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Given a sequence of kernels Φ n for which the operators T n f = Φ n * f converge a.e. in all L p (R) spaces, p > 1, a perturbation method is provided with the property that the modified convolution operators converge pointwise only in selective spaces.

Convolution operators induced by approximate identities and pointwise convergence in 𝐿_{𝑝}(ℝ) spaces