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A rational approximation scheme is exhibited for a set A which consists of a finite union of compact subintervals of the real line. This rational approximation scheme provides a characterization of the analytic functions on A which generalizes S. N. Bernstein's characterization of the analytic functions on [-1, 1].doi:10.1090/s0002-9939-1977-0433086-6 fatcat:duk5u7tcyngk3imtv37pvu2uju