By Waclaw Sierpinski, I. N. Sneddon, M. Stark

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Indeed, since ((2KN11 = 1, all N , we have that Ilf- Ilf- sn(flIlp Now we take a closer look at sn(2KN, x); it equals ~ K *N2Dn(x) = aN(2Dn,x), x E T. e. rr2)In n, n large. 1) cannot hold and convergence in norm cannot occur. ” This is motivated by noting that - 03 S n U XI = 1 Xl-n,n](k)Ckeikr k=-a and is achieved by studying a new object, the conjugate operator. 3. THE CONJUGATE MAPPING To each f E L( T ) ,f - CcjeiJr,we associate the trigonometric series ‘jJ-i)(sgn j )c,eijt where 1 0 -1 if j > 0, if j = 0, if j < 0.

Ii) implies (iii). 3. 4). (iii) implies (iv). We polarize (iii). We then take A = 1 and A = i to obtain the desired conclusion. (iv) implies (i). By (iv), with x = y there, we get that lly112 = 0, or y = 0, which is a contradiction. + How do these results apply to L2(T)? 8 above holds. Therefore, (ii)-(iv) of that theorem, as well as the rest of the results in this section also hold. 6) and 2. 6), for the case p = 2. On the other hand, the case p = 00 was ruled out, even with L"( T) replaced by C ( T), by the results in Section 2 of Chapter I.

More precisely for F i n D' let DF be defined by DF( u ) = F ( - D u ) , u E C"( T). 6). More generally, for an integer k let D k F ( u )= F ( ( - l ) k D k u ) , tl E C"( T). , differentiation is continuous in D'. Although the product of two distributions cannot be defined in general, we can always multiply a distribution F and a C" function u. The product uF is then the distribution given by (uF)(u ) = F( uu), u E C"( T ) . It is readily seen that with this definition Leibnitz's rule holds, namely, D( uF) = (Du)F + uDE We may also introduce the notion of Fourier series for F E D'.