On Fourier Approximation of Periodic Functions and Their Conjugates Belonging to the Weighted Lipschitz Class

Taking into consideration that the superposition of two summability methods is superior to the individual one, in this paper we present the calculations of the degree of approximation of functions and their conjugates belonging to the weighted Lipschitz class by a trigonometric polynomial generated by the application of product means \(\mathcal {C}^{1}.\mathcal {T}\) ${\mathcal{C}}^1.\mathcal{T}$ on their trigonometric Fourier series and conjugate series, respectively. Here we also reduce some conditions imposed on the increasing function \(\Psi (t)\) $\mathrm{\Psi }(t)$ . Further, with the help of an example, we demonstrate the application of the results in the circumstances when the Fourier series of the function has the Gibbs phenomenon. We also provide a few corollaries that follow directly from our results.