Exponential approximation of functions in Lebesgue spaces with Muckenhoupt weight

Using a transference result, several inequalities of approximation by entire functions of exponential type in $\mathcal{C}(\mathbf{R})$, the class of bounded uniformly continuous functions defined on $\mathbf{R}:=\left( -\infty ,+\infty \right)$, are extended to the Lebesgue spaces $L^{p}\left( \mathbf{\varrho }dx\right)$ $1\leq p<\infty$ with Muckenhoupt weight $\mathbf{\varrho }$ ($1\leq p<\infty$). This gives us a different proof of Jackson type direct theorems and Bernstein-Timan type inverse estimates in $L^{p}\left( \mathbf{\varrho }dx\right)$. Results also cover the case $p=1$.