A multidimensional Wiener-Wintner theorem and spectrum estimation
Authors: Benedetto John J. 
Published: June 2012

Sufficient conditions are given for a bounded positive measure

μ
\mu

on

R

d

{\mathbb {R}^d}

to be the power spectrum of a function

φ
\varphi

. Applications to spectrum estimation are made for the cases in which a signal

φ
\varphi

is known or its autocorrelation

P
ϕ

{P_\phi }

is known. In the first case, it is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral StartAbsoluteValue ModifyingAbove f With caret left-parenthesis gamma right-parenthesis EndAbsoluteValue squared d mu Subscript phi Baseline left-parenthesis gamma right-parenthesis equals limit Underscript upper R right-arrow normal infinity Endscripts StartFraction 1 Over StartAbsoluteValue upper B left-parenthesis upper R right-parenthesis EndAbsoluteValue EndFraction integral Underscript upper B left-parenthesis upper R right-parenthesis Endscripts StartAbsoluteValue f asterisk phi left-parenthesis t right-parenthesis EndAbsoluteValue squared d t comma">

∫

|

f
^

(
γ
)

|

2

d

μ
ϕ

(
γ
)
=

lim

R
→
∞

1

|

B
(
R
)

|

∫

B
(
R
)

|

f
∗
φ
(
t
)

|

2

d
t
,

\int {|\hat f(\gamma )|^2}d{\mu _\phi }(\gamma )= \lim \limits _{R \to \infty }\,\frac {1}{|B(R )|}\,\int _{B(R)} |f \ast \varphi (t)|^2\;dt,

\]
where

P
^

φ

=

μ
φ

{\hat P}_{\varphi }= {\mu _\varphi }

,

B
(
R
)

B(R)

is the

d
d

-dimensional ball of radius

R
R

, and

f
f

ranges through a prescribed function space. In the second case, an example, which is a variant of the Tomas-Stein restriction theorem, is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="for-all f element-of upper L Superscript 1 Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis intersection upper L Superscript p Baseline left-parenthesis double-struck upper R Superscript d Baseline right-parenthesis comma left-parenthesis integral Underscript sigma-summation Underscript d minus 1 Endscripts Endscripts StartAbsoluteValue ModifyingAbove f With caret left-parenthesis theta right-parenthesis EndAbsoluteValue squared d mu Subscript d minus 1 Baseline left-parenthesis theta right-parenthesis right-parenthesis Superscript 1 slash 2 Baseline less-than-or-equal-to left-parenthesis one half parallel-to mu Subscript d minus 1 Superscript logical-or Baseline parallel-to right-parenthesis left-parenthesis parallel-to f parallel-to Subscript 1 Baseline plus parallel-to f parallel-to right-parenthesis comma">

∀
f
∈

L
1

(

R

d

)

∩

L
p

(

R

d

)
,

(

∫

∑

d
−
1

|

f
^

(
θ
)

|

2

d

μ

d
−
1

(
θ
)
)

1

/

2

≤

(

1
2

∥

μ

d
−
1

∨

∥

p
′

1

/

2

)

(
∥
f

∥

1

+
∥
f

∥

p

)

,

\forall f \in {L^1}({\mathbb {R}^d})\, \cap \,{L^p}({\mathbb {R}^d}), \\ \left (\int _{\sum _{d - 1}} | \hat f(\theta )|^2 d {\mu _{d - 1}}(\theta ) \right )^{1/2} \leq \left (\frac {1}{2}\parallel \mu _{d - 1}^{\vee } \parallel _{p’}^{1/2} \right )\;\left (\parallel f {\parallel _{1}} + \parallel f{\parallel _{p}} \right ),

\]
where

1
≤
p
>
2
d

/

(
d
+
1
)

1 \leq p > 2d/(d + 1)

and the power spectrum

μ

d
−
1

{\mu _{d - 1}}

is the compactly supported restriction of surface measure to the unit sphere

∑

d
−
1

⊆

R

^

d

\sum \nolimits _{d - 1} { \subseteq } \;{{\hat {\mathbb {R}}}^d}

.