Treffer: The gradient iteration for approximation in reproducing kernel Hilbert spaces

The paper is devoted to the problem of the normal solution to an equation \(Lf=z\), where \(L: \mathcal{F}\rightarrow Z\) is a linear bounded operator, \(Z\subseteq \mathbb{R}^N\) and \(\mathcal{F}\) is a reproducing kernel Hilbert space (RKHS) of functions \(f(x)\) on a field \(X\). The last means that for a given \(x_i\) the functional \(L_i{f}\equiv{f(x_i)}\) is a bounded linear one. According to the known Riesz theorem such a functional has the canonical representation \(L_i{f}=\langle{\psi_i,f}\rangle\), where \(\psi_i\in\mathcal{F}\), and \(\langle \cdot,\cdot\rangle\) is the inner product in \(\mathcal{F}\). This representation can be fulfilled via the reproducing kernel (RK) \(k(x_1,x_2)\) of the RKHS, and the problem of a function approximation is reduced to the system of algebraic equations with matrix \(K=\{k(x_i,x_j), i,j= 1,\dots,N\}\). In the case of large \(N\) the matrix \(K\) become ill-conditioned, and this obstacle can be overcome by some iterative technique. This is the main object of the paper, and authors propose the gradient method for minimization of the Tikhonov regularization functional \(\| Lf-z\| ^2_Z+\rho\| f\| ^2_{\mathcal{F}}\). The iterations have the form \[ f^n=L^*c^n=\sum^N_{i=1}{c^n_i{k(x_i.\cdot)}}, \quad c^{n+1}=c^{n}-\gamma^{n}[(LL^{*}+\rho I)c^{n}-z]. \] Convergence of the iterations to the normal solution of the approximation problem is proved on the base of spectral properties of the RK. In addition to the author's review of the works on numerical applications of the Riesz's representation theorem to problems of observations treatment, we note on our earlier paper: \textit{V. K. Gorbunov} [Zh. Vychisl. Mat. Mat. Fiz. 25, No. 2, 210-223 (1985; Zbl 0578.65132)], and the last paper: \textit{V. K. Gorbunov} and \textit{V. V. Petrishchev} [Comput. Math. Math. Phys. 43, No. 8, 1099-1108 (2003; Zbl 1077.65085)]. Here specific RKHSs of Sobolev type were considered, a technique of RK construction was developed, and solution methods for function approximation, ill-posed and singular classes of integral and differential equations were created.