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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 *L*^{1} norm of

the n-fold convolution, ∥ g_{n}

∥*L*^{1} =

∫_{-∞}^{∞}|g_{n}(x)|d*x*,

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, g_{1}(x) =g(x).

Consideration is restricted here to those *g(x)* which

simultaneously satisfy the following four Assumptions I: *g(x) εL ^{1}∩L^{s1}*, for

some

II:

III: There is

only one point,

supremum. i.e,

IV:

|ĝ(t)|

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(t_{0})|^{n}{

Σ_{ℓ=0}^{L}c_{2ℓ}(1/n)^{ℓ}

+ o((1/n)^{L}) }

as *n → ∞*, where the coefficients

*c _{ℓ}* =

1/√2 π |K

∫

S

*S _{0}(γ) = 1*, and

m!(

Σ

[Σ

with

= Σ

Σ

{(1/√K

(ln(ĝ))

is the monic Hermite

polynomial of degree m. Here, Σ^{'} indicates summation over

all r-tuples

where the

all nonnegative integers which satisfy simultaneously the two

conditions Σ

= r

j=2,3,4,…,2+p

lim_{n → ∞}

n^{p+1}{∥ g_{n} ∥_{L1}

/|ĝ(t_{0})|^{n} -1 }

=c_{p+1} =

{Im(K_{3+p})}^{2}/2(3+p)!(K_{2})^{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} → L^{1}*

has ∥ T

∥ g_n ∥

∥

T_{g}^{n} ∥^{1/n}/|ĝ(t_{0})| - 1 = b(1/n) + (½b^{2} + c)(1/n)^{2} + o((1/n)^{2}).

Here, the constants *b=ln(c _{0})= ¼ln(1 + (Im(K_{2})/Re(K_{2}))^{2})* and

Jeffery McNeal (Advisor)

Bogdan Baishanski (Other)

Vitaly Bergelson (Other)

Gerald Edgar (Other)

Bogdan Baishanski (Other)

Vitaly Bergelson (Other)

Gerald Edgar (Other)

Stey, G. (2007). Asymptotic expansion for the *L*^{1} Norm of N-Fold convolutions. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

Stey, George. "Asymptotic expansion for the *L*^{1} Norm of N-Fold convolutions." Electronic Thesis or Dissertation. Ohio State University, 2007. OhioLINK Electronic Theses and Dissertations Center. 25 Sep 2017.

Stey, George "Asymptotic expansion for the *L*^{1} Norm of N-Fold convolutions." Electronic Thesis or Dissertation. Ohio State University, 2007. https://etd.ohiolink.edu/