By Choudhary P.

ISBN-10: 8189473956

ISBN-13: 9788189473952

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**Extra info for A practical approach to linear algebra**

**Example text**

2. 2 is still unknown. 2 are the best possible since faster growth assumptions on P and Q will make the theorem incorrect. 1 can be proved on jRn if we replace the pointwise estimates on f and by integral conditions. 2, sn - I whe re the fun ctions P and Q are such that . log P (r ) . log Q ().. ) hm sup 2 = hm sup 2 = O. ->- 00 i, and when ab = i, ).. 40 1. Euclidean Spaces where p and q are such that their L 2 -norms on and Q(r) respectively. Ixl = r do not grow faster than per) This theorem is proved in [61] where the authors have also proved analogues of Hardy's theorem for the Dunk!

X) where A and B are real, symmetric matrices and P is a polynomial. (l Iyl)-N d xdy < 00 ; jR" jR" (iii) f f If(x)llf(y)lel(x ,Y)I(l + Ixl)- ~(l + Iyl)-~dxdy < 00 . jR" jR" Proof. If f is as in the proposition with A symmetric , then by diagonalising A we see that A has to be positive definite since f is square integrable. 2) where Q is another polynomial with deg P = deg Q = m. It is easy to show that (i) implies (iii). 3) 00. jR" jR" Whenlxl2: 3lylorlyl2: 3Ixltheintegralisfinite,sincethenlx-Ye 2: c(lxI 2+IYI2) for some constant c > O.

Rr 2a 1 }. < - , a> - 2 Suppose F(I;) is holomorphic in Q and continuous up to the boundary and satisfies the estimates W(I;)I :s M on oQ, W(I;)I :s K ell;Jtl , fJ < a on Q. Then W(I;)I < M on the whole of Q . Proof. Let E > 0 and choose y such that fJ < y < a. Consider the function F€ (I;) = e-€I;Y F(I;) so that IF€(I;)I = e-€I I;IY cos(ye)IF(I;)I . On the boundary lines 0 = ±~ we have cos(yO) > 0 since y < a . Therefore, on these lines IF€(I;)I :s W(I;)I :s M. 4. Hardy's theorem on jRn 21 which goes to 0 as R ~ 00 since f3 < y.

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