Making the case for Cauchy transforms

Making the case for Cauchy transforms
William T. Ross
University of Richmond
William T. Ross (University of Richmond) Making the case for Cauchy transforms 1 / 42

Prolog
William T. Ross (University of Richmond) Making the case for Cauchy transforms 2 / 42

Cauchy transform of a measure µ
dµ(ζ)
ζ − z
William T. Ross (University of Richmond) Making the case for Cauchy transforms 3 / 42

Theorem (Cauchy - 1831)
If f is analytic on {|z| < 1} and continuous on {|z| 1}, then
f (z) =
2π
0
f (eiθ )
1 − e−iθ dθ
z 2π .
f (z) = 1
f (ζ)
2πi |ζ|=1 ζ − z dζ
William T. Ross (University of Richmond) Making the case for Cauchy transforms 4 / 42

Theorem (Cauchy - 1831)
If f is analytic on {|z| < 1} and continuous on {|z| 1}, then
f (z) =
2π
0
f (eiθ )
1 − e−iθ dθ
z 2π .
f (z) = 1
f (ζ)
2πi |ζ|=1 ζ − z dζ
William T. Ross (University of Richmond) Making the case for Cauchy transforms 4 / 42

2π
f (z) =
0
f (eiθ )
1 − e−iθ dθ
z 2π
Privalov, Morera, Plemelj, and Sokhotski considered
(Kµ)(z) =
2π
0
1
1 − e −iθ z dµ(θ),
where µ is a measure on [0, 2π]. Actually, they considered
(Kµ)(z) =
2π
0
1
1 − e−iθ dF (θ), F ∈ BV [0, 2π].
z
K = {Kµ : µ is a measure on [0, 2π]}
William T. Ross (University of Richmond) Making the case for Cauchy transforms 5 / 42

2π
f (z) =
0
f (eiθ )
1 − e−iθ dθ
z 2π
Privalov, Morera, Plemelj, and Sokhotski considered
(Kµ)(z) =
2π
0
1
1 − e −iθ z dµ(θ),
where µ is a measure on [0, 2π]. Actually, they considered
(Kµ)(z) =
2π
0
1
1 − e−iθ dF (θ), F ∈ BV [0, 2π].
z
K = {Kµ : µ is a measure on [0, 2π]}
William T. Ross (University of Richmond) Making the case for Cauchy transforms 5 / 42

2π
f (z) =
0
f (eiθ )
1 − e−iθ dθ
z 2π
Privalov, Morera, Plemelj, and Sokhotski considered
(Kµ)(z) =
2π
0
1
1 − e −iθ z dµ(θ),
where µ is a measure on [0, 2π]. Actually, they considered
(Kµ)(z) =
2π
0
1
1 − e−iθ dF (θ), F ∈ BV [0, 2π].
z
K = {Kµ : µ is a measure on [0, 2π]}
William T. Ross (University of Richmond) Making the case for Cauchy transforms 5 / 42

Example
µ = δ0 (point mass at θ = 0).
δ0(A) =
(Kδ0)(z) =
2π
0
1, if 0 ∈ A;
0, if 0 ∈ A.
1
1 − e−iθ z dδ0 = 1
1 − z
William T. Ross (University of Richmond) Making the case for Cauchy transforms 6 / 42

Example
µ = δ0 (point mass at θ = 0).
δ0(A) =
(Kδ0)(z) =
2π
0
1, if 0 ∈ A;
0, if 0 ∈ A.
1
1 − e−iθ z dδ0 = 1
1 − z
William T. Ross (University of Richmond) Making the case for Cauchy transforms 6 / 42

Example
µ = δ0 (point mass at θ = 0).
δ0(A) =
(Kδ0)(z) =
2π
0
1, if 0 ∈ A;
0, if 0 ∈ A.
1
1 − e−iθ z dδ0 = 1
1 − z
William T. Ross (University of Richmond) Making the case for Cauchy transforms 6 / 42

Example
µ = ∞
j=1 cjδθj where ∞
j=1 |cj| < ∞. Then
Kµ(z) =
∞
j=1
cj
1 − e −iθj z .
William T. Ross (University of Richmond) Making the case for Cauchy transforms 7 / 42

What I am going to talk about:
Boundary values of Kµ
Mapping properties of the transformation µ → Kµ
Which analytic functions f can be written as f = Kµ?
Distribution function for Kµ, i.e., m(|Kµ| > y)
William T. Ross (University of Richmond) Making the case for Cauchy transforms 8 / 42

What I am going to talk about:
Boundary values of Kµ
Mapping properties of the transformation µ → Kµ
Which analytic functions f can be written as f = Kµ?
Distribution function for Kµ, i.e., m(|Kµ| > y)
William T. Ross (University of Richmond) Making the case for Cauchy transforms 8 / 42

What I am going to talk about:
Boundary values of Kµ
Mapping properties of the transformation µ → Kµ
Which analytic functions f can be written as f = Kµ?
Distribution function for Kµ, i.e., m(|Kµ| > y)
William T. Ross (University of Richmond) Making the case for Cauchy transforms 8 / 42

What I am going to talk about:
Boundary values of Kµ
Mapping properties of the transformation µ → Kµ
Which analytic functions f can be written as f = Kµ?
Distribution function for Kµ, i.e., m(|Kµ| > y)
William T. Ross (University of Richmond) Making the case for Cauchy transforms 8 / 42

What I am going to talk about:
Boundary values of Kµ
Mapping properties of the transformation µ → Kµ
Which analytic functions f can be written as f = Kµ?
Distribution function for Kµ, i.e., m(|Kµ| > y)
William T. Ross (University of Richmond) Making the case for Cauchy transforms 8 / 42

What I’m not going to talk about:
Cauchy transforms of measures in the plane
Topologies on K
Multipliers of K (gKµ = Kν)
Operators on K
William T. Ross (University of Richmond) Making the case for Cauchy transforms 9 / 42

What I’m not going to talk about:
Cauchy transforms of measures in the plane
Topologies on K
Multipliers of K (gKµ = Kν)
Operators on K
William T. Ross (University of Richmond) Making the case for Cauchy transforms 9 / 42

What I’m not going to talk about:
Cauchy transforms of measures in the plane
Topologies on K
Multipliers of K (gKµ = Kν)
Operators on K
William T. Ross (University of Richmond) Making the case for Cauchy transforms 9 / 42

What I’m not going to talk about:
Cauchy transforms of measures in the plane
Topologies on K
Multipliers of K (gKµ = Kν)
Operators on K
William T. Ross (University of Richmond) Making the case for Cauchy transforms 9 / 42

What I’m not going to talk about:
Cauchy transforms of measures in the plane
Topologies on K
Multipliers of K (gKµ = Kν)
Operators on K
William T. Ross (University of Richmond) Making the case for Cauchy transforms 9 / 42

Boundary values
William T. Ross (University of Richmond) Making the case for Cauchy transforms 10 / 42

2π 1
(Kµ)(z) =
0 1 − e−itz dµ(t)
lim
r→1−(Kµ)(reiθ ) =??
William T. Ross (University of Richmond) Making the case for Cauchy transforms 11 / 42

(Kµ)(z) =
=
(1 − r)(Kµ)(re it ) =
µ = c1δθ1 + c2δθ2 + · · · + cnδθn
2π
1
0 1 − e−iθ z dµ(θ)
c1 c2
cn
1 − e−iθ1z +
1 − e−iθ2z + · · · +
1 − e−iθnz (1 − r)c1
1 − e
(1 − r)c2
(1 − r)cn
−iθ1re
+ it −iθ2re
+ · · · + it −iθnreit 1 − e
1 − e
William T. Ross (University of Richmond) Making the case for Cauchy transforms 12 / 42

(Kµ)(z) =
=
(1 − r)(Kµ)(re it ) =
µ = c1δθ1 + c2δθ2 + · · · + cnδθn
2π
1
0 1 − e−iθ z dµ(θ)
c1 c2
cn
1 − e−iθ1z +
1 − e−iθ2z + · · · +
1 − e−iθnz (1 − r)c1
1 − e
(1 − r)c2
(1 − r)cn
−iθ1re
+ it −iθ2re
+ · · · + it −iθnreit 1 − e
1 − e
William T. Ross (University of Richmond) Making the case for Cauchy transforms 12 / 42

(Kµ)(z) =
=
(1 − r)(Kµ)(re it ) =
µ = c1δθ1 + c2δθ2 + · · · + cnδθn
2π
1
0 1 − e−iθ z dµ(θ)
c1 c2
cn
1 − e−iθ1z +
1 − e−iθ2z + · · · +
1 − e−iθnz (1 − r)c1
1 − e
(1 − r)c2
(1 − r)cn
−iθ1re
+ it −iθ2re
+ · · · + it −iθnreit 1 − e
1 − e
William T. Ross (University of Richmond) Making the case for Cauchy transforms 12 / 42

(Kµ)(z) =
=
(1 − r)(Kµ)(re it ) =
µ = c1δθ1 + c2δθ2 + · · · + cnδθn
2π
1
0 1 − e−iθ z dµ(θ)
c1 c2
cn
1 − e−iθ1z +
1 − e−iθ2z + · · · +
1 − e−iθnz (1 − r)c1
1 − e
(1 − r)c2
(1 − r)cn
−iθ1re
+ it −iθ2re
+ · · · + it −iθnreit 1 − e
1 − e
William T. Ross (University of Richmond) Making the case for Cauchy transforms 12 / 42

(1 − r)(Kµ)(re it ) =
Thus
(1 − r)c1
1 − re
(1 − r)c2
(1 − r)cn
+ + · · · +
i(t−θ1) 1 − rei(t−θ2) 1 − rei(t−θn) lim
r→1−(1 − r)(Kµ)(reit
cj, if t = θj;
) =
0, otherwise.
lim
r→1 − |(Kµ)(reit )| = +∞ if t = θj
William T. Ross (University of Richmond) Making the case for Cauchy transforms 13 / 42

(1 − r)(Kµ)(re it ) =
Thus
(1 − r)c1
1 − re
(1 − r)c2
(1 − r)cn
+ + · · · +
i(t−θ1) 1 − rei(t−θ2) 1 − rei(t−θn) lim
r→1−(1 − r)(Kµ)(reit
cj, if t = θj;
) =
0, otherwise.
lim
r→1 − |(Kµ)(reit )| = +∞ if t = θj
William T. Ross (University of Richmond) Making the case for Cauchy transforms 13 / 42

(1 − r)(Kµ)(re it ) =
Thus
(1 − r)c1
1 − re
(1 − r)c2
(1 − r)cn
+ + · · · +
i(t−θ1) 1 − rei(t−θ2) 1 − rei(t−θn) lim
r→1−(1 − r)(Kµ)(reit
cj, if t = θj;
) =
0, otherwise.
lim
r→1 − |(Kµ)(reit )| = +∞ if t = θj
William T. Ross (University of Richmond) Making the case for Cauchy transforms 13 / 42

For a general measure µ,
lim
r→1−(1 − r)(Kµ)(reit ) = lim
r→1− Thus, by taking µ =
∞
2 −n δθn , θn dense in [0, 2π]
n=1
on a dense subset of the circle.
lim
r→1− |(Kµ)(reit )| = +∞
1 − r
dµ(θ) = µ({t})
1 − rei(t−θ) William T. Ross (University of Richmond) Making the case for Cauchy transforms 14 / 42

For a general measure µ,
lim
r→1−(1 − r)(Kµ)(reit ) = lim
r→1− Thus, by taking µ =
∞
2 −n δθn , θn dense in [0, 2π]
n=1
on a dense subset of the circle.
lim
r→1− |(Kµ)(reit )| = +∞
1 − r
dµ(θ) = µ({t})
1 − rei(t−θ) William T. Ross (University of Richmond) Making the case for Cauchy transforms 14 / 42

For a general measure µ,
lim
r→1−(1 − r)(Kµ)(reit ) = lim
r→1− Thus, by taking µ =
∞
2 −n δθn , θn dense in [0, 2π]
n=1
on a dense subset of the circle.
lim
r→1− |(Kµ)(reit )| = +∞
1 − r
dµ(θ) = µ({t})
1 − rei(t−θ) William T. Ross (University of Richmond) Making the case for Cauchy transforms 14 / 42

f (z) =
∞
j=1
cj
1 − e −iθj z ,
∞
|cj| < ∞
In a way Poincaré knew about this behavior back in 1883 while creating
non-continuable functions.
..... but needed a longer proof since the Lebesgue dominated convergence
theorem was not discovered yet.
William T. Ross (University of Richmond) Making the case for Cauchy transforms 15 / 42
j=1

Theorem (Smirnov - 1929)
For almost every (Lebesgue measure) t ∈ [0, 2π],
Kµ(e it ) = lim
r→1 −(Kµ)(reit )
exists and is finite. Moreover, for each 0 < p < 1,
and in fact 2π
0
2π
|Kµ(e
0
it )| p dt < ∞
|Kµ(e it )| p dt = O( 1
1 − p ).
William T. Ross (University of Richmond) Making the case for Cauchy transforms 16 / 42

Theorem (Smirnov - 1929)
For almost every (Lebesgue measure) t ∈ [0, 2π],
Kµ(e it ) = lim
r→1 −(Kµ)(reit )
exists and is finite. Moreover, for each 0 < p < 1,
and in fact 2π
0
2π
|Kµ(e
0
it )| p dt < ∞
|Kµ(e it )| p dt = O( 1
1 − p ).
William T. Ross (University of Richmond) Making the case for Cauchy transforms 16 / 42

Theorem (Smirnov - 1929)
For almost every (Lebesgue measure) t ∈ [0, 2π],
Kµ(e it ) = lim
r→1 −(Kµ)(reit )
exists and is finite. Moreover, for each 0 < p < 1,
and in fact 2π
0
2π
|Kµ(e
0
it )| p dt < ∞
|Kµ(e it )| p dt = O( 1
1 − p ).
William T. Ross (University of Richmond) Making the case for Cauchy transforms 16 / 42

Theorem (Fatou - 1906)
For almost every (Lebesgue measure) t ∈ [0, 2π],
lim
r→1−(Kµ)(reit ) − lim
r→1−(Kµ)(1r eit ) = Dµ(t)
µ([t − h, t + h])
Dµ(t) = lim
h→0 2h
William T. Ross (University of Richmond) Making the case for Cauchy transforms 17 / 42

Theorem (Fatou - 1906)
For almost every (Lebesgue measure) t ∈ [0, 2π],
lim
r→1−(Kµ)(reit ) − lim
r→1−(Kµ)(1r eit ) = Dµ(t)
µ([t − h, t + h])
Dµ(t) = lim
h→0 2h
William T. Ross (University of Richmond) Making the case for Cauchy transforms 17 / 42

Theorem (Privalov - 1919)
For almost every (Lebesgue measure) t ∈ [0, 2π],
lim
r→1−(Kµ)(reit ) + lim
r→1−(Kµ)(1r eit 2π 1
) = 2P.V .
dµ(θ).
0 1 − ei(θ−t)
1
dµ(ζ), |z| < 1
ζ − z
1
dµ(ζ), |ξ| = 1
ζ − ξ
William T. Ross (University of Richmond) Making the case for Cauchy transforms 18 / 42

Theorem (Privalov - 1919)
For almost every (Lebesgue measure) t ∈ [0, 2π],
lim
r→1−(Kµ)(reit ) + lim
r→1−(Kµ)(1r eit 2π 1
) = 2P.V .
dµ(θ).
0 1 − ei(θ−t)
1
dµ(ζ), |z| < 1
ζ − z
1
dµ(ζ), |ξ| = 1
ζ − ξ
William T. Ross (University of Richmond) Making the case for Cauchy transforms 18 / 42

Theorem (Privalov - 1919)
For almost every (Lebesgue measure) t ∈ [0, 2π],
lim
r→1−(Kµ)(reit ) + lim
r→1−(Kµ)(1r eit 2π 1
) = 2P.V .
dµ(θ).
0 1 − ei(θ−t)
1
dµ(ζ), |z| < 1
ζ − z
1
dµ(ζ), |ξ| = 1
ζ − ξ
William T. Ross (University of Richmond) Making the case for Cauchy transforms 18 / 42

Mapping properties
William T. Ross (University of Richmond) Making the case for Cauchy transforms 19 / 42

There are a multitude of results which talk about the mapping properties
of µ → Kµ.
William T. Ross (University of Richmond) Making the case for Cauchy transforms 20 / 42

Theorem (M. Riesz - 1927)
If 1 < p < ∞ and g(θ) ∈ L p [0, 2π] and
then
f (z) =
2π
0
g(θ)
1 − e−iθ dθ
z 2π ,
2π
sup |f (re
0

Theorem (M. Riesz - 1927)
If 1 < p < ∞ and g(θ) ∈ L p [0, 2π] and
then
f (z) =
2π
0
g(θ)
1 − e−iθ dθ
z 2π ,
2π
sup |f (re
0

We began this talk with the Cauchy integral formula
f (z) =
2π
0
f (eiθ )
1 − e−iθ dθ
z 2π
f analytic on {|z| < 1} and continuous on {|z| 1}.
Theorem (M. Riesz - 1931)
If f is analytic on {|z| < 1} and
2π
sup |f (re
0

We began this talk with the Cauchy integral formula
f (z) =
2π
0
f (eiθ )
1 − e−iθ dθ
z 2π
f analytic on {|z| < 1} and continuous on {|z| 1}.
Theorem (M. Riesz - 1931)
If f is analytic on {|z| < 1} and
2π
sup |f (re
0

An extension of the CIF
f (z) =
2π
Ul’yanov (1956), Aleksandrov (1981):
0
f (eiθ )
1 − e−iθ dθ
z 2π
f (e
f (z) = lim
A→∞ |f |

An extension of the CIF
f (z) =
2π
Ul’yanov (1956), Aleksandrov (1981):
0
f (eiθ )
1 − e−iθ dθ
z 2π
f (e
f (z) = lim
A→∞ |f |

An extension of the CIF
f (z) =
2π
Ul’yanov (1956), Aleksandrov (1981):
0
f (eiθ )
1 − e−iθ dθ
z 2π
f (e
f (z) = lim
A→∞ |f |

Distribution function
William T. Ross (University of Richmond) Making the case for Cauchy transforms 24 / 42

We know that the function
is defined for almost every θ.
(Kµ)(e iθ ) = lim
r→1 −(Kµ)(reiθ )
What can we say about the distribution function
y → m(|Kµ| > y)?
m(|Kµ| > y) = 1
2π Leb. meas. of {θ : |(Kµ)(eiθ )| > y}
William T. Ross (University of Richmond) Making the case for Cauchy transforms 25 / 42

We know that the function
is defined for almost every θ.
(Kµ)(e iθ ) = lim
r→1 −(Kµ)(reiθ )
What can we say about the distribution function
y → m(|Kµ| > y)?
m(|Kµ| > y) = 1
2π Leb. meas. of {θ : |(Kµ)(eiθ )| > y}
William T. Ross (University of Richmond) Making the case for Cauchy transforms 25 / 42

We know that the function
is defined for almost every θ.
(Kµ)(e iθ ) = lim
r→1 −(Kµ)(reiθ )
What can we say about the distribution function
y → m(|Kµ| > y)?
m(|Kµ| > y) = 1
2π Leb. meas. of {θ : |(Kµ)(eiθ )| > y}
William T. Ross (University of Richmond) Making the case for Cauchy transforms 25 / 42

For any f (θ) ∈ L 1 [0, 2π], we have
Note that
m(|f | > y) 1
2π
|f (θ)|
y 0
dθ
2π
yχ |f |>y |f |
William T. Ross (University of Richmond) Making the case for Cauchy transforms 26 / 42

Example
|Kδ0(e iθ )| =
1
|1 − e iθ |
1 1
=
2 | sin(θ/2)|
m(|Kδ0| > y) = 2
π sin−1 ( 1 1
) = O(
2y y )
William T. Ross (University of Richmond) Making the case for Cauchy transforms 27 / 42

Example
|Kδ0(e iθ )| =
1
|1 − e iθ |
1 1
=
2 | sin(θ/2)|
m(|Kδ0| > y) = 2
π sin−1 ( 1 1
) = O(
2y y )
William T. Ross (University of Richmond) Making the case for Cauchy transforms 27 / 42

Boole’s idea:
Suppose c1, · · · , cn > 0 and a1 < a2 < · · · < an. Define
g(x) = c1
+
x − a1
c2
+ · · · +
x − a2
cn
x − an
m1(g > y) =?
William T. Ross (University of Richmond) Making the case for Cauchy transforms 28 / 42

Boole’s idea:
Suppose c1, · · · , cn > 0 and a1 < a2 < · · · < an. Define
g(x) = c1
+
x − a1
c2
+ · · · +
x − a2
cn
x − an
m1(g > y) =?
William T. Ross (University of Richmond) Making the case for Cauchy transforms 28 / 42

g(x) = c1
+
x − a1
c2
+ · · · +
x − a2
cn
x − an
William T. Ross (University of Richmond) Making the case for Cauchy transforms 29 / 42

Theorem (Boole - 1857)
Suppose c1, · · · , cn > 0, a1 < a2 < · · · < an and
Then
g(x) = c1
+
x − a1
c2
+ · · · +
x − a2
cn
.
x − an
m1(g > y) = 1
y (c1 + c2 + · · · + cn)
William T. Ross (University of Richmond) Making the case for Cauchy transforms 30 / 42

Proof of Boole’s theorem (not the original):
William T. Ross (University of Richmond) Making the case for Cauchy transforms 31 / 42

g(x) = c1
+
x − a1
c2
+ · · · +
x − a2
cn
x − an
m1(g > y) = (s1 − a1) + (s2 − a2) + · · · + (sn − an)
Consider the polynomial
p(x) =
n
(x − aj) 1 − g(x)
y
j=1
The roots of this poly are s1, s2, · · · , sn.
p(x) = x n ⎛
n
− ⎝ aj + 1
y
j=1
Viète’s formula ⇒
n
sj =
j=1
n
j=1
cj
n
j=1
⎞
⎠ x n−1 + · · ·
aj + 1
y
William T. Ross (University of Richmond) Making the case for Cauchy transforms 32 / 42
n
j=1
cj

g(x) = c1
+
x − a1
c2
+ · · · +
x − a2
cn
x − an
m1(g > y) = (s1 − a1) + (s2 − a2) + · · · + (sn − an)
Consider the polynomial
p(x) =
n
(x − aj) 1 − g(x)
y
j=1
The roots of this poly are s1, s2, · · · , sn.
p(x) = x n ⎛
n
− ⎝ aj + 1
y
j=1
Viète’s formula ⇒
n
sj =
j=1
n
j=1
cj
n
j=1
⎞
⎠ x n−1 + · · ·
aj + 1
y
William T. Ross (University of Richmond) Making the case for Cauchy transforms 32 / 42
n
j=1
cj

g(x) = c1
+
x − a1
c2
+ · · · +
x − a2
cn
x − an
m1(g > y) = (s1 − a1) + (s2 − a2) + · · · + (sn − an)
Consider the polynomial
p(x) =
n
(x − aj) 1 − g(x)
y
j=1
The roots of this poly are s1, s2, · · · , sn.
p(x) = x n ⎛
n
− ⎝ aj + 1
y
j=1
Viète’s formula ⇒
n
sj =
j=1
n
j=1
cj
n
j=1
⎞
⎠ x n−1 + · · ·
aj + 1
y
William T. Ross (University of Richmond) Making the case for Cauchy transforms 32 / 42
n
j=1
cj

g(x) = c1
+
x − a1
c2
+ · · · +
x − a2
cn
x − an
m1(g > y) = (s1 − a1) + (s2 − a2) + · · · + (sn − an)
Consider the polynomial
p(x) =
n
(x − aj) 1 − g(x)
y
j=1
The roots of this poly are s1, s2, · · · , sn.
p(x) = x n ⎛
n
− ⎝ aj + 1
y
j=1
Viète’s formula ⇒
n
sj =
j=1
n
j=1
cj
n
j=1
⎞
⎠ x n−1 + · · ·
aj + 1
y
William T. Ross (University of Richmond) Making the case for Cauchy transforms 32 / 42
n
j=1
cj

g(x) = c1
+
x − a1
c2
+ · · · +
x − a2
cn
x − an
m1(g > y) = (s1 − a1) + (s2 − a2) + · · · + (sn − an)
Consider the polynomial
p(x) =
n
(x − aj) 1 − g(x)
y
j=1
The roots of this poly are s1, s2, · · · , sn.
p(x) = x n ⎛
n
− ⎝ aj + 1
y
j=1
Viète’s formula ⇒
n
sj =
j=1
n
j=1
cj
n
j=1
⎞
⎠ x n−1 + · · ·
aj + 1
y
William T. Ross (University of Richmond) Making the case for Cauchy transforms 32 / 42
n
j=1
cj

g(x) = c1
+
x − a1
c2
+ · · · +
x − a2
cn
x − an
m1(g > y) = (s1 − a1) + (s2 − a2) + · · · + (sn − an)
Consider the polynomial
p(x) =
n
(x − aj) 1 − g(x)
y
j=1
The roots of this poly are s1, s2, · · · , sn.
p(x) = x n ⎛
n
− ⎝ aj + 1
y
j=1
Viète’s formula ⇒
n
sj =
j=1
n
j=1
cj
n
j=1
⎞
⎠ x n−1 + · · ·
aj + 1
y
William T. Ross (University of Richmond) Making the case for Cauchy transforms 32 / 42
n
j=1
cj

Theorem (Kolmogorov - 1925)
m(|Kµ| > y) = O
1
, y → ∞.
y
William T. Ross (University of Richmond) Making the case for Cauchy transforms 33 / 42

Kolmogorov says
lim sup ym(|Kµ| > y) < ∞.
y→∞
Theorem (Hruschev-Vinogradov - 1981)
Theorem (Poltoratski - 1996)
1
lim ym(|Kµ| > y) =
y→∞ π µs
yχ |Kµ|>y · m → 1
π µs weak-* as y → ∞
William T. Ross (University of Richmond) Making the case for Cauchy transforms 34 / 42

Kolmogorov says
lim sup ym(|Kµ| > y) < ∞.
y→∞
Theorem (Hruschev-Vinogradov - 1981)
Theorem (Poltoratski - 1996)
1
lim ym(|Kµ| > y) =
y→∞ π µs
yχ |Kµ|>y · m → 1
π µs weak-* as y → ∞
William T. Ross (University of Richmond) Making the case for Cauchy transforms 34 / 42

Kolmogorov says
lim sup ym(|Kµ| > y) < ∞.
y→∞
Theorem (Hruschev-Vinogradov - 1981)
Theorem (Poltoratski - 1996)
1
lim ym(|Kµ| > y) =
y→∞ π µs
yχ |Kµ|>y · m → 1
π µs weak-* as y → ∞
William T. Ross (University of Richmond) Making the case for Cauchy transforms 34 / 42

Geometric conditions
William T. Ross (University of Richmond) Making the case for Cauchy transforms 35 / 42

Q: If f is analytic on D = {|z| < 1}. When is f = Kµ?
William T. Ross (University of Richmond) Making the case for Cauchy transforms 36 / 42

Theorem (Herglotz - 1911)
If f (D) is contained in a half-plane, then f = Kµ.
William T. Ross (University of Richmond) Making the case for Cauchy transforms 37 / 42

Theorem (Cima-Bourdon - 1986)
If f (D) is contained in a region which omits two oppositely pointing rays,
then f = Kµ.
William T. Ross (University of Richmond) Making the case for Cauchy transforms 38 / 42

Example
f (z) = 2z 1
=
1 − z2 1 − z
− 1
1 + z = K(δ0 − δπ)(z)
William T. Ross (University of Richmond) Making the case for Cauchy transforms 39 / 42

Example
f (z) = 2z 1
=
1 − z2 1 − z
− 1
1 + z = K(δ0 − δπ)(z)
William T. Ross (University of Richmond) Making the case for Cauchy transforms 39 / 42

Example
f (z) = 2z 1
=
1 − z2 1 − z
− 1
1 + z = K(δ0 − δπ)(z)
William T. Ross (University of Richmond) Making the case for Cauchy transforms 39 / 42

The normalized Cauchy transform
William T. Ross (University of Richmond) Making the case for Cauchy transforms 40 / 42

Consider the normalized Cauchy transform
K(fdµ)
K(dµ)
lim
r→1− K(fdµ)(reiθ )
K(dµ)(reiθ = lim
) r→1− (1 − r)K(fdµ)(reiθ )
(1 − r)K(dµ)(reiθ )
= f (eiθ )µ({θ})
µ({θ})
= f (e iθ )
William T. Ross (University of Richmond) Making the case for Cauchy transforms 41 / 42

Consider the normalized Cauchy transform
K(fdµ)
K(dµ)
lim
r→1− K(fdµ)(reiθ )
K(dµ)(reiθ = lim
) r→1− (1 − r)K(fdµ)(reiθ )
(1 − r)K(dµ)(reiθ )
= f (eiθ )µ({θ})
µ({θ})
= f (e iθ )
William T. Ross (University of Richmond) Making the case for Cauchy transforms 41 / 42

Theorem (Poltoratski - 1993)
lim
r→1− K(fdµ)(reiθ )
K(dµ)(reiθ ) = f (eiθ ) µs-a.e. θ.
William T. Ross (University of Richmond) Making the case for Cauchy transforms 42 / 42