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In three recent notes [1], [2], [3] I have discussed uniform convergence by polynomials (in the complex variable) to a given function as a consequence of convergence in the mean of those polynomials to the given function, and also convergence in the mean of one order as a consequence of convergence in the mean of a lower order. The present note contains analogs of those results, but now for approximation by bounded analytic functions. As a first illustration of the new results, we have

Approximation by bounded analytic functions: Uniform convergence as implied by mean convergence