Divergence of interpolation processes on sets of the second category

Abstract

C([0, 1]) is the space of real continuous functions f(x) on [0, 1] and ω(δ) is a majorant of the modulus of continuity ω(f, δ), satisfying the condition\(\mathop {\overline {\lim } }\limits_{n \to \infty } \omega (1/n) \ln n = \infty \). A solution is given to a problem of S. B. Stechkin: for any matrix\(\mathfrak{M}\) of interpolation points there exists an f(x) ɛ c([0, 1]), ω (f, δ) = o{ω(δ)} whose Lagrange interpolation process diverges on a set ℰ of second category on [0, 1].