Fourier—Price coefficients of class GM and best approximations of functions in the Lorentz space Lpθ[0, 1), 1<p<+∞, 1<θ<+∞

Abstract

For polynomials in the Price system, we establish an inequality of different metrics in the Lorentz spaces. Applying this inequality, we prove a Hardy–Littlewood theorem for the Fourier–Price series with GM sequences of coefficients in the two-parameter Lorentz spaces and in the Nikol’skii–Besov spaces with a Price basis. We also study the behavior of the best approximations of functions by Price polynomials in the metric of the Lorentz space.