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Download e-book for iPad: Abelian Harmonic Analysis, Theta Functions and Functional by L. Auslander, R. Tolimieri

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By L. Auslander, R. Tolimieri

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We have 0)1 n)0M>7) = (* - t)n/n l, n = 1 , 2, . . §1. A GENERALIZATION OF THE TAYLOR FORMULA 29 so that the order of the derivative of p is reduced by one. To this end, let 1 f x~< i c+7fc dC (C + lu) whence (*/«)-< dC dt. (C + l u ) Then {x/t)-< dÇ < ■ " ........ 5), respectively. Letting k = 1, 2 , . . 10) Ro(

As we said above, our investigation will be carried out for the basic interval (0,ti]. has been studied in Badalyan (1962) and will be studied in Chapter 6 , so that we will not tackle this case here. 20 ) does not converge anywhere outside the interval ( 0 , u]. , for 7 ^ < xv, v = 1, 2 , . . , 0 < x < 1, and a fortiori for 7 „ = 0 ( 1/), v = fco, ko + 1, . . ) the series ( 2 . 20') with radius of convergence R = u. 210 ^ ( £ , 7 ) = ( - 1) * « " 7* ' v=l Ju ... 22 ) ^ - ¿ / oIE £ (C + Iv) l > 1.

29) (2 2 Q "1 ^n(*n. 7 o. i, 0 2 , . . „ 2iri J c I=I z + 7«/ * ( * + 7 i) ’ 44 2. 31). 32) 0 <®y < h + l 2+ h, where (setting yo e - 1) h = a rvo ” iy + ßp 71 J r iy + Ju f/ n I - yo Ün? I 'V + 'Tv vtj—O = 2. ]• Let us estimate each of these integrals. To estimate /1, observe that n iy + ßp v = 2 iy + lv =n( y 2 + ilv —7 i )2 '\1/2 = n f i - ^ L = J Î ) y2 + it ) y2+ii ) ÜV Consider separately the two possibilities: a) limn_>oo 7n = °°> and b) 7 ^ < A, v = 1, 2 , . . , 0 < A < 00 . In case a) take n large enough for the inequality (71/4) Y^yo<7V<7n()-hv) > \og{l/e) to hold; then obviously I\ < C\e, where c\ > 0 does not depend on n.

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