Let K be a regular (in the sense of pluripotential theory) compact set in Cⁿ and let V_K(z) denote its pluricomplex Green function with a logarithemic singularity at ¥. Then, with probability 1, a sequence of random polynomials {f_a} gives the pluricomplex Green function via the formula

Some recent results concerning approximation on Riemann surfaces will be presented. These will include generalizations to Riemann surfaces of a theorem of A. G. Vitushkin on uniform approximation by rational (meromorphic) functions, and of results by T. W. Gamelin and J. B. Garnett on bounded pointwise approximation.

This is joint work with Jiang B.

When A and B are countable dense subsets of R, it is a well-known result of Cantor that A and B are order-isomorphic. A theorem of K. F. Barth and W. J. Schneider states that the order-isomorphism can be taken to be very smooth, in fact the restriction to R of an entire function. J. E. Baumgartner showed that consistently 2^À₀ > À₁ and any two subsets of R having À₁ points in every interval are order-isomorphic. However, U. Abraham, M. Rubin and S. Shelah produced a ZFC example of two such sets for which the order-isomorphism cannot be taken to be smooth. A useful variant of Baumgartner's result for second category sets was established by S. Shelah. He showed that it is consistent that 2^À₀ > À₁ and second category sets of cardinality À₁ exist while any two sets of cardinality À₁ which have second category intersection with every interval are order-isomorphic. In this paper, we show that the order-isomorphism in Shelah's theorem can be taken to be the restriction to R of an entire function. Moreover, using an approximation theorem of L. Hoischen, we show that given a nonnegative integer n, a nondecreasing surjection g :R ® R of class Cⁿ and a positive continuous function e: R ® R, we may choose the order-isomorphism f so that for all i = 0,1,...,n and for all x Î R, |Dⁱ f(x)-Dⁱ g(x)| < e(x).

Let p be a polynomial of degree n. Let m_s : = n^-1 sin^-2( sp/(2n) ), s odd, 1 £ s < n, and m_n = (1-(-1)ⁿ ) / (4n). By using m_s and x_j = cos(jp/n), j integer, we define the linear functionals

The following inequality is established:

Remark 1 In view of

Remark 2 The inequality obtained is a refinement of the well known inequality:

References

[1]

R. J. Duffin and A. C. Schaeffer, A refinement of an inequality of the brothers Markoff. Trans. Amer. Math. Soc. 50(1941), 517-528.

Let A,B be the two 2×2 matrices

Let D be the unit disc of the complex plane C. We prove that for any polynomial p of degree at most n

This is joint work with Dimiter Dryanov.

Firstly, we approximate the Riemann zeta function by meromorphic functions for which the Riemann hypothesis fails. Secondly, we approximate arbitrary holomorphic functions by linear combinations of translates of the Riemann zeta function. The first result is joint work with E. S. Zeron. The second work is joint with N. N. Tarkhanov.

Functions from finite sets to R^m occur very frequently in applied problems. If m=1, then there is a standard definition of monotonicity, and it is often useful to break up the function into (approximately) monotone segments. For the case of higher dimensional range spaces, there is no order structure, so we instead consider breaking up the functions into segments of bounded curvature. This leads to the problem of determining a bound on the curvature of a segment in a computationally efficient way, which we do by a recursive formula. We compare with quasi-linear fitting, obtained by a least-squares method, and find that the curvature method is more robust, in particular, with respect to deletion of data points.

Joint work with Daniel Lemire, UQAM.

There are several uniqueness theorems for functions in the classical Dirichlet space. But, a necessary and sufficient condition (like the Blaschke condition for Hardy spaces) for this space is not yet available. We will discuss a new uniqueness theorem.

This is a joint work with Thomas Ransford and Abdellatif Bourhim.

We prove a quantitative form of the classical Denjoy-Carleman theorem on quasi-analytic classes. As an application, we derive an extension of Carleman's theorem on the unique determination of probability measures by their moments.

Joint work with Isabelle Chalendar, Laurent Habsieger and Jonathan Partington.

Let K₁, K₂ be compact sets in the complex plane C. We say that K₁, K₂ is a fusion pair if there exists some constant a = a(K₁,K₂) with the following property: for all rational functions r₁, r₂ and for each compact set K Ì C there is a rational function r which fulfills ||r_j-r||_KjÈK £ a·||r₁-r₂||_K simultaneously for j=1,2.

Alice Roth proved in 1976 that K₁,K₂ is a fusion pair if the sets K₁, K₂ are disjoint. If K₁ÇK₂ = Æ is not required we have of course to replace ||r₁-r₂||_K above by ||r₁-r₂||_KÈ(K1ÇK2). But in general K₁, K₂ is no fusion pair (examples are due to Gauthier and Gaier).

Under rather natural topological restrictions (especially that K₁,K₂ have no common interior points) we can characterize the fusion pairs by the simple condition that ¶K₁ ǶK₂ = ¶(K₁ÈK₂).

We use the Carleson-Sobolev estimates for an operator valued solution of the heat equation to give the regularity of the solution and its application to the Weierstrass approximation theorem.

There are several criteria to decide whether a continuous function F(z) defined from a compact set K Ì Cⁿ into C can be approximated by holomorphic rational functions (P/Q)(z). Now, any rational function (P/Q)(z) can be seen as a holomorphic function defined from an open set of Cⁿ into the Riemann sphere S². And we can even generalise the concept of rational function to consider holomorphic functions with range into the complex projective space CPⁿ.

We may then ask about the rational approximation of continuous functions G(z) defined from a compact set K Ì Cⁿ into CPⁿ. We want to show, in this talk, that there is essentially one extra necessary condition on G(z) to be approximated by rational functions: Function G must be homotopically trivial.