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The problem of estimating a message, , which is a sample function from a continuous Gaussian random process is considered. The message to be estimated may be contained in the transmitted signal in a nonlinear manner. The signal is corrupted by additive noise before observation. The received waveform is available over some observation interval . We want to estimate over the same interval. Instead of considering explicit estimation procedures, we find bounds on how well any procedure The principle results are as follows: 1) a lower bound on the mean-square estimation error. This bound is a generalization of bounds derived previously by Cramer, Rao, and Slepian for estimating finite sets of parameters. 2) The bound is evaluated for several practical examples. Possible extension and applications are discussed briefly.