A transplantation theorem for Jacobi series in weighted Hardy spaces
The author studies the Jacobi series over the set of functions \[ \phi_n^{(\alpha,\beta)}(t)=c_n(1-\cos t)^{\alpha/2+1/4}(1+\cos t)^{\beta/2+1/4} P_n^{(\alpha,\beta)}(\cos t), \] where \(c_n>0\) is a normalizing constant, \(\alpha,\beta>-1\) and \(P_n\) are Jacobi polynomials. The set \(\{\phi_n\}\) is known to be the orthonormal basis in \(L^2(0,\pi)\). Let \(L^2_{a,b}\) be the weighted \(L^2(w)\) with the weight function \(w(t)=t^a(\pi-t)^b\). It was proved by R. Askey and later extended by B. Muckenhoupt that the norm of any Fourier series over \(\phi_n\) is equivalent to the norm of cos-Fourier series with the same Fourier coefficients in \(L^2_{a,b}\). The author establishes the similar result in weighted Hardy spaces \(H^p_{a,b}\) with the parameters subject to certain relations.
scientific article; zbMATH DE number 2081960