We investigate $L_2$-approximation problems in the worst case setting in the weighted Hilbert spaces $H\left(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}\right)$ with weights $R_{d,\mathit{\alpha },\mathit{\gamma }}$ under parameters $1\ge {\gamma }_1\ge {\gamma }_2\ge \cdots \ge 0$ and $1<{\alpha }_1\le {\alpha }_2\le \cdots$. Several interesting weighted Hilbert spaces $H\left(K_{R_{d,\mathit{\alpha },\mathit{\gamma }}}\right)$ appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss tractability of $L_2$-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on d and ${\epsilon }^{-1}$. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and $(t_1,t_2)$-weak tractability ($(t_1,t_2)$-WT) for all $t_1>1$ and $t_2>0$ in terms of the introduced weights under the absolute error criterion or the normalized error criterion.

We investigate \texorpdfstring{$L_2$}{}-approximation problems in the worst case setting in the weighted Hilbert spaces \texorpdfstring{$H(K_{R_{d,\a,\g}})$}{} with weights \texorpdfstring{$R_{d,\a,{\bm \ga}}$}{} under parameters \texorpdfstring{$1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $1<\az_1\le \az_2\le \cdots$}{}. Several interesting weighted Hilbert spaces \texorpdfstring{$H(K_{R_{d,\a,\g}})$}{} appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. Multivariate approximation; information complexity; tractability; weighted Hilbert spacesWe discuss tractability of \texorpdfstring{$L_2$}{}-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on \texorpdfstring{$d$}{} and \texorpdfstring{$\va^{-1}$}{}. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and \texorpdfstring{$(t_1,t_2)$}{}-weak tractability (\texorpdfstring{$(t_1,t_2)$}{}-WT) for all \texorpdfstring{$t_1>1$}{} and \texorpdfstring{$t_2>0$}{} in terms of the introduced weights under the absolute error criterion or the normalized error criterion.

We study L_2-approximation problems in the weighted Hilbert spaces in the worst case setting. Three interesting weighted Hilbert spaces appear in this paper, whose weights are equipped with two positive parameters $\ga_j$ and $\az_j$ for $j=1,\,2,\dots,\,d$. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss the exponential convergence-(t,s)-weak tractability (EC-(t,s)-WT) of these L_2-approximation problems under the absolute or normalized error criterion. In particular, we obtain the sufficient and necessary conditions for EC-(1,1)-WT and EC-(t,1)-WT with t

We study the multivariate $L_2$-approximation problem ${\mathrm{APP}}_d$ defined over a Banach space in the average case setting. The space is equipped with a zero-mean Gaussian measure with a weighted covariance kernel, which depends on parameter sequences $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\beta }={\left\{{\beta }_j\right\}}_{j\in \mathbb{N}}$ with $1<{\alpha }_1\le {\alpha }_2\le \cdots$ and $1\ge {\beta }_1\ge {\beta }_2\ge \cdots >0$. In this paper, two interesting weighted covariance kernels are considered, which model the importance of the covariance kernels. Under the absolute error criterion or the normalized error criterion, we discuss $(s,t)$-weak tractability of the $L_2$-approximation problem $\mathrm{APP}={\left\{{\mathrm{APP}}_d\right\}}_{d\in \mathbb{N}}$ from a Banach space whose zero-mean Gaussian measure has the above two weighted covariance kernels for some positive numbers s and t in the average case setting. Here, $(s,t)$-weak tractability means how the information complexity behaves as a function of $d^t$ and ${\epsilon }^{-s}$ for large dimension d and small threshold $\epsilon$. In particular, for all $s>0$ and $t\in (0,1)$, we find the matching sufficient and necessary condition on the parameter sequences $\mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j\in \mathbb{N}}$ and $\mathit{\beta }={\left\{{\beta }_j\right\}}_{j\in \mathbb{N}}$ to obtain $(s,t)$-weak tractability under the absolute error criterion or the normalized error criterion in the average case setting. We describe $(s,t)$-weak tractability by the matching sufficient and necessary condition, which reflects symmetry.

We study $L_2$-approximation problems in the weighted Hilbert spaces in the worst case setting. Three interesting weighted Hilbert spaces appear in this paper, whose weights are equipped with two positive parameters ${\gamma }_j$ and ${\alpha }_j$ for $j\in \mathbb{N}$. We consider algorithms using the class of arbitrary linear functionals. We discuss the exponential convergence-$(t,s)$-weak tractability of these $L_2$-approximation problems. In particular, we obtain the sufficient and necessary conditions on the weights for exponential convergence-weak tractability and exponential convergence-$(t,1)$-weak tractability with $t<1$.

We study $L_2$-approximation problems $\text{APP}_d$ in the worst case setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameter sequences ${\bm \ga}=\{\ga_j\}$ and $\a=\{\az_j\}$ of positive real numbers $1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $\frac1 2<\az_1\le \az_2\le \cdots$. We consider the minimal worst case error $e(n,\text{APP}_d)$ of algorithms that use $n$ arbitrary continuous linear functionals with $d$ variables. We study polynomial convergence of the minimal worst case error, which means that $e(n,\text{APP}_d)$ converges to zero polynomially fast with increasing $n$. We recall the notions of polynomial, strongly polynomial, weak and $(t_1,t_2)$-weak tractability. In particular, polynomial tractability means that we need a polynomial number of arbitrary continuous linear functionals in $d$ and $\va^{-1}$ with the accuracy $\va$ of the approximation. We obtain that the matching necessary and sufficient condition on the sequences ${\bm \ga}$ and $\a$ for strongly polynomial tractability or polynomial tractability is $$\dz:=\liminf_{j\to\infty}\frac{\ln \ga_j^{-1}}{\ln j}>0,$$ and the exponent of strongly polynomial tractability is $$p^{\text{str}}=2\max\big\{\frac 1 \dz, \frac 1 {2\az_1}\big\}.$$