$L^p$-Convergence of Fourier-Heckman-Opdam Expansions

We study the $L^p$-convergence of Fourier expansions in terms of non-symmetric Heckman-Opdam polynomials of type $A_1$. Using kernel estimates and duality arguments, we prove that the partial sums converge in $L^p([-\pi ,\pi ],d{m}_k)$ for $$2-\frac{1}{k+1}<p<2+\frac{1}{k}\mathrm{.}$$