# Thomas BloomUniversity of Toronto | U of T · Department of Mathematics

Thomas Bloom

PhD

## About

85

Publications

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1,214

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## Publications

Publications (85)

We correct the calculation of the Monge-Amp\`ere measure of a certain extremal plurisubharmonic function for the complex Euclidean ball in C^2.

We continue the study in a previous work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P-$pluripotential-theoretic notions. As an important preliminary step, w...

In their seminal paper, Berman and Boucksom exploited ideas from complex geometry to analyze asymptotics of spaces of holomorphic sections of tensor powers of certain line bundles $L$ over compact, complex manifolds as the power grows. This yielded results on weighted polynomial spaces in weighted pluripotential theory in $\Bbb{C}^d$. Here, motivat...

We discuss the Siciak-Zaharjuta extremal function of pluripotential theory for the unit ball in C^d for spaces of polynomials with the notion of degree determined by a convex body P. We then use it to analyze the approximation properties of such polynomial spaces, and how these may differ depending on the function f to be approximated.

We develop potential theory including a Bernstein-Walsh type estimate for functions of the form $p(g(z))q(f(z))$, where $p,q$ are polynomials and $f,g$ are holomorphic. For $g(z)=z$, such functions arise in the study of certain ensembles of probability measures and in this case we can further extend the
theory leading to probabilistic results such...

We give a survey of recent results, due mainly to the authors, concerning
Bernstein-Markov type inequalities and connections with potential theory.

We prove asymptotics of one-point correlation functions and derive a large
deviation principle for biorthogonal ensembles associated to probability
density functions $Prob_k$ of the form $$\frac{1}{Z_k}\prod_{i<j}|z_i-z_j|\cdot
\prod_{i<j}|z_i^{\theta}-z_j^{\theta}|\cdot \exp \Big(-k[Q(z_0)+\cdots +
Q(z_k)]\Big)d\nu (z_0)\cdots d\nu (z_k)$$ on $K^{...

We derive a general large deviation principle for a canonical sequence of probability measures, having its origins in random matrix theory, on unbounded sets \(K\) of \(\mathbb {C}\) with weakly admissible external fields \(Q\) and very general measures \(\nu \) on \(K\) . For this we use logarithmic potential theory in \(\mathbb {R}^{n}\) , \(n\ge...

For Y a subset of the complex plane, a @b ensemble is a sequence of probability measures Prob"n","@b","Q on Y^n for n=1,2,... depending on a positive real parameter @b and a real-valued continuous function Q on Y. We consider the associated sequence of probability measures on Y where the probability of a subset W of Y is given by the probability th...

There is a natural pluripotential-theoretic extremal function V_{K,Q}
associated to a closed subset K of C^m and a real-valued, continuous function Q
on K. We define random polynomials H_n whose coefficients with respect to a
related orthonormal basis are independent, identically distributed
complex-valued random variables having a very general dis...

For Y a subset of the complex plane,a beta ensemble is a sequence of
probability measures on Y^n for n=1,2,3...depending on a real-valued continuous
function Q and a real positive parameter beta.We consider the associated
sequence of probability measures on Y where the probability of a subset W is
given by the probability that at least one coordina...

For d nonpolar compact sets K_1,...,K_d in the complex plane, d admissible
weights Q_1,...,Q_d, and a positive semidefinite d x d interaction matrix C
with no zero column, we define natural discretizations of the associated
weighted vector energy of a d-tuple of positive measures \mu=(\mu_1,...,\mu_d)
where \mu_j is supported in K_j and has mass r_...

We discuss a notion of the energy of a compactly supported measure in
\mathbbCn \mathbb{C}^n for n > 1 which we show is equivalent to that defined by Berman, Boucksom, Guedj and Zeriahi. This generalizes the classical notion
of logarithmic energy of a measure in the complex plane
\mathbbC \mathbb{C} ; i.e., the case n = 1.
KeywordsPluripotenti...

We calculate the transfinite diameter for the real unit ball Bd:= {x ∈ ℝd: |x| ≤ 1} and the real unit simplex.

We give the asymptotics for D-optimal (equivalently G-optimal) designs on a
compact (possibly complex) design space.

We update the state of the subject approximately 20 years after the
publication of a previous article on this topic. This report is mostly a
survey, with a sprinkling of assorted new results throughout.

Almost surely in an Angelesco ensemble,the normalized counting measure of a
random point converges weak* to the equilibrium measure.This result, for
orthogonal polynomial ensembles, is well-known.

We give a new proof of the result,originating in work of Voiculescu,that the logarithmic energy of a planar measure is a triple limit of volumes. Comment: 15 pages

We prove a version of asymptotics of Christoffel functions with varying weights for a general class of sets E in, and measures μ on the complex plane ℂ. This class includes all regular measures μ in the sense of Stahl-Totik [18] on regular compact sets E in ℂ and even allows varying weights. Our main theorems cover some known results for subsets of...

Using recent results of Berman and Boucksom (arXiv: 0807. 0035), we show that for a nonpluripolar compact set K⊂ℂd
and an admissible weight function w=e
−φ
, any sequence of optimal measures converges weak-star to the equilibrium measure μ
K,φ
of (weighted) pluripotential theory for K,φ.

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite
diameter, and weighted transfinite diameter for sets inℂ
N
. An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann.
337 (2007), 729–738) which relates the Robin function and t...

Let E be a compact subset of CN and w ≥ 0 an admissible weight function on E. To (E,w) we associate a canonical circular set ℤ ⊂ CN+1. We obtain precise relations between the weighted pluricomplex Green function and weighted equilibrium measure of (E,w) and the pluricomplex Green function and equilibrium measure of ℤ. These results, combined with a...

For a regular, compact, polynomially convex circled set K in C^2, we construct a sequence of pairs {P_n,Q_n} of homogeneous polynomials in two variables with deg P_n = deg Q_n = n such that the sets K_n: = {(z,w) \in C^2 : |P_n(z,w)| \leq 1, |Q_n(z,w)| \leq 1} approximate K and the normalized counting measures {\mu_n} associated to the finite set {...

For a regular compact set $K$ in $C^m$ and a measure $\mu$ on $K$ satisfying the Bernstein-Markov inequality, we consider the ensemble $P_N$ of polynomials of degree $N$, endowed with the Gaussian probability measure induced by $L^2(\mu)$. We show that for large $N$, the simultaneous zeros of $m$ polynomials in $P_N$ tend to concentrate around the...

Let K be a regular (in the sense of pluripotential theory) compact set in ℂⁿ and let VK(z) denote its pluricomplex Green function with a logarithmic singularity at ∞. Then, with probability 1, a sequence of random polynomials {fα} (linear combination of monomials of lexicographic order ≤ α) gives the pluricomplex Green function via the formula for...

Let K ⊂ CN be compact and let w be a nonnegative, uppersemicontinuous function on K with {z ∈ K: w(z) > 0} nonpluripolar. Let Q := - log w and define the weighted pluricomplex Green function V*K,Q(z) = lim sup ζ→z VK,Q(ζ) where VK,Q(z) := sup{u(z): u plurisubharmonic in CN, u(z) ≤ log+ |z| + C, u ≤ Q on K} (C depends on u). If w ≡ 1; i.e., Q ≡ 0, w...

Given an irreducible algebraic curves A in ℂN, let m d be the dimension of the complex vector space of all holomorphic polynomials of degree at most d restricted to A. Let K be a nonpolar compact subset of A, and for each d = 1, 2, ..., choose md points {A dj} j=1,...,md in K. Finally, let Adbe the d-th Lebesgue constant of the array {Adj}i i.e., A...

Let {r(n)} be a sequence of rational functions (deg r(n) less than or equal to n) that converge rapidly in measure to an analytic function f on an open set in C-N. We show that {r(n)} converges rapidly in capacity to f on its natural domain of definition W-f (which, by a result of Goncar, is an open subset of C-N). In particular, for f meromorphic...

Given compact sets E and F in [open face C]n (n [gt-or-equal, slanted] 1) related by
F = q−1(E) where q is
a polynomial map, we are interested in the general problem of comparing minimal
polynomials for E with minimal polynomials for F. Let α be an n-multi-index of
length d. We define the classes of polynomials [open face P](α)
:= zα + [open f...

Given a function f, uniform limit of analytic polynomials on a compact, regular set E⊂CN, we relate analytic extension properties of f to the location of the zeros of the best polynomial approximants to f in either the uniform norm on E or in appropriate Lq norms. These results give multivariable versions of one-variable results due to Blatt–Saff,...

Given a sequence fEjg of Borel subsets of a given non-pluripolar Borel set E in the unit ball B in CN with E B, we show that the rel- ative capacities C(Ej) converge to C(E) if and only if the relative (global) extremal functions u Ej (V Ej ) converge pointwise to u E (V E ). This is used to prove a sucient mass-density condition on a nite positive...

We prove a number of results concerning the transfinite diameter of compact sets in ℂn. These results give explicit relations between the transfinite diameter and basic quantities of pluripotential theory.

We prove a number of results concerning the transfinite diameter of compact sets in ℂn. These results give explicit relations between the transfinite diameter and basic quantities of pluripotential theory. © 1999 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.

LetE⊂nbe compact, regular, and polynomially convex with pluricomplex Green functionVE. Given a sequence of polynomials {pj}j=1, 2, …, the first result is a condition for (j→∞(log |pj(z)|/deg(pj)))* to equalVEon n−E. The condition involves the Robin function ofEand the highest order homogeneous terms of thepjand generalizes one-variable results of B...

We show that a convex totally real compact set in Cn admits an extremal array for Kergin interpolation if and only if it is a totally real ellipse. (An array is said to be extremal for K when the corresponding sequence of Kergin interpolation polynomials converges uniformly ton K) to the interpolated function as soon as it is holomorphic on a neigh...

. Let fS t g be a sequence of interpolation schemes in R n of degree d (i.e. for each S t one has unique interpolation by a polynomial of total degree d) and total order l. Suppose that the points of S t tend to 0 2 R n as t ! 1 and the Lagrange-Hermite interpolants, H S t , satisfy lim t!1 H S t (x ff ) = 0 for all monomials x ff with jffj = d + 1...

We study orthogonal polynomials in ℂn following the one-variable approach of H. Stahl and V. Totik in their book "General Orthogonal Polynomials." We define the concept for a finite, positive, Borel measure μ on ℂn to have regular asymptotic distribution (denoted μ ∈ Reg). We show that if supp(μ) is regular (in the sense of pluripotential theory) t...

Let D be a C-convex domain in C n . Let \(\{A_{dj}\}, \ j = 0,\ldots,d\) , and d = 0,1,2, ..., be an array of points in a compact set \(K \subset D\) . Let f be holomorphic on \(\overline D\) and let K d (f) denote the Kergin interpolating polynomial to f at A d0 ,... , A dd . We give conditions on the array and D such that \(\lim_{d\to\infty} \|K_...

Suppose that, forn≥1,$$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} f(x_{jn} )} $$ is aninterpolatory integration rule of numerical integration, that is,$$I_n [f]: = \int\limits_{ - 1}^1 {P(x)dx,} degree(P)< n.$$ Suppose, furthermore, that, for each continuousf:[−1, 1]→R,$$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - 1}^1 {f(x)dx.}...

Letw be a “nice” positive weight function on (−∞, ∞), such asw(x)=exp(−⋎x⋎α) α>1. Suppose that, forn≥1,$$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} } f(x_{jn} )$$ is aninterpolatory integration rule for the weightw: that is for polynomialsP of degree ≤n-1,$$I_n [P]: = \int\limits_{ - \infty }^\infty {P(x)w(x)dx.} $$ Moreover, suppose that the sequenc...

We investigate which types of asymptotic distributions can be generated by the knots of convergent sequences of interpolatory integration rules. It will turn out that the class of all possible distributions can be described exactly, and it will be shown that the zeros of polynomials that are orthogonal with respect to varying weight functions are g...

For CHI a finite subset of C(n) we let chi-x(closed-integral) denote the Kergin polynomial interpolant of the function closed-integral at CHI. Let A1,A2,...be a sequence of points in C(n). Suppose that to every finite subset A of the sequence a polynomial P(A) of degree (card(A)-1) is assigned satisfying the compatibility relations chi-c(P(A)) = ch...

C ( I n ) \mathcal {C}({I^n}) denotes the Banach space of continuous functions on the unit n n -cube, I n {I^n} , in R n {{\mathbf {R}}^n} . Let { a i } \{ {a^i}\} , i = 0 , 1 , 2 , … , i = 0,1,2, \ldots , , be a countable collection of n n -tuples of positive real numbers satisfying lim i a j i = + ∞ {\operatorname {lim}_i}a_j^i = + \infty for j =...

LetA
dj be a triangular array in a compact setXC
n
. Forf analytic in a neighborhood ofX, letL
d
(f) denote the Lagrange interpolant tof at staged of the array. In the caseX is locally regular, we construct a continuous function satisfying the complex Monge-Ampre equation onC
n
–X, such that iff is analytic onR forR>1 then, for someB>0, we have...

Let $m$ be a real $\mathscr{C}^\infty$ hypersurface of an open subset of $C^3$ and let $p \in M$. Let $a^1(M,p)$ denote the maximal order of contact of a one-dimensional complex submanifold of a neighborhood of $p$ in $C^3$ with $M$ at $p$. Let $c^1(M,p)$ denote the $\sup\{m \in Z\mid$ for all tangential holomorphic vector fields $L$ with $L(p) \ne...

## Projects

Project (1)

This project aims to develop a weighted pluripotential theory
arising from polynomials associated to a convex body $P\subset (R^+)^d$.