Asymptotic expansion for the L^1 Norm of N-Fold convolutions

by Stey, George C.

Abstract (Summary)
An asymptotic expansion of nonnegative powers of 1/n is obtained which describes the large-n behavior of the L^1 norm of the n-fold convolution, ? gn ?L^1 = ?-??|gn(x)|dx, of an integrable complex-valued function, g(x), defined on the real line, where, gn+1(x) = ?-?? g(x-y)gn(y)dy, g1(x) =g(x). Consideration is restricted here to those g(x) which simultaneously satisfy the following four Assumptions I: g(x) ?L^1?Ls1, for some s1 > 1, II: xjg(x)?L^1, (j=1,2,3,…), III: There is only one point, t= t0, at which |?(t)| attains its supremum. i.e, |?(t)| < |h?(t0)|= sups??|h?(s)|,for all t ? t0, IV: |?(t)|(2)|t=t0 < 0, where ?(t) denotes the Fourier transform of g(x). We obtain the following ewline Theorem: Let g(x) satisfy simultaneously Assumptions I,II,III,IV above, and let L be an arbitrary positive integer, then ? gn ? L^1 = |?(t0)|n{ ??=0Lc2?(1/n)? + o((1/n)L) } as n ? ?, where the coefficients c? = 1/?2 ? |K2 ?-??e{-?^2Re(1/2k2)}, S2?(?)d?,(? = 0,1,2,3,...), where S0(?) = 1, and Sr(?) = ?m=1r m!(1/2m) ?'(m1,m2,…,mr),m?j=1r [?j_1=1j Qj-j1(?)Q?j1(?)]mj/mj!, with Q0(?)=1, Qr(?) = = ?m=1rHe2m+r(-?/?K2) ?'(m1,m2,…,mr),m?j=1r {(1/?K2)2+jK2+j/(2+j)!}mj/mj! (r=1,2,3,…) and Kj = (-i)j (ln(?))(j)(t0), (j=2,3,4,…), and where the Hem(u) is the monic Hermite polynomial of degree m. Here, ?^{'} indicates summation over all r-tuples (m1,m2,…,mr) where the mj run over all nonnegative integers which satisfy simultaneously the two conditions ?j=1r,mj = m and ?j=1r j,mj = r. It is proved that in the special case where the Kj , j=2,3,4,…,2+p are all real, then limn ? ? np+1{? gn ?L^1 /|?(t0)|n -1 } =cp+1 = {Im(K3+p)}^2/2(3+p)!(K2)3+p As an application of the above Theorem, it is observed that for a g(x) satisfying,I,II,III,IV above, the corresponding convolution operator Tg : L^1 ? L^1 has ? Tgn ? = ? g_n ?L^1, so that as n ? ?, ? Tgn ?1/n/|?(t0)| - 1 = b(1/n) + (½b^2 + c)(1/n)^2 + o((1/n)^2). Here, the constants b=ln(c0)= ¼ln(1 + (Im(K2)/Re(K2))^2) and c = c1/c0.
Bibliographical Information:


School:The Ohio State University

School Location:USA - Ohio

Source Type:Master's Thesis

Keywords:n fold convolution norm asymptotic expansion


Date of Publication:01/01/2007

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