论文标题
在繁殖内核希尔伯特空间中，n-最佳内核近似的足够条件
A sufficient condition for n-Best Kernel Approximation in Reproducing Kernel Hilbert Spaces
论文作者
论文摘要
We show that if a reproducing kernel Hilbert space $H_K,$ consisting of functions defined on ${\bf E},$ enjoys Double Boundary Vanishing Condition (DBVC) and Linear Independent Condition (LIC), then for any preset natural number $n,$ and any function $f\in H_K,$ there exists a set of $n$ parameterized multiple kernels $ {\ tilde {k}} _ {w_1}，\ cdots，{\ tilde {k}} _ {w_n}，w_k \ in {\ bf e}，k = 1，\ cdots，n，n，n，$ and offimation n ofight（n n n n n offor） \ [\ | f- \ sum_ {k = 1}^n c_k {\ tilde {k}} _ {w_k} \ | = \ | = \ | = \ | = \ | f- \ | f- \ sum_ {k = 1} {\ bf e}，d_k \ in {\ bf r} \（{\ rm或} \ {\ bf c}），k = 1，\ cdots，n \}。 通过应用本文定理，我们表明，耐力的空间和伯格曼空间以及设备光盘中的所有加权伯格曼空间都具有$ n $ best的近似值。在强壮的空间案例中，这给出了经典结果的新证明。基于获得的结果，我们进一步证明了$ n $ best球形泊松内核近似与有限能量在实际电视上的功能的存在。
We show that if a reproducing kernel Hilbert space $H_K,$ consisting of functions defined on ${\bf E},$ enjoys Double Boundary Vanishing Condition (DBVC) and Linear Independent Condition (LIC), then for any preset natural number $n,$ and any function $f\in H_K,$ there exists a set of $n$ parameterized multiple kernels ${\tilde{K}}_{w_1},\cdots,{\tilde{K}}_{w_n}, w_k\in {\bf E}, k=1,\cdots,n,$ and real (or complex) constants $c_1,\cdots,c_n,$ giving rise to a solution of the optimization problem \[ \|f-\sum_{k=1}^n c_k{\tilde{K}}_{w_k}\|=\inf \{\|f-\sum_{k=1}^n d_k{\tilde{K}}_{v_k}\|\ |\ v_k\in {\bf E}, d_k\in {\bf R}\ ({\rm or}\ {\bf C}), k=1,\cdots,n\}.\] By applying the theorem of this paper we show that the Hardy space and the Bergman space, as well as all the weighted Bergman spaces in the unit disc all possess $n$-best approximations. In the Hardy space case this gives a new proof of a classical result. Based on the obtained results we further prove existence of $n$-best spherical Poisson kernel approximation to functions of finite energy on the real-spheres.
