Polynomial Approximation by Doubly Periodic Weierstrass Functions on Disjoint Segments in the LP Metric

Let s k , 1 ≤ k ≤ m , m ≥ 2, be disjoint segments lying in a parallelogram Q . Denote by ℘ ( z ) a doubly periodic Weierstrass function with the fundamental parallelogram Q . Let f k : s k → ℂ be functions, and let f k ′ ∈ L p k ( s k ), 1 ≤ k ≤ m , 1 < p k < ∞. Consider the Green function G ( z ) of the domain C \ ⋃ k = 1 m s k with the pole at infinity and define L h = def ζ : ζ ∈ C \ ⋃ k = 1 m s k , G ζ = log 1 + h , h > 0 ; ρ h ζ = def dist ζ , L h . Theorem . There exist polynomials P n ( u , v ), deg P n ≤ n , n = 1, 2, · · · , such that ∑ k = 1 m ∫ s k f k ζ - P n ℘ ζ , ℘ ′ ζ ρ 1 n ζ p k d ζ ≤ c .