The problem of estimation for bandlimited signals is investigated. The desired estimate is based on a finite number of time samples taken from a measurement interval of length T(,M). With no additional information, the estimate is based on these samples alone. When the signal is bandlimited with normalized bandwidth (beta), this provides additional information which is used to improve the estimate by extrapolating the signal. The effectiveness of this extrapolation is limited to an interval of length T(,L) (DBLTURN) T(,M)/2(beta). A survey of various bandlimited estimation methods is made. These methods do not make explicit use of the effective extrapolation limit T(,L). A new estimation method for bandlimited signals is then described which does incorporate the interval T(,L) into the problem formulation. For many bandlimited signals, this new method yields better estimation than the previous methods. Estimation error bounds and examples are presented in comparing the methods. Estimation with inexact data is considered. Sensitivity of the bandlimited estimation methods to errors in the data is discussed. Estimation with smoothing is utilized in reducing the effects of this noise. A recursive algorithm for the calculation of the new estimation method is described. The estimates are computed using discrete prolate spheriodal sequences. A Fortran program implementing the new bandlimited estimation method is presented.