By Furui S., Sandhi M.M. (eds.)

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Antoni Zygmund proposed conjecture (5) as my thesis problem. I will indicate how the An generalized derivatives arose in my investigations. x/ exists. x/ h term can be removed, we will have essentially a generalized L1 fifth derivative of F . So, on the one hand, substitute h ! 2h, on the other hand, just multiply the equation by 2, and then subtract the second equation from the first. x 2h/ Remarks on Various Generalized Derivatives 33 We have more or less arrived at a fifth generalized L1 derivative with no excess, but the bi are f 4; 2; 1; 1; 2; 4g and so that derivative is not a Dn .

0; m ! x/ ! 1; m ! x/ dx D 1; gives 1 km f ! P. O. x; 0/ (for the case n > 1 minor changes give us the same result). Some Non Standard Applications of the Laplace Method 47 Hence, a Calderón–Zygmund partition can be done using Whitney type cubes fQk g, decomposing the given function f > 0, and G given as in (9), as f D f1 C f2 ; where f1 is the “good part,” f1 Ä ; a. e. x; G/ > 0, where PG D kD1 5Qk ; (5Q is the five times dilation of Q from its center). n/ different Q This leads to a genuine generalization of the Natanson lemma in this context.

N 4 4 n Now look at each of the above four implications generically. In other words, ask whether the converse of an implication can fail at each point of a set of positive measure. The first implication is still irreversible; but the other three all have a converse at almost every point. More specifically, in 1954 H. W. Oliver gave an example of a function f having a second Peano derivative at every point of a certain set E of positive measure although f 00 fails to exist at every point of E [Ol].

### Advances in Speech Signal Processing by Furui S., Sandhi M.M. (eds.)

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