Vector-valued L p-convergence of orthogonal series and Lagrange interpolation. Abstract We give necessary and sufficient conditions for interpolation inequalities of the type considered by Marcinkiewicz and Zygmund to be true in the case of Banach space-valued polynomials and Jacobi weights and nodes. We also study the vector-valued expansion problem of L p-functions in terms of Jacobi polynomi...

This paper deduces exponential matrix concentration from a Poincaré inequality via short, conceptual argument. Among other examples, this theory applies to matrix-valued functions of uniformly log-concave random vector. The proof relies on the subadditivity inequalities and chain rule for trace Dirichlet form. It also uses symmetrization technique avoid difficulties associated with direct exten...

We use spectral methods of automorphic forms to establish a holomorphic projection operator for tensor products of vector-valued harmonic weak Maass forms and vector-valued modular forms. We apply this operator to discover simple recursions for Fourier series coefficients of Ramanujan's mock theta functions.

After introducing the concept of functional dissipativity Dirichlet problem in a domain ??RN for systems partial differential operators form ?h(Ahk(x)?k) (Ahk(x) being m×m matrices with complex valued L? entries), we find necessary and sufficient conditions two-dimensional Lamé system. As an application our theory provide two regularity results displacement vector N-dimensional equilibrium prob...