Transplantation Theorems for Ultraspherical Polynomials in Re H 1 and BMO

We consider the uniformly bounded orthonormal system of functions $$ u_n^{(\l)}(x)= \varphi_n^{(\lambda)}(\cos x)(\sin x)^\lambda, \qquad x\in 0,\pi], $$ where $\{\varphi_n^{(\lambda)}\}_{n=0}^\infty \,\, (\lambda > 0)$ is the normalized system of ultraspherical polynomials. R. Askey and S. Wainger proved that the $L^p$-norm $(1 < p < \infty)$ of any linear combination of the first $N+1$ functions $u_n^{(\lambda)}(x)$ is equivalent to the $L^p$-norm of the even trigonometric polynomial of degree $N$ with the same coefficients. This theorem fails if $p=1 $ or $p=\infty.$ Studying these limiting cases, we prove (for $0 < \lambda < 1$) similar transplantation theorems in $\mbox{Re } H^1$ and $\mbox{BMO}.$