For 1≤ p,q≤∞, the Nikolskii factor for a trigonometric polynomial T_𝐚 is defined by N_p,q(T_𝐚)= T_𝐚_q/T_𝐚_p, T_𝐚(x)=a_1+∑ ^n_k=1(a_2k√(2)cos kx+a_2k+1√(2)sin kx). We study this average Nikolskii factor for random trigonometric polynomials with independent N(0,σ ^2) coefficients and obtain the exact orders. For 1≤ p< q<∞, the average Nikolskii factor is of order n^0 (i.e., constant), as compared to the worst case bound of order n^1/p-1/q, and for 1≤ p< q=∞, the average Nikolskii factor is of order (ln n)^1/2 as compared to the worst case bound of order n^1/p. We also give the generalization to random multivariate trigonometric polynomials.

We investigate average case tractability of approximation of additive random fields with marginal random processes corresponding to the Korobov kernels for the non-homogeneous case. We use the absolute error criterion (ABS) or the normalized error criterion (NOR). We show that the problem is always polynomially tractable for ABS or NOR, and give sufficient and necessary conditions for strong polynomial tractability for ABS or NOR.

This paper is devoted to discussing the linear tensor product problems in the worst case setting. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate algebraic (s, t)-weak tractability (ALG-(s, t)-WT) under the absolute error criterion in the case λ_1 > 1, where λ_1 is the square of the univariate maximal singular value. We solve the problem by giving the necessary and sufficient conditions for ALG-(s, t)-WT on univariate singular values and fill the gap left open.

In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in L^2(𝕊^d) for Sobolev spaces H^α ,β(𝕊^d) with logarithmic perturbation on the unit sphere 𝕊^d in ℝ^d+1 . First we obtain strong equivalences of the approximation numbers for H^α ,β(𝕊^d) with α >0 , which gives a clue to Open problem 3 as posed by Krieg and Vybíral in [31]. Second, for the optimal quadrature errors for H^α ,β(𝕊^d) , we use the “fooling" function technique to get lower bounds in the case α >d/2 , and apply Hilbert space structure and Vybíral’s theorem about Schur product theory to obtain lower bounds in the case α =d/2, β >1/2 of small smoothness, which confirms the conjecture as posed by Grabner and Stepanyuk in [16] and solves Open problem 2 in [31]. Finally, we employ the Weighted least squares operators and the least squares quadrature rules to obtain approximation theorems and quadrature errors for H^α ,β(𝕊^d) with α >d/2 or α =d/2, β >1/2 , which are order optimal.

Let Λ⊆Rd be a discrete uniformly separated subset. In the unweighted case and p=2, Landau obtained the necessary conditions for sampling and interpolation of functions in Paley–Wiener space in terms of the upper and lower uniform densities of Λ. In this paper, we generalize the above results to the weighted case, and give some necessary density conditions for Lp weighted sampling and interpolating sets for all 0

We consider multivariate approximation problems in the average case setting with a zero mean Gaussian measure whose covariance kernel is a periodic Gevrey kernel. We investigate various notions of algebraic tractability and exponential tractability, and obtain necessary and sufficient conditions in terms of the parameters of the problem.

In this paper, a Riesz variational characterization of the first-order Sobolev space W_p^1(𝕊^d) , d

For $1\le p,q\le \infty$, the Nikolskii factor for a diffusion polynomial $P_{\bf a}$ of degree at most $n$ is defined by $$N_{p,q}(P_{\bf a})=\frac{\|P_{\bf a}\|_{q}}{\|P_{\bf a}\|_{p}},\ \ P_{\bf a}({\bf x})=\sum_{k:\lambda_{k}\leq n}a_{k}\phi_{k}({\bf x}),$$ where ${\bf a}=\{a_k\}_{\lambda_k\le n}$, and $\{(\phi_k,-\lambda_k^2)\}_{k=0}^\infty$ are the eigenpairs of the Laplace-Beltrami operator $\Delta_{\mathbb M}$ on a closed smooth Riemannian manifold $\mathbb M$ with normalized Riemannian measure. We study this average Nikolskii factor for random diffusion polynomials with independent $N(0,\sigma^{2})$ coefficients and obtain the exact orders. For $1\leq p

For 1≤ p,q≤∞, the Nikolskii factor for a diffusion polynomial P_ a of degree at most n is defined by N_p,q(P_ a)=P_ a_q/P_ a_p, P_ a( x)=∑_k:λ_k≤ na_kϕ_k( x), where a={a_k}_λ_k≤ n, and {(ϕ_k,-λ_k^2)}_k=0^∞ are the eigenpairs of the Laplace-Beltrami operator Δ_𝕄 on a closed smooth Riemannian manifold 𝕄 with normalized Riemannian measure. We study this average Nikolskii factor for random diffusion polynomials with independent N(0,σ^2) coefficients and obtain the exact orders. For 1≤ p<q<∞, the average Nikolskii factor is of order n^0 (i.e., constant), as compared to the worst case bound of order n^d(1/p-1/q), and for 1≤ p<q=∞, the average Nikolskii factor is of order (ln n)^1/2 as compared to the worst case bound of order n^d/p.

This paper is devoted to discussing the weighted linear tensor product problems in the worst case setting. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate exponential (s, t)-weak tractability (EXP-(s, t)-WT) with max(s,t)<1 and exponential uniform weak tractability (EXP-UWT) under the absolute or normalized error criterion. We solve the problem by filling the remaining gaps left open on EXP-tractability. That is, we obtain necessary and sufficient conditions for EXP-(s, t)-WT with max(s, t) < 1 and for EXP-UWT.

In this paper, we obtain some exact L_2 Bernstein-Markov inequalities for generalized Hermite and Gegenbauer weight. More precisely, we determine the exact values of the extremal problem M_n^2(L_2(W_λ), D):=sup_0≠ p∈𝒫_n∫_I| D p(x)|^2W_λ(x) dx/∫_I| p(x)|^2W_λ(x) dx,λ>0, where 𝒫_n denotes the set of all algebraic polynomials of degree at most n, D is the differential operator given by D={ d/ dx or𝒟_λ, if W_λ(x)=|x|^2λe^-x^2 and I=ℝ, (1-x^2)^1/2 d/ dx or(1-x^2)^1/2 𝒟_λ, ifW_λ(x):=|x|^2λ(1-x^2)^μ-1/2,μ>-1/2 andI=[-1,1], and 𝒟_λ is the univariate Dunkl operator, i.e., 𝒟_λ f(x)=f'(x)+λ(f(x)-f(-x))/x. Furthermore, the corresponding extremal polynomials are also obtained.

Given a sequence of Marcinkiewicz–Zygmund inequalities in L_2 on a compact space, Gröchenig (J Approx Theory 257:105455, 2020) discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all 1≤ p≤∞ , we develop weighted least ℓ _p approximation induced by a sequence of Marcinkiewicz–Zygmund inequalities in L_p on a compact smooth Riemannian manifold 𝕄 with normalized Riemannian measure (typical examples are the torus and the sphere). In this paper we derive corresponding approximation theorems with the error measured in L_q, 1≤ q≤∞ , and least quadrature errors for both Sobolev spaces H_p^r(𝕄), r>d/p generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces B_p,γ^r(𝕄), 0<γ≤∞ , r>d/p defined by best ”polynomial” approximation. Finally, we discuss the optimality of the obtained results by giving sharp estimates of sampling numbers and optimal quadrature errors for the aforementioned spaces.

We study multivariate approximation of periodic functions in the worst case setting with the error measured in the L∞ norm. We consider algorithms that use standard information Λstd consisting of function values or general linear information Λall consisting of arbitrary continuous linear functionals. We investigate equivalences of various notions of algebraic and exponential tractability for Λstd and Λall under the absolute or normalized error criterion, and show that the power of Λstd is the same as the one of Λall for various notions of algebraic and exponential tractability. Our results can be applied to weighted Korobov spaces and Korobov spaces with exponential weights. This gives a special solution to Open Problem 145 as posed by Novak and Woźniakowski (2012) [40].

In this paper, various inequalities for Paley-Wiener space with doubling (or A∞) weights are proved in the multivariate setting. This includes Bernstein-type, Schur-type, Plancherel-Pólya-type, Logvinenko-Sereda-type, weak Remez-type, and Nikolskii-type inequalities, etc.

Let f be a function defined on the real line, and T-f be the corresponding superposition operator which maps hh to T-f(h) , i.e., T-f(h)=f degrees h . In this article, the sufficient and necessary conditions such that T-f maps periodic Holder-Lipschitz spaces H-p(alpha) into itself with 0<alpha<1/p and 1/p<alpha<1 , where alpha- is the smoothness index, are shown. Our result in the case 0<alpha<1p may be the first result about the superposition operator problems of smooth function space containing unbounded functions.

Consider the numerical integration Int_𝕊^d,w(f)=∫_𝕊^df( x)w( x) dσ( x) for weighted Sobolev classes BW_p,w^r(𝕊^d) with a Dunkl weight w and weighted Besov classes BB_γ^Θ(L_p,w(𝕊^d)) with the generalized smoothness index Θ and a doubling weight w on the unit sphere 𝕊^d of the Euclidean space ℝ^d+1 in the deterministic and randomized case settings. For BW_p,w^r(𝕊^d) we obtain the optimal quadrature errors in both settings. For BB_γ^Θ(L_p,w(𝕊^d)) we use the weighted least ℓ_p approximation and the standard Monte Carlo algorithm to obtain upper estimates of the quadrature errors which are optimal if w is an A_∞ weight in the deterministic case setting or if w is a product weight in the randomized case setting. Our results show that randomized algorithms can provide a faster convergence rate than that of the deterministic ones when p>1. Similar results are also established on the unit ball and the standard simplex of ℝ^d.

In this note,we obtain the sufficient and necessary conditions for the inclusions ΛφBV⊂BV(q,δ),and BV(q,δ)CΛφBV is false when φ satisfies the Δ2 condition.

Let Lp,w,1≤p<∞, denote the weighted Lp space of functions on the unit ball Bd with a doubling weight w on Bd. The Markov factor for Lp,w of a polynomial P is defined by ‖|∇P|‖p,w‖P‖p,w, where ∇P is the gradient of P. We investigate the worst case Markov factors for Lp,w and prove that the degree of these factors is at most 2. In particular, for the Gegenbauer weight wμ(x)=(1−|x|2)μ−1/2,μ≥0, the exponent 2 is sharp. We also study the average case Markov factor for L2,w on random polynomials with independent N(0,σ2) coefficients and obtain that the upper bound of the average (expected) Markov factor is order degree to the 3/2, as compared to the degree squared worst case upper bound.

We consider the numerical integrationINTd(f)=∫Bdf(x)wμ(x)dx for the weighted Sobolev classes BWp,μr and the weighted Besov classes BBτr(Lp,μ) in the randomized case setting, where wμ,μ≥0, is the classical Jacobi weight on the ball Bd, 1≤p≤∞, r>(d+2μ)/p, and 0<τ≤∞. For the above two classes, we obtain the orders of the optimal quadrature errors in the randomized case setting are n−r/d−1/2+(1/p−1/2)+. Compared to the orders n−r/d of the optimal quadrature errors in the deterministic case setting, randomness can effectively improve the order of convergence when p>1.

We study multivariate approximation in the average case setting with the error measured in the weighted L2 norm. We consider algorithms that use standard information Λstd consisting of function values or general linear information Λall consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for Λstd and Λall for the absolute error criterion, and show that the power of Λstd is the same as that of Λall for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 116-118 and almost solve Open Problem 115 as posed by Novak and Woźniakowski (2012) [25].