Convergence Properties of Generalized Fourier Series on a Parallel Hexagon Domain

Abstract

A new Rogosinski-type kernel function is constructed using kernel function of partial sums $S_n(f;t)$ of generalized Fourier series on a parallel hexagon domain $Ω$ associating with three-direction partition. We prove that an operator $W_n(f;t)$ with the new kernel function converges uniformly to any continuous function $f(t) ∈ C_∗(Ω)$ (the space of all continuous functions with period $Ω$) on $Ω$. Moreover, the convergence order of the operator is presented for the smooth approached function.