Related papers: Cardinal Interpolation With General Multiquadrics:…
This paper studies the cardinal interpolation operators associated with the general multiquadrics, $\phi_{\alpha,c}(x) = (\|x\|^2+c^2)^\alpha$, $x\in\mathbb{R}^d$. These operators take the form $$\mathscr{I}_{\alpha,c}\mathbf{y}(x) =…
In this paper, interpolation by scaled multi-integer translates of Gaussian kernels is studied. The main result establishes $L_p$ Sobolev error estimates and shows that the error is controlled by the $L_p$ multiplier norm of a Fourier…
If $f\in \{f\in L^p(\mathbb{R}): f(x)=\int_{-\pi}^{\pi}e^{ix\xi}d\beta(\xi), \beta\in B.V.([-\pi,\pi]) \}$, then $f$ is determined by its samples on the integers by taking an appropriate limit. Specifically, $\| f - L_{\phi_\alpha}f…
A new generalization of multiquadric functions $\phi(x)=\sqrt{c^{2d}+||x||^{2d}}$, where $x\in\mathbb{R}^n$, $c\in \mathbb{R}$, $d\in \mathbb{N}$, is presented to increase the accuracy of quasi-interpolation further. With the restriction to…
A new generalization of shifted thin plate splines $$\varphi(x)=(c^{2d}+||x||^{2d})\log\left(c^{2d}+||x||^{2d}\right),\qquad x\in\mathbb{R}^n, d\in \mathbb{N}, c>0$$ is presented to increase the accuracy of quasi-interpolation further. With…
We prove interpolation estimates between Morrey-Campanato spaces and Sobolev spaces. These estimates give in particular concentration-compactness inequalities in the translation-invariant and in the translation- and dilation-invariant case.…
The interpolation of couples of separable Hilbert spaces with a function parameter is studied. The main properties of the classic interpolation are proved. Some applications to the interpolation of isotropic H\"ormander spaces over a closed…
Cocompactness is a useful weaker counterpart of compactness in the study of imbeddings between function spaces. In this paper we show that subcritical continuous imbeddings of fractional Sobolev spaces and Besov spaces over \mathbb{R}^{N}…
In this paper, applied strictly monotonic increasing scaled maps, a kind of well-conditioned linear barycentric rational interpolations are proposed to approximate functions of singularities at the origin, such as $x^\alpha$ for $\alpha \in…
We explicitly describe all Hilbert function spaces that are interpolation spaces with respect to a given couple of Sobolev inner product spaces considered over $\mathbb{R}^{n}$ or a half-space in $\mathbb{R}^{n}$ or a bounded Euclidean…
Convergence rates for $L_2$ approximation in a Hilbert space $H$ are a central theme in numerical analysis. The present work is inspired by Schaback (Math. Comp., 1999), who showed, in the context of best pointwise approximation for radial…
Given $E_0, E_1, F_0, F_1, E$ rearrangement invariant function spaces, $a_0$, $a_1$, $b_0$, $b_1$, $b$ slowly varying functions and $0< \theta_0<\theta_1<1$, we characterize the interpolation spaces $$(\overline{X}^{\mathcal…
Interpolation inequalities play an important role in the study of PDEs and their applications. There are still some interesting open questions and problems that related to integral estimates and regularity of solutions to the elliptic…
Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters…
This paper constructs unique compactly supported functions in Sobolev spaces that have minimal norm, maximal support, and maximal central value, under certain renormalizations. They may serve as optimized basis functions in interpolation or…
A \Riesz-basis sequence for $L_2[-\pi,\pi]$ is a strictly increasing sequence $X:=(x_j)_{j\in\mathbb{Z}}$ in $\mathbb{R}$ such that the set of functions $\left(e^{-ix_j(\cdot)}\right)_{j\in\mathbb{Z}}$ is a Riesz basis for $L_2[-\pi,\pi]$.…
We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions $\widetilde{\varphi}_j$ and trigonometric polynomials $\varphi_j$. The class of such operators…
This paper studies the influence of scaling on the behavior of Radial Basis Function interpolation. It focuses on certain central aspects, but does not try to be exhaustive. The most important questions are: How does the error of a…
Polynomial approximation is studied in the Sobolev space $W_p^r(w_{\alpha,\beta})$ that consists of functions whose $r$-th derivatives are in weighted $L^p$ space with the Jacobi weight function $w_{\alpha,\beta}$. This requires…
This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert…