Decomposition of spaces of distributions induced by tensor product bases (Q447900)

For \(0<p<\infty\) let NEWLINE\[NEWLINE \|f \|_p = \Big( \int_{[-1,1]^d} |f(x)|^p \, w_{\alpha, \beta} (x) \, dx \Big)^{1/p} NEWLINE\]NEWLINE where NEWLINE\[NEWLINE w_{\alpha, \beta} (x) = \prod^d_{i=1} (1- x_i)^{\alpha_i} (1+ x_i )^{\beta_i} NEWLINE\]NEWLINE is the typical weight in connection with the \(d\)-dimensional tensor product of Jacobi polynomials. The authors construct in the context of these tensor products rapidly decreasing kernels and building blocks, called needlets, and related multivariate \(C^\infty\) cutoff functions. On this basis the authors introduce in Definition 7.1 weighted Triebel-Lizorkin spaces \(F^{s, \rho}_{p,q} (w_{\alpha, \beta})\) where \(s,\rho \in \mathbb{R}\), \(0<p<\infty\), \(0<q\leq \infty\) and in Definition 8.1 weighted Besov spaces \(B^{s, \rho}_{p,q} (w_{\alpha, \beta})\) where \(s,\rho \in \mathbb{R}\) and \(0<p,q \leq \infty\). It is the main aim of this paper to study these spaces. Some applications are given to nonlinear approximations and to product spaces where the underlying domains are products of cubes, balls, spheres and other domains.