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We extend to locally compact abelian groups, Fejer's theorem on pointwise convergence of the Fourier transform. We prove that lim φ * f(y) = f (y) almost everywhere for any function f in the space (L, l)(G) (hence in L(G)), 2 ≤ p ≤ ∞, where {φ} is Simon's generalization to locally compact abelian groups of the summability Fejer Kernel. Using this result, we extend to locally compact abelian groups a theorem of F. Holland on the Fourier transform of unbounded measures of type q.
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