On Jackson-type inequalities generated by the (k,n)-Fourier transform on the real line

This study aims to achieve some results in approximating functions, analogous to Jackson’s theorems, tied to the generalized (k, n)-Fourier transform, on the real line \(\mathbb {R}\) $\mathbb{R}$ . The proofs are carried out in the space \(f \in L^{2}(\textrm{d}\mu _{k, n})\) $f\in L^2(\text{d}{\mu }_{k,n})$ , focusing on some approximation theorems for the relations between sequences of approximation and appropriate measures of smoothness using generalized translation operators. As an application, we present a characterisation of a generalized Lipschitz spaces, known as the Titchmarsh Theorem.