On multiple orthogonal polynomials

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