Abstract
We study the class of q-Fourier multiplier operators T-m,T-a := F-q(m(a)F(q)(f))(x), a > 0, and we give a new Calderon's reproducing formula using the theory of Fourier transform; thus, we use the theory of reproducing kernels to give best approximation of these operators and Calderon's reproducing formula of the associated extremal function. As an application, we provide a numerical solution of the Poisson equation, using q-calculus which generalizes the Gauss-Kronrod method studied given in [20] in one-dimensional space.