Regarding the \(p\)-norms of radial basis interpolation matrices
The main result of the paper consists in establishing the upper bound for \(|A^{- 1}|_p\), \(p\geq 1\), where \[ A= (\varphi(x_j- x_k))^n_{j, k= 1}\qquad (\varphi: \mathbb{R}^d\to \mathbb{R}) \] is an interpolation matrix of basis function approximation \[ S(x)= \sum^n_{j= 1} y_j \varphi(x- x_j),\qquad x\in \mathbb{R}^d;\quad y_i\text{-real}. \] This bound depends on \(\varphi\), the dimension \(d\) and the minimal distance between points \((x_j)^n_{j= 1}: \min_{j\neq k} |x_j- x_k|_2\). Previously, this estimates was only known for \(p= 2\). Using this main result for the interpolation matrix \[ A_n= (\varphi(j- k))^n_{j, k= 1}\quad\text{when} \quad \varphi(x)= (x^2+ c^2)^{1/2} \] is a Hardy multiquadratic form, it is proved that \(\sup_n|A^{- 1}_n|_\infty\) is finite. In the case when \(\varphi\) is Gaussian, it is established that for bi-infinite symmetric Toeplitz matrices \[ E= (\varphi(j- k))_{j, k\in L^d}= (\exp(- \lambda|j- k|^2_2)_{j, k\in L^d} \] the following property \(|E^{- 1}|_p= |E^{- 1}|_2\) is true for all \(p\geq 1\). This property is valid also when \(\varphi\) is a tensor product of even absolutely integrable (in the sense of Pólya) periodic functions. In establishing these results there are used the theorem of Demko, Moss, and Smith on the inverse of a bounded symmetric positive definite matrix, Riesz' theorem on convexity, statements from the theory of Toeplitz matrices.
scientific article; zbMATH DE number 700843
scientific article