Convergence in Variation for a Homothetic Modulus of Smoothness in Multidimensional Setting

摘要

In order to get approximation results for linear and nonlinear convolution integral operators in BV-spaces, it is crucial to study convergence of the modulus of continuity. In the case of the modulus of continuity ω(f, δ) defined by means of the classical variation, it is well known that, if f is absolutely continuous, then ω(f, δ) → 0, as δ → 0~+. The purpose of this paper is to extend the above result to the frame of BV~φ((R_0~+)~N) for the modulus of smoothness ω~φ(f, δ) := sup_(|1-t|)<δV~φ[τ_t f - f], where τ_t f(x) = f(xt) is the homothetic operator and V~φ[f] is the multidimensional φ-variation introduced in [4].