# Wavefronts, light rays and caustic associated with the refraction of a plane
wave by a conospherical lens

José Israel Galindo-Rodríguez<EMAIL_ADDRESS>Facultad de Ciencias
Físico Matemáticas de la Benemérita Universidad Autónoma de Puebla, Apartado
Postal 1152, 72001, Puebla, Pue., México.  Gilberto Silva-Ortigoza  Facultad
de Ciencias Físico Matemáticas de la Benemérita Universidad Autónoma de
Puebla, Apartado Postal 1152, 72001, Puebla, Pue., México.

###### Abstract

The aim of the present work is to introduce a lens whose faces are a conical
surface and a spherical surface. We illuminate this lens by a plane wavefront
and its associated refracted wavefronts, light rays and caustic are computed.
We find that the caustic has two branches. The first is constituted by two
segments of a line, one part of this caustic is real and the other one
virtual. The second branch of the caustic is a two-dimensional surface with a
singularity of the cusp ridge type. It is important to remark that the two
branches of the caustic are disconnected. Because of this property we believe
that using this optical element one could generate a scalar optical
accelerating beam in the region where the caustic is a two-dimensional surface
of revolution, and at the same time a scalar optical beam with similar
properties to the Bessel beam of zero order in the region were the real
caustic is a segment of a line along the optical axis.

Geometrical Optics, Lenses, Caustics.

##  I Introduction

In 1954 McLeod [1] introduced a new class of optical elements, these were
called axicons. There are different kinds of axicons, but the most important
one is a glass cone. At that time one of its possible applications was in a
telescope [2]. In this case, the usual spherical objective is replaced by a
cone. This axicon telescope is in focus for targets from a foot or so to
infinity without the necessity of moving any parts. It can be used to view
simultaneously two or more small sources placed along the line of sight.

One of the most important properties of the axicon lens is its caustic, which
is a focal line when it is illuminated by a plane wave evolving along the axis
of symmetry of the lens. Because of this property, the axicon lens has been
used for different purposes. In optical coherence tomography, axial and
lateral resolutions are determined by the source coherence length and the
numerical aperture of the sampling lens, respectively. Whereas axial
resolution can be improved by use of a broadband light source, there is a
trade-off between lateral resolution and focusing depth when conventional
optical elements are used. Ding et al. [3] incorporated an axicon lens into
the sample arm of an interferometer to overcome this limitation. Burvall et
al. [4] presented the design of a cemented double-lens axicon made from
spherical surfaces only. This axicon lens is based on the deliberate use of
the spherical aberration of the surfaces and has a narrow, extended line focus
of relatively constant width. Golub [5] had introduced the fraxicon, this
element consists of concentric prism-like grooves with an apex angle equal to
that of a bulk axicon and whose action relies on refraction. Cabrini et al.
[6] by using an axicon lens showed that it is possible to make efficient
optical tweezers. Bélanger and Rioux [7] combined an axicon and a lens to form
an optical system producing a ring-shaped pattern to show that when a
lens–axicon combination is illuminated by a Gaussian beam, the transverse
distribution of the focal ring is also a Gaussian distribution. Tanaka and
Yamamoto [8] studied the aberrations produced by an axicon and a lens. They
found that the beam spot of an axicon is affected only by astigmatism and not
by coma, while the point created by a lens is blurred by both astigmatism and
coma. This implies that in the scanning optical system that incorporates an
axicon it is easy to correct the aberrations and achieves high-resolution and
high contrast imaging with a wide field of view. Zheng et al. [9] demonstrated
experimentally an efficient electron axicon lens using a magnetic vortex. In
particular, the axicon provides the most efficient method for realizing
generally diffractionless beams [10].

It is important to remark that there are few works on the axicon lens using
geometrical optics [1, 2, 11, 12]. Rayces [11] studied, in a qualitative way,
the formation of axicon images. To be more precise, he considered a simple
refracting axicon illuminated by a point light source, placed on and off the
optical axis, and described the form of the caustic associated with the
refracted light rays. By using the property that the axicon, when the point
light source is on the optical axis, can be regarded as a double prism, he
concluded that the caustic has two sheets; one is a two-dimensional surface,
and the other is a segment of a line. For the case in which the point light
source is placed off the optical axis, he explained that the sheet of the
caustic corresponding to the segment of a line unfolds into a two-dimensional
surface such that its transversal section has four cusp singularities.

The aim of the present work is to introduce a lens whose faces are a conical
surface and a spherical surface. We illuminate this lens by a plane wavefront
and its associated refracted wavefronts, light rays and caustic are computed.
We find that the caustic has two branches. The first is constituted by two
segments of a line, one part of this caustic is real and the other one
virtual. The second branch of the caustic is a two-dimensional surface with a
singularity of the cusp ridge type. It is important to remark that the two
branches of the caustic are disconnected. Because of this property we believe
that using this optical element one could generate a scalar optical
accelerating beam in the region where the caustic is a two-dimensional surface
of revolution, and at the same time a scalar optical beam with similar
properties to the Bessel beam of zero order in the region were the real
caustic is a segment of a line along the optical axis.

To this end, in section 2 we review the refraction of a plane wavefront by the
plano spherical and the axicon lenses. That is, we assumed that we have a
plane wavefront evolving along the $\hat{z}$ direction and it is refracted by
a plano spherical lens. Then, we compute the corresponding refracted
wavefronts, light rays and caustic. Furthermore, in this section we also
obtain analogous results for the axicon lens. In section 3 we study the
refraction of a plane wave evolving along the $z$-axis by the conospherical
lens. To be more precise, we obtain the refracted wavefronts, light rays and
the corresponding caustic. We find that the caustic has three components, one
of them is virtual, and the other two are real. The virtual caustic is a
segment of a line along the $z$-axis and the two real branches of the caustic
are a segment of a line along the optical axis and a two-dimensional surface
of revolution about the $z$-axis, which has a singularity of the cusp ridge
type. Finally, we present the conclusions.

##  II The plano spherical and the axicon lenses

In this section we review the computation of the wavefronts, light rays and
caustics associated with the refraction of a plane wavefront by a plano
spherical and axicon lenses. These results allow us to introduce, in the next
section, a conospherical lens. If we have a region of the free space filled
out with two optical media with constant refractive indices, $n_{1}$ and
$n_{2}$, separated by an interface given locally by a position vector
$\mathbf{r}(\rho,\phi)$, then a plane wavefront evolving in the direction
$\hat{z}$, in the optical medium with refractive index $n_{1}$, is refracted
by the interface in the direction $\hat{\mathbf{R}}$. Thus, using the
refraction law and the reference plane wavefront
$\mathbf{s}=\rho(\cos\phi\hat{x}+\sin\phi\hat{y})$, a direct computation shows
that the refracted wavefronts and light rays, are given by

$\displaystyle\mathbf{X}$ $\displaystyle=$
$\displaystyle\mathbf{r}+l\hat{\mathbf{R}},$ (1) $\displaystyle l$
$\displaystyle=$
$\displaystyle\frac{\mathcal{C}}{n_{2}}-\gamma|\mathbf{r}-\mathbf{s}|,$ (2)
$\displaystyle\hat{\mathbf{R}}$ $\displaystyle=$
$\displaystyle\gamma\hat{z}+\Omega\hat{\mathbf{N}},$ (3) $\displaystyle\Omega$
$\displaystyle=$
$\displaystyle-\gamma(\hat{z}\cdot\hat{\mathbf{N}})+\sqrt{1-\gamma^{2}\left[1-(\hat{z}\cdot\hat{\mathbf{N}})^{2}\right]},$
(4)

where $\mathcal{C}$ is a real parameter, $\hat{\mathbf{N}}$ is the unit vector
to the interface and $\gamma\equiv n_{1}/n_{2}$. It is important to remark
that for fixed values of $\rho$ and $\phi$ equation (1) describes a particular
refracted light ray, while for a fixed value of $\mathcal{C}$ equations
(1)-(3) describe a particular refracted wavefront.

From a mathematical point of view equation (1) describes a map between two
subsets of ${\cal R}^{3}$, where ($\rho$, $\phi$, $l$) are local coordinates
of the domain space and ($X$, $Y$, $Z$), with
$\mathbf{X}=X\hat{x}+Y\hat{y}+Z\hat{z}$, are local coordinates of the target
space. By definition, the focal region associated with the refracted light
rays is the caustic set determined by the map (1). The caustic set of a
general map is computed by using the following:

Definition: Let h: ${\cal M}$ $\rightarrow$ ${\cal N}$ be a differentiable
map, with ${\cal M}$ and ${\cal N}$ differentiable manifolds. The set of
points in ${\cal M}$ where h is not locally one-to-one is referred to as its
critical set, and the image of the critical set is referred to as caustic set
of $h$ [13, 14, 15].

In accordance with this definition the critical set of the map (1) is computed
from the following condition

$J\equiv\left(\frac{\partial\mathbf{X}}{\partial\phi}\right)\cdot\left[\frac{\partial\mathbf{X}}{\partial
l}\times\frac{\partial\mathbf{X}}{\partial\rho}\right]=H_{2}l^{2}+H_{1}l+H_{0}=0,$
(5)

where

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\hat{\mathbf{R}}\cdot\left[\left(\frac{\partial\mathbf{r}}{\partial\rho}\right)\times\left(\frac{\partial\mathbf{r}}{\partial\phi}\right)\right],$
$\displaystyle H_{1}$ $\displaystyle=$
$\displaystyle\hat{\mathbf{R}}\cdot\left[\left(\frac{\partial\mathbf{r}}{\partial\rho}\right)\times\left(\frac{\partial\hat{\mathbf{R}}}{\partial\phi}\right)+\left(\frac{\partial\hat{\mathbf{R}}}{\partial\rho}\right)\times\left(\frac{\partial\mathbf{r}}{\partial\phi}\right)\right],$
$\displaystyle H_{2}$ $\displaystyle=$
$\displaystyle\hat{\mathbf{R}}\cdot\left[\left(\frac{\partial\hat{\mathbf{R}}}{\partial\rho}\right)\times\left(\frac{\partial\hat{\mathbf{R}}}{\partial\phi}\right)\right].$
(6)

If $H_{2}\neq 0$, like in the plano spherical lens, the critical set has two
branches given by

$\displaystyle l_{\pm}=\frac{-H_{1}\pm\sqrt{H_{1}^{2}-4H_{2}H_{0}}}{2H_{2}}.$
(7)

And the caustic set, which is obtained substituting equation (7) into equation
(1), can be written in the following manner

$\displaystyle\mathbf{X}_{\pm}(\rho,\phi)=\mathbf{r}+\left(\frac{-H_{1}\pm\sqrt{H_{1}^{2}-4H_{2}H_{0}}}{2H_{2}}\right)\hat{\mathbf{R}}.$
(8)

However, if $H_{2}=0$ and $H_{1}\neq 0$, like in the axicon lens, then the
critical set has only one branch given by

$l_{c}=\frac{-H_{0}}{H_{1}}.$ (9)

And the caustic set, which is obtained substituting equation (9) into equation
(1), has only one branch given by

$\displaystyle\mathbf{X}_{c}(\rho,\phi)=\mathbf{r}-\left(\frac{H_{0}}{H_{1}}\right)\hat{\mathbf{R}}.$
(10)

###  II.1 The plano spherical lens

Now we assume that we have a plano spherical lens with refractive index $n$ in
the free space. Using the polar coordinates $\rho$ and $\phi$, for this
particular case we have

$\displaystyle\mathbf{r}$ $\displaystyle=$
$\displaystyle\rho\hat{\rho}+\sqrt{a^{2}-\rho^{2}}\hat{z},$ (11)
$\displaystyle\hat{\rho}$ $\displaystyle=$
$\displaystyle\cos\phi\,\hat{x}+\sin\phi\,\hat{y},$ (12)

where $0\leq\rho\leq a$ and $0\leq\phi\leq 2\pi$, with $a$ being the radius of
the plane face. Using (11) and (12) a direct computation shows that

$\displaystyle\hat{\mathbf{N}}=\frac{1}{a}\left(\rho\hat{\rho}+\sqrt{a^{2}-\rho^{2}}\hat{z}\right).$
(13)

The plane wavefronts evolving in the direction $\hat{z}$ arrive to the plane
surface of the lens and are transmitted without any reflection, then they
arrive to the spherical surface and are refracted. Therefore, using equations
(1)-(4) and (11)-(13), a direct computation shows that the refracted
wavefronts and light rays can be written in the following manner

$\mathbf{X}=\left[\frac{a^{2}+l(\sqrt{a^{2}-\gamma^{2}\rho^{2}}-\gamma\sqrt{a^{2}-\rho^{2})}}{a^{2}}\right]\rho\hat{\rho}\\\
+\left[\frac{a^{2}\sqrt{a^{2}-\rho^{2}}+l(a\gamma+\sqrt{a^{2}-\rho^{2}})(\sqrt{a^{2}-\gamma^{2}\rho^{2}}-\gamma\sqrt{a^{2}-\rho^{2})}}{a^{2}}\right]\hat{z},$
(14)

where

$l=\frac{\mathcal{C}}{n_{2}}-\gamma\sqrt{a^{2}-\rho^{2}}.$ (15)

It is important to remark that equation (14) is correct for
$a^{2}-\gamma^{2}\rho^{2}\geq 0$. That is, $0\leq\rho\leq\rho_{m}=a/\gamma$.

Finally, another direct computation shows that the two branches of the caustic
region associated with the refracted light rays by the plane spherical lens
can be written in the following manner:

$\displaystyle\mathbf{X}_{-}$ $\displaystyle=$
$\displaystyle\frac{\gamma^{2}\rho^{3}}{a^{2}}\hat{\rho}+\frac{\gamma\left[(a^{2}-\gamma^{2}\rho^{2})^{3/2}+\gamma(a^{2}-\rho^{2})^{3/2}\right]}{a^{2}\left(\gamma^{2}-1\right)}\hat{z},$
(16) $\displaystyle\mathbf{X}_{+}$ $\displaystyle=$
$\displaystyle\frac{\gamma\left(\sqrt{a^{2}-\gamma^{2}\rho^{2}}+\gamma\sqrt{a^{2}-\rho^{2}}\right)}{\gamma^{2}-1}\hat{z}.$
(17)

Observe that $\mathbf{X}_{-}$ corresponds to a two-dimensional surface of
revolution about the $z$-axis with a degenerate singularity of the cusp type,
while $\mathbf{X}_{+}$ corresponds to a segment of a line along the axis of
symmetry of the optical system (see Figure 1).

Figure 1: Here we present on the plane $y=0$: (a) some incident plane
wavefronts and the incident marginal light rays (Green), the plano spherical
lens (Black), some refracted wavefronts and the refracted marginal light rays
(Blue) and the two branches of the caustic (Red), (b) some incident light rays
(Green), the plano spherical lens (Black), some refracted light rays (Blue)
and the two branches of the caustic (Red), (c) superposition of the plots (a)
and (b). To obtain these plots we use (14), (16) and (17) with $a=1cm$ and
$\gamma=1.5$.

###  II.2 The plano conical lens

Figure 2: Here we present on the plane $y=0$: (a) some incident plane
wavefronts and the incident marginal light rays (Green), the plano conical
lens (Black), some refracted wavefronts and the refracted marginal light rays
(Blue) and the caustic (Red), (b) some incident light rays (Green), the plano
conical lens (Black), some refracted light rays (Blue) and the caustic (Red),
(c) superposition of the plots (a) and (b). To obtain these plots we use (20)
and (22) with $a=1cm$, $\alpha=\pi/3$ and $\gamma=1.5$.

Now we assume that we have a plano conical lens with refractive index $n$ in
the free space. Using the polar coordinates $\rho$ and $\phi$, for this
particular case we have:

$\mathbf{r}=\rho\hat{\rho}+(a-\rho)\cot\alpha\hat{z},$ (18)

where $0\leq\rho\leq a$, $0\leq\phi\leq 2\pi$ and $0\leq\alpha\leq\pi/2$, with
$a$ being the radius of the plane face. Using (12) and (18) a direct
computation shows that

$\hat{\mathbf{N}}=\cos\alpha\hat{\rho}+\sin\alpha\hat{z}.$ (19)

From equations (1)-(4), (12), (18) and (19), a direct computation shows that
the refracted wavefronts and light rays when a plane wavefront is coming in
the $\hat{z}$ direction can be written in the following manner

$\displaystyle\mathbf{X}$ $\displaystyle=$
$\displaystyle\left[\rho+l\cos\alpha\left(\sqrt{1-\gamma^{2}\cos^{2}\alpha}-\gamma\sin\alpha\right)\right]\hat{\rho}$
(20)
$\displaystyle+\left\\{(a-\rho)\cot\alpha+l\left[\gamma+\sin\alpha\left(\sqrt{1-\gamma^{2}\cos^{2}\alpha}-\gamma\sin\alpha\right)\right]\right\\}\hat{z},$

where

$l=\frac{\mathcal{C}}{n_{2}}-\gamma(a-\rho)\cot\alpha.$ (21)

It is important to remark that equation (20) is correct for
$1-\gamma^{2}\cos\alpha\geq 0$. That is, $\arccos(1/\gamma)\leq\alpha<\pi/2$.

Finally, another direct computation shows that the caustic, associated with
the refracted light rays by the plane conical lens, has only one branch; which
can be written in the following manner:

$\mathbf{X}_{c}=\left[\cot\alpha(a-\rho(\sin\phi+\cos\phi))-\frac{\rho}{\Omega}\sec\alpha\left(\gamma+\Omega\sin\alpha\right)\right]\hat{z},$
(22)

where

$\Omega=\sqrt{1-\gamma^{2}\cos^{2}\alpha}-\gamma\sin\alpha.$ (23)

Observe that $\mathbf{X}_{c}$ corresponds to a segment of line along the axis
of symmetry of the optical system (see Figure 2).

##  III The conospherical lens

Figure 3: Here we depict an incident plane wavefront and the conospherical
lens with its corresponding parameters. The first surface is a cone given by
(24) and the second surface is spherical and is given by (25), $a$ is the
radius of the spherical surface and $\alpha$ is the aperture angle of the
conical surface. The black region corresponds to the conospherical lens.

In this section we compute the refraction of a plane wavefront by a
conospherical lens, see Figure 3. That is, we assume that a certain region of
the free space is filled out with three optical media of refractive indices
$n_{0}$, $n_{1}$ and $n_{2}$ respectively. The interface between the first and
second media is a conical surface given by

$\mathbf{r}_{1}=\rho\hat{\rho}+(a-\rho\cot\alpha)\hat{z},$ (24)

while the interface between the second and third media is a spherical surface
given by

$\displaystyle\mathbf{r}_{2}$ $\displaystyle=$
$\displaystyle\rho_{s}\hat{\rho}_{s}+\sqrt{a^{2}-\rho_{s}^{2}}\hat{z},$ (25)
$\displaystyle\hat{\rho_{s}}$ $\displaystyle=$
$\displaystyle\cos\phi_{s}\hat{x}+\sin\phi_{s}\hat{y},$ (26)

where $\rho_{s}=\rho_{s}(\rho)$ and $\phi_{s}=\phi_{s}(\rho)$, are determined
in the following manner. First, following the procedure used in subsection
2.2, we compute the refraction of the incoming plane wavefront by the conical
surface of the lens given by $\mathbf{r}_{1}$. In this case a direct
computation shows that the refracted light rays and wavefronts by this first
surface of the lens can be written in the following manner

$\displaystyle\mathbf{X}_{1}$ $\displaystyle=$
$\displaystyle\left[\rho+l_{1}\cos\alpha\left(\sqrt{1-\gamma_{0}^{2}\cos^{2}\alpha}-\gamma_{0}\sin\alpha\right)\right]\hat{\rho}$
(27)
$\displaystyle+\left\\{a-\rho\cot\alpha+l_{1}\left[\gamma_{0}+\sin\alpha\left(\sqrt{1-\gamma_{0}^{2}\cos^{2}\alpha}-\gamma_{0}\sin\alpha\right)\right]\right\\}\hat{z},$

where

$l_{1}=\frac{C_{1}}{n_{1}}-\gamma_{0}\left(a-\rho\cot\alpha\right).$ (28)

It is important to remark that for fixed values of $\rho$ and $\phi$, the
parameter $C_{1}$ labels the points on a particular refracted light ray, while
for fixed values of the parameter $C_{1}$, $\mathbf{X}_{1}$ describes a
particular refracted wavefront. Observe that the points on the refracted light
rays are also labeled by the parameter $l_{1}$. Now, we take the refracted
light rays that arrive at the second spherical interface given by
$\mathbf{r}_{2}$. To determine the relationship between the two sets of polar
coordinates ($\rho$, $\phi$) and ($\rho_{s}$, $\phi_{s}$), from equations (25)
and (27) we obtain the parametrization of the spherical surface of the lens
$\mathbf{r}_{2}$ in terms of the polar coordinates ($\rho$, $\phi$) that
parametrize the points on the first conical surface of the lens
$\mathbf{r}_{1}$. To be more specific, we follow the refracted light rays by
the first surface of the lens then when they arrive to the second surface of
the lens we obtain the following relationship $\mathbf{X}_{1}=\mathbf{r}_{2}$,
which determines the relationship we are looking for. This condition is
equivalent to

$\displaystyle\left[\rho+l_{1}\cos\alpha\left(\sqrt{1-\gamma_{0}^{2}\cos^{2}\alpha}-\gamma_{0}\sin\alpha\right)\right]\cos\phi=\rho_{s}\cos\phi_{s},$
(29)
$\displaystyle\left[\rho+l_{1}\cos\alpha\left(\sqrt{1-\gamma_{0}^{2}\cos^{2}\alpha}-\gamma_{0}\sin\alpha\right)\right]\sin\phi=\rho_{s}\sin\phi_{s},$
(30) $\displaystyle
a-\rho\cot\alpha+l_{1}\left[\gamma_{0}+\sin\alpha\left(\sqrt{1-\gamma_{0}^{2}\cos^{2}\alpha}-\gamma_{0}\sin\alpha\right)\right]=\sqrt{a^{2}-\rho_{s}^{2}}.$
(31)

That is,

$\displaystyle\phi_{s}$ $\displaystyle=$ $\displaystyle\phi,$ (32)
$\displaystyle l_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{a^{2}-\rho_{s}^{2}}+\rho\cot\alpha-a}{\gamma_{0}+\sin\alpha\left(\sqrt{1-\gamma_{0}^{2}\cos^{2}\alpha}-\gamma_{0}\sin\alpha\right)},$
(33)

and

$\displaystyle A\rho_{s}^{2}+B\rho_{s}+C=0,$ (34)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{\sec^{2}\alpha(2\gamma_{0}\Omega_{1}\sin\alpha+\gamma_{0}^{2}+\Omega_{1}^{2})}{\Omega_{1}^{2}},$
$\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{2(\Omega_{1}+\gamma_{0}\csc\alpha)[\rho(\gamma_{0}+\Omega_{1}\csc\alpha)-a\Omega_{1}\cos\alpha]\sec\alpha\tan\alpha}{\Omega_{1}^{2}},$
$\displaystyle C$ $\displaystyle=$
$\displaystyle-\frac{\rho(\gamma_{0}+\Omega_{1}\csc\alpha)(\rho(\gamma_{0}+\Omega_{1}\csc\alpha)-2a\Omega_{1}\cos\alpha)\sec^{2}\alpha}{\Omega_{1}^{2}},$
$\displaystyle\Omega_{1}$ $\displaystyle=$
$\displaystyle\sqrt{1-\gamma_{0}^{2}\cos^{2}\alpha}-\gamma_{0}\sin\alpha,$
(35)

with $\gamma_{0}\equiv n_{0}/n_{1}$. Therefore, the two solutions for
$\rho_{s}$ are explicitly given by

$\rho_{s}=\rho_{\pm}=\frac{\sin\alpha\left\\{(\Omega_{1}+\gamma_{0}\csc\alpha)[a\Omega_{1}\cos\alpha-\rho(\gamma_{0}+\Omega_{1}\csc\alpha)]\pm\csc\alpha\sqrt{\Delta}\right\\}}{\gamma_{0}^{2}+2\gamma_{0}\Omega_{1}\sin\alpha+\Omega_{1}^{2}},$
(36)

with

$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle(\Omega_{1}+\gamma_{0}\csc\alpha)^{2}[a\Omega_{1}\cos\alpha-\rho(\gamma_{0}+\Omega_{1}\csc\alpha)]^{2}\sin^{2}\alpha-\rho(\gamma_{0}+\Omega_{1}\csc\alpha)$
(37)
$\displaystyle\times[\rho(\gamma_{0}+\Omega_{1}\csc\alpha)-2a\Omega_{1}\cos\alpha](\gamma_{0}^{2}+2\gamma_{0}\Omega_{1}\sin\alpha+\Omega_{1}^{2}).$

Until this moment we have computed the refraction of the incoming plane
wavefront evolving along the $z$-axis by the first conical surface of the lens
and we have obtained the parameterization of the second spherical surface of
the lens in terms of the polar coordinates ($\rho$, $\phi$), which label the
points on the first conical surface of the lens. Thus, the parametrization of
the second spherical surface that we have to use to compute the second
refraction of the light rays is given by

$\mathbf{r}_{2}=\rho_{s}\hat{\rho}+\sqrt{a^{2}-\rho_{s}^{2}}\hat{z},$ (38)

where $\rho_{s}$ is given by (36).

Therefore, the refracted light rays and wavefronts by the second surface of
the lens (38) can be written in the following manner

$\displaystyle\mathbf{X}_{2}$ $\displaystyle=$
$\displaystyle\mathbf{r}_{2}+\left[\frac{\mathcal{C}_{2}}{n_{2}}-\gamma_{0}\gamma_{1}\left|\mathbf{r}_{1}-\mathbf{s}\right|-\gamma_{1}\left|\mathbf{r}_{2}-\mathbf{r}_{1}\right|\right]\hat{\mathbf{R}}_{2},$
(39) $\displaystyle\hat{\mathbf{R}}_{2}$ $\displaystyle=$
$\displaystyle\gamma_{1}\hat{\mathbf{I}}_{2}+\Omega_{2}\hat{\mathbf{N}}_{2},$
(40) $\displaystyle\Omega_{2}$ $\displaystyle=$
$\displaystyle-\gamma_{1}(\hat{\mathbf{I}}_{2}\cdot\hat{\mathbf{N}}_{2})+\sqrt{1-\gamma_{1}^{2}\left[1-(\hat{\mathbf{I}}_{2}\cdot\hat{\mathbf{N}}_{2})^{2}\right]},$
(41)

where

$\displaystyle\hat{\mathbf{I}}_{2}$ $\displaystyle=$
$\displaystyle\Omega_{1}\cos\alpha\hat{\rho}+\left(\gamma+\Omega_{1}\sin\alpha\right)\hat{z},$
(42) $\displaystyle\hat{\mathbf{N}}_{2}$ $\displaystyle=$
$\displaystyle\frac{\rho_{s}}{a}\hat{\rho}_{s}+\frac{\sqrt{a^{2}-\rho_{s}^{2}}}{a}\hat{z}.$
(43)

Using equations (40)-(43) a direct computation shows that

$\displaystyle\hat{\mathbf{R}}_{2}$ $\displaystyle=$
$\displaystyle\left(\gamma_{1}\Omega_{1}\cos\alpha+\frac{\Omega_{2}\rho_{s}}{a}\right)\hat{\rho}+\left(\gamma_{1}(\gamma_{0}+\Omega\sin\alpha)+\frac{\Omega_{2}\sqrt{a^{2}-\rho_{s}^{2}}}{a}\right)\hat{z},$
(44) $\displaystyle\Omega_{2}$ $\displaystyle=$
$\displaystyle-\frac{\gamma_{1}\rho_{s}\Omega_{1}\cos\alpha}{a}-\frac{\gamma_{1}\sqrt{a^{2}-\rho_{s}^{2}}(\gamma_{0}+\Omega_{1}\sin\alpha)}{a}+\frac{\sqrt{\Delta_{2}}}{a},$
(45) $\displaystyle\Delta_{2}$ $\displaystyle=$ $\displaystyle
a^{2}-\gamma_{1}\left(a^{2}-\left(\rho_{s}\Omega_{1}\cos\alpha+\sqrt{a^{2}-\rho_{s}^{2}}(\gamma_{0}+\Omega_{1}\sin\alpha\right)^{2}\right).$
(46)

Since $\mathbf{s}=\rho\hat{\rho}$, finally using equations (24), (38) and (44)
another direct computation shows that equation (39) can be written in the
following manner

$\displaystyle\mathbf{X}_{2}$ $\displaystyle=$
$\displaystyle\left[\rho_{s}+l_{2}\left(\gamma_{1}\Omega_{1}\cos\alpha+\frac{\Omega_{2}\rho_{s}}{a}\right)\right]\hat{\rho}$
(47)
$\displaystyle+\left[\sqrt{a^{2}-\rho_{s}^{2}}\left(1+\frac{l_{2}\Omega_{2}}{a}\right)+l_{2}\gamma_{1}\left(\gamma_{0}+\Omega_{1}\sin\alpha\right)\right]\hat{z},$

where

$l_{2}=\frac{\mathcal{C}_{2}}{n_{2}}-\gamma_{0}\gamma_{1}(a-\rho\cot\alpha)-\gamma_{1}\sqrt{(\rho-\rho_{s})^{2}+\left(\sqrt{a^{2}-\rho_{s}^{2}}-a+\rho\cot\alpha\right)^{2}}.$
(48)

In conclusion, equation (47) provides both the light rays and wavefronts
refracted by the second spherical surface of the lens. Since (47) describes a
map between two subsets of ${\cal R}^{3}$, where ($\rho$, $\phi$, $l_{2}$) are
local coordinates in the domain space and ($X_{2}$, $Y_{2}$, $Z_{2}$), with
$\mathbf{X}_{2}=X_{2}\hat{x}+Y_{2}\hat{y}+Z_{2}\hat{z}$, are local coordinates
in the target space. Therefore, the critical set associated with this map is
computed from the following condition

$J\equiv\left(\frac{\partial\mathbf{X}_{2}}{\partial\phi}\right)\cdot\left[\frac{\partial\mathbf{X}_{2}}{\partial
l_{2}}\times\frac{\partial\mathbf{X}_{2}}{\partial\rho}\right]=0.$ (49)

Since

$\displaystyle\frac{\partial\mathbf{X}_{2}}{\partial\phi}$ $\displaystyle=$
$\displaystyle D\hat{\phi},$ (50)
$\displaystyle\frac{\partial\mathbf{X}_{2}}{\partial l_{2}}$ $\displaystyle=$
$\displaystyle E\hat{\rho}+F\hat{z},$ (51)
$\displaystyle\frac{\partial\mathbf{X}_{2}}{\partial\rho}$ $\displaystyle=$
$\displaystyle G\hat{\rho}+H\hat{z},$ (52)

where

$\displaystyle D$ $\displaystyle=$
$\displaystyle\rho_{s}+l_{2}\left(\gamma_{1}\Omega_{1}\cos\alpha+\frac{\Omega_{2}\rho_{s}}{a}\right),$
(53) $\displaystyle E$ $\displaystyle=$
$\displaystyle\gamma_{1}\Omega_{1}\cos\alpha+\frac{\Omega_{2}\rho_{s}}{a},$
(54) $\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{\Omega_{2}\sqrt{a^{2}-\rho_{s}^{2}}}{a}+\gamma_{1}\left(\gamma_{0}+\Omega_{1}\sin\alpha\right),$
(55) $\displaystyle G$ $\displaystyle=$
$\displaystyle\rho_{s}^{\prime}+l_{2}\frac{(\Omega_{2}\rho_{s})^{\prime}}{a},$
(56) $\displaystyle H$ $\displaystyle=$ $\displaystyle
l_{2}\left(\frac{\Omega_{2}\sqrt{a^{2}-\rho_{s}^{2}}}{a}\right)^{\prime},$
(57)

and the prime denotes the derivative with respect to the coordinate $\rho$.
Therefore, the critical set of the map (47) is determined from the following
condition

$J=D(FG-EH)=0,$ (58)

which implies that the two branches of the critical set are given by $D=0$ and
$FG=EH$. That is

$\displaystyle l_{2}$ $\displaystyle=$ $\displaystyle
l_{21}=\frac{-a\rho_{s}}{a\gamma_{1}\Omega_{1}\cos\alpha+\Omega_{2}\rho_{s}},$
(59) $\displaystyle l_{2}$ $\displaystyle=$ $\displaystyle
l_{22}=\frac{aF\rho_{s}^{\prime}}{E(\Omega_{2}\sqrt{a^{2}-\rho_{s}^{2}})^{\prime}-F(\Omega_{2}\rho_{s})^{\prime}}.$
(60)

To determine the two branches of the caustic region we substitute (59) and
(60) into (47). Thus, we obtain

$\displaystyle\mathbf{X}_{c21}$ $\displaystyle=$
$\displaystyle\left[\sqrt{a^{2}-\rho_{s}^{2}}\left(1+\frac{l_{21}\Omega_{2}}{a}\right)+l_{21}\gamma_{1}\left(\gamma_{0}+\Omega_{1}\sin\alpha\right)\right]\hat{z},$
(61) $\displaystyle\mathbf{X}_{c22}$ $\displaystyle=$
$\displaystyle\left[\rho_{s}+l_{22}\left(\gamma_{1}\Omega_{1}\cos\alpha+\frac{\Omega_{2}\rho_{s}}{a}\right)\right]\hat{\rho}$
(62)
$\displaystyle+\left[\sqrt{a^{2}-\rho_{s}^{2}}\left(1+\frac{l_{22}\Omega_{2}}{a}\right)+l_{22}\gamma_{1}\left(\gamma_{0}+\Omega_{1}\sin\alpha\right)\right]\hat{z}.$

Observe that $\mathbf{X}_{c21}$ describes a segment of a line along the
$z$-axis and $\mathbf{X}_{c22}$ describes a revolution surface about the
$z$-axis. In Figure 6 we find that $\mathbf{X}_{c21}$ corresponds to two
disconnected segments of line, one is real and the other one is virtual.

##  IV A particular example of a conospherical lens

Figure 4: Here we plot $\rho_{s}=\rho_{\pm}$ given by (36). Figure 5: In this
plot we present some incident light rays (green), the conical surface (black),
the corresponding refracted light rays inside the lens (magenta) and the
spherical surface (black).

In this section we apply the general results to a particular conospherical
lens characterized by $a=1$, $\gamma_{0}=2/3$, $\alpha={\rm arccot}\,(1/3)$,
$\gamma=3/2$. That is, we are assuming that we have a conospherical lens with
refractive index $n_{1}=3/2$ in free space ($n_{0}=1$ and $n_{2}=1$).
Furthermore, from (14) we find that in this case
$0\leq\rho\leq\rho_{m}=a/\gamma_{0}=2/3$. Using these particular values for
the parameters and equation (36) in Figure 4 we present the plots of
$\rho_{s}$. From this plot we find that the right expression for $\rho_{s}$ is
given by $\rho=\rho_{-}$. Therefore, the refracted light rays inside the lens
are given by

Figure 6: Here we present on the plane $y=0$: (a) some incident plane
wavefronts and the incident marginal light rays (Green), the plano spherical
lens (Black), some refracted wavefronts and the refracted marginal light rays
(Blue) and the two branches of the caustic (Red), (b) some incident light rays
(Green), the plano spherical lens (Black), some refracted light rays (Blue)
and the two branches of the caustic (Red), (c) superposition of the plots (a)
and (b). To obtain these plots we use equations (24), (38), (47), (61) and
(62).

$\textbf{X}_{i}=\mathbf{r}_{1}+\sigma(\mathbf{X}_{1-}-\mathbf{r}_{1}),$ (63)

where $0\leq\sigma\leq 1$,

$\displaystyle\mathbf{X}_{1-}$ $\displaystyle=$
$\displaystyle\left[\rho+\cos\alpha\left(\sqrt{1-\gamma_{0}^{2}\cos^{2}\alpha}-\gamma_{0}\sin\alpha\right)\right]\hat{\rho}$
(64)
$\displaystyle+\left\\{a-\rho\cot\alpha+l_{\rho}\left[\gamma_{0}+\sin\alpha\left(\sqrt{1-\gamma_{0}^{2}\cos^{2}\alpha}-\gamma_{0}\sin\alpha\right)\right]\right\\}\hat{z},$

and

$\displaystyle l_{\rho}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{a^{2}-\rho_{-}^{2}}+\rho\cot\alpha-a}{\gamma_{0}+\sin\alpha\left(\sqrt{1-\gamma_{0}^{2}\cos^{2}\alpha}-\gamma_{0}\sin\alpha\right)}.$
(65)

Thus, using equations (63)-(65) in Figure 5 we present some incident light
rays, the conospherical lens and the corresponding refracted light rays inside
the lens. In Figure 6 we present on the plane $y=0$: (a) some incident
wavefronts and the incident marginal light rays, the conospherical lens, the
corresponding refracted wavefronts leaving the lens and the branches of the
caustic. In (b) we present some incident light rays, the conospherical lens,
the corresponding refracted light rays leaving the lens and the branches of
the caustic. Finally, in (c) we show the superposition of the plots (a) and
(b). From these plots it is clear that the caustic, for this case, has three
branches two of them are real and one virtual. The virtual branch of the
caustic is behind the lens and the real ones are in front the lens. One of the
real branches corresponds to a segment of a line along the $z$-axis, which is
similar to the caustic generated by the axicon lens. The second real branch of
the caustic is a two-dimensional surface of revolution about the $z$-axis with
a singularity of the cusp ridge type. Finally, using equations (24), (38),
(47), (61) and (62) in Figure 7 we present (a) the conospherical lens, some
refracted wavefronts leaving the lens and the branches of the caustic, in (b)
we present the conospherical lens, some refracted light rays leaving the lens
and the branches of the caustic. Finally, in (c) we show the superposition of
the plots (a) and (b). In these plots we show the two real branches of the
caustic, which correspond to a segment of a line along the $z$-axis and a two
dimensional surface of revolution about the $z$-axis with a singularity of the
cusp ridge type.

Figure 7: In this plot we present: (a) the conospherical lens (gray), some
refracted wavefronts leaving the lens (blue) and the branches of the caustic
(red). (b) the conospherical lens (gray), some refracted light rays leaving
the lens (blue) and the branches of the caustic (red). Finally, in (c) we
present the superposition of the plots (a) and (b).

##  V Conclusions

In this work we have introduced the conospherical lens; that is, a lens whose
faces are a conical surface and a spherical surface. Then we illuminated this
lens with a plane wavefront and its associated refracted wavefronts, light
rays and caustic were computed. We found that the caustic has two branches.
The first is constituted by two segments of a line, one part of this caustic
is real and the other one virtual. The second branch of the caustic is a two-
dimensional surface with a singularity of the cusp ridge type. It is important
to remark that the two branches of the caustic are disconnected, see Figure 6,
and because of this property we believe that using this optical element one
could generate a scalar optical accelerating beam in the region where the
caustic is a two-dimensional surface of revolution, and at the same time a
scalar optical beam with similar properties to the Bessel beam of zero order
in the region were the real caustic is a segment of a line along the optical
axis. That is, this optical element could be used to generate a scalar optical
beam such that in a region is accelerating and in the other behaves like the
Bessel beam of zero order. These properties are under study and the results
will be reported elsewhere.

###### Acknowledgements.

Gilberto Silva-Ortigoza acknowledges financial support from SNI.

## References

## References

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PREDICTION THEORY FOR STATIONARY FUNCTIONAL TIME SERIES

N. H. BINGHAM

Abstract
We survey aspects of prediction theory in infinitely many dimensions, with a
view to the theory and applications of functional time series.

Keywords Cramér representation, Kolmogorov isomorphism theorem, Verblunsky
coefficients, Szegő’s theorem, Szegő alternative, Beurling-Lax-Halmos theorem,
functional time series, functional principal components

2010 Mathematics Subject Classification: Primary 60-02 Secondary 62-02

1\. Introduction

This paper continues the theme of the author’s earlier surveys [Bin1] on
prediction theory for one-dimensional time series, [Bin2] on the finite-
dimensional case, and (with Badr Missaoui, [BinM]) touching briefly on the
infinite-dimensional case, our theme here. Our motivation is partly
mathematical interest and completeness, partly the vigorous development of
functional data analysis (FDA; [RamS1,2], [HorK], [FerV]) made possible by the
explosive growth in computer power, data storage and data handling.
We begin in §2 with the Kolmogorov Isomorphism Theorem (KIT), on which
everything rests. In §3 we turn to Verblunsky coefficients, Szegő’s theorem
and the Wold decomposition, followed in §4 by the Szegő alternative and
factorization, all themes familiar from e.g. [Bin1,2]. Section 5 is on the
Beurling-Lax-Halmos theorem and inner functions. Complements follow in §6, in
particular (§6.1) the deterministic case and (§6.2) model spaces. We close in
§7 with some open questions, with which the area abounds.
The mathematics here involves vectorial integration (see e.g. [Rud2, Ch. 12]).
As with FDA, implementation involves discretization (‘calculus is continuous,
calculation is discrete’), and so in principle reduces the setting to a
finite-dimensional one, which can be handled by the finite-dimensional methods
of e.g. [Bin2]. We prefer to use the infinite-dimensional context suggested by
the nature of the data. See [DetKA] for more on this.

2\. The Kolmogorov Isomorphism Theorem

We confine ourselves throughout to stationary processes, that is, those whose
distributions are invariant under time-shifts. One can work in discrete or
continuous time, depending on preference, context or the data available. As
the term ‘time series’ indicates, the first is the traditional one, and we
shall follow it here. Then the time set is the integers $\mathbb{Z}$, with
(Pontryagin-van Kampen) dual [Pon, Ch. 6] the torus $\mathbb{T}$ (equivalently
$\mathbb{R}/2\pi\mathbb{Z}$; one can pass between these by
$\theta\leftrightarrow e^{i\theta}$, and we shall do this at will). In the
second case, the time-set is the real line $\mathbb{R}$, with dual
$\mathbb{R}$ also. Stationarity is a strong condition, which yields
correspondingly strong conclusions; we turn later to how it may be relaxed
(§6.4). For a monograph treatment, see Nikolskii [Nik1], whose sub-title
‘Spectral function theory’ gives a hint of the mathematics involved (see also
[Nik2]).
We write our process (or time series) as $x=\\{x_{n}:n\in\mathbb{Z}\\}$, where
the $x_{n}$ are in $\mathbb{R}$, ${\mathbb{R}}^{d}$, a Hilbert space $\cal H$
or a Banach space $\cal B$, depending on context (one can work more generally;
see e.g. [BinM, §5.5, §5.6]). While our data consists of functions, as in our
title, these functions will always belong to function spaces, which are at
least topological vector spaces, the elements of which we will call vectors as
usual. So vectors here are infinite-dimensional unless otherwise stated.
Covariances are matrices in the finite-dimensional case (as in multivariate
analysis in statistics, and in [Bin2]). But here, covariances are (linear)
operators, hence the crucial role of operator theory in what follows. We will
work in a Hilbert space $\cal H$ unless otherwise stated.
Write the time-shift $n\mapsto n+1$ as $U$. Then $U$ is unitary, and generates
a unitary group, $\cal U$. Being unitary, $U$ is normal, and so the spectral
theorem for the unitary case (Stone’s theorem) applies ([Rud2, Ch. 12]; [Sto
VIII.2], [RieN, §109], [DunS, X.2]). This involves integration over
$\mathbb{T}$ with respect to a resolution of the identity (projection-valued
measure, in Mackey’s terminology [Mac])
$E=\\{E(\theta):\theta\in\mathbb{T}\\}$: $E(\mathbb{T})=1$, and

$U^{n}=\int_{\mathbb{T}}e^{in\theta}dE(\theta),\qquad n\in\mathbb{Z}$ $None$

(as all our integrals will be over the torus $\mathbb{T}$, we omit the
$\mathbb{T}$ below). Note that as $|e^{in\theta}|=1$ and $E(\mathbb{T})=1$,
for $x\in\cal H$ $(U)$ gives

$\|Ux\|=\|\int e^{i\theta}dE(\theta)x\|\leq\|x\|:$

$U$ is a contraction.
We now have

$x_{n}=U^{n}x_{0}=\int e^{in\theta}dE(\theta)x_{0}=\int
e^{in\theta}dY(\theta),\qquad n\in\mathbb{Z},$ $None$

say, the Cramér representation; the random measure $Y$ is the Cramér measure
or representing measure [Cra1,2, CraL]. This is orthogonally scattered [Mas1]:
the masses of disjoint sets are orthogonal.
Take $\cal H$ a separable Hilbert space, $B(\cal H)$ the algebra of all
bounded linear operators on $\cal H$, $T(\cal H)$ the trace class operators in
$B(\cal H)$. The Kolmogorov Isomorphism Theorem in the Hilbert case is due to
Mandrekar and Salehi [ManS]; here we follow Kakihara [Kak1-4]. Write
$X:=L_{0}^{2}(\Omega,\cal H)$ for the Hilbert space of all $\cal H$-valued
(strong) random variables with mean 0 and finite second moment: with
$(\Omega,{\cal F},\mathbb{P})$ the probability space,
$\mathbb{E}[x]:=\int_{\Omega}x(\omega)d\mathbb{P}(\omega)=0$,

$\|x\|^{2}_{X}:=\int_{\Omega}\|x(\omega)\|^{2}_{\cal H}\
d\mathbb{P}(\omega)=\mathbb{E}\ [\ \|x(\omega)\|^{2}_{\cal H}]<\infty,$

and so with inner product on $X$

$(x,y)_{X}:=\int_{\Omega}(x(\omega),y(\omega))_{\cal H}\
d\mathbb{P}(\omega)=\mathbb{E}[(x,y)_{\cal H}].$

The Gramian operator, or Gramian, is the $T(\cal H)$-valued inner product
$[.,.]_{X}$ defined by

$([x,y]_{X}\phi,\psi)_{H}=\int_{\Omega}(x(\omega),\phi)_{\cal
H}(\psi,y(\omega))_{\cal H}\ d\mathbb{P}(\omega)$ ${\ \ \ \ \ \ \ \ \ \
}=\mathbb{E}[(x,\phi)_{\cal H}(\psi,y)_{\cal H}]\quad(x,y\in
X,\phi,\psi\in\cal H).$ $None$

In the finite-dimensional case, this reduces to a doubly-indexed set of
scalars, which gives a matrix, the Gramian matrix [HorJ, §7.2], which is
positive definite; so too is the Gramian operator likewise. Symbolically,

$[x,y]_{X}=\int_{\Omega}x(\omega)\otimes\overline{y(\omega)}d\mathbb{P}(\omega)=\mathbb{E}[x\otimes\overline{y}],$

where

$(\phi\otimes\overline{\psi}){\phi}^{\prime}:=({\phi}^{\prime},\psi)_{\cal
H}\phi,\qquad\phi,{\phi}^{\prime},\psi\in{\cal H}.$

Now $X$ is both
(i) a left $B(\cal H)$-module under the module action

$(a,x)\mapsto a.x=ax\qquad(a\in B({\cal H}),x\in{\cal H}),$

(ii) a Hilbert space with Gramian $[.,.]_{X}$:
$X$ is a normal Hilbert $B(\cal H)$-module in the sense of [Kak2]. (For
background on Hilbert modules, see Lance [Lan]. Both Hilbert modules and
Gramians go back here at least to Masani’s comments on Wiener’s work on
prediction [Mas2].)
We may now without ambiguity abbreviate $[.,.]_{X}$ (whose values are non-
random operators) to $[.,.]$. With it, we can define the spectral measure, or
control measure,

$F(.):=[Y(.),Y(.)]$ $None$

(operator-valued, indeed Gramian-valued), and the operator covariance
function,

$\Gamma(m,n):=[x(m),x(n)].$

The process $x=\\{x(n)\\}$ is called operator stationary (or just stationary)
if its operator covariance function (or just covariance function) is a
function of $m-n$ only, $\tilde{\Gamma}(m-n)=\Gamma(m,n)$, say. We assume
stationarity unless otherwise stated.
As in the matrix case above, the Gramian is positive definite (Gretsky [Gret,
§3]). One has [Kak3] the spectral representation

$\tilde{\Gamma}(n)=\int e^{in\theta}dF(\theta).$ $None$

This is the operator version of Herglotz’s theorem (or Bochner’s theorem); cf.
[Bin1,2] in finite dimensions and van Delft and Eichler [vDelE] in this
setting.
Finally, one has the Kolmogorov isomorphism theorem ([Kak3, p.104; [ManS])]:
as in the scalar and matrix cases (see e.g. [Bin1,2]),

$x(n)\leftrightarrow e^{in.}I,\qquad n\in\mathbb{Z},$ $None$

with $I$ the identity operator (here the time domain is on the left, the
frequency domain on the right).
Below, we shall need (where it exists) the spectral density $f$, the Radon-
Nikodym derivative of the spectral measure $F$ with respect to normalised
Lebesgue measure $d\theta/2\pi$ on $\mathbb{T}$ (operator-valued, as $F$ is).

3\. Verblunsky coefficients; Szegő’s theorem; Wold decomposition

Verblunsky’s theorem
In the scalar case [Bin1], the distribution of a (discrete-time, complex-
valued) stationary sequence may be described (encoded, parametrised) by a
sequence $\alpha=({\alpha}_{n})_{0}^{\infty}$ of complex numbers
${\alpha}_{n}\in\mathbb{D}$, the unit disc of $\mathbb{C}$, the Verblunsky
doefficients (there are several other names; see [Bin1], [Sim1]). This is
Verblunsky’s theorem. The relevant theory depends heavily on orthogonal
polynomials on the unit circle (OPUC), due originally to Szegő, the theme of
Simon’s books [Sim1,2] ([Sim3] deals with orthogonal polynomials on the line
and the cirle together; these correspond to continuous and to discrete time).
In the finite-dimensional ($\ell$-vector, $\ell\times\ell$ matrix) case, the
relevant theory is matrix orthogonal polynomials on the unit circle (MOPUC)
[Bin2]. The process also has a Verblunsky parametrisation, but now the
${\alpha}_{n}$ are ($\ell\times\ell$) matrices, of norm $\|\alpha_{n}\|<1$.
These encode the stationary processes, as before: Verblunsky’s theorem. See
Damanik, Pushnitski and Simon [DamPS, (3.10), Th. 3.12].

Szegő’s theorem
For stationary processes, the key result in both cases is Szegő’s theorem,
which relates to the influence of the remote past. This may be absent, e.g.,
bathwater forgetting its thermal history as it thermalises; total, e.g.
tempered steel, whose thermal history is locked in; or present with a partial
influence, e.g. the climate-weather interplay, where (on a time-scale, in
years, short enough to neglect climate change) climate is permanent, while
weather is (again on a suitable time-scale, in days) temporary and
unpredictable, indeed chaotic.
Szegő’s theorem in the scalar case tells us when the influence of the remote
past is not total. The condition for this is Szegő’s condition

$\log f\in L_{1}(\mathbb{T}).$ $None$

This is extended to the finite-dimensional setting by Derevyagin, Holtz,
Khrushchev and Tyaglov [DerHKT, Th. 28, Th. 29]: with ${\dagger}$ for the
adjoint (following their notation here), det and tr for determinant and trace,

$\prod_{0}^{\infty}\det(1-{\alpha}_{k}{\alpha}_{k}^{{\dagger}})=\exp\int\hbox{tr}\
\log f(\theta)d\theta/2\pi$ $None$

(the Kolmogorov-Szegő formula; see e.g. [Bin1, §4]), and Szegő’s condition –
that the right here is positive – holds iff

$\sum_{0}^{\infty}\|{\alpha}_{k}^{{\dagger}}{\alpha}_{k}\|<\infty.$

(extending earlier results of Delsarte, Genin and Kamp [DelGK, Th. 18, 19]).
Note that $(KSz)$ involves the determinants of the matrices ${\alpha}_{n}$,
and (despite the Fuglede-Kadison determinant, [BleL]) these do not extend to
the infinite-dimensional case in general.
Payen [Pay] makes a thorough study of the Hilbert-valued case. There ([Pay,
II.]; [BinM]) he gives an infinite-dimensional form of Szegő’s theorem in
terms of factorization (§4 below).

Wold decomposition
The past at time $n$, the remote past and the remote future of the process
$x=(x(n))$ are the closed linear subspaces spanned by the random variables
below:

$H(x,n):=\bigvee(x(k):k<n),\quad H(x,-\infty):=\bigcap\
(\bigvee(x,n):n\in\mathbb{Z}),$ $H(x,+\infty):=\bigcup\
(\bigvee(x,n):n\in\mathbb{Z}).$

The process is called deterministic if all three are equal, purely non-
deterministic (pnd) if the second is trivial. The Wold decomposition [Kak2]
splits the process $x=(x(n))$ into a deterministic and a purely non-
deterministic component,

$x=x_{d}+x_{p},$

which are Gramian orthogonal:

$[x_{d}(m),x_{p}(n)]=0\quad(m,n\in\mathbb{Z}).$

The Cramér measure $Y$ of the process also splits, into a sum of absolutely
continuous and singular components, $Y_{a}$ and $Y_{s}$, and similarly for the
spectral measures:

$Y=Y_{s}+Y_{a},\qquad F=F_{s}+F_{a}.$

In one dimension, one has Wold-Cramér concordance: regarded as processes in
their own right, the ‘good’ process $x_{p}$ has Cramér and spectral measures
$Y_{a}$, $F_{a}$, while the ‘bad’ one $x_{d}$ has $Y_{s}$, $F_{s}$ (see [Bin1]
for details and references). In the matrix case, Wold-Cramér concordance holds
in the full-rank case but not in general ([Bin2]; see e.g. Payen [Pay,
Remarque 8, 376-7]). The vector case is studied in [Kak3], where such
concordance is shown to be preserved under dilation from the stationary case
here to the harmonisable case ($\S$ 6.4).
The ‘good’ component $x_{p}$ in the Wold decomposition is a moving average of
the products of the innovations (new randomness) at past times and the
matrices appearing in the Taylor expansion of the Szegő function (‘analytic
square root’ of the spectral density matrix), as we shall see below.
In the Hilbert-valued case, spectral criteria for the process to be purely
non-deterministic are given by Kallianpur and Mandrekar [KalM].

4\. Szegő alternative; factorization

Szegő alternative
In one dimension, one has a clean split, the Szegő alternative, between ‘good’
and ‘bad’ cases. In the first, there is a genuine innovation (input of new
randomness) in each time-step from $n$ to $n+1$. The size of this new input is
measured by the prediction error variance ${\sigma}^{2}$, the minimum variance
attainable by all linear predictors based on the present and (finite sections
of) the past of the next future value $x_{n+1}$. This ‘good’ case can only
happen if there is an absolutely continuous component $F_{a}$ to $F$, that is,
if the spectral density $f$ is not a.e. zero. When this is so, it happens if
and only if the Szegő condition $(Sz)$ holds, and then ${\sigma}^{2}$ is the
geometric mean of $\log f$ (see e.g. [Bin1]), so $(KSz)$ becomes

${\sigma}^{2}=\exp\\{\int\log f(\theta)d\theta/2\pi\\}>0.$

Note that the singular component of $F$ (if present) plays no role here.
In the Wold decomposition, $x$ is the sum of a ‘bad’ part $x_{d}$ ($\S 3$),
and the ‘good’ part $x_{p}$, a moving average of the innovations over the past
to date; this contains a factor $\sigma$, and so is absent if $\sigma=0$
[Bin1]. In terms of the Verblunsky coefficients,

${\sigma}^{2}={\prod}_{1}^{\infty}(1-|{\alpha}_{n}|^{2}),$

so $(Sz)$ holds iff $\sigma>0$ iff the product converges, i.e. iff

$\alpha=({\alpha}_{n})_{0}^{\infty}\in{\ell}_{2}(\mathbb{N}).$

Then (and only then), one can define the Szegő function

$h(z):=\exp\Bigl{(}\frac{1}{2}\int\Bigl{(}\frac{e^{i\theta}+z}{e^{i\theta}-z}\Bigr{)}\log
f(\theta)\ d\theta/2\pi\Bigr{)}\qquad(z\in D).$ $None$

This is an outer function, in the Hardy space $H^{2}(\mathbb{D})$ ([Dur];
[Sim1]; [Bin1]). It is analytic in $\mathbb{D}$, and zero-free there [Sim1,
Th. 2.4.1]. As (from Fatou’s theorem: [GarMR, Th. 1.10], [Rud1, Th. 17.10])

$\int\Bigl{(}\frac{e^{i\theta}+z}{e^{i\theta}-z}\Bigr{)}\log w(\theta)\
d\theta/2\pi\to\log f(\phi)\ \ a.e.\ \ \quad(z=re^{i\phi},\quad r\uparrow 1),$

its square has radial limit $f$ a.e. on $\mathbb{T}$ :

$\lim|h(z)|^{2}=f(\theta)\ \ a.e.\ \ (z=re^{i\theta},\ r\uparrow 1).$

It may thus be regarded as the analytic square root of the spectral density
$f$.
In the matrix case, we refer to [Bin2, §5] for details and references. Szegő’s
theorem (the matrix forms of $(Sz)$, $(KSz)$ above) hold; see [DerHKT, Th.
28]. As noted above, this involves determinants, which do not extend to the
infinite-dimensional case.

Factorization
In the scalar case, existence of the Szegő function (analytic square root of
the spectral density), the Szegő condition $(Sz)$, and
$\alpha\in{\ell}_{2}(\mathbb{N})$ (above) are equivalent. One seeks a matrix
and a vector version of this. So one seeks to factorise a matrix spectral
density into its analytic square root times its adjoint (denoted by $\ast$
here, to conform to the sources below),

$F=\Phi{\Phi}^{*}.$

There is an extensive theory here, due mainly to Wiener and Masani and to
Helson and Lowdenslager. We note that the coefficient matrices ${\Phi}_{n}$ in
the Taylor expansion

$\Phi(z)=\sum_{0}^{\infty}{\Phi}_{n}z^{n}$

appear in the prediction-error matrix (and operator, in the vector case,
below). See e.g. Masani [Mas2, p.278], Whittle [Whi] (for the scalar case, see
e.g. [GreSz, §10.8]). Payen [Pay, II.8 Prop. 9] shows that in the Hilbert case
factorization is possible iff the spectral density decomposes as

$f(\theta)=\sum_{n}{\rho}_{n}(\theta)Q_{n}(\theta),$ $None$

where the operators $Q_{n}$ are projections onto one-dimensional subspaces and
the ${\rho}_{n}$ are scalar functions satisfying Szegő’s condition $(Sz)$.
For factorizations in the operator case, we refer to [RosR Ch. 6]. We also
mention briefly the approach of Power [Pow]. We make the restrictive
assumption that the spectral density $f$ is essentially bounded (i.e. is
bounded, after excluding some null set on the torus $\mathbb{T}$). Then

$f=hh^{*}+g,$

where $g$ is a positive operator and $h$ is analytic and outer. Here $g$ can
be taken to be minimal. A Wold(-Zasuhin) decomposition is obtained. The
prediction-error operator $G(f)$ of $f$ is obtained as

$G(f)=G(hh^{*})=(QhQ)(QhQ)*,$

where $Q$ is an orthogonal projection; we refer to [Pow] for details.
As Power points out, his method gives $g$ and $h$ as functions of $f$, while
the methods of Sz.-Nagy and Foiaş ([SzNFBK]; §5 below) do not. But this
applies only to the (‘good’ and) ‘bad’ behaviour encoded in the spectral
density, the absolutely continuous component in the Lebesgue decomposition of
the spectral measure. We turn now to how to address the ‘bad’ behaviour
encoded in the other two components.

5\. The Beurling-Lax-Halmos theorem; inner functions

So far as prediction theory is concerned, it is the ‘good’ part of the process
(or its spectral measure) that matters. Nevertheless, the Wold decomposition
shows us that to understand the structure of a stationary process we need to
look at the ‘bad’ part also.
Recall the outer function of $(OF)$, §4 above. The term is due to Beurling
[Beu], as is the corresponding term inner function. These correspond
respectively to the ‘good’ and ‘bad’ parts above. When we pass from the time
domain (Wold decomposition) to the frequency (or spectral) domain by the
Kolmogorov Isomorphism Theorem, we obtain the factorization into outer and
inner factors in Hardy spaces $H^{2}(\cal H)$. For the scalar case (where the
Hardy space is written $H^{2}$) there are good accounts in [Dur], [Gar] and
[Hof] (see also [Rud1, Ch. 17]). For the vector case, see [SzNFBK, III, V],
[Nik1, I.7, XI.3], [RosR, Ch. 4-6], [Hof, 114-116].
The remote past (the ‘bad’ part above) is invariant under the time-shift. The
invariant subspaces are exactly those given by multiplication in $H^{2}(\cal
H)$ by an inner function ([SzNFBK, V, Th. 3.3]; [Nik1, I.7, XI.3]; [RosR,
1.12]). This is the Beurling-Lax-Halmos theorem, due to Beurling (dimension
$d=1$), Lax [Lax] ($1<d<\infty$) and Halmos [Hal] ($d\leq\infty$). So study of
the remote past reduces to study of inner functions, $u$ say. These have a
rich structure; they factorise into a unimodular constant, a Blaschke product,
and an integral factor as in $(OF)$ but with a singular rather than an
absolutely continuous measure. See §6.1,2 below.
Halmos’s approach [Hal] is via the concept of a wandering subspace; these
correspond to the innovations (new randomness) in the process. For an isometry
$V$ on a Hilbert space $\cal H$, call a subspace $L$ of $\cal H$ wandering if
for all distinct integers $m,n$ $V^{m}L$ and $V^{n}L$ are orthogonal. Then
[SzNFBK, §1.1] if

$M_{+}(L):={\oplus}_{0}^{\infty}\ V^{n}L,$

one has

$L=M_{+}(L)\ominus VM_{+}(L).$

Call an isometry $V$ on $\cal H$ (which will be $U$ of §2 for us) a unilateral
shift if $\cal H$ has a wandering subspace $L$ with $M_{+}(L)=\cal H$. Then
$L$, called generating for $V$, is uniquely determined by $V$, indeed $L=\cal
H\ominus V\cal H$. One has the Wold decomposition ([SzNBK, Th. 1.1], [RosR,
§1.3]): $\cal H$ decomposes into the orthogonal sum ${\cal H}={{\cal
H}_{0}}\oplus{{\cal H}_{1}}$ such that the ${\cal H}_{i}$ reduce ${\cal H}$
(each is mapped onto itself by $V$), $V|{\cal H}_{0}$ is unitary and $V|{\cal
H}_{1}$ is a unilateral shift. This decomposition is unique; indeed,

${{\cal H}_{0}}=\bigcap_{0}^{\infty}V^{n}{\cal H},\quad{{\cal
H}_{1}}=M_{+}(L),\ \hbox{where}\ L={\cal H}\ominus V{\cal H}.$

Either ${{\cal H}_{0}}$ or ${{\cal H}_{1}}$ may be absent ($=\\{0\\}$).

6\. Complements

1\. The deterministic case
When the Szegő condition $(Sz)$ fails (e.g., when the density is absent –
$F_{a}=0$ in the notation of §3), the entire process is ‘bad’, and consists
entirely of ‘echoes of the remote past’. In the scalar case (for simplicity):
the variance ${\sigma}_{n}^{2}$ of the best linear predictor based on the last
$n$ readings decreases to ${\sigma}^{2}=0$. The interesting question of how
fast was addressed long ago by Rosenblatt [Ros]. The question is still
topical; see Babayan, Ginovyan and Taqqu [BabGT].
2\. Model spaces
Again in the scalar case first for simplicity: the (unilateral, forward) shift
$S$ (time $n\mapsto n+1$) is represented on the Hardy space $H^{2}$ by

$Sf=zf\qquad(f=f(z)\in H^{2}).$

This has adjoint the backward shift $S^{*}$:

$S^{*}f=\frac{f-f(0)}{z}.$

This follows from the Taylor series for $f(z)=\sum_{0}^{\infty}a_{n}z^{n}$ in
each case. Note that $f\in H^{2}$ and $a=(a_{n})\in{\ell}_{2}(\mathbb{N})$ are
equivalent, and give the ‘Hardy-Hilbert space’ (see de Branges [dBra],
Martinez-Avendano and Rosenthal [MarAR] for monograph treatments).
By the Beurling(-Lax-Halmos) theorem, the invariant subspaces of $S$ are
$uH^{2}$ for the inner functions $u$. Similarly, the invariant subspaces for
$S^{*}$ are the orthogonal complements of these, written

${\cal K}_{u}:=(uH^{2})^{\perp}.$

These are called model spaces (‘model spaces are the invariant subspaces of
the backward shift’). To explain the terminology, we quote [GarMR, 105] ‘The
term model space originates in the theory of model operators, developed by
Sz.-Nagy and Foiaş, where it is shown that certain types of Hilbert-space
contractions are unitarily equivalent to the compressions of the unilateral
shift to a model space. This underscores the importance of model spaces in
developing concrete, function-theoretic realizations of abstract Hilbert space
operators.’ For background, see the two classic sources [SzNFBK] (‘first
edition [SzNF]’) and Nikolskii [Nik1] cited in §5, plus Rosenblum and Rovnyak
[RosR] and the two-volume [Nik2], and the two recent treatments by Garcia,
Mashreghi and Ross [GarMR] cited above (see also [GarR]) and Agler, McCarthy
and Young [AglMY]. (Note that different authors use the term compression
differently [GarMR, Remark 9.2].) See §5 for references specific to the
vector-valued case $H^{2}(\cal H)$, our main concern here, to which we add
[Nik2, §4.8.8] and [NikV].
3\. Compressions and dilations
For $A,B$ operators on spaces ${\cal A}\subset{\cal B}$, $B$ is a dilation of
$A$ if

$A^{n}=pr\ B^{n}\quad\hbox{for all}\ n\in\cal N,$

where $pr$ is projection [SzNBK, p.10]. Then $A$ is a compression of $B$
([GarMR, Def. 9.1]; [SzNBK] does not use the term compression).
The basic result here is the Sz.-Nagy-Foiaş dilation theorem: if $H$ is a
Hilbert space and $T$ a linear contraction on it, there exists a larger
Hilbert space $\cal H$ and a unitary operator $U$ on $\cal H$ with $U$ a
dilation of $T$ and

${{\cal H}}={\bigvee}(U^{n}H:n\in\mathbb{N})$

(such a dilation is called minimal). For the extensive theory here, see e.g.
[SzNFBK, Ch. I], [Nik1, Introductory Lecture, Lecture III], [GarMR, Ch. 9].
4\. Harmonizability
This concept, due to Loève [Loe] in 1948, addresses the need to relax the
strong assumption of stationarity, which one cannot expect to hold exactly in
practice. It has been studied and extended by Karhunen, Cramér and others. As
the covariance function now needs two arguments, $s$ and $t$ say, rather than
one, $s-t$, the spectral measure now becomes a ‘bimeasure’. For details and
references, see e.g. [Rao1,2], [Kak1-4]. To summarise [Rao2, 292]:
strongly harmonisable $\subset$ weakly harmonizable $\subset$ Karhunen
$\subset$ Cramér.
Note [Kak3, Th. 2] that a process is weakly harmonizable iff it has a
stationary dilation.
5\. Banach spaces and beyond
Such extensions were addressed briefly in [BinM, §5.5]. We refer there for
further detail, and to e.g. Chobanyan and Weron [ChoW], Weron [Wer], Miamee
and Salehi [MiaS] and Klotz and Riedel KloR]. For a Banach-space version of
the Wold decomposition, see Faulkner and Honeycutt [FauH].
6\. Operator-valued processes
For an approach via dilation theory and operator models, see Makagon and
Salehi [MakS, §2] and the references cited there. The vector- and operator-
valued cases are developed together in [RosR, Ch. 4-6].
7\. The multivariate case
See e.g. [Kak3, §6] for a multivariate vector-valued treatment.
8\. High- and infinite-dimensional probability and statistics.
For high-dimensional treatments, see e.g. Vershynin [Ver] for probability,
Wainwright [Wai] for statistics. For infinite-dimensional statistics, see e.g.
[Giné and Nickl [GinN].
9\. Specifically infinite-dimensional phenomena.
We have been concerned here with vector-valued Hardy space theory (inner-outer
function factorization, etc.), in finitely many dimensions [Bin2], or
infinitely many, our main concern here. There are results in the area which do
not extend to the infinite-dimensional case. For example, see Treil [Tre1,2]
for the (operator) corona problem.
10\. The Levinson-Durbin algorithm
For this classical algorithm for the computation of the best linear predictor
based on the last $n$ data points, see e.g. Brockwell and Davis [BroD, §5.2],
[Bin1] in the scalar case, [Bin2] in the matrix case. It is worth noting that
numerical improvements have been made here (the split Levinson algorithm); see
Delsarte and Genin [DelG].
11\. Karhunen-Loève expansion
The Karhunen-Loève expansion [GinN, Th. 2.6.10] may be combined with the
Cramér representation to give a dynamicized form of the expansion, which has
been used for modelling of functional time series. See e.g. Antoniadis and
Sapatinas [AntS] (who considered El Niño), [AntPS], Panaretos and Tavakoli
[PanT1,2].
12\. Functional data analysis; prediction; change-points
For some recent developments in these areas, see e.g. Aue, Norinho and Hörmann
[AueNH], Dette, Kokot and Aue [DetKA].
13\. Smoothness of functions
Our data are functions, drawn from function spaces, often Hilbert spaces $\cal
H$. These may (and typically do) have some smoothness properties. One of the
most basic is continuity of point evaluation, $x\mapsto f(x)$ ($f\in\cal H$).
By the Riesz representation theorem, this is the condition for $\cal H$ to be
a reproducing-kernel Hilbert space (RKHS): to have a reproducing kernel
$k(.,.)$ such that, with $k_{x}(.):=k(.,x)$,

$(f,k_{x})=(f(.),k(.,x))_{\cal H}=f(x)\qquad(f\in{\cal H}).$

Such spaces are common in the Hardy-space setting above. For example, the
Hardy-Hilbert space $H^{2}$ is a RKHS with kernel

$k_{\lambda}(z)=1/(1-\overline{\lambda}z)\qquad(\lambda,z\in\mathbb{D})$

[GarMR, Prop. 3.3], and so are the model spaces ${\cal K}_{u}$ above, with
kernel (with the $u$ here to be understood in the notation)

$k_{\lambda}(z)=\frac{1-\overline{u(\lambda})u(z)}{1-\overline{\lambda}z}\qquad(\lambda,z\in\mathbb{D})$

[GarMR, §5.5]. For the extensive theory of RKHS, and applications to
probability and statistics, see e.g. Berlinet and Thomas-Agnan [BerTA].

7\. Questions

We close with some questions arising from the work surveyed above.
Q1. Does the work of [DamPS] (§3) on Verblunsky’s theorem extend to infinitely
many dimensions?
Q2. Does the work of [DerHKT] (§3) on Szegő’s theorem extend to infinitely
many dimensions?
Q3. Is there an infinite-dimensional version of Szegő’s theory of orthogonal
polynomials on the unit circle (OPUC, [Sim1], [Bin1]; MOPUC, [Bin2])? In this
regard, see [GarMR, Example 8.3]. This involves the Szegő (or Toeplitz)
conjugation, which is familiar from the Szegő recursion of OPUC and MOPUC.
Q4. Can the assumption of essentially bounded spectral density in [Pow], §4,
be relaxed (or dropped)?
Q5. Can the assumption of having a scalar multiple in [SzNFBK, V.6,7], §4, be
relaxed (or dropped)?.

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Mathematics Department, Imperial College, London SW7 2EZ, UK;
<EMAIL_ADDRESS>


e\backslash f$, we have that $L\backslash e=N\backslash e$. Note also that
$N\backslash e=N^{\prime}\backslash e=N^{\prime\prime}\backslash e$, so
$L\backslash e=N^{\prime\prime}\backslash e$. The same argument shows that
$L\backslash f=N^{\prime\prime}\backslash f$.

Suppose $L\neq N^{\prime\prime}$. There must exist a set $A$ which is a non-
spanning circuit in $N^{\prime\prime}$ and independent in $L$ or vice versa.
The previous results tell us that $e,f\in A$, as otherwise $A$ would have the
same rank in both matroids.

Suppose $A$ is a non-spanning circuit in $L$. Say that the points in
$E(L)-\\{e,f\\}$ were added in the order $e_{1},....,e_{t}$. Let $e_{j}$ be
the largest element of $A$ according to this ordering, and let $e_{j}\in
H_{k}$. As $e_{j}$ was freely placed, $A$ must span
$\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{k},x_{k+1}\\}\rangle$. This
means that $(A-H_{k})\cup(X\cup\\{x_{0},...,x_{n-1}\\})-\\{x_{k},x_{k+1}\\})$
spans the same set as $A$. Suppose the last element added to $A$ before those
in $H_{k}$ is $e_{l}\in H_{j}$ where $j<k$. Then
$(A-H_{k})\cup(X\cup\\{x_{0},...,x_{n-1}\\})-\\{x_{k},x_{k+1}\\}$ spans an
element from $H_{j}$, and by construction, as every element in the set above
was added before $H_{j}$, we see that this set spans
$(X\cup\\{x_{0},...,x_{n-1}\\})-\\{x_{j},x_{j+1}\\}$. Thus $A$ spans both
$\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{k},x_{k+1}\\}\rangle$ and
$\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{j},x_{j+1}\\}\rangle$. As
shown in the previous lemma, if $H_{j}$ and $H_{k}$ are inconsecutive, $A$
will have rank $r+1$ and be spanning. $H_{j}$ and $H_{k}$ thus must be
consecutive in order for $A$ to be non-spanning. Take a dependent subset of
$H_{j}\cup H_{k}$ in $L$. As this subset does not include $e$ nor $f$, it has
the same rank in $L\backslash e$. Likewise, the rank of the subset in
$N^{\prime\prime}$ has the same rank in $N^{\prime\prime}\backslash e$. As we
have already shown $L\backslash e=N^{\prime\prime}\backslash e$, we have that
any dependent subset of $H_{j}\cup H_{k}$ in $L$ is also dependent in
$N^{\prime\prime}$. This contradicts the assumption that $A$ is independent in
$N^{\prime\prime}$. If there is no point contained in a set $H_{j}$ where
$j<k$, in order for $A$ to be a circuit, $A$ must be equal to
$H_{k}\cup\\{e,f\\}$, where $k\notin\\{0,i\\}$, as any subset of this is
independent in $L$, as proved in the next lemma.

######  Lemma 7.8.

$H_{k}\cup\\{e,f\\}$ is a circuit of $L$ for all
$k\in\\{1,\ldots,i-1,i+1,\ldots,n\\}$.

###### Proof.

Consider $H_{k}\cup\\{e,f\\}$. Recall that $p$ was added freely to $\langle
X\cup\\{x_{1},\ldots,x_{i}\\}\rangle$ while $p^{\prime}$ was added freely to
$\langle X\cup\\{x_{i+1},\ldots,x_{n-1},x_{0}\\}\rangle$. $H_{k}$ is contained
in $\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{k},x_{k+1}\\}\rangle$. When
$i\leq k$, this subspace spans $\langle X\cup\\{x_{1},\ldots,x_{i}\\}\rangle$
and so spans $p$. When $i\geq k$, this subspace spans $\langle
X\cup\\{x_{i+1},\ldots,x_{n-1},x_{0}\\}\rangle$ and so spans $p^{\prime}$. As
$e$ and $f$ were freely placed on the line spanned by $p$ and $p^{\prime}$, in
either case we have that $H_{k}\cup\\{e,f\\}\in cl(H_{k}\cup\\{e\\})$ and so
$H_{k}\cup\\{e,f\\}$ is dependent.

Suppose $H_{k}$ is dependent for some $k$. Then at some point in constructing
$L$, we would have added a point $g$ to already chosen elements of $H_{k}$ so
that the point was contained in $cl(H_{k}-g)$, i.e. contained in
$\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{k},x_{k+1}\\}\rangle$. This
contradicts each of the $r-1$ points of $H_{k}$ being freely placed in the
rank $r-1$ space
$\langle(X\cup\\{x_{o},\ldots,x_{n-1}\\})-\\{x_{k},x_{k+1}\\}\rangle$. Thus
$H_{k}$ is independent.

Now suppose $H_{k}\cup\\{e\\}$ is dependent. Then $e\in cl(H_{k})$ – that is,
$e\in\langle(X\cup\\{x_{0},\ldots,x_{n-1}\\}-\\{x_{k},x_{k+1}\\}\rangle$.
However, $e$ was freely placed on the line spanned by $p$ and $p^{\prime}$.
Thus if $H_{k}$ spans $e$, it must span this line. As the line itself is free
in the matroid, for this to happen, $H_{k}$ must be spanning – contradiction.
Likewise, $H_{k}\cup\\{f\\}$ is also independent. We have shown that every
subset of $H_{k}\cup\\{e,f\\}$ is independent, meaning that
$H_{k}\cup\\{e,f\\}$ must be a circuit. ∎

Thus $H_{k}\cup\\{e,f\\}$ is dependent in $L$. We have shown that it is also
dependent in $N^{\prime\prime}$, so again have a contradiction to $A$ being
independent in $L$. The same argument shows that if $A$ is dependent in
$N^{\prime\prime}$, $A$ is also dependent in $L$. Thus $L=N^{\prime\prime}$. ∎

We constructed $N$ to be representable, so $N$ must satisfy every Kinser
inequality. In particular, it must be contained inside $\mathcal{K}_{n+1}$.
Next we have shown that if we relax a single circuit-hyperplane of $N$, the
resulting matroid $N^{\prime}$ has a bad family for $\mathcal{K}_{n+1}$. We
will now show that $N^{\prime}$ is in fact an excluded minor for
$\mathcal{K}_{n+1}$ – that is, we will show that each proper minor of
$N^{\prime}$ is representable and thus is contained in $\mathcal{K}_{n+1}$.

First suppose that $x\in H_{j}$ where $j\neq 0$. Let
$N^{\prime\prime}=N^{\prime}$ with the circuit-hyperplane $H_{j}\cup\\{e,f\\}$
relaxed. By [6, Proposition 3.3.5], $N^{\prime\prime}\backslash
x=N^{\prime}\backslash x$. As $N^{\prime\prime}$ is $\mathbb{K}$-representable
by Theorem 7.7, and representability is preserved under minors,
$N^{\prime}\backslash x$ is $\mathbb{K}$-representable. Say
$l\in\\{0,\ldots,n-1\\}-\\{0,j\\}$. Now let $N^{\prime\prime}=N^{\prime}$ with
$H_{l}\cup\\{e,f\\}$ relaxed. Also by [6, Proposition 3.3.5], we have that
$N^{\prime\prime}/x=N^{\prime}/x$, and so $N^{\prime}/x$ is
$\mathbb{K}$-representable.

Next, suppose $x\in H_{0}$. As $N^{\prime}=N$ with the circuit-hyperplane
$H_{0}\cup\\{e,f\\}$ relaxed, we have that $N^{\prime}\backslash x=N\backslash
x$, so $N^{\prime}\backslash x$ is $\mathbb{K}$-representable. Let
$N^{\prime\prime}=N^{\prime}$ with $H_{i}\cup\\{e,f\\}$ relaxed. We have that
$N^{\prime\prime}/x=N^{\prime}/x$, so $N^{\prime}/x$ is
$\mathbb{K}$-representable.

Now suppose $x$ is equal to $e$. As $e$ and $f$ were freely placed on the line
spanned by $p$ and $p^{\prime}$, the same argument as follows works for $x=f$.
We have that $N^{\prime}\backslash e=N\backslash e$, so $N^{\prime}\backslash
e$ is $\mathbb{K}$-representable.

Finally, consider $N^{\prime}/e$. Take some $z\in H_{0}$. Recall that
$N^{\prime}=N$ with the circuit-hyperplane $H_{0}\cup\\{e,f\\}$ relaxed. Note
that $N^{\prime}/e$ is obtained from $N/e$ by relaxing $H_{0}\cup\\{f\\}$.
This gives us that $N^{\prime}/e\backslash z=N/e\backslash z$, as deleting $z$
effectively undoes the relaxation. As $N$ is $\mathbb{K}$-representable, and
thus $N/e\backslash z$ is $\mathbb{K}$-representable, $N^{\prime}/e\backslash
z$ is also $\mathbb{K}$-representable. Let $Z\subseteq E(N^{\prime}/e)$ be
such that $z\notin Z$ and $z\in cl_{N^{\prime}/e}(Z)$. $N^{\prime}/e$ is a
relaxation of $N/e$ which can only affect closures in so far as that some may
contain additional elements in $N/e$, so $z\in cl_{N/e}(Z)$. This implies that
$z\in cl_{N}(Z\cup\\{e\\})$ by [6, Proposition 3.1.11]. Due to the way $H_{0}$
was constructed, we thus have that $\langle
Z\cup\\{e\\}\rangle\supseteq(X\cup\\{x_{0},\ldots,x_{n-1}\\})-\\{x_{0},x_{1}\\}$.
As we have that $z\in cl_{N}(Z\cup\\{e\\})$ and all elements of $H_{0}$ are
freely placed in the relevant subspace, $Z\cup\\{e,f\\}$ must thus also span
every other element of $H_{0}$. As $e$ and $f$ were freely placed on the line
spanned by $p$ and $p^{\prime}$, we also have that $f\in
cl_{N}(Z\cup\\{e\\})$. Thus, in $N$, $H_{0}\cup\\{e,f\\}$ is contained in
$\langle Z\cup\\{e\\}\rangle$. As $H_{0}\cup\\{e,f\\}$ is a circuit-
hyperplane, this implies that either $Z\cup\\{e\\}$ is spanning in $N$ or that
$Z\cup\\{e\\}=H_{0}\cup\\{e,f\\}$. If $Z\cup\\{e\\}=H_{0}\cup\\{e,f\\}$, we
have a contradiction to the assumption that $z\notin Z$. We thus have that
$Z\cup\\{e\\}$ is spanning in $N$. This means that $Z\cup\\{e\\}$ is also
spanning in $N^{\prime}$, and, as $r(N^{\prime}/e)=r(N^{\prime})-1$, that $Z$
is spanning in $N^{\prime}/e$. We have that $z$ is only in the closure of a
subset of $N^{\prime}/e$ when that subset spans $N^{\prime}/e$ – that is, we
have shown that $z$ is freely placed in $N^{\prime}/e$. Thus $N^{\prime}/e$ is
a free extension of $N^{\prime}/e\backslash z$ by $z$. As
$N^{\prime}/e\backslash z$ is $\mathbb{K}$-representable and this fact is
preserved under free extentions, we have that $N^{\prime}/e$ is
$\mathbb{K}$-representable.

We have now shown that every minor of $N^{\prime}$ is
$\mathbb{K}$-representable and so contained in $\mathcal{K}_{n+1}$, making
$N^{\prime}$ an excluded minor for $\mathcal{K}_{n+1}$. This completes the
proof of Theorem 7.1. ∎

##  Chapter 8 Conjectures

Finally, we give some conjectures on the hierachy of the Kinser classes.

######  Conjecture 8.1.

Let $n>5$. $\mathcal{K}_{n}\neq\mathcal{K}_{n}^{*}$.

As shown in the previous chapter, verifying that a matroid satisfies a Kinser
inequality is very difficult. Given the amount of difficulty involved in
proving that the fifth Kinser class is not dual closed, proving this result in
general would involve an even greater amount of work. Based on that case,
however, we give a strengthening of the above conjecture.

######  Conjecture 8.2.

$\mathrm{Kin}(n)^{-}\in\mathcal{K}_{n}^{*}-\mathcal{K}_{n}$

One further question about the structure of the hierarchy is how each dual
class sits within the previous Kinser class. There are two possibilities here,
and we conjecture that the following is true.

######  Conjecture 8.3.

Let $n>4$. $\mathcal{K}_{n+1}^{*}\subseteq\mathcal{K}_{n}$.

The following conjecture is true if Conjecture 8.3 is as well.

######  Conjecture 8.4.

$\mathcal{K}_{\infty}^{*}=\mathcal{K}_{\infty}$

To see that this follows from Conjecture 8.3, assume
$M\in\mathcal{K}_{\infty}$, but $M\not\in\mathcal{K}_{\infty}^{*}$. Then there
exists an integer $n$ such that $M\not\in\mathcal{K}_{n}^{*}$. However, this
contradicts Conjecture 8.3, which gives that
$M\in\mathcal{K}_{n+1}\subseteq\mathcal{K}_{n}^{*}$.

Assuming these conjectures to hold true, we have a final diagram of the
hierarchy.

Figure 8.1: Kinser classes (4)

Now we will consider two classes of matroids which we conjecture satisfy every
Kinser inequality.

######  Definition 8.5.

Let $G$ be an abelian group. Take a complete graph and add a loop to very
vertex, replace every edge with a parallel class of $|G|$ edges. Call this
graph $H$. Orient every edge which is not a loop so that parallel edges have
the same direction. Bijectively label each parallel class with the elements of
$G$, and label loops with non-identities. Let $C$ be a cycle of $H$. Consider
the product of group labels in $C$, taken in cyclic order, where if an edge is
oriented against the cyclic order we take the inverse of its label instead. If
the result is the identity, call $C$ _positive_. Otherwise, call $C$
_negative_. Take a Dowling geometry $H$. There exists a matroid which has
$E(H)$ as its ground set, and set of circuits equal to the positive cycles of
$H$ and minimal connected subgraphs that contain two negative cycles. Call
this matroid a _Dowling geometry_.

Take a field $\mathbb{F}$. Recall that $\mathbb{F}^{\times}$ is the
multiplicative group consisting of the non-zero elements of $\mathbb{F}$.

######  Lemma 8.6.

[6, Theorem 6.10.10] Take a Dowling geometry of rank $r$ over a finite group
$G$. This matroid is representable over a field $\mathbb{F}$ if and only if
$G$ is isomorphic to a subgroup of $\mathbb{F}^{\times}$.

In this case, the Dowling matroid satisfies every Kinser inequality. We also
have that if $G$ is a finite subgroup of the multiplicative group of a field,
then $G$ is cyclic by [1]. The following conjecture is thus open when $G$ is
both finite and non-cyclic.

######  Conjecture 8.7.

A Dowling geometry satisfies every Kinser inequality.

Now we will consider matroids which are representable over skew partial
fields. All of the following definitions and results can be found in [7].

######  Definition 8.8.

A _skew partial field_ is a pair $(R,G)$ where $R$ is a ring, and $G$ is a
subgroup of the group of units of $R$, such that $-1\in G$.

######  Definition 8.9.

Let $R$ be a ring, and let $E$ be a finite set. An _R-chain group_ on $E$ is a
subset $C\subseteq R^{E}$ such that, for all $f,g\in C$ and $r\in R$,

  * i.

$0\in C$,

  * ii.

$f+g\in C$,

  * iii.

$rf\in C$

The elements of $C$ are called _chains_ , and the _support_ of a chain
$c=\\{c_{1},\ldots,c_{e}\\}\in C$ is

$||c||=\\{i\in E\ |\ c_{i}\neq 0\\}$

######  Definition 8.10.

A chain $c\in C$ is _elementary_ if $c\neq 0$ and there is no $c^{\prime}\in
C-\\{0\\}$ with $||c^{\prime}||\subset||c||$.

######  Definition 8.11.

Let $G$ be a subgroup of the group of units of $R$. A chain $c\in C$ is _G-
primitive_ if $c\in(G\cup\\{0\\})^{E}$.

######  Definition 8.12.

Let $\mathbb{P}=(R,G)$ be a skew partial field, and $E$ a finite set. A
_$\mathbb{P}$ -chain group_ on $E$ is an $R$-chain group $C$ on $E$ such that
every elementary chain $c\in C$ can be written as $c=rc^{\prime}$ for some
$G$-primitive chain $c^{\prime}\in C$ and some $r\in R$.

######  Lemma 8.13.

Let $\mathbb{P}=(R,G)$ be a skew partial field, and let $C$ be a
$\mathbb{P}$-chain group on $E$. Then $\mathcal{C}^{*}=\\{||c||\ |\ c\in C,\
c\ \text{is elementary}\\}$ is the set of cocircuits of a matroid on $E$.

A matroid $M$ is said to be _$\mathbb{P}$ -representable_ if there exists a
$\mathbb{P}$-chain group $C$ such that $M=M(C)$. If a matroid is representable
and thus satisfies every Kinser inequality, it is representable over a skew
partial field.

######  Conjecture 8.14.

Take a matroid $M$ which is $\mathbb{P}$-representable. $M$ satisfies every
Kinser inequality.

## Bibliography

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