By Charles J. Mozzochi

ISBN-10: 3540054758

ISBN-13: 9783540054757

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**Example text**

Also, Cn(m ) is related to Cn(~ ) is related to Sn(~), the n th partial sum of the Fourier series of f over ~. I) That is, for m = (O, 2~) Sn-1 = cn and ISnl - ]Sn_ll = O(C n ). is the technical basis for the above relations. ) Lemma Let ~(t) ¢ C2(~), I~I = 2~'2 "v Then we can r e p r e s e n t ~ (t). (*) where (i + p2)l~pI Proof. ~t} ~ (t) = Z ~ (o, (m~x I~I + 2 -2v maxm l~''l)- By a change of variables, We choose polynomials 2~). I (t) <. const. t ~ ~, = pl,P2 Pl(t) [-2~,O) (t) (O,2~) P2(t) [2~,4~] ¢ c 2 ([-2~,4~]) , ~(k) (-2~) = ~(k) t = 2"~, we may assume such that satisfies (4~) = O, k = O, I, 2.

K k -p m Xk < 2 b k y mF. Proof. ~. a. ~, ~ o i v ~. 7) Remark. It is immediate by the definition of Ak(X) that if x c Xk, then there exists a dyadic interval ~ each ~ ~ Xk with x ~ 9. X k we consider its three left dyadic neighbors ~ and its three right dyadic neighbors Let X = w 1 2 3 ~r,~r,~r 12 ,~ For 3 all of lengthlm I ~Iuw2u~3u~U~ U~ U~ r r r ~ ~ If ~ is located too close to either 2~ or -2~ , then some or all of the three left or right dyadic neighbors may not exist. If this situation occurs, simply delete the missing terms from the expression for X~ .

44) IS~o(X;XF;~°*°)l = lSnl (x; XF;m~){ we have + 0 ( L m b o mo- lY) 43 Case 3. 41). o it is u n d e r s t o o d = m = i we h a v e that m* * o = ml We c o n t i n u e u n t i l Case (3) o c c u r s or u n t i l y i e l d an i n t e r v a l ~* 3+1 so small t h a t n~+ 1J Cases (1) and (2) = O. " APPENDIX A. V. I) f(t) x-t Theorem. 2) (a,b). Let jb a (a,b) and dt = If f ¢ limit e + O+ a 1 L (a,b), then f(t)dt + x-t I (Hf) exists f(t) x+¢ x-t (x) almost (a,b). Remark. I) see product L 1 (a,b). h c inequality Since that For an explanation (x) + i (Hv) [15] page 132.

### On the Pointwise Convergence of Fourier Series by Charles J. Mozzochi

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