Metric entropy of integration operators and small ball probabilites for the Brownian sheet
The authors consider the \(d-\)dimensional integration operator \( T_{d}:L_{2}([0,1]^{d})\rightarrow C([0,1]^{d})\) defined as \[ T_{d}f(x_{1},x_{2,}...,x_{d}):=\int_{0}^{1}.... \int_{0}^{1}f(y_{1},...,y_{d})dy_{1}...dy_{d}, f\in L_{2}([0,1]^{d}). \] The main result concerning \(T_{d}\) is the following estimate for its Kolmogorov and entropy numbers. {Theorem 1.} There exists constants \(C_{1}\) and \(C_{2\text{ }}\) such that \[ d_{k}(T_{d})\leq \frac{C_{1}}{k}(1+\log k)^{d-1/2}\text{ and } e_{k}(T_{d})\leq \frac{C_{2}}{k}(1+\log k)^{d-1/2} \] hold for all \(k\geq 1.\)
scientific article; zbMATH DE number 1389110
scientific article