Asymptotic expansion for the L1 norm of N-fold convolutions
by Stey, George Carl, Ph.D., THE OHIO STATE UNIVERSITY, 2007, 70 pages; 3388779
 Abstract: 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, [special characters omitted] |gn(x)| dx, of an integrable complex-valued function, g(x), defined on the real line, where, gn+1( x) = [special characters omitted] 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)&egr; L1 ∩ [special characters omitted], for some s1 > 1, II: x jg(x)&egr; L1, (j = 1, 2, 3,…), III: There is only one point, t = t0, at which |ĝ( t)| attains its supremum, i.e., |ĝ( t)| < |ĝ(t0)| = [special characters omitted]|ĝ(s)|, for all t ≠ t0, IV: |ĝ(t)| (2)[special characters omitted] < 0, where ĝ(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 [special characters omitted] as n → ∞, where the coefficients cℓ=12p&vbm0; K2&vbm0;-∞ ∞e&cubl0;-g2R e&parl0;12K2&parr0;&cubr0; S2ℓg dg,ℓ=0,1,2,3,&ldots; ,where S0(γ) = 1, and Sr(γ) = [special characters omitted]m![special characters omitted] [special characters omitted] [special characters omitted]/mj!, with Q0(γ) = 1, Qr(γ) == [special characters omitted] (r = 1, 2, 3,…) and Kj = (−i)j(ln (ĝ))(j )(t0), (j = 2, 3, 4,…), and where the Hem(u) is the monic Hermite polynomial of degree m. Here, Σ′ indicates summation over all r-tuples (m1, m2, …, mr) where the mj run over all nonnegative integers which satisfy simultaneously the two conditions [special characters omitted] = m and [special characters omitted] = 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&cubl0;∥gn∥ L1/&vbm0; g&d4;&parl0;t0&parr0;&vbm0; n-1&cubr0;=cp+1=&cubl0;Im&parl0;K 3+p&parr0;&cubr0;223+ p!&parl0;K2&parr0;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 → L1 has [special characters omitted], so that as n → ∞, ∥Tng∥ 1/n/&vbm0;g&d4; &parl0;t0&parr0;&vbm0;-1=b1/n +&parl0;12b2+c&parr0; 1/n2+o&parl0;1/ n2&parr0;. Here, the constants b = ln( c0) = ¼ln(1 + (Im( K2)/Re(K2)) 2) and c = c1/ c0. Thus, when ImK2 ≠ 0 the convergence of [special characters omitted] to the spectral radius of Tg is less rapid, than when ImK2 = 0.

 Adviser Jeffrey D. McNeal School THE OHIO STATE UNIVERSITY Source DAI/B 70-12, p. , Jan 2010 Source Type Dissertation Subjects Mathematics Publication Number 3388779
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