Polyharmonic approximation on the sphere

Approximation on the complex sphere

We develop new elements of harmonic analysis on the complex sphere on the basis of which Bernstein's, Jackson's and Kolmogorov's inequalities are established. We apply these results to get order sharp estimates of $m$-term approximations. The results obtained is a synthesis of new results on classical orthogonal polynomials, harmonic analysis on manifolds and geometric properties of Euclidean spaces.

Weighted Approximation of functions on the unit sphere

The direct and inverse theorems are established for the best approximation in the weighted $L^p$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups. The theorems are stated using a modulus of smoothness of higher order, which is proved to be equivalent to a $K$-functional defined using the power of the spherical $h$-Laplacian. Furthermore, similar results are also established for weighted approximation on the unit ball and on the simplex of $\RR^d$.

Moduli of Smoothness and Approximation on the Unit Sphere and the Unit Ball

A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a $K$-functional, defined via partial derivatives in Euler angles. The set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding $K$-functionals on the unit ball, which are used to characterize the best approximation by polynomials on the ball.

Best Polynomial Approximation on the Unit Sphere and the Unit Ball

This is a survey on best polynomial approximation on the unit sphere and the unit ball. The central problem is to describe the approximation behavior of a function by polynomials via smoothness of the function. A major effort is to identify a correct gadget that characterizes smoothness of functions, either a modulus of smoothness or a $K$- functional, the two of which are often equivalent. We will concentrate on characterization of best approximations, given in terms of direct and converse theorems, and report several moduli of smoothness and $K$-functionals, including recent results that giv…

Approximation numbers of Sobolev and Gevrey type embeddings on the sphere and on the ball -- Preasymptotics, asymptotics, and tractability

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 $G^{α,β}\ (α,β>0)$ on the sphere $\Bbb S^d$ and on the ball $\Bbb B^d$, where the approximation error is measured in the $L_2$-norm. We obtain preasymptotics, asymptotics, and strong equivalences of the above approximation numbers as a function in $n$ and the dimension $d$. We emphasis that all equivalence constants in the above preasymptotics and asymptotics are independent of the dimension $d$ a…

Approximation by boolean sums of Jackson operators on the sphere

This paper concerns the approximation by the Boolean sums of Jackson operators $\oplus^rJ_{k,s}(f)$ on the unit sphere $\mathbb S^{n-1}$ of $\mathbb{R}^{n}$. We prove the following the direct and inverse theorem for $\oplus^rJ_{k,s}(f)$: there are constants $C_1$ and $C_2$ such that \begin{equation*} C_1\|\oplus^rJ_{k,s}f-f\|_p \leq ω^{2r}(f,k^{-1})_p \leq C_2 \max_{v\geq k}\|\oplus^rJ_{k,s}f-f\|_p \end{equation*} for any positive integer $k$ and any $p$th Lebesgue integrable functions $f$ defined on $\mathbb S^{n-1}$, where $ω^{2r}(f,t)_p$ is the modulus of smoothness of degree $2r$ of $f$. W…

On filtered polynomial approximation on the sphere

This paper considers filtered polynomial approximations on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$, obtained by truncating smoothly the Fourier series of an integrable function $f$ with the help of a "filter" $h$, which is a real-valued continuous function on $[0,\infty)$ such that $h(t)=1$ for $t\in[0,1]$ and $h(t)=0$ for $t\ge2$. The resulting "filtered polynomial approximation" (a spherical polynomial of degree $2L-1$) is then made fully discrete by approximating the inner product integrals by an $N$-point cubature rule of suitably high polynomial degree of precision, giving …

The strong converse inequality for de la Vallée Poussin means on the sphere

This paper discusses the approximation by de la Vallée Poussin means $V_nf$ on the unit sphere. Especially, the lower bound of approximation is studied. As a main result, the strong converse inequality for the means is established. Namely, it is proved that there are constants $C_1$ and $C_2$ such that \begin{eqnarray*} C_1ω(f,\frac{1}{\sqrt n})_p \leq \|V_{n}f-f\|_p \leq C_2ω(f,\frac{1}{\sqrt n})_p \end{eqnarray*} for any $p$-th Lebesgue integrable or continuous function $f$ defined on the sphere, where $ω(f,t)_p$ is the modulus of smoothness of $f$.

关于具有有界拉普拉斯算子的函数在球面上的最佳逼近及Bohr–Favard型估计

On best approximation and estimates of Bohr--Favard type on the sphere for functions with bounded Laplacian

该研究确定了球面上具有有界拉普拉斯算子且一阶球谐分量消失的函数的L^∞范数估计的最佳常数，发现S^2、S^3上n≥2时该常数严格小于对应Bohr–Favard常数，还给出了不同维度下常数的表达式及渐近公式。

Distributed filtered hyperinterpolation for noisy data on the sphere

Problems in astrophysics, space weather research and geophysics usually need to analyze noisy big data on the sphere. This paper develops distributed filtered hyperinterpolation for noisy data on the sphere, which assigns the data fitting task to multiple servers to find a good approximation of the mapping of input and output data. For each server, the approximation is a filtered hyperinterpolation on the sphere by a small proportion of quadrature nodes. The distributed strategy allows parallel computing for data processing and model selection and thus reduces computational cost for each serve…