Related papers: Optimal quadrature errors and sampling numbers for…
In this paper, we investigate optimal linear approximations ($n$-approximation numbers ) of the embeddings from the Sobolev spaces $H^r\ (r>0)$ for various equivalent norms and the Gevrey type spaces $G^{\alpha,\beta}\ (\alpha,\beta>0)$ on…
This paper deals with the construction of an optimal quadrature formula for the approximation of Fourier integrals in the Sobolev space $L_2^{(1)}[a,b]$ of non-periodic, complex valued functions which are square integrable with first order…
The paper studies Sard's problem on construction of optimal quadrature formulas in the space $W_2^{(m,0)}$ by Sobolev's method. This problem consists of two parts: first calculating the norm of the error functional and then finding the…
In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space $L_2^{(m)}(0,1)$is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first…
The Sobolev space $H^{s}(\mathbb{R}^{d})$, where $s > d/2$, is an important function space that has many applications in various areas of research. Attributed to the inertia of a measuring instrument, it is desirable in sampling theory to…
We are concerned with the numerical integration of functions from the Sobolev space $H^{r,\text{mix}}([0,1]^d)$ of dominating mixed smoothness $r\in\mathbb{N}$ over the $d$-dimensional unit cube. In 1976, K. K. Frolov introduced a…
In the Sobolev space $L_2^{(m)}(0,1)$ optimal quadrature formulas with the nodes (1.5) are investigated. For optimal coefficients explicit form are obtained and norm of the error functional is calculated. In particular, by choosing…
In this paper in the space $L_2^{(m)}(0,1)$ the problem of construction of optimal quadrature formulas is considered. Here the quadrature sum consists on values of integrand at nodes and values of first derivative of integrand at the end…
In this paper we study the worst-case error of numerical integration on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, for certain spaces of continuous functions on $\mathbb{S}^{d}$. For the classical Sobolev spaces…
Consider the numerical integration $${\rm Int}_{\mathbb S^d,w}(f)=\int_{\mathbb S^d}f({\bf x})w({\bf x}){\rm d}\sigma({\bf x}) $$ for weighted Sobolev classes $BW_{p,w}^r(\mathbb S^d)$ with a Dunkl weight $w$ and weighted Besov classes…
Using tools from the theory of operator ideals and s-numbers, we develop a general approach to transfer estimates for $L_2$ -approximation of Sobolev functions into estimates for $L_\infty$-approximation, with precise control of all…
The Sobolev space $H^{\varsigma}(\mathbb{R}^{d})$, where $\varsigma > d/2$, is an important function space that has many applications in various areas of research. Attributed to the inertia of a measurement instrument, it is desirable in…
Function values are, in some sense, "almost as good" as general linear information for $L_2$-approximation (optimal recovery, data assimilation) of functions from a reproducing kernel Hilbert space. This was recently proved by new upper…
In this paper, we study the approximation problem for functions in the Gaussian-weighted Sobolev space $W^\alpha_p(\mathbb{R}^d, \gamma)$ of mixed smoothness $\alpha \in \mathbb{N}$ with error measured in the Gaussian-weighted space…
In the present paper, optimal quadrature formulas in the sense of Sard are constructed for numerical integration of the integral $\int_a^be^{2\pi i\omega x}\varphi(x)d x$ with $\omega\in \mathbb{R}$ in the Sobolev space $L_2^{(m)}[a,b]$ of…
We study optimal quadrature formulas for arbitrary weighted integrals and integrands from the Sobolev space $H^1([0,1])$. We obtain general formulas for the worst case error depending on the nodes $x_j$. A particular case is the computation…
Let $L_{q,\mu},\, 1\le q<\infty, \ \mu\ge0,$ denote the weighted $L_q$ space with the classical Jacobi weight $w_\mu$ on the ball $\Bbb B^d$. We consider the weighted least $\ell_q$ approximation problem for a given…
In this paper, we deal with several aspects of the universal Frolov cubature method, that is known to achieve optimal asymptotic convergence rates in a broad range of function spaces. Even though every admissible lattice has this favorable…
This paper investigates the numerical approximation of integrals for functions in fractional Gaussian Sobolev spaces $W^s_{p}(\mathbb{R}^d,\gamma)$ with dominating mixed smoothness defined via kernel related to the fractional…
We prove an optimal order error bound in the discrete $H^2(\Omega)$ norm for finite difference approximations of the first boundary-value problem for the biharmonic equation in $n$ space dimensions, with $n \in \{2,\dots,7\}$, whose…