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Miklos Schweitzer 1979_8

Source: sequence of periodical continuous functions

January 28, 2009
functionintegrationreal analysisreal analysis unsolved

Problem Statement

Let Kn(n=1,2,…) K_n(n=1,2,\ldots) be periodical continuous functions of period 2π 2 \pi, and write kn(f;x)=∫02πf(t)Kn(x−t)dt. k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt . Prove that the following statements are equivalent: (i) ∫02π∣kn(f;x)−f(x)∣dx→0  (n→∞) \int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty) for all f∈L1[0,2π] f \in \mathcal{L}_1[0,2 \pi]. (ii) kn(f;0)→f(0) k_n(f;0) \rightarrow f(0) for all continuous, 2π 2 \pi-periodic functions f f. V. Totik
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