Related papers: Multivariate exact and falsified sampling approxim…
Several new inequalities for moduli of smoothness and errors of the best approximation of a function and its derivatives in the spaces $L_p$, $0<p<1$, are obtained. For example, it is shown that for any $0<p<1$ and $k,\,r\in \mathbb{N}$ one…
In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange…
A generalized sampling theorem for frequency localized signals is presented. The generalization in the proposed model of sampling is twofold: (1) It applies to various prefilters effecting a "soft" bandlimitation, (2) an approximate…
We present an approximation scheme for functions in three dimensions, that requires only their samples on the Cartesian grid, under the assumption that the functions are sufficiently concentrated in both space and frequency. The scheme is…
Approximation properties of quasi-projection operators $Q_j(f,\varphi, \widetilde{\varphi})$ are studied. Such an operator is associated with a function $\varphi$ satisfying the Strang-Fix conditions and a tempered distribution…
In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We derive formulas for the uniform approximation…
We study in this paper the function approximation error of multivariate linear extrapolation. The sharp error bound of linear interpolation already exists in the literature. However, linear extrapolation is used far more often in…
A function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is referred to as a Sparse Additive Model (SPAM), if it is of the form $f(\mathbf{x}) = \sum_{l \in \mathcal{S}}\phi_{l}(x_l)$, where $\mathcal{S} \subset [d]$, $|\mathcal{S}| \ll d$.…
In a previous paper [Adcock & Huybrechs, 2019] we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but…
We study computational methods for the approximation of special functions recurrent in geometric function theory and quasiconformal mapping theory. The functions studied can be expressed as quotients of complete elliptic integrals and as…
A sharper estimate for the summatory Euler phi function $\sum_{n \leq x} \varphi(n)$ is presented in this work. It improves the established estimate in the current mathematical literature. In addition, an estimate for its reciprocal…
In this paper, we investigate the convergence properties of Fourier partial sums associated with general orthonormal systems, focusing on functions that belong to specific differentiable function classes. While classical Fourier analysis…
In our previous publications we have introduced the cosine product-to-sum identity [17] $$ \prod\limits_{m = 1}^M {\cos \left( {\frac{t}{{{2^m}}}} \right)} = \frac{1}{{{2^{M - 1}}}}\sum\limits_{m = 1}^{{2^{M - 1}}} {\cos \left( {\frac{{2m -…
In this work, we consider the approximate reconstruction of high-dimensional periodic functions based on sampling values. As sampling schemes, we utilize so-called reconstructing multiple rank-1 lattices, which combine several preferable…
In this paper, we introduce Mellin-Steklov exponential samplingoperators of order $r,r\in\mathbb{N}$, by considering appropriate Mellin-Steklov integrals. We investigate the approximation properties of these operators in continuousbounded…
The purpose of this article is to provide new error estimates for a popular type of SBF approximation on the sphere: approximating by linear combinations of Green's functions of polyharmonic differential operators. We show that the $L_p$…
In some applications, one is interested in reconstructing a function $f$ from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we…
We study approximation properties of linear sampling operators in the spaces $L_p$ for $1\le p<\infty$. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of…
A function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is a Sparse Additive Model (SPAM), if it is of the form $f(\mathbf{x}) = \sum_{l \in \mathcal{S}}\phi_{l}(x_l)$ where $\mathcal{S} \subset [d]$, $|\mathcal{S}| \ll d$. Assuming $\phi$'s,…
Fourier extensions have been shown to be an effective means for the approximation of smooth, nonperiodic functions on bounded intervals given their values on an equispaced, or in general, scattered grid. Related to this method are two…