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In this paper, a new class of Bayesian lower bounds is proposed. Derivation of the proposed class is performed via projection of each entry of the vector-function to be estimated on a Hilbert subspace of L2. This Hilbert subspace contains linear transformations of elements in the domain of an integral transform, applied on functions used for computation of bounds in the Weiss-Weinstein class. The integral transform generalizes the traditional derivative and sampling operators, used for computation of existing performance lower bounds, such as the Bayesian Cramér-Rao, Bayesian Bhattacharyya, and Weiss-Weinstein bounds. It is shown that some well-known Bayesian lower bounds can be derived from the proposed class by specific choice of the integral transform kernel. A new lower bound is derived from the proposed class using the Fourier transform kernel. The proposed bound is compared with other existing bounds in terms of signal-to-noise ratio (SNR) threshold region prediction in the problem of frequency estimation. The bound is shown to be computationally manageable and provides better prediction of the SNR threshold region, exhibited by the maximum a posteriori probability (MAP) and minimum-mean-square-error (MMSE) estimators.