In Memory of V. K. Dzyadyk: Talent, Integrity, and the Love of Mathematics
Vladyslav Kyrylovych Dzyadyk (18 February 1919–26 October 1998) helped shape a major Ukrainian school of approximation theory. When I think about his legacy, what matters most to me is his talent, his integrity, and the love of mathematics that he passed on to his students.
There is a particular pleasure in doing mathematics: putting a formula on the page, changing it, noticing a relation, and gradually seeing why something must be true. What I value in Igor Shevchuk and his students is their sincere love of this work. They do not treat mathematics as a career ladder. They enjoy writing formulas, almost as an artist enjoys drawing, and finding out why those formulas are true. Here I see Dzyadyk’s legacy: love of mathematics, together with the honesty and patience needed to turn an attractive calculation into a proof.
A teacher’s influence lives in such habits. It can be felt long after a student has moved to another country, entered another field, or begun teaching students of their own.
I would like to remember him here by following one of his ideas from the classical Favard problem to a problem on the two-dimensional sphere. The accompanying notes, Favard–Dzyadyk S²: a
detailed exposition in English (https://drive.google.com/file/d/1Nqa0oY1GAlPojXDqyaqlinfNRBh4TEIK/view), contain the arguments. This post explains the connection, states the main result, and indicates what remains to be done.
Dzyadyk and the fractional Favard problem
The classical Favard problem asks for the exact error of best trigonometric approximation on classes of periodic functions with a prescribed derivative bound. For positive integer orders, the foundational results belong to Favard and, independently, Akhiezer and Krein. Passing from integer derivatives to fractional derivatives in the sense of Weyl brought a substantial new difficulty: the relevant kernels no longer had the elementary structure of the integer-order Bernoulli kernels.
Dzyadyk solved the problem for orders between zero and one in 1953 and obtained a general solution for arbitrary positive orders in 1959. His work connected the approximation of these kernels with absolute monotonicity and developed ways of controlling their intersections with approximating polynomials.
The idea relevant here is that a representation of a kernel through functions with controlled derivatives can determine the sign of an interpolation error. If that sign also annihilates the approximating space, it certifies best integral approximation. An extremal problem then becomes an explicit integral.
The spherical notes follow this principle. The operator changes: on the sphere we use spectral powers of the positive Laplace–Beltrami operator. The connection with Dzyadyk is a connection of methods, rather than an identification of this operator with the periodic Weyl derivative.
A Ukrainian school, with branches around the world
Dzyadyk, a student of S. M. Nikol’skii, led the Department of Function Theory at the Institute of Mathematics in Kyiv from 1963 to 1990. His student Igor Shevchuk continued and developed this tradition through research and teaching at Kyiv University. Their joint book, Theory of Uniform Approximation of Functions by Polynomials, is one lasting expression of that continuity.
The influence extends through teaching, supervision, and collaboration. Maryna Viazovska, Andriy Bondarenko, and Danylo Radchenko belong to the younger generation nurtured by this Kyiv environment and by Shevchuk’s guidance. In recent years, Andriy Prymak’s talent has found particularly fruitful expression in his work on major problems of convex and discrete geometry. Their subjects and individual paths differ, but I recognize in their work the same affection for mathematics itself.
The branches have grown in different directions. Bondarenko, Radchenko, and Viazovska proved the optimal asymptotic bounds for spherical designs, resolving the Korevaar–Meyers conjecture. Viazovska’s work on sphere packing and related extremal and Fourier interpolation problems earned her the 2022 Fields Medal.
A striking recent example is Andriy Prymak’s joint work with Andrii Arman, Andriy Bondarenko, Fedor Nazarov, and Danylo Radchenko,
Small Volume Bodies of Constant Width (https://arxiv.org/abs/2405.18501), published in International Mathematics Research Notices in 2025. They answered a question of Oded Schramm by explicitly constructing, in every sufficiently large dimension \(d\), convex bodies of constant width \(2\) whose volume is less than \(0.9^d\) times the volume of the unit ball. These bodies have the same width as the ball in every direction, yet an exponentially smaller volume.
I think of this as part of a living Ukrainian mathematical tree. Dzyadyk and Shevchuk are among its strong roots; the branches now reach across countries and subjects. Its growth owes much to teachers who gave their students both demanding mathematics and a reason to love it. The prizes are welcome recognition. The daily pleasure of thinking, calculating, and understanding is what keeps the tree alive.
The result on the two-dimensional sphere
Let \(S^2\) be the unit sphere and set \(L=-\Delta_{S^2}\). On spherical harmonics of degree \(\ell\), the spectral power \(L^s\) acts by multiplication by \([\ell(\ell+1)]^s\). Here \(s>0\) is arbitrary, so the differential order is \(2s\).
Write \(\Pi_{n-1}\) for the spherical polynomials of degree less than \(n\), and \(E_{n-1}(f)_p\) for best approximation from this space in \(L^p(S^2)\).
Define
\[
A_s=\frac{4\Gamma(s+\tfrac12)}{\sqrt\pi\,\Gamma(s)}
\bigl(1-2^{-2s-1}\bigr)\zeta(2s+1).
\]
The main functional consequence is the following inequality, initially for smooth functions:
\[
E_{n-1}(f)_p
\le \frac{A_s}{n^{2s}}\,E_{n-1}(L^s f)_p,
\qquad s>0,\quad n\ge1,\quad 1\le p\le\infty.
\]
There is also a Bohr–Favard estimate. If all spherical harmonic components of \(f\) of degrees less than \(n\) vanish, then
\[
\|f\|_\infty
\le \frac{A_s}{n^{2s}}\,\|L^s f\|_\infty.
\]
Both follow from a more precise kernel result. With ordinary surface measure, of total area \(4\pi\), the mean-zero inverse kernel is
\[
G_s(x)=\frac1{4\pi}\sum_{\ell=1}^{\infty}
\frac{2\ell+1}{[\ell(\ell+1)]^s}P_\ell(x).
\]
The series is understood in the Abel sense where necessary; the resulting kernel belongs to \(L^1[-1,1]\) for every \(s>0\). Set
\[
F(s,n)=2\pi\inf_{\deg p<n}\int_{-1}^{1}|G_s(x)-p(x)|\,dx.
\]
For every positive order, the unique best polynomial is the interpolant at the same nodes,
\[
x_j=\cos\frac{j\pi}{n+1},\qquad j=1,\ldots,n.
\]
Thus the interpolation latitudes are independent of \(s\). The error has the sign of the Chebyshev polynomial \(U_n\), giving
\[
F(s,n)=2\pi\int_{-1}^{1}G_s(x)\operatorname{sgn}U_n(x)\,dx.
\]
Put \(K=n+1\) and
\[
R_a(z)=\frac{(1-z)^{-a}+(1+z)^{-a}}2.
\]
The quantitative conclusion is a two-sided estimate:
\[
\frac{A_s}{K^{2s}}<F(s,n)
\le\frac{A_s}{K^{2s}}R_{2s+1}(K^{-1}),
\qquad
F(s,n)\le\frac{A_s}{n^{2s}}.
\]
In particular,
\[
F(s,n)=\frac{A_s}{(n+1)^{2s}}
\bigl(1+O_s((n+1)^{-2})\bigr).
\]
This is a two-sided statement about the kernel approximation error. The functional inequalities above are consequences with explicit constants; their optimality is a separate question.
What the transfer requires
The essential spherical step is a positive integral representation of \(G_s\) through Poisson kernels. It proves positivity of every derivative of positive order with respect to \(x\). After adding a constant, the kernel is absolutely monotone. Algebraic interpolation then supplies the error sign, and an elementary orthogonality identity supplies best \(L^1\) approximation.
The same representation also leads to a positive integral for \(F(s,n)\), from which the bounds follow. What the notes add to our earlier integer-order discussion is a unified argument for every real \(s>0\), common interpolation nodes, and explicit two-sided estimates valid for every \(n\ge1\). The positivity representation and its quantitative analysis require work; once they are available, the interpolation argument is classical.
Sharpness, refinements, and further questions
For \(s=1\), the coefficient is
\[
A_1=\frac74\zeta(3).
\]
Alexey Solyanik’s preprint, On best approximation and estimates of Bohr–Favard type on the sphere for functions with bounded Laplacian (https://arxiv.org/abs/2610.14811), establishes sharpness of the Bohr–Favard result on \(S^2\), including this best coefficient of \(n^{-2}\). It also makes a significant distinction: for each fixed \(n\ge2\), on \(S^2\) and \(S^3\), the optimal constants for best uniform approximation are strictly smaller than the corresponding Bohr–Favard constants.
For the chronology of this discussion, our integer-order spherical estimates were already recorded in the
blog note dated 4 October 2026 (https://kryakin.blogspot.com/2026/10/bookpart6final.html), before the first version of Solyanik’s preprint on 10 October. Our discussion did not establish sharpness of the functional estimates; credit for the sharpness result just mentioned belongs to Solyanik.
Several questions remain beyond the scope of the accompanying notes:
• For general fractional \(s\), determine the optimal Bohr–Favard and best-approximation constants, and the relation between them.
• Refine the finite-degree bounds and obtain further asymptotic terms for \(F(s,n)\).
• Extend the explicit analysis to higher-dimensional spheres. Zonal integration there has weight \((1-x^2)^{(d-2)/2}\). Even dimensions give integer powers of this weight, while odd dimensions give half-integer powers. The positivity approach is available in both settings; the explicit weighted extremal-polynomial problem changes. The particularly simple nodes on \(S^2\) should not be assumed to persist unchanged.
These are invitations to further work. We leave the remaining sharpness questions to interested readers and future investigations.
The detailed exposition is available in the linked English PDF,
Favard–Dzyadyk S² (https://drive.google.com/file/d/1Nqa0oY1GAlPojXDqyaqlinfNRBh4TEIK/view).
Dzyadyk’s methods still lead to results. But his deeper legacy is also present whenever someone takes genuine pleasure in writing a formula, insists on an honest proof, and helps a younger mathematician experience that same joy. Talent, integrity, and love of mathematics can pass from one generation to the next. That is how a mathematical school stays alive.
Further reading on the school and its work
•
Dzyadyk: biography and scientific work, Institute of Mathematics, Kyiv (https://imath.kiev.ua/~funct/staff/Dzyadyk-ua.html).
•
Dzyadyk’s students, Institute of Mathematics, Kyiv (https://www.imath.kiev.ua/~funct/dzyadyk/pupils.html).
•
National Research Foundation of Ukraine: Viazovska and her teachers (https://nrfu.org.ua/en/uncategorized/herald-3/).
•
Bondarenko, Radchenko, and Viazovska: Optimal asymptotic bounds for spherical designs (https://annals.math.princeton.edu/2013/178-2/p02).
•
International Mathematical Union: Viazovska’s 2022 Fields Medal citation (https://www.mathunion.org/fileadmin/IMU/Prizes/Fields/2022/IMU_Fields22_Viazovska_citation.pdf).
•
Arman, Bondarenko, Nazarov, Prymak, and Radchenko: Small Volume Bodies of Constant Width, IMRN 2025, no. 4, rnaf020 (https://doi.org/10.1093/imrn/rnaf020).
