Local Estimates for Jacobi Polynomials

It is shown that if α, β ≥ − 12 , then the orthonormal Jacobi polynomials p (α,β) n fulfill the local estimate |p n (t)| ≤ C(α, β) ( √ 1− x+ 1 n ) α+ 2 ( √ 1 + x+ 1 n ) β+ 2 for all t ∈ Un(x) and each x ∈ [−1, 1], where Un(x) are subintervals of [−1, 1] defined by Un(x) = [x− φn(x) n , x+ φn(x) n ]∩[−1, 1] for n ∈ N and x ∈ [−1, 1] with φn(x) = √ 1− x2+ 1 n . Applications of the local estimate are given at the end of the paper.