By Sigurdur Helgason (auth.)

ISBN-10: 1489967656

ISBN-13: 9781489967657

ISBN-10: 1489967672

ISBN-13: 9781489967671

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Example text

R whence S(ep) = S(ep) = O. A (lS) Proof. 1, The other relation is proved in the same manner. 6) for distributions. 4. supp Ißt TE &' (Fl) we can consider the "conwlution" (f x CP)(~) = Jf(y)cp(~-y)dy, Rn ~ E pIl, where for by -y. ~-y denotes the translation of the hyperplane ~ Then * "cp. (f x cp)v = f In fact, if ~ o is any hyperplane through (f = x CP)v(x) = Idk J 0, tdklnf(y)CP(X + k'~o - f(x - y)CP(y + k·~o)dy y)dy * ~) (x).

Is a function pn. 4. Let A be!!! defined cf = (Ai? ' f ~ E (37). n) I Then 26 Ch. I where as before c = (-471) Proof. ~ r(Z) ::r:- . r(Z) By the inversion fomula for the Fourier transfonn and by (4) (271)-nJdw~ (~e-iSPf(w,p)dP)eis(X,w)sn-lds f(x) Sn-1 which Yle -00 wri te as f(x) = (271) /\ Using f(-w,-p) f(x) (38) 0 = -nln-lF(w,x)dw = (271) -nfSn-l 1 (~~(w,x) ,.. f(w,p) + F(-w,x)) dw this gives the formula = ~(271)-nfn_~foolsln-leiS(X,W)dsf:-iSP f(w,p)dp S _00 _00 If n is odd the absolute value on s can be dropped.

Denoting closure, and I the identity. If Tg Proof. Dg0 2 = = (g,g) (Tg,g) = g then since = (g,T*g) the inequalities are equal. ßg - T*g112 so T*g = g. ts 11 T*II = 11 TU ~ 1 we have ~ ßgn ßT*gll ~ ngn 2 so all terms in Hence = ßgll 2 - (g,T*g) - (T*g,g) + nT*gll2 = 0 I - T and is equivalent to I - T* have the sarre 001 space.

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The Radon Transform by Sigurdur Helgason (auth.)

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