On the convergence of interpolation process with minimal norm in the Hilbert space (Q2739978)

Let \(X,Y\) be real Hilbert spaces (\(X\) is separable), let \(\Pi_{n}\) be the set of continuous operator polynomials \(P_{n}:X\to Y\) of order \(n\). Let \(\{x_{i}\}_{1}^{m}\), \(\{Bx_{i}\}_{1}^{m}\subset X\), \(\vec y= (y_1,\ldots,y_{m})\in Y^{m}\) be given, where \(B\) is a correlation operator of measure \(\mu\) on \(X\). The authors prove that \(\lim_{m\to \infty}\|P_{n}-P_{n}^{*}\|_{H}=0\), where \(P_{n}(x)\in \Pi_{n}\); \(y_{i}=P_{n}(Bx_{i}),\;i=0,\ldots,N\); \(\|\cdot\|_{H}\) is a norm in \(\Pi_{n}\); \(P_{n}^{*}(x)= \left\langle \vec y,V_{m}^{+}\sum_{k=0}^{n} \{(x,x_{i})_{X}^{k}\}_{1}^{m}\right\rangle\) is the interpolation operator polynomial of minimal norm; \(V_{m}^{+}\) is the pseudoinverse matrix to matrix \(V_{m}=\|\sum_{k=0}^{n}(Bx_{i}, x_{j})_{X}^{k}\|_{i,j=1}^{m}\); \((\cdot,\cdot)\) is the inner product in \(Y\); \(\langle\vec a,\vec b\rangle=\sum_{i=1}^{m}a_{i}b_{i}\).