In this paper, a new class of lower bounds on the mean-square-error (MSE) of unbiased estimators of deterministic parameters is proposed. Derivation of the proposed class is performed by approximating each entry of the vector of estimation error in a closed Hilbert subspace of L2. This Hilbert subspace is spanned by a set of linear combinations of elements in the domain of an integral transform of the likelihood-ratio function. It is shown that some well known lower bounds on the MSE of unbiased estimators, can be derived from this class by inferring the integral transform. A new lower bound is derived from this class by choosing the Fourier transform. The bound is computationally manageable and provides better prediction of the signal-to-noise ratio (SNR) threshold region, exhibited by the maximum-likelihood estimator. The proposed bound is compared with other existing bounds in term of threshold SNR prediction in the problem of single tone estimation.