Mellin convolution

The pedestrian approach to polydispersity that has been demonstrated up to here is an extension to three dimensions of the well-known rigorous treatment of polydispersity in one dimension by means of the Mellin convolution (Eq. (8.85), p. 168). 

The treatments of Kochendorfer, Porod, and Warren-Averbach identify superposition with the mathematical operation of a convolution. While this is true for translational superposition, for dilational superposition it is a coarse approximation that is only valid for small polydispersity. In the latter case the convolution must be replaced by the Mellin convolution (Eq. (8.85), p. 168) governed by a dilation factor distribution and the structure of the reference crystal, the structure of each observed crystal is generated by affine dilation of the reference crystal (Stribeck [2]). 

Kochendorfer addresses superposition of symmetrical lattice line ghosts originating from the dilation - although it is readily shown that dilatation superposition (Mellin convolution) always causes the observed profile to become asymmetrical [125,126]. 

If we assume that the shape of each disc is circular, the CLD of an uncorrelated hard-disc fluid is the Mellin convolution of the intrinsic chord distribution, gc r 2), of an ideal disc of diameter 1 and the diameter distribution, ho (D) which characterizes the structure. The definition of the Mellin convolution (Titchmarsh [202], S. 53 Marichev [203] ... 

GENERALIZED INTEGRAL TRANSFORMATIONS, A.H. Zemanian. Graduate-level study of recent generalizations of the Laplace, Mellin, Hankel, K. Weierstrass, convolution and other simple transformations. Bibliography. 320pp. 5H x 8H. 65375-7 Pa. 7.95... 

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