Extremalpolynome in der L¹- und L²-Norm auf Zwei Disjunkten Intervallen

Abstract

Let α, α ∈ (-1,+1) with α ≦3β and let P_n be a polynomial of degree n with leading coefficient one which deviates least from zero on [-1, α] ∪ [β, 1] with respect to the L¹-norm. We show that P_n can be represented as a product of two polynomials which are orthogonal with respect to weight functions vanishing on (α, β). For the recursion coefficients of those orthogonal polynomials a recurrence relation is given.