论文标题
关于Mercer定理中均匀收敛的速度
On the speed of uniform convergence in Mercer's theorem
论文作者
论文摘要
经典Mercer的定理声称，紧凑型集合的连续积极确定的内核$ k（{\ Mathbf X}，{\ MathBf y}）$可以表示为$ \ sum_ {i = 1}^\inftyλ_ix_i（{\ mathbf x} $ n pher $ \ {（λ_i，ϕ_i）\} $是相应积分运算符的eigenvalue-eigenVector对。众所周知，这种无限表示会均匀地汇合到内核$ k $。我们根据特征值的衰减速度估算了这种收敛的速度，并证明，$ $ k $的第一个$ n $在$ 2M $ $ times乘以$ k $的第一个$ n $条款为$ \ Mathcal {o} \ big（\ sum_（\ \ sum_（\ \ sum_ = n+1}或$ \ MATHCAL {O} \ big（（（\ sum_ {i = n+1}^\inftyλ^2_i）^{\ frac {m} {m} {2m+n}}} \ big）$。最后，我们证明了结果的某些应用在具有连续根和其他力量的整体运营商的光谱交流中。
The classical Mercer's theorem claims that a continuous positive definite kernel $K({\mathbf x}, {\mathbf y})$ on a compact set can be represented as $\sum_{i=1}^\infty λ_iϕ_i({\mathbf x})ϕ_i({\mathbf y})$ where $\{(λ_i,ϕ_i)\}$ are eigenvalue-eigenvector pairs of the corresponding integral operator. This infinite representation is known to converge uniformly to the kernel $K$. We estimate the speed of this convergence in terms of the decay rate of eigenvalues and demonstrate that for $2m$ times differentiable kernels the first $N$ terms of the series approximate $K$ as $\mathcal{O}\big((\sum_{i=N+1}^\inftyλ_i)^{\frac{m}{m+n}}\big)$ or $\mathcal{O}\big((\sum_{i=N+1}^\inftyλ^2_i)^{\frac{m}{2m+n}}\big)$. Finally, we demonstrate some applications of our results to a spectral charaterization of integral operators with continuous roots and other powers.
