On LpLp multiple orthogonal polynomials

Denote by PnPn the space of real algebraic polynomials of degree at most n−1n−1 and consider a multi-index n≔(n1,…,nd)∈Nd,d≥1, of length |n|≔n1+⋯+nd. Then given the nonnegative weight functions wj∈L∞[a,b],1≤j≤dwj∈L∞[a,b],1≤j≤d, the polynomial Q∈P|n|+1∖{0} is called a multiple orthogonal polynomial relative to n and the weights wj,1≤j≤dwj,1≤j≤d, if ∫[a,b]wj(x)xkQ(x)dμ=0,0≤k≤nj−1,1≤j≤d. The above orthogonality relations are equivalent to the conditions for the L2L2 multiple best approximation ‖Q‖L2(wj)≤‖Q−g‖L2(wj),∀g∈Pnj,1≤j≤d. The existence of multiple L2L2 orthogonal polynomials easily follows from the solvability of the above linear system. The analogous question for the multiple best LpLp approximation, i.e., the existence of an extremal polynomial Qp∈P|n|+1∖{0} satisfying ‖Qp‖Lp(wj)≤‖Qp−g‖Lp(wj),∀g∈Pnj,1≤j≤d, poses a more difficult nonlinear problem when 1≤p≤∞,p≠21≤p≤∞,p≠2. In this paper we shall address this question and verify the existence and uniqueness of multiple LpLp orthogonal polynomials under proper conditions.