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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.
}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19300.html} }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.