate orthogonality properties of the extremal functions associated with the Kolmogorov, Gelfand, and linear n-widths of A p in L q ðm; EÞ; 1pqoN; qpp: r