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Asymptotic expansion for the L1 Norm of N-Fold convolutions
Stey, George C.

2007, Doctor of Philosophy, Ohio State University, Mathematics.

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)
εL1∩Ls1
, for
some s1 > 1,

II: xjg(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)|= supsεℝ|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 ∥ gnL1 = |ĝ(t0)|n{
Σℓ=0Lc2ℓ(1/n)
+ o((1/n)L) }
as n → ∞, where the coefficients



c =
1/√2 π |K2
-∞e{-γ2Re(1/2k2)},
S2ℓ(γ)dγ,(ℓ = 0,1,2,3,...), where




S0(γ) = 1, and Sr(γ) = Σm=1r
m!(1/2m)
Σ'(m1,m2,…,mr),mΠj=1r
j_1=1j Qj-j1(γ)Q̄j1(γ)]mj/mj!
,

with Q0(γ)=1, Qr(γ) =
= Σm=1rHe2m+r(-γ/√K2)
Σ'(m1,m2,…,mr),mΠj=1r
{(1/√K2)2+jK2+j/(2+j)!}mj/mj!

(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 Σj=1r,mj = m and
Σj=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{∥ gnL1
/|ĝ(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 ∥ Tgn ∥ =
∥ g_n ∥L1, so that as n → ∞,




Tgn1/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.



Jeffery McNeal (Advisor)
Bogdan Baishanski (Other)
Vitaly Bergelson (Other)
Gerald Edgar (Other)

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Stey, G. (2007). Asymptotic expansion for the L1 Norm of N-Fold convolutions. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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Stey, George. "Asymptotic expansion for the L1 Norm of N-Fold convolutions." Electronic Thesis or Dissertation. Ohio State University, 2007. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

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Stey, George "Asymptotic expansion for the L1 Norm of N-Fold convolutions." Electronic Thesis or Dissertation. Ohio State University, 2007. https://etd.ohiolink.edu/

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