A transplantation theorem for ultraspherical polynomials at critical index

We investigate the behaviour of Fourier coefficients with respect to the sy stem of ultraspherical polynomials. This leads us to the study of the "boun dary" Lorentz space L-lambda corresponding to the left endpoint of the mean convergence interval. The ul- traspherical coefficients {C-n((lambda)) (f) } of L-lambda-functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReHl. Namely, we prove that for any f is an element of L-lambda the series Sigma (infinity)(n=1) c(n)((lambda)) (f) cos n theta is the Fourier series of some function phi is an element of Re H1 with parallel to phi parallel to Re H-1 less than or equal to c parallel tof parallel to l(lambda).