Woula Themistoclakis

Woula Themistoclakis, Marc Van Barel

The paper concerns the uniform polynomial approximation of a function $f$, continuous on the unit Euclidean sphere of ${\mathbb R}^3$ and known only at a finite number of points that are somehow uniformly distributed on the sphere. First we focus on least squares polynomial approximation and prove that the related Lebesgue constants w.r.t. the uniform norm grow at the optimal rate. Then, we consider delayed arithmetic means of least squares polynomials whose degrees vary from $n-m$ up to $n+m$, being $m=\lfloor θn\rfloor$ for any fixed parameter $0<θ<1$. As $n$ tends to infinity, we prove that these polynomials uniformly converge to $f$ at the near-best polynomial approximation rate. Moreover, for fixed $n$, by using the same data points we can further improve the approximation by suitably modulating the action ray $m$ determined by the parameter $θ$. Some numerical experiments are given to illustrate the theoretical results.

Woula Themistoclakis, Marc Van Barel

We investigate the uniform approximation provided by least squares polynomials on the unit Euclidean sphere $\mathbb{S}^q$ in $\mathbb{R}^{q+1}$, with $q\ge 2$. Like any other polynomial projection, the study concerns the growth, as the degree $n$ tends to infinity, of the associated Lebesgue constant, i.e., of the uniform norm of the least squares operator. If the least squares polynomial of degree $n$ is based on a set of points, which are nodes of a positive weighted quadrature rule of degree of exactness $2n$, then we state two different sufficient conditions for having an optimal Lebesgue constant that increases with $n$ at the minimal projections order. Hence, under our assumptions least squares and hyperinterpolation polynomials provide a comparable approximation with respect to the uniform norm.

Donatella Occorsio, Giuliana Ramella, Woula Themistoclakis

We present a new image scaling method both for downscaling and upscaling, running with any scale factor or desired size. The resized image is achieved by sampling a bivariate polynomial which globally interpolates the data at the new scale. The method's particularities lay in both the sampling model and the interpolation polynomial we use. Rather than classical uniform grids, we consider an unusual sampling system based on Chebyshev zeros of the first kind. Such optimal distribution of nodes permits to consider near--best interpolation polynomials defined by a filter of de la Vallée Poussin type. The action ray of this filter provides an additional parameter that can be suitably regulated to improve the approximation. The method has been tested on a significant number of different image datasets. The results are evaluated in qualitative and quantitative terms and compared with other available competitive methods. The perceived quality of the resulting scaled images is such that important details are preserved, and the appearance of artifacts is low. Competitive quality measurement values, good visual quality, limited computational effort, and moderate memory demand make the method suitable for real-world applications.