From Dzyadyk to the “Old Kents”

From Dzyadyk to the “Old Kents”
How a mathematical school keeps finding new paths

There’s life in the old Kents yet!

The epigraph hides two affectionate jokes. The English expression “There’s life in the old dog yet” celebrates someone who retains their energy and can still surprise us. Here the old dog has become the old Kents: Vladimir Andrievskii and Fedor Nazarov are colleagues at Kent State University. For a Russian-speaking reader familiar with Odessa slang, kent also means a friend or pal. These particular pals are two experienced mathematicians who found each other, and some excellent problems.

Their story began long before their collaboration. It has a Ukrainian line of succession: Vladislav Dzyadyk, his student Vladimir Belyi, and Belyi’s student Vladimir Andrievskii. What interests us is how the substance of the work changed from one generation to the next. What survived from the original vision, and what became possible through new tools?

Dzyadyk developed polynomial approximation in the complex plane with the geometry of the underlying set built into the error estimate itself. On an interval, approximation near an endpoint behaves differently from approximation in the middle. A complicated region raises the same question: what local scale corresponds to the degree of a polynomial? The answer involves nearby level curves of the exterior Green function. To construct approximations, Dzyadyk developed special polynomial kernels.

Belyi gave this programme new geometric strength. A 1989 survey by Belyi, Golub, and Shevchuk describes Dzyadyk’s suggestion in the early 1970s that the Donetsk school of mapping theory should help develop direct approximation theorems further. Conformal invariants, quasiconformal mappings, and precise distance-distortion estimates entered the picture. The boundaries allowed in the theory became progressively less smooth.

Andrievskii continued the programme, extending the classes of sets and investigating the link between geometry and constructive properties of functions. All three names appear on the monograph Conformal Invariants in Constructive Theory of Functions of Complex Variable. A shared research programme had acquired several generations of contributors.

Later, Fedor Nazarov joined this story: a mathematician trained in St Petersburg, a student of Victor Havin, and a Salem Prize laureate. His collaboration with Andrievskii shows the geometric experience of the Ukrainian school meeting another strong tradition in analysis. Two of their joint papers make this encounter particularly tangible.

The first asks how to make a monic polynomial small on a complicated region. Monic means that its leading coefficient is one; otherwise we could simply multiply the polynomial by a tiny number. On a quasidisk \(K\), the authors construct polynomials whose zeros lie in the set itself and which satisfy

\[
\|p_n\|_K\le C(K)\operatorname{cap}(K)^n.
\]

Capacity supplies the natural exponential scale, while the factor \(C(K)\) does not grow with the degree. The result covers even the region enclosed by the Koch snowflake. An important historical detail: Andrievskii obtained the bound without the restriction on zeros in 2017. The joint paper, published in 2019, gives a new construction and the stronger conclusion about their location.

The beauty of the construction lies in replacing a continuous distribution by groups of points whose first few moments agree with it. Taking logarithms of products of distances makes initial error terms cancel exactly. Geometry then keeps the total remainder under control. Accuracy comes from preserving the right information about the distribution.

The second paper, published in 2022, studies Leja nodes. Each new point maximizes the product of its distances to all previous points. Old nodes stay in place, which is convenient for successive computations. But how reliable is interpolation from such a table? Could small errors in the data be amplified disastrously?

Andrievskii and Nazarov bound the Lebesgue constants through the behaviour of the Green function near a real compact set. A crucial freedom is to change the reference point while estimating a long product. Combined with node separation, this yields polynomial bounds on broad classes of sets. In 2023, Totik substantially used their idea to prove general subexponentiality on complex compact sets of positive capacity.

The details are developed in the illustrated account Arranging Points. Its four chapters begin with elementary problems: an optimal quadratic, two points replacing continuous mass, and interpolation from three values. The teaching examples are proved completely; the specialist lemmas are identified explicitly. It can be read one chapter at a time, with pauses for thought.

Here is a living branch of Ukrainian mathematics. Dzyadyk’s vision continues through new geometric settings, new ways of choosing points, and new proofs. A school can be recognized by the questions its successors learn to pose and solve afresh. And the old Kents can still surprise us.

Original sources:  

the historical survey by Belyi, Golub, and Shevchuk; 

the quasidisk paper; 

the Leja-node paper.

 Full references are included in the longer account.