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The equivalent problems of extrapolation and spectral estimation are formulated as deterministic estimation problems for the class of bandlimited signals which are concentrated in a known time interval. Estimators are derived which minimize the maximum magnitude of the error over the class of bandlimited, time-concentrated signals. The dependence of this minimum error on bandwidth, degree of time-concentration, and measurement errors (noise) is discussed. Practical examples are presented using discrete prolate spheroidal sequences to compute spectral estimation and extrapolation operators.