ABSTRACT. Let X C Rn and let K be a trace class operator on L2(X) with corresponding kernel K(x,y) € L2(X x X). An integral formula for tr K, proven by Duflo for continuous kernels, is generalized for arbitrary trace class kernels. This formula is shown to be equivalent to one involving the factorization of K into a product of Hilbert-Schmidt operators. The formula and its derivation yield two new necessary conditions for traceability of a Hilbert-Schmidt kernel, and these conditions are also shown to be sufficient for positive operators. The proofs make use of the boundedness of the HardyLittlewood maximal function on L2(R").

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The FBI has recently adopted a standard for the compression of digitized 8-bit gray-scale fingerprint images. The standard is based on scalar quantization of a 64-subband discrete wavelet transform decomposition of the images, followed by Huffman coding. Novel features of the algorithm include the use of symmetric boundary conditions for transforming finite-length signals and a subband decomposition tailored for fingerprint images scanned at 500 dpi. The standard is intended for use in conjunction with ANSI/NBS-CLS 1-

Group lifting structures are introduced to provide an algebraic framework for studying lifting factorizations of two-channel perfect reconstruction FIR filter banks. The lifting factorizations generated by a group lifting structure are characterized by abelian groups of lower and upper triangular lifting matrices, an abelian group of unimodular gain scaling matrices, and a set of base filter banks. Examples of group lifting structures are given for linear phase lifting factorizations of the two nontrivial classes of two-channel linear phase FIR filter banks, the whole-and half-sample symmetric classes, including both the reversible and irreversible cases. This covers the lifting specifications for whole-sample symmetric filter banks in Parts 1 and 2 of the ISO/IEC JPEG 2000 still image coding standard. The theory is used to address the uniqueness of lifting factorizations. With no constraints on the lifting process, it is shown that lifting factorizations are highly nonunique. When certain hypotheses developed in the paper are satisfied, however, lifting factorizations generated by a group lifting structure are shown to be unique. A companion paper applies the uniqueness results proven in this paper to the linear phase group lifting structures for whole-and half-sample symmetric filter banks.

Let K be a trace class operator on L 2 (X, Jί, μ) with integral kernel K(x, y) € L 2 (X x X, μ x μ). An averaging process is used to define K on the diagonal in X x X so that the trace of K is equal to the integral of K(x, x), generalizing results known previously for continuous kernels. This formula is also shown to hold for positive-definite Hilbert-Schmidt operators, thus giving necessary and sufficient conditions for the traceability of positive integral kernels. These results make use of Doob's maximal theorem for martingales and generalize previous results obtained by the author using HardyLittlewood maximal theory when IcR".