Asymptotic expansion for the L¹ Norm of N-Fold convolutions

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, ∥ g_n ∥L¹ = ∫_-∞^∞|g_n(x)|dx, of an integrable complex-valued function, g(x), defined on the real line, where, g_n+1(x) = ∫_-∞^∞ 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) εL¹∩L^s₁, for some s₁ > 1, II: x^jg(x)εL¹, (j=1,2,3,…), III: There is only one point, t= t₀, at which |ĝ(t)| attains its supremum. i.e, |ĝ(t)| < |hĝ(t₀)|= sup_sεℝ|hĝ(s)|,for all t ≠ t₀, IV: |ĝ(t)|⁽²⁾|_t=t0 < 0,

where ĝ(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 ∥ g_n ∥ _L¹ = |ĝ(t₀)|ⁿ{ Σ_ℓ=0^Lc_2ℓ(1/n)^ℓ + o((1/n)^L) } as n → ∞, where the coefficients

c_ℓ = 1/√2 π |K₂ ∫_-∞^∞e^{{-γ2Re(1/2k₂)}}, S_2ℓ(γ)dγ,(ℓ = 0,1,2,3,...), where

S₀(γ) = 1, and S_r(γ) = Σ_m=1^r m!(^1/2_m) Σ^'_{(m1,m2,…,mr),m}Π_j=1^r [Σ_{j_1=1}^j Q_j-j1(γ)Q̄_j1(γ)]^m_j/m_j!, with Q₀(γ)=1, Q_r(γ) = = Σ_{m=1^rHe2m+r(-γ/√K2}) Σ^'_{(m1,m2,…,mr),m}Π_j=1^r {(1/√K₂)^2+jK_2+j/(2+j)!}^m_j/m_j! (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 Σ_j=1^r,m_j = m and Σ_j=1^r j,m_j = r. It is proved that in the special case where the K_j , j=2,3,4,…,2+p are all real, then

lim_{n → ∞} n^p+1{∥ g_n ∥_L¹ /|ĝ(t₀)|ⁿ -1 } =c_p+1 = {Im(K_3+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 T_g : L¹ → L¹ has ∥ T_gⁿ ∥ = ∥ g_n ∥_L¹, so that as n → ∞,

∥ T_gⁿ ∥^1/n/|ĝ(t₀)| - 1 = b(1/n) + (½b² + 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.