We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an $\epsilon$-approximation for the $d$-variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on $d$ and logarithmically on ${\epsilon }^{-1}$. The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on $d$ and logarithmic dependence on ${\epsilon }^{-1}$, we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential $(s,t)$-weak tractability with $max(s,t)\ge 1$. For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential $(s,t)$-weak tractability with $max(s,t)<1$ is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).

我们研究在某些加权张量积希尔伯特空间上定义的紧凑线性算子的逼近。信息复杂度定义为获得$\epsilon$-的近似值 $d$变量问题，该问题完全根据权重和单变量奇异值确定。指数可延展性意味着信息复杂性受多项式依赖的某个函数的限制$d$ 和对数地 ${\epsilon }^{-\mathrm{1个}}$。Hickernell等人研究了相应的未加权问题。（2020）对指数可延展性有许多负面结果。本文研究的产品权重改变了这种情况。取决于多项式形式的依赖$d$ 和对数依赖 ${\epsilon }^{-\mathrm{1个}}$，我们研究指数强多项式，指数多项式，指数拟多项式和指数 $（s，Ť）$-具有弱的易处理性 $最高（s，Ť）\ge \mathrm{1个}$。对于所有这些指数可伸缩性概念，我们在权重和单变量奇异值上建立了必要且充分的条件，对于这些条件，确实有可能实现相应的指数可伸缩性概念。指数的情况$（s，Ť）$-具有弱的易处理性 $最高（s，Ť）<\mathrm{1个}$留待将来研究。本文使用了在Hickernell等人中获得的一些一般结果。（2020）和Kritzer和Woźniakowski（2019）。