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We review our previous studies of truncated Mellin moments of partondistributions. We show in detail the derivation of the evolution equation fordouble truncated moments. The obtained splitting function has the same rescaledform as in a case of the single truncated moments. We apply the truncatedmoments formalism to QCD analyses of the spin structure functions of thenucleon, $g_1$ and $g_2$. We generalize the Wandzura-Wilczek relation in termsof the truncated moments and find new sum rules. We derive the DGLAP-likeevolution equation for the twist-2 part of $g_2$ and solve it numerically. Wefind also useful relations between the truncated and untruncated moments.

Truncated Mellin moments: Useful relations and implications for the spin structure function $g_2$