Schur test

Abstract: By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on $$\mathcal {H}^2$$ , the space of Dirichlet series with square summable coefficients, for the inducing symbol $$\varphi (s)=c_1+c_{q}q^{-s}$$ where $$q\ge 2$$ is a fixed integer. We also give an estimate on the approximation numbers of such an operator.

Abstract: Given a function $$f$$ on the positive half-line $${\mathbb R}_+$$ and a sequence (finite or infinite) of points $$X=\{x_k\}_{k=1}^\omega $$ in $${\mathbb R}^n$$ , we define and study matrices $${\mathcal S}_X(f)=[f(\Vert x_i-x_j\Vert )]_{i,j=1}^\omega $$ called Schoenberg’s matrices. We are primarily interested in those matrices which generate bounded and invertible linear operators $$S_X(f)$$ on $$\ell ^2({\mathbb N})$$ . We provide conditions on $$X$$ and $$f$$ for the latter to hold. If $$f$$ is an $$\ell ^2$$ -positive definite function, such conditions are given in terms of the Schoenberg measure $$\sigma _f$$ . Examples of Schoenberg’s operators with various spectral properties are presented. We also approach Schoenberg’s matrices from the viewpoint of harmonic analysis on $${\mathbb R}^n$$ , wherein the notion of the strong $$X$$ -positive definiteness plays a key role. In particular, we prove that each radial $$\ell ^2$$ -positive definite function is strongly $$X$$ -positive definite whenever $$X$$ is a separated set. We also implement a “grammization” procedure for certain positive definite Schoenberg’s matrices. This leads to Riesz–Fischer and Riesz sequences (Riesz bases in their linear span) of the form $${\mathcal F}_X(g)=\{g(\cdot -x_j)\}_{x_j\in X}$$ for certain radial functions $$g\in L^2({\mathbb R}^n)$$ .