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We study the convergence of the Laurent polynomials of Lagrange interpolation on the unit circle for continuous functions satisfying a condition about their modulus of continuity. The novelty of the result is that now the nodal systems are more general than those constituted by the n roots of complex unimodular numbers and the class of functions is wider than the usually studied. Moreover, some consequences for the Lagrange interpolation on [−1, 1] and the Lagrange trigonometric interpolation are obtained.

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