多重级数乘子算子的熵数估计
Estimates for entropy numbers of multiplier operators of multiple series
摘要：
研究了概率空间上有界于$L^{\infty}$中的抽象完备规范正交系$\{ϕ_{\textbf{m}}\}_{\textbf{m}\in \mathbb{N}^d_0}$的多重级数的一般Fourier乘子算子熵数的渐近性态.规范正交系可以是$ϕ_{\textbf{m}}(\textbf{x}) = ϕ_{m_1}^{(1)}(x_1)\cdots ϕ_{m_d}^{(d)}(x_d)$类型，其中每个$\left\{ϕ_{l}^{(j)}\right\}_{l\in \mathbb{N}_0}$是规范正交系，标准正交系统对于每个$j$可以是不同的，例如，它可以是Vilenkin系统、实球上的Walsh系统或单位圆上的三角系统。利用由规范正交系构造的Levy范数平均建立了熵数的一般下界。这些结果被应用到特定的乘子算子的熵数的上界和下界，在特定的情况下，生成的集合的可微和无限可微函数，在通常的意义上，并在并矢意义上。结果表明，这些估计在各种重要情况下都是有阶的。
英文摘要：
The asymptotic behavior for entropy numbers of general Fourier multiplier operators of multiple series with respect to an abstract complete orthonormal system $\{ϕ_{\textbf{m}}\}_{\textbf{m}\in \mathbb{N}^d_0}$ on a probability space and bounded in $L^{\infty}$, is studied. The orthonormal system can be of the type $ϕ_{\textbf{m}}(\textbf{x}) = ϕ_{m_1}^{(1)}(x_1)\cdots ϕ_{m_d}^{(d)}(x_d)$, where each $\left\{ϕ_{l}^{(j)}\right\}_{l\in \mathbb{N}_0}$ is an orthonormal system, that can be different for each $j$, for example, it can be a Vilenkin system, a Walsh system on a real sphere or the trigonometric system on the unit circle. General upper and lower bounds for the entropy numbers are established by using Levy means of norms constructed using the orthonormal system. These results are applied to get upper and lower bounds for entropy numbers of specific multiplier operators, which generate, in particular cases, sets of finitely and infinitely differentiable functions, in the usual sense and in the dyadic sense. It is shown that these estimates have order sharp in various important cases.
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