# Registered Data

## [01074] Approximation Theory, Approximation Methods and Applications (ATAMA)

**Session Date & Time**:- 01074 (1/3) : 5B (Aug.25, 10:40-12:20)
- 01074 (2/3) : 5C (Aug.25, 13:20-15:00)
- 01074 (3/3) : 5D (Aug.25, 15:30-17:10)

**Type**: Proposal of Minisymposium**Abstract**: Approximation theory is a subject that serves as an important bridge between pure and applied mathematics. It has become a very important branch of mathematics and is of fundamental support of many new disciplines and research areas. The proposed minisymposium aims to merge together active researchers in the following topics: polynomial inequalities in the multivariate real and complex fields, pluripotential numerics, kernel-based approximation, generalized sampling type operators and exponential sampling. Special attention will be given to applications, modeling, as well as computational and numerical aspects in approximation.**Organizer(s)**: Leokadia Bialas-Ciez, Stefano De Marchi**Classification**:__41Axx__,__65Dxx__,__42A10__**Speakers Info**:- Stefano De Marchi (University of Padova)
- Mariusz Bieniek (Maria Curie-Skłodowska University in Lublin)
- Leszek Plaskota (University of Warsaw)
- Maciej Klimek (Uppsala University)
- Mieczyslaw Mastylo (Adam Mickiewicz University in Poznan)
- Janin Jaeger (Justus-Liebig University)
- Francesco Marchetti (University of Padova)
- Elisabeth Larsson (Uppsala University)
- Giovanni Migliorati (Sorbonne University)
- Michael Slawinski (Memorial University of Newfoundland)

**Talks in Minisymposium**:**[04256] Approximation Theory, Approximation Methods and Applications: an introduction****Author(s)**:**Stefano De Marchi**(University of Padova)

**Abstract**: In this initial talk, I will briefly summarize why we have proposed such a mini-symposium, why approximation theory is still an important research subject for numerical and analytical analysts, and why approximation theory can help industrial applications. I will also outline the main contributions provided by all the speakers invited to this mini-symposium.

**[04857] On the quality of adaptive methods for numerical approximation****Author(s)**:**Leszek Plaskota**(University of Warsaw)

**Abstract**: Methods for numerical approximation generally fall into two categories: nonadaptive algorithms and adaptive algorithms. By ‘adaptive’ we mean that in its successive steps the algorithm uses information about the underlying function obtained from the previous steps. If the function possesses some singularities and is otherwise smooth, then adaption is necessary to restore the right convergence rate. For globally smooth functions adaptive algorithms can essentially lower asymptotic constants. We present recent quantitative results on the subject.

**[04984] Interpolation on the sphere using series kernels****Author(s)**:**Janin Jäger**(KU Eichstätt-Ingolstadt)- Simon Hubbert (Birkbeck University of London)

**Abstract**: We study the use of series kernels for interpolation and approximation of data on a d- dimensional sphere. Kernels in series representation allow us to easily deduce geometric properties of the kernel, strict positive definiteness of the kernel, smoothness of the approximant and approximation error estimates. We will show how series representations can be derived for restrictions of radial basis functions from the surrounding Euclidean space to the sphere and state the explicit expansion for the generalised Wendland functions.

**[05019] Monte Carlo approximation of non-autonomous Julia sets****Author(s)**:**Maciej Klimek**(Uppsala University)

**Abstract**: In the the metric space of compact, pluriregular and polynomially convex subsets of $\mathbb{C}^N$, both finite and infinite families of proper polynomial mappings generate a variety of Julia type compact sets, known also as the composite Julia sets. These sets can be interpreted as attractors of generalized iterated function systems. Since stochastic approximation of composite Julia sets is viable, some of those sets can be visualized with the help of Monte Carlo methods.

**[05064] (β,γ)-Chebyshev functions and points****Author(s)**:**Francesco Marchetti**(University of Padova)

**Abstract**: $(\beta,\gamma)$-Chebyshev functions are a generalization of classical Chebyshev polynomials of the first kind. They consist of a family of orthogonal functions on a subset of $[-1,1]$, which indeed satisfies a three-term recurrence formula and complies with various properties of classical families of orthogonal polynomials. Moreover, for certain configurations of parameters $\beta$ and $\gamma$, the roots of $(\beta,\gamma)$-Chebyshev functions contained in the corresponding orthogonality interval lead to a stable polynomial interpolation process.

**[05085] Optimal scaling of radial basis function approximations****Author(s)**:**Elisabeth Larsson**(Uppsala University)- Boštjan Mavrič (Institute of Metals and Technology)
- Andreas Michael (Uppsala University)
- Ulrika Sundin (Uppsala University)

**Abstract**: We revisit the question about which shape parameter to use in radial basis function approximations. We compute the flat shape parameter limit of a Gaussian interpolant as an expansion in even powers of the shape parameter and illustrate how the correction terms cancel the dominant part of the error for smooth solution functions. We also provide a closed form expression that explains the shape of the error curves in terms of the shape parameter.

**[05108] On empirical adequacy of approximations within mathematical models****Author(s)**:**Michael A Slawinski**(Memorial University)

**Abstract**: We discuss a phenomenological model formulated to study the power applied by a cyclist on a velodrome. The dissipative forces we consider are air resistance, rolling resistance, lateral friction and drivetrain resistance. Also, the power is used to increase the kinetic and potential energy. Following derivations and justifications of expressions that constitute this mathematical model, we discuss them in the context of measurements.

**[05151] Projection Constants for Spaces of Multivariate Polynomials****Author(s)**:**Mieczysław Mastyło**(Adam Mickiewicz University, Poznań)

**Abstract**: We study the projection constant of spaces of polynomials with the supremum norm over the unit ball of finite dimensional Banach sequence space including the Hilbert space. We consider the action of topological groups over these spaces of polynomials and provide some integral formulas for their projection constants. This cycle of ideas leads to the asymptotic behaviour of the projection constants of spaces of Dirichlet polynomials and the space of nuclear operators on $\ell_2^n$.

**[05170] Stable high-order randomized cubatures for integration in arbitrary dimension****Author(s)**:**Giovanni Migliorati**(Sorbonne Université)

**Abstract**: We present cubature formulae for the integration of functions in arbitrary dimension and arbitrary domain. These cubatures are exact on a given finite-dimensional subpace $V_n$ of $L^2$ of dimension $n$, they are stable with high probability and are constructed using $m$ pointwise evaluations of the integrand function with $m$ proportional to $n \log n$. For these cubatures we provide a convergence analysis showing that the expected cubature error decays as $m^{-1/2}$ times the $L^2$ best approximation of the integrand function in $V_n$.

**[05186] On technical considerations of velodrome track design****Author(s)**:**Theodore Stanoev**(Memorial University of Newfoundland)

**Abstract**: We present a novel approach to velodrome track design. The mathematical model uses differential geometry to form a three-dimensional ruled surface. The track is comprised of straight lines, the arcs of circles, and connecting transition curves, whose features are derived from the Frenet-Serret relations. Symmetric and asymmetric designs are obtained using least-squares optimization. The formulation may be used to design velodrome tracks of a wide variety of track geometry specifications.