


Approximation Theory / Théorie d'approximation Org: Richard Fournier and/et Paul Gauthier (Montreal)
 THOMAS BLOOM, University of Toronto, 100 St. George Street, Toronto, ON M1M 3W5
Random Polynomials and Green Functions

Let K be a regular (in the sense of pluripotential theory) compact
set in C^{n} 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

æ è

\varlimsup_{a} 
1
deg(f_{a})

logf_{a}(z) 
ö ø

*

= V_{K} (z) for all z Î C^{n} 

In the onedimensional case, this result may be used to generalize a
result of ShiffmanZelditch on the limiting normalized distributon of
zeroes of random polynomials.
 ANDRÉ BOIVIN, Univerity of Western Ontario
Approximation on Riemann surfaces: recent results

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.
 MAXIM BURKE, University of Prince Edward Island, Charlottetown, PE C1A 4P3
Entire functions mapping uncountable dense sets of reals onto
each other monotonically

When A and B are countable dense subsets of R, it is a
wellknown result of Cantor that A and B are orderisomorphic. A
theorem of K. F. Barth and W. J. Schneider states that the
orderisomorphism 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} > À_{1} and any two subsets
of R having À_{1} points in every interval are
orderisomorphic. However, U. Abraham, M. Rubin and S. Shelah
produced a ZFC example of two such sets for which the
orderisomorphism 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} > À_{1} and second category sets of cardinality À_{1} exist
while any two sets of cardinality À_{1} which have second
category intersection with every interval are orderisomorphic. In
this paper, we show that the orderisomorphism 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 C^{n} and a positive continuous
function e: R ® R, we may
choose the orderisomorphism f so that for all i = 0,1,...,n and
for all x Î R, D^{i} f(x)D^{i} g(x) < e(x).
 DIMITER DRYANOV, Concordia University, 7141 Sherbrooke Street West, Montreal,
QC H4B 1R6
A Refinement of an Inequality of R. J. Duffin and
A. C. Schaeffer

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
L_{j} (p) = 
n å
s=1, s odd

m_{s} 
æ è

p(x_{j})  
p(x_{j+s}) + p(x_{js})
2

ö ø



for j = 0,1,...,n.
The following inequality is established:

max
x Î [1,1]

p^{¢}(x)  £ n 
max
0 £ j £ n

L_{j}(p) . 

Remark 1
In view of

n å
s=1, s odd

m_{s} = 
n
2



and by making use of the sets
T_{o} : = {x_{j} : j odd, 0 £ j £ n}, T_{e} : = {x_{j} : j even, 0 £ j £ n} 

we have:


max
x Î [1,1]

 p^{¢}(x)  

£ n 
max
0 £ j £ n

 L_{j}(p)  
  (1) 
 
£ 
n^{2}
2


max
x Î C_{o}, y Î C_{e}

p(x)p(y) £ n^{2} 
max
0 £ j £ n

p(x_{j}). 
  (2) 
Remark 2
The inequality obtained is a refinement of the well known inequality:

max
x Î [1,1]

p^{¢}(x)  £ n^{2} 
max
0 £ j £ n

p(x_{j}) 

which is due to Duffin and Schaeffer [1].
References
 [1]

R. J. Duffin and A. C. Schaeffer,
A refinement of an inequality of the brothers Markoff.
Trans. Amer. Math. Soc. 50(1941), 517528.
 SERGE DUBUC, Université de Montréal, C.P. 6128, Succ. Centreville,
Montréal, Québec H3C 3J7
The joint spectral radius of a family of matrices

Let A,B be the two 2×2 matrices
we define the subset of the plane W = {(l,m) :r(A,B) < 1} where r(A,B) is the joint spectral radius of
{A,B}. We discuss the approximation of W.
 RICHARD FOURNIER, CRM, Université de Montréal, C.P. 6128, Succ. Centreville, Montréal, Québec H3C 3J7
A New Inequality for Polynomials

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

max
q Î R


ê ê

p(e^{iq})  p(e^{iq})
e^{iq}  e^{iq}

ê ê

£ n 
max
0 £ j £ n


ê ê

p(e^{ijp/n}) + p(e^{ijp/n})
2

ê ê

. 

We shall also discuss how this inequality is related to classical
results or Bernstein and Markov and to more recent ones due to Duffin
and Schaeffer, Frappier, Rahman and Ruscheweyh.
This is joint work with Dimiter Dryanov.
 PAUL GAUTHIER, Université de Montréal, Centreville, Montréal, QC
H3C 3J7
Approximation of and by the Riemann zeta function

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.
 DAN KUCEROVSKY, University of New Brunswick
Quasimonotone sequences

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 quasilinear fitting, obtained by a
leastsquares 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.
 JAVAD MASHREGHI, Université Laval
Zeros of functions in the Dirichlet space

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.
 THOMAS RANSFORD, Université Laval, Dép. de mathématiques, Québec
(QC), G1K 7P4
A DenjoyCarleman maximum principle

We prove a quantitative form of the classical DenjoyCarleman theorem
on quasianalytic 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.
 GERALD SCHMIEDER, Universitaet Oldenburg, Fak. V, Inst. f. Math., 26111
Oldenburg, Germany
Extension of the Fusion Lemma

Let K_{1}, K_{2} be compact sets in the complex plane C. We
say that K_{1}, K_{2} is a fusion pair if there exists some
constant a = a(K_{1},K_{2}) with the following property: for all
rational functions r_{1}, r_{2} and for each compact set K Ì C there is a rational function r which fulfills
r_{j}r_{KjÈK} £ a·r_{1}r_{2}_{K} simultaneously for
j=1,2.
Alice Roth proved in 1976 that K_{1},K_{2} is a fusion pair if the sets
K_{1}, K_{2} are disjoint. If K_{1}ÇK_{2} = Æ is not required
we have of course to replace r_{1}r_{2}_{K} above by
r_{1}r_{2}_{KÈ(K1ÇK2)}. But in general K_{1}, K_{2} is no
fusion pair (examples are due to Gauthier and Gaier).
Under rather natural topological restrictions (especially that K_{1},K_{2} have no common interior points) we can characterize the fusion
pairs by the simple condition that ¶K_{1} Ç¶K_{2} = ¶(K_{1}ÈK_{2}).
 JIE XIAO, Memorial University of Newfoundland, St. John's, NL A1C 5S7
The Heat Equation II: Regularity and Approximation

We use the CarlesonSobolev 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.
 EDUARDO ZERON, Cinvestav (Math), Apartado postal 14740, Mexico DF, 07000,
Mexico
Homotopical obstructions to rational approximation

There are several criteria to decide whether a continuous function
F(z) defined from a compact set K Ì C^{n} 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 C^{n} into
the Riemann sphere S^{2}. And we can even generalise the concept of
rational function to consider holomorphic functions with range into
the complex projective space CP^{n}.
We may then ask about the rational approximation of continuous
functions G(z) defined from a compact set K Ì C^{n}
into CP^{n}. 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.
 PING ZHOU, St. Francis Xavier University, Antigonish, NS B2G 2W5
Divided differences in construction of multivariate Padé
approximants

We explicitly construct multivariate Padé approximants to some
functions of the form
F(x_{1},x_{2},...,x_{m}) = 
¥ å
n=0

f(n) 
å
j_{1}+j_{2}+¼ +j_{m}=n

x_{1}^{j1} x_{2}^{j2} ¼x_{m}^{jm}, 

where f(n) is a certain function of n, by using divided
differences. Examples include the multivariate exponential
function
E(x_{1},x_{2},...,x_{m}) : = 
¥ å
j_{1},j_{2},...,j_{m}=0


x_{1}^{j1} x_{2}^{j2} ¼x_{m}^{jm}
(j_{1}+j_{2}+¼ +j_{m}+1)!

, 

the multivariate logarithm function
L(x_{1},x_{2},...,x_{m}) : = 
¥ å
j_{1},j_{2},...,j_{m}=0


x_{1}^{j1} x_{2}^{j2} ¼x_{m}^{jm}
j_{1}+j_{2}+¼+j_{m}+1

, 

and others.

