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] |g_n(x)| dx, of an integrable complex-valued function, g(x), defined on the real line, where, g_n+1( x) = [special characters omitted] g(x − y)g_n( 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 S_r(γ) = [special characters omitted]m![special characters omitted] [special characters omitted] [special characters omitted]/m_j!, with Q₀(γ) = 1, Q_r(γ) == [special characters omitted] (r = 1, 2, 3,…) and K_j = (−i)^j(ln (ĝ))^{(j )}(t₀), (j = 2, 3, 4,…), and where the He_m(u) is the monic Hermite polynomial of degree m. Here, Σ^′ indicates summation over all r-tuples (m₁, m₂, …, m_r) where the m_j 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 K_j, j = 2, 3, 4, …, 2 + p are all real, then limn→∞ n^p+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 T_g : 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 T_g is less rapid, than when ImK₂ = 0.