Nikolskii-type inequalities for shift invariant function spaces Academic Article uri icon

abstract

  • Let V n be a vectorspace of complex-valued functions defined on R of dimension n + 1 over ℂ We say that V n is shift invariant (on ℝ) if f ∈ V n implies that f a ∈ V n for every a ∈ ℝ, where f a(x) := f(x - a) on R. In this note we prove the following. Theorem. Let V n ⊂ C[a, b] be a shift invariant vectorspace of complex-valued functions defined on ℝ of dimension n + 1 over ℂ. Let p ∈(0, 2]. Then ∥f∥ L∞[a+δ,b-δ] ≤2 2/p2 (n+1/δ) 1/p ∥f∥ Lp[a,b]for every f ∈ V n and δ ∈ δ ∈ (0, 1/2 (b - a)).

author list (cited authors)

  • Borwein, P., & Erdélyi, T.

citation count

  • 9

publication date

  • June 2006