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Asymptotic expansion for the L¹ Norm of N-Fold convolutions

by Stey, George Carl.

Abstract (Summary)
An asymptotic expansion of nonnegative powers of 1/n is obtained which describes the large-n behavior of the L1 norm of the n-fold convolution, ? gn ?L1= ? ? ?? |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)? L1 ? Ls1, for some s1 > 1 , II: xjg(x) ? L1, (j = 1, 2, 3, ...), III: There is only one point, t = t0, at which |ˆg(t)| attains its supremum. i.e,|ˆg(t)| < |ˆg(t0)| = sups?R|ˆg(s)|,for all t ?= t0, IV: |ˆg(t)|(2) |t=t0 < 0, where ˆg(t) denotes the Fourier transform of g(x). We obtain the following Theorem: Let g(x) satisfy simultaneously Assumptions I,II,III,IV above, and let L be an arbitrary positive integer, then ? gn ?L1= |ˆg(t0)|n{?L 1 ?=0c?( n)? + o(( 1 n)L)} as n ? ?, where the coefficients 1 c? = ?2?|K2| ? ? ?? e{??2 Re( 1 2K2 )} S2?(?)d?,(? = 0, 1, 2, 3, ...), where S0(?) = 1, and Sr(?) = ?r m=1m!(1/2 m )? ? (m1,m2,...,mr),m?r j=1[?j j1=1Qj?j1(?) ¯ Qj1(?)]mj /mj!, with Q0(?) = 1 , Qr(?) = = ?r m=1He2m+r( ?? ? K2 )?? (m1,m2,...,mr),m?rj=1{( 1 ? K2 )2+j K2+j (2+j)!}mj /mj! (r = 1, 2, 3, ...) and Kj = (?i)j(ln(ˆg))(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 ii simultaneously the two conditions ?r j=1 mj = m and ?r j=1j 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 ?L1 /|ˆg(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 : L1 ? L1 has ? T n g ?=? gn ?L1, so that as n ? ?, ? T n g ?1/n /|ˆg(t0)| ? 1 = b(1/n) + ( 1 2b2 + c)(1/n)2 + o((1/n)2). Here, the constants b = ln(c0) = 1 4ln(1 + (Im(K2)/Re(K2))2) and c = c1/c0. Thus, when ImK2 ?= 0 the convergence of ? T n g ?1/n to the spectral radius of Tg is less rapid, than when ImK2 = 0 iii
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School:The Ohio State University

School Location:USA - Ohio

Source Type:Master's Thesis

Keywords:convolutions mathematics asymptotic expansions

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