A novel noninvasive second-harmonic-generation tomographic method of mapping the structure of animal tissues by use of 100-fs laser pulses at 625nm is described. Subsurface structures were measured with this approach, which is potentially a symmetry-sensitive tool for optical histological reconstruction.

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The Hardy operator Ta on a tree Γ is defined byProperties of Ta as a map from L p (Γ) into itself are established for 1 ≤ p ≤ ∞. The main result is that, with appropriate assumptions on u and v, the approximation numbers an(Ta) of Ta satisfy ( * ) lim n→∞ nan(Ta) = αp Γ |uv|dt for a specified constant αp and 1 < p < ∞. This extends results of Naimark, Newman and Solomyak for p = 2. Hitherto, for p = 2, (*) was unknown even when Γ is an interval. Also, upper and lower estimates for the l q and weak-l q norms of {an(Ta)} are determined.

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