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We introduce the Shepard-Bernoulli operator as a combination of the Shepard operator with a new univariate interpolation operator: the generalized Taylor polynomial. Some properties and the rate of convergence of the new combined operator are studied and compared with those given for classical combined Shepard operators. An application to the interpolation of discrete solutions of initial value problems is given.doi:10.1090/s0025-5718-06-01894-1 fatcat:c7hud26rqvhhhbmrcl7f27g5mq