# Revisiting mean-square approximation by polynomials in the unit disk

Bartosz Malman  Division of Mathematics and Physics, Mälardalen University,
Västerås, Sweden<EMAIL_ADDRESS>
###### Abstract.

For a positive finite Borel measure $\mu$ compactly supported in the complex
plane, the space $\mathcal{P}^{2}(\mu)$ is the closure of the analytic
polynomials in the Lebesgue space $L^{2}(\mu)$. According to Thomson’s famous
result, any space $\mathcal{P}^{2}(\mu)$ decomposes as an orthogonal sum of
pieces which are essentially analytic, and a residual $L^{2}$-space. We study
the structure of this decomposition for a class of Borel measures $\mu$
supported on the closed unit disk $\overline{\mathbb{D}}$ for which the part
$\mu_{\mathbb{D}}$, living in the open disk $\mathbb{D}$, is radial and
decreases at least exponentially fast near the boundary of the disk. For the
considered class of measures, we give a precise form of the Thomson
decompsition. In particular, we confirm a conjecture of Kriete and MacCluer
from 1990, which gives an analog to Szegö’s classical theorem.

##  1\. Introduction

The classical and highly influential theorem of Szegö states that the
divergence of a logarithmic integral, namely

(1) $\int_{\mathbb{T}}\log w\,d\textit{m}=-\infty,$

is necessary and sufficient for the set $\mathcal{P}$ of analytic polynomials
to be dense in the Lebesgue space $L^{2}(\mu)$, where $d\mu=w\,d\textit{m}$.
Here $w$ is a non-negative Borel measurable function (a weight) on the unit
circle $\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$, $m$ is the Lebesgue measure on
$\mathbb{T}$, and $L^{2}(\mu)$ is the space of measurable functions $f$
satisfying the integrability condition

$\|f\|^{2}_{L^{2}(\mu)}:=\int_{\mathbb{T}}|f|^{2}w\,d\textit{m}<\infty.$

Szegö’s result has important consequences and applications in operator and
spectral theory, complex analysis, and theory of stochastic processes, for
instance.

This paper continues a line of research dealing with the following question:
what is the closure of $\mathcal{P}$ in $L^{2}(\mu)$ if the measure
$d\mu=d\mu_{\mathbb{T}}=w\,d\textit{m}$ appearing in (1) is enlarged to
include also a part on the open disk $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$?
If $\mu=\mu_{\mathbb{D}}+\mu_{\mathbb{T}}$, with the two pieces living on
$\mathbb{D}$ and $\mathbb{T}$ respectively, then we expect to need more than
the condition (1) if, for instance, we want to conclude that any function in
$L^{2}(\mu_{\mathbb{T}})$ lies in the closure of $\mathcal{P}$ in
$L^{2}(\mu)$.

There is an extensive literature dealing with this question, and results in
this direction have interesting applications in other parts of analysis. We
will review some of the previous results and applications below. The main
result of the present paper is maintheoremSuperExp below. It is a structural
result on the span of the analytic polynomials in $L^{2}(\mu)$, in the case
when the part of $\mu$ living on the unit disk decreases sufficiently fast
near the boundary of $\mathbb{D}$. It can be seen as an analog to Szegö’s
theorem.

###  1.1. Background and some notation

The set of analytic polynomials $\mathcal{P}$ consists of functions of the
form $p(z)=\sum_{n=0}^{N}p_{n}z^{n}$ where $z\in\mathbb{C}$ is the complex
variable. By $\mathcal{P}^{t}(\mu)$ we denote the norm-closure of
$\mathcal{P}$ in the usual Lebesgue space $L^{t}(\mu)$. Here $t$ is a finite
positive number, and $\mu$ is a finite, positive Borel measure which is
compactly supported in the plane. We shall for the most part be dealing with
the case $t=2$ (but some of our results will be applicable in the case $t\neq
2$ also) and with measures $\mu$ supported on the closed unit disk
$\overline{\mathbb{D}}=\\{z\in\mathbb{C}:|z|\leq 1\\}$ of a particular
structure. Before specializing to our intended context, we give a brief review
of some facets of the general theory.

Thomson in [12] solved a long-standing problem: $\mathcal{P}^{t}(\mu)\neq
L^{t}(\mu)$ if and only if there exists an open set $U$ consisting of points
of bounded evaluation for $\mu$, in the sense that

$|p(\lambda)|^{t}\leq C\int|p|^{t}d\mu$

holds for $p\in\mathcal{P}$ and some $C>0$ independent of $p$ and $\lambda\in
U$. More precisely, there exists a decomposition of the measure
$\mu=\sum_{i}\mu_{i}$, each piece living on a different subset of the support
of $\mu$, such that

(2) $\mathcal{P}^{t}(\mu)=\Big{(}\oplus_{i\geq
1}\mathcal{P}^{t}(\mu_{i})\Big{)}\oplus L^{t}(\mu_{0})$

where for the pieces corresponding to $i\geq 1$ there exists an open simply
connected domain $U_{i}$ consisting of points of bounded evaluation for
$\mu_{i}$. These pieces are also irreducible in the sense that they do not
contain any non-trivial characteristic functions. For a Borel set $A$, the
characteristic function $1_{A}$ is defined, as usual, by

$1_{A}(x)=\begin{cases}1,\quad x\in A\\\ 0,\quad x\not\in A\end{cases}$

and it is trivial for $\mu_{i}$ if it equals identically $0$ or $1$ almost
everywhere with respect to $\mu_{i}$. In concrete applications, the pieces
$\mathcal{P}^{t}(\mu_{i})$ are often easily identified as spaces of analytic
functions on the domain $U_{i}$.

###  1.2. The considered class of measures

Thomson’s theorem is a cornerstone of the theory of $\mathcal{P}^{t}(\mu)$
spaces. However, it is hard to apply in specific examples appearing in the
applications in which the exact identification of the pieces $\mu_{i}$ in (2)
is necessary. Such applications are found, for instance, in Fourier analysis
and the theory of the Cauchy integral operator ([6], [10]) and also in the
theory of de Branges-Rovnyak spaces ([8], [9]). In these applications, the
related space $\mathcal{P}^{2}(\mu)$ splits into two pieces:
$\mathcal{P}^{2}(\mu_{1})$, which is a space of analytic functions on
$\mathbb{D}$, and a full space $L^{2}(\mu_{0})$, with $\mu_{0}$ supported on
the circle $\mathbb{T}$. This will also be the case in our context.

If $F$ is a Borel set, then by $\mu_{F}$ we will denote the restriction of
$\mu$ to $F$. The Lebesgue space $L^{t}(\mu_{F})$ can be regarded as a closed
subspace of $L^{t}(\mu)$ in the obvious way. The measures $\mu$ in our study
will live on the closed disk $\overline{\mathbb{D}}=\mathbb{D}\cup\mathbb{T}$,
and be composed of two pieces $\mu=\mu_{\mathbb{D}}+\mu_{\mathbb{T}}$. The
part $\mu_{\mathbb{T}}$, living on the circle $\mathbb{T}$, will be a general
absolutely continuous measure

(3) $d\mu_{\mathbb{T}}=w\,d\textit{m}$

where $w$ is a non-negative Borel measurable function, and $d\textit{m}$ is
the Lebesgue measure (arclength measure) on $\mathbb{T}$, normalized by the
condition $\textit{m}(\mathbb{T})=1$. We assume the global divergence of the
logarithmic integral (1), since it is known from elementary Hardy space theory
that convergence in (1) means that $\mathcal{P}^{t}(\mu)$ is essentially the
image of the Hardy space $H^{t}$ under a multiplication operator. In our main
result we shall show how instead the local integrability of $\log w$ shapes
the structure of (2).

The part $\mu_{\mathbb{D}}$ living on $\mathbb{D}$ will be of the form

(4) $d\mu_{\mathbb{D}}(z)=G(1-|z|)dA(z),$

where $G$ is some increasing function defined for $x$ between $0$ and $1$ and
satisfying $G(0)=0$, and $dA=\pi^{-1}dxdy$ is the normalized area measure on
$\mathbb{D}$. We will assume one lower bound on the rate of decrease of $G$ as
$x$ tends to zero, and one upper bound. Namely, we require that $G$ has at
least exponential decay, and we express this by

(ExpDec) $\liminf_{x\to 0^{+}}\,x\log(1/G(x))>0.$

The upper bound will be an integrability condition

(LogLogInt) $\int_{0}^{1}\log\log(1/G(x))\,dx<\infty.$

For the condition (LogLogInt) to make sense, we must of course have that
$\log(1/G(x))>0$, or in other words $G(x)<1$. This we can always assume by
reshaping the function $G$ on $(c,1]$ for some positive $c$. Such an operation
introduces an equivalent norm on $\mathcal{P}^{t}(\mu)$, and does not affect
any of our results.

###  1.3. Main result

Let $\mu$ be given by

(5) $\displaystyle d\mu$ $\displaystyle=d\mu_{\mathbb{D}}+d\mu_{\mathbb{T}}$
$\displaystyle=G(1-|z|)dA(z)+w\,d\textit{m}.$

Throughout the paper the measure $\mu$ will have this form. The main theorem
states, roughly speaking, that the two assumptions (ExpDec) and (LogLogInt)
place us in the correct range of functions $G$ for which the structure of
$\mathcal{P}^{2}(\mu)$ is determined by local integrability of the logarithm
of $w$ on intervals $I$ (or, in other words, arcs) of $\mathbb{T}$.

######  Definition 1.1.

Let $E\subseteq\mathbb{T}$ be the carrier set of the weight $w$:

(6) $E:=\\{x\in\mathbb{T}:w(x)>0\\}$

and define the residual set of $w$ to consist of those $x\in E$ which are not
contained in any (say, open) interval $I$ on which $\log w$ is integrable:

(7) $F:=\Big{\\{}x\in E:\int_{I}\log w\,d\textit{m}=-\infty\text{ whenever
$x\in I$}\Big{\\}}.$

It is not hard to verify that the residual set $F$ is (up to a set of
$m$-measure zero) the complement in $E$ of a countable union of intervals on
which $\log w$ is integrable.

The following is our structure theorem for the considered class of
$\mathcal{P}^{2}(\mu)$-spaces, and the main result of the paper.

######  Theorem A.

maintheoremSuperExp Let $\mu$ have the form (5), with $G$ being an increasing
function which is continuously differentiable and satisfies the conditions
(ExpDec) and (LogLogInt). Let $F\subseteq\mathbb{T}$ be the residual set of
$w$ given by (7), and decompose $\mu$ as

$\mu=(\mu_{\mathbb{D}}+\mu_{\mathbb{T}\setminus F})+\mu_{F},$

where $\mu_{\mathbb{T}\setminus F}$ and $\mu_{F}$ are the restrictions of
$\mu_{\mathbb{T}}$ to the sets $\mathbb{T}\setminus F$ and $F$, respectively.
Then

(8)
$\mathcal{P}^{2}(\mu)=\mathcal{P}^{2}(\mu_{\mathbb{D}}+\mu_{\mathbb{T}\setminus
F})\oplus L^{2}(\mu_{F}),$

where $\mathcal{P}^{2}(\mu_{\mathbb{D}}+\mu_{\mathbb{T}\setminus F})$ is
irreducible.

The span of the functions $G$ to which maintheoremSuperExp applies is large,
and starts roughly with $G(x)=\exp(-c/x)$ for any $c>0$ (inclusive), and ends
at $G(x)=\exp\exp(-c/x)$, which falls just short of the integrability
assumption (LogLogInt). This range is seen to be rather sharp by consideration
of previous work of Khrushchev and Volberg, which we discuss below. An
absolutely sharp dichotomy is very hard to achieve because of the various
needed regularity conditions on $G$ which appear (and seem to be necessary) in
the different works. However, the differentiability assumption on $G$ is
introduced mainly for convenience. It can be avoided at the cost of more
technical proofs and estimates, and in any case, some other regularity
assumption.

We remark that the part of the theorem which asserts the inclusion
$L^{2}(\mu_{F})\subset\mathcal{P}^{2}(\mu)$ holds also for any finite $t\neq
2$ (see FwSplittingOffProp). The proof of irreducibility of the other piece in
(8) uses an adaptation from [7] of a Hilbert space argument.

The piece $\mathcal{P}^{2}(\mu_{\mathbb{D}}+\mu_{\mathbb{T}\setminus F})$ can
be identified with a space of analytic functions on $\mathbb{D}$ in which the
analytic polynomials are dense. The deep work of Aleman, Richter and Sundberg
in [1] explains the boundary behaviour, zero sets, and other properties of the
functions contained in this space.

###  1.4. Work of Kriete and MacCluer

The present paper is inspired by results and conjectures in [7], where Kriete
and MacCluer study the so-called splitting problem for measures of form
similar to the ones appearing here. Splitting is said to occur for a measure
$\mu$ of the form (5) if the the space $\mathcal{P}^{t}(\mu)$ decomposes into
orthogonal pieces living on $\mathbb{D}$ and $\mathbb{T}$, respectively. The
method of Kriete and MacCluer is based on estimations of certain composition
operators on Bergman spaces, and is much different from ours. It allowed them
to establish results which shed light on what the definitive structure of
$\mathcal{P}^{t}(\mu)$ might be.

Among other interesting results, they noted that logarithmic integrability of
$w$ on an interval prohibits splitting if $G$ satisfies (LogLogInt) (we re-use
parts of their argument in the proof of maintheoremSuperExp), and they found
also that a certain change of behaviour occurs, roughly, at (ExpDec). Namely,
results of [7] indicate that behaviour of $\log w$ on intervals is key if
(ExpDec) is satisfied, while if this limit is zero, then more complicated sets
play a role. Based on these observations, they pose two conjectures (see
Section 9 in [7]). One of these conjectures is that splitting occurs if
$\int_{I}\log w\,d\textit{m}=-\infty$ for every interval $I$ and if $G$
satisfies a slightly stronger version of (ExpDec). Their condition on the
pervasive non-integrability of $\log w$ is nothing more than the statement
that $F=E$ in (7). Thus our main result confirms Conjecture 2 from [7, Section
9].

######  Theorem B.

KrieteMaccluerConjectureProof Assume that $G$ satisfies (ExpDec) and that

$\int_{I}\log w\,d\textit{m}=-\infty$

for every interval $I$ of $\mathbb{T}$. Then

$\mathcal{P}^{t}(\mu)=\mathcal{P}^{t}(\mu_{\mathbb{D}})\oplus
L^{t}(\mu_{\mathbb{T}}).$

The analogy with Szegö’s theorem is clear. If we add a piece
$\mu_{\mathbb{D}}$ to our measure $\mu_{\mathbb{T}}=wd\textit{m}$, then we
need a stronger condition than (1) to conclude that
$L^{t}(\mu_{\mathbb{T}})\subset\mathcal{P}^{t}(\mu)$.

Note that our method allows for the conclusion even in the non-Hilbertian
setting $t\neq 2$ (see FwSplittingOffProp). Kriete and MacCluer proved
KrieteMaccluerConjectureProof in the special case when $w$ is a characteristic
function and $t=2$ (see [7, Corollary 7.1]).

Kriete and MacCluer studied also the setting in which the limit in (ExpDec) is
zero. The methods developed in the present paper can be applied to this
situation also, and one can reach some sharp versions of results from [7].
They propose in [7, Section 9] a corresponding Conjecture 1 on conditions for
splitting in this setting, expecting behavior similar to the one proved here
in KrieteMaccluerConjectureProof in the case when (ExpDec) holds, but where
intervals are replaced by more complicated sets (more precisely Beurling-
Carleson sets, defined below). It is not immediately clear if the present
method can give a conclusive answer to that problem. This question, and more
broadly the setting in which (ExpDec) is violated, is planned to be studied in
a future work. The resolution of the conjecture would have implications for
the theory of approximations in de Branges-Rovnyak spaces (see [8], [9]).

###  1.5. Theorems of Khrushchev and Volberg, sharpness of the main result

Both maintheoremSuperExp and KrieteMaccluerConjectureProof are close to
optimal, in the sense that integrability of the logarithm of $w$ on intervals
does not determine the structure of $\mathcal{P}^{t}(\mu)$ for natural choices
of weights $G$ which just barely fail to satisfy (ExpDec) and (LogLogInt).

To be more precise, just below the cutoff (ExpDec) one can find the functions

(9) $G(x)=\exp(-c/x^{\alpha})$

for $\alpha\in(0,1)$ and $c>0$. For such $G$ the structure of
$\mathcal{P}^{2}(\mu)$ can not be determined from the integrability properties
of $\log w$ restricted to intervals. This is a consequence of fundamental
results of Khrushchev from [6]. His theory can be applied to certain measures
of the form (5) in which the weight $w=1_{E}$ is a characteristic function of
some set $E$. In fact, let $E$ a closed subset of $\mathbb{T}$ which has
positive measure and satisfies the $\alpha$-Beurling-Carleson condition

(10) $\sum_{\ell\in L}|\ell|^{\alpha}<\infty,$

where $L$ is the family of open intervals $\ell$ which form the complement of
$E$ in $\mathbb{T}$, and $|\ell|$ is the length of $\ell$. It is possible to
produce such a set which additionally contains no intervals, so that trivially
the integral of $\log w=\log 1_{E}$ diverges over any interval. But, in
contrast to maintheoremSuperExp, Khrushchev’s results can be applied to
conclude that for $G$ given by (9) and $d\mu=G(1-|z|)dA(z)+1_{E}d\textit{m}$,
the space $\mathcal{P}^{2}(\mu)$ is irreducible. His results apply to general
weights $G$ for which the limit in (ExpDec) is zero instead of positive, but
some regularity assumptions are always needed. In particular, Khrushchev’s
method requires the integrability of $\log 1/G$.

The paper [6] also presents implicitly a typical application in complex
function theory of a structure theorem for $\mathcal{P}^{2}(\mu)$. For
instance, assume that $G(z)=(1-|z|)^{\beta}$ for some large $\beta>0$,
$w=1_{E}$ is the characteristic function of some set $E$, and that we are in
the situation that

(11) $\mathcal{P}^{2}(\mu)\neq\mathcal{P}^{2}(\mu_{\mathbb{D}})\oplus
L^{2}(\mu_{\mathbb{T}}),$

i.e., splitting does not occur. Then there must exist a function

$f=f_{\mathbb{D}}+f_{\mathbb{T}}\in\mathcal{P}^{2}(\mu_{\mathbb{D}})\oplus
L^{2}(\mu_{\mathbb{T}})$

which is orthogonal to the monomials
$\\{z^{n}\\}_{n=0}^{\infty}=\\{z^{n}_{\mathbb{D}}+z^{n}_{\mathbb{T}}\\}_{n=0}^{\infty}$
in $\mathcal{P}^{2}(\mu_{\mathbb{D}})\oplus L^{2}(\mu_{\mathbb{T}})$. This
orthogonality means that the positive Fourier coefficients
$\widehat{(f_{\mathbb{T}})}_{n}$ of the function $f_{\mathbb{T}}$, which lives
only on $E$, are equal to
$-\int_{\mathbb{D}}\overline{z}^{n}f_{\mathbb{D}}(z)(1-|z|)^{\beta}dA(z)$.
Therefore, by the Cauchy-Schwarz inequality, these coefficients admit a bound

(12) $|\widehat{(f_{\mathbb{T}})}_{n}|\leq
C\sqrt{\int_{\mathbb{D}}|z|^{2n}(1-|z|)^{\beta}dA(z)}\simeq\frac{1}{(1+n)^{\gamma}},\quad
n\geq 0$

where $\gamma=(1+\beta)/2$. If $E$ would happen to be nowhere dense and
contain no intervals, then the fact that $f_{\mathbb{T}}$ lives only on $E$
implies that this function is very irregular, in the sense of being far from
smooth. So $f_{\mathbb{T}}$ satisfies two conflicting properties: it is
irregular, and yet obeys a strong spectral decay condition indicated by (12).
Of course, the point is that the spectral decay is only one-sided. Khrushchev
showed in [6] that closed sets $E$ which support such a function are precisely
the Beurling-Carleson sets, and he did it essentially by solving a splitting
problem for a class of $\mathcal{P}^{t}(\mu)$-spaces. Beurling-Carleson sets
are those closed sets which satisfy a weaker version of (10), namely

$\sum_{\ell\in L}|\ell|\log(1/|\ell|)<\infty.$

For more information regarding Beurling-Carleson sets, and their applications
in analysis, one can consult the recent article [5]. The already mentioned
Conjecture 1 of Kriete and MacCluer from [7] is related to a definitive
generalization of Khrushchev’s results discussed in this paragraph, to the
weighted context $w\neq 1_{E}$.

Sharpness at the other end, at the condition (LogLogInt), follows from the
work of Volberg. Essentially, if the condition (LogLogInt) is not satisfied by
$G$, then local considerations of integrability of $\log w$ play no role at
all, and the global divergence or convergence of the integral of $\log w$ on
the whole of $\mathbb{T}$ is the only interesting parameter. Namely,
$\mathcal{P}^{2}(\mu)$ of the form (5) with $G$ not satisfying (LogLogInt),
splits if and only if (1) is satisfied, similarly to how Szegö’s classical
theorem works. As usual, a regularity condition on $G$ is necessary. The claim
follows from Volberg’s theorem on quasianalytic functions, presented in
English for instance in [13], and also in [3]. Volberg’s theorem asserts that
a function $h\in L^{2}$ which satisfies a very strong unilateral spectral
decay estimate

$|\widehat{h}_{n}|\leq Ce^{-M(n)},\quad n=1,2,3\ldots$

for some positive sequence $\\{M(n)\\}_{n\geq 1}$ satisfying

$\sum_{n\geq 1}\frac{M(n)}{n^{2}}=\infty,$

should have a summable logarithm:
$\int_{\mathbb{T}}\log|h|\,d\textit{m}>-\infty$. To derive the splitting
statement from Volberg’s theorem, one mimics the orthogonality and spectral
decay argument in the previous paragraph where the application of Khrushchev’s
theorem is presented. See [7] and references therein for further details of
the proof.

###  1.6. Structure of the rest of the paper

In the preliminary Section 2 we set some conventions, and review a few
background results from operator theory which will be used in the proofs.
Section 3 forms the core of the paper and is concerned with the identification
of the largest $L^{t}$-summand appearing in $\mathcal{P}^{t}(\mu)$. This part
contains the main technical constructions of the paper, and it ends with a
proof of KrieteMaccluerConjectureProof. In Section 4 we mainly re-use, extend
and specialize some ideas and techniques already appearing in the literature,
mainly coming from the Kriete and MacCluer paper [7], in order to prove the
irreducibility of the first summand in (8). This completes the proof of
maintheoremSuperExp.

###  1.7. Acknowledgement

The author would like to sincerely thank Adem Limani for many insightful
discussions and ideas concerning the content of this work.

##  2\. Preliminaries and conventions

For clarity of exposition, we include in this preliminary section a brief
discussion of two rather peripheral issues, and set two conventions. We also
recall the structure of subspaces of $L^{t}$-spaces on the circle which are
invariant for multiplication by analytic polynomials.

###  2.1. $H^{\infty}$ as a subset of $\mathcal{P}^{t}(\mu)$

The space $H^{\infty}$ consists of functions which are analytic in the unit
disk $\mathbb{D}$ and which are uniformly bounded there. Any function $h\in
H^{\infty}$ is well known to admit an extension to the circle $\mathbb{T}$.
This extension is defined, only up to a subset of m-measure zero, by the
radial limits

(13) $h(z)=\lim_{r\to 1^{+}}h(rz),\quad z\in\mathbb{T}.$

Using this extension, we can consider $h=h_{\mathbb{D}}+h_{\mathbb{T}}$ as a
Borel measurable function on $\overline{\mathbb{D}}$ and as an element of the
spaces $L^{t}(\mu)=L^{t}(\mu_{\mathbb{D}})+L^{t}(\mu_{\mathbb{T}})$ whenever
the measure $\mu$ has the structure (5). To be more precise, we may choose a
Borel set of full measure on $\mathbb{T}$ on which the radial limit in (13)
exists, and we can set $h_{\mathbb{T}}$ to zero elsewhere.

It is easy to see that this extension of $h$ lies in $\mathcal{P}^{t}(\mu)$.
Indeed, the dilation $h_{r}(z):=h(rz)$, for $z\in\overline{\mathbb{D}}$ and
$r\in(0,1)$, is holomorphic in a neighbourhood of $\overline{\mathbb{D}}$, and
so can be approximated uniformly by analytic polynomials. Thus
$h_{r}\in\mathcal{P}^{t}(\mu)$ for any finite positive $t$. By (13) and the
dominated convergence theorem it is clear that, as $r$ tends to $1$, the
functions $h_{r}$ converge in the norm of $L^{t}(\mu)$ to the function
$h=h_{\mathbb{D}}+h_{\mathbb{T}}$, which hence lies in $\mathcal{P}^{t}(\mu)$.
By this argument, we see that the closure of $H^{\infty}$ in $L^{t}(\mu)$ is
the same as the closure of $\mathcal{P}$ in $L^{t}(\mu)$, both of these
closures being equal to $\mathcal{P}^{t}(\mu)$. In the proofs below we shall
be working with the more flexible class $H^{\infty}$ instead of $\mathcal{P}$.

If $\mathcal{P}^{t}(\mu)$ is not itself irreducible, then there will exist
many elements of the space which all have a common restriction to
$\mathbb{D}$. Therefore, whenever we work with a function $h\in H^{\infty}$
considered as an element of a space $\mathcal{P}^{t}(\mu)$, we always mean
that the part $h_{\mathbb{T}}$ of $h$ which lives on $\mathbb{T}$ is defined
by the radial boundary values of $h$, as above.

###  2.2. Carrier sets of measurable functions

If $f$ is a member of $L^{t}(\mu_{\mathbb{T}})=L^{t}(w\,d\textit{m})$ then of
course this function is only well-defined up to a set of $m$-measure zero, and
thus so is the set

(14) $\\{x\in\mathbb{T}:|f(x)|>0\\}.$

A set $E$ will be a a carrier set of $f$ if it is a Borel set on which
$|f(x)|>0$ up to a set of $m$-measure zero, and such that $f(x)=0$ on the
complement of $E$, again up to a set of measure zero. The exact choice of a
representative of the carrier set will never play a role, since all our
measures on $\mathbb{T}$ will be absolutely continuous with respect to m. Note
specifically that the carrier set in (14) differs from the usual support of
the measure $fd\textit{m}$.

###  2.3. Multiplication-invariant subspaces of Lebesgue spaces on the circle

Consider the usual Lebesgue space $L^{t}(\textit{m})$, say for $t>1$. The
structure of the lattice of invariant subspaces of the shift operator:

$f(z)\mapsto zf(z),\quad f\in L^{t}(\textit{m})$

is well-known. If $\mathcal{S}\subset L^{t}(\mathbb{T})$ is closed and
invariant for the shift (or equivalently, invariant under multiplication by
the analytic polynomials in $\mathcal{P}$), then $\mathcal{S}$ equals either a
space of the form

$L^{t}(m_{F}):=\\{f\in L^{t}(\textit{m}):f\equiv 0\text{ on
}\mathbb{T}\setminus F\\}$

for some measurable subset $F\subset\mathbb{T}$, or it equals

$qH^{t}:=\\{qh:h\in H^{t}\\}$

where $q$ is a unimodular function on $\mathbb{T}$ and $H^{t}$ is the usual
Hardy space of functions in $L^{t}(m)$ which have a non-negative Fourier
spectrum. A proof can be found in [4, Lectures II and IV]. We shall have a
need for a result which is an easy consequence of this.

######  Lemma 2.1.

beurlingwienerlemma Let $t>1$, $w\in L^{1}(m)$ be a non-negative function with
carrier set $E$, and $\mu=wd\textit{m}$. Let $g\in L^{t}(\mu)$ be a non-zero
function such that

(15) $\int_{\mathbb{T}}\log(|g|^{t}w)\,d\textit{m}=-\infty.$

Then the smallest closed subspace $\mathcal{S}\subset L^{t}(\mu)$ which
contains $g$ and is invariant under multiplication by analytic polynomials
equals

$\mathcal{S}=\\{f\in L^{t}(\mu):f\equiv 0\text{ on }E\setminus
F\\}=L^{t}(\mu_{F}),$

where

$F=\\{x\in E:g(x)\neq 0\\}$

is the carrier set of $g$.

###### Proof.

The map $U:L^{t}(\mu)\to L^{t}(\textit{m}_{E})$ which maps $f\in L^{t}(\mu)$
to the function $Uf=fw^{1/t}$ is a surjective isometry between the spaces. The
mapping sends $\mathcal{S}$ to a closed subspace $U\mathcal{S}$ of
$L^{t}(\textit{m}_{E})\subset L^{t}(\textit{m})$ which is invariant under
multiplication by analytic polynomials. By the structure theorem for such
invariant subspaces discussed above, $U\mathcal{S}$ equals either a space of
the form $L^{t}(m_{F})$, for some measurable subset $F\subset\mathbb{T}$ (in
fact, $F\subset E$), or it equals $qH^{t}$. If we are in the second case, then
since $g\in\mathcal{S}$, we have that $qh=gw^{1/t}$ for some non-zero $h\in
H^{t}$. It is well known that non-zero $h\in H^{t}$ implies that $|h|=|qh|$
has an integrable logarithm, so we immediately arrive at a contradiction to
(15). It follows that $U\mathcal{S}=L^{t}(\textit{m}_{F})$ for some measurable
set $F$. It is not hard to see that $F$ must be of the form presented above. ∎

We remark also a corollary to beurlingwienerlemma which will be useful to keep
in mind.

######  Corollary 2.2.

cor22 If $\mu$ is as in (5) and $\mathcal{P}^{t}(\mu)$ contains the
characteristic function of a set $F\subset\mathbb{T}$, then it also contains
the characteristic function of any measurable subset of $F$.

###### Proof.

If $\mathcal{P}^{t}(\mu)$ contains $1_{F}$, then by beurlingwienerlemma it
contains $L^{t}(\mu_{F})$, so in particular it contains the characteristic
functions of any subset of $F$. ∎

##  3\. Identifying the non-analytic summand

The goal of this section is to show that
$L^{t}(\mu_{F})\subset\mathcal{P}^{t}(\mu)$, where $F$ is the residual set in
(7). This will already imply KrieteMaccluerConjectureProof.

###  3.1. Reduction to a problem of real variable analysis

We start by showing that our problem can be solved if we can construct a
sequence of real-valued functions with certain properties. This reduction is
detailed in reductionLemma below. The proof is somewhat lengthy but it is
fairly straight-forward, and uses only basic functional analysis, measure
theory and the structure of shift invariant subspaces of $L^{t}(\textit{m})$
discussed earlier in Section 2.

For a measurable and real-valued function $f$ on $\mathbb{T}$ the Poisson
integral $P_{f}$, appearing in $(iv)$ below, is defined as usual by

$\displaystyle P_{f}(z)$
$\displaystyle:=\int_{\mathbb{T}}\frac{1-|z|^{2}}{|x-z|^{2}}f(x)\,d\textit{m}(x)$
$\displaystyle=\operatorname{Re}\int_{\mathbb{T}}\frac{x+z}{x-z}f(x)\,d\textit{m}(x),\quad
z\in\mathbb{D}.$

######  Lemma 3.1.

reductionLemma Let $G$ be a function which satisfies the condition (ExpDec),
$w\in L^{1}(m)$ be a non-negative measurable function, and $F$ be a subset of
$E=\\{x\in\mathbb{T}:w(x)>0\\}$ which is of positive Lebesgue measure. Assume
that for each sufficiently large positive number $N$ there exists a bounded
real-valued function $f_{N}$ defined on $\mathbb{T}$ and a corresponding set
$A_{N}\subset\mathbb{T}$ such that the following five conditions hold:

  1. (i)

$\int_{\mathbb{T}}f_{N}\,d\textit{m}=0,$

  2. (ii)

$f_{N}(x)\leq-N$ on $F\cap A_{N}$,

  3. (iii)

$|F\setminus A_{N}|$ tends to zero as $N$ tends to $+\infty$,

  4. (iv)

the Poisson integral $P_{f_{N}}$ satisfies the bound
$P_{f_{N}}(z)\leq\frac{C}{1-|z|}$ for some $C>0$ independent of $N$,

  5. (v)

$\int_{\mathbb{T}}\exp(f_{N})w\,d\textit{m}\leq C$, for some $C>0$ independent
of $N$.

In that case, we have that $L^{t}(\mu_{F})\subset\mathcal{P}^{t}(\mu)$, with
$\mu$ as in (5), for any finite $t>0$.

###### Proof.

Let $M=M(N)$ be a positive number lesser than $N$, which tends to $+\infty$ as
$N$ tends to $+\infty$, and such that the ratio $N/M$ tends to $+\infty$ as
$M$ and $N$ tend to $+\infty$. By replacing $f_{N}$ with $M^{-1}f_{N}$ we do
not affect the assumptions $(i)$ or $(iii)$ in the statement of the lemma. The
assumption $(ii)$ is only slightly changed: we have that $M^{-1}f_{N}\leq-N/M$
on $F\cap A_{N}$, which of course tends to zero uniformly on $F\cap A_{N}$, as
$N$ tends to $+\infty$. However, the assumption $(iv)$ is improved to

(16) $P_{f_{N}}(z)\leq\frac{C_{N}}{1-|z|}$

for a constant $C_{N}$ $(=C/M)$ which tends to zero as $N\to+\infty$, and
$(v)$ is improved to $\exp(f_{N})\in L^{t}(\mu_{\mathbb{T}})$ for any $t>0$,
as long as $N$ (and consequently $M$) is large enough, with the corresponding
$L^{t}(\mu_{\mathbb{T}})$-norm being bounded uniformly in $N$ for any fixed
$t$. So a simple re-labeling of the subscripts in the family $\\{f_{N}\\}_{N}$
gives us a new family for which $(i)$, $(ii)$ and $(iii)$ hold, and also the
mentioned improved versions of $(iv)$ and $(v)$ are satisfied.

With this modification of the functions $f_{N}$, we construct the outer
functions

(17)
$g_{N}(z)=\exp\Big{(}\int_{\mathbb{T}}\frac{x+z}{x-z}f_{N}(x)d\textit{m}(x)\Big{)},\quad
z\in\mathbb{D}$

each of which is a bounded analytic functions in the unit disk. These
functions are contained in $\mathcal{P}^{t}(\mu)$, in the sense explained in
Section 2. Moreover, by the second assumption in the lemma statement and by
the well-known properties of boundary behaviour of outer functions, we also
have

(18) $|g_{N}(x)|=\exp(f_{N}(x))\leq\exp(-N)$

for almost every $x\in F\cap A_{N}$.

Recalling the assumpions on $G$ in (ExpDec), let

$\inf_{x\in(0,1]}x\log(1/G(x))=d>0.$

Using this, and the inequality in (16), we obtain for $z\in\mathbb{D}$ a
uniform growth estimate:

(19) $\displaystyle|g_{N}(z)|$ $\displaystyle=\exp\Big{(}P_{f_{N}}(z)\Big{)}$
$\displaystyle\leq\exp\Big{(}\frac{C_{N}}{1-|z|}\Big{)}$
$\displaystyle\leq\exp\Big{(}\frac{C_{N}}{d}\log(1/G(1-|z|))\Big{)}$
$\displaystyle=\Big{(}G(1-|z|)\Big{)}^{-C_{N}/d}.$

Since $C_{N}$ is eventually small enough to ensure the inequality

$\frac{tC_{N}}{d}\leq 1,$

we have from (19) the norm bound

(20) $\sup_{N}\int_{\mathbb{D}}|g_{N}(z)|^{t}G(1-|z|)dA(z)<\infty.$

Let us suppose for the moment that $t>1$. The remarks made at the beginning of
the proof imply that, without loss of generality, we can assume that the
functions $\exp(f_{N})$ are uniformly bounded in the norm of the reflexive
space $L^{t}(\mu_{\mathbb{T}})$. In particular, by fixing a definitive
sequence of numbers $N$ tending to $+\infty$ and passing to a subsequence, we
can assume that the functions $\exp(f_{N})$ converge weakly in the space
$L^{t}(\mu_{\mathbb{T}})$ to some function $g\in L^{t}(\mu_{\mathbb{T}})$.
Since this works for arbitrary $t$, we can even pass to further subsequence
and obtain that $g\in L^{t_{*}}(\mu_{\mathbb{T}})$ for some $t_{*}>t$. It is
not hard to see that $g$ vanishes almost everywhere on $F$. Indeed, if $s$ is
the Hölder conjugate exponent to $t$ and $r$ is any function in
$L^{s}(\mu_{\mathbb{T}})$ living only on the set $F$, then

$\displaystyle\int_{\mathbb{T}}gr\,d\mu_{\mathbb{T}}$
$\displaystyle=\lim_{N\to+\infty}\int_{\mathbb{T}}g_{N}r\,d\mu_{\mathbb{T}}$
(21) $\displaystyle=\lim_{N\to+\infty}\int_{F\cap
A_{N}}g_{N}r\,d\mu_{\mathbb{T}}+\int_{F\setminus
A_{N}}g_{N}r\,d\mu_{\mathbb{T}}.$

For the first term in the limit above, we can use the Cauchy-Schwarz
inequality and the boundary value equality in (18) to obtain the bound

$\Big{|}\int_{F\cap
A_{N}}g_{N}r\,d\mu_{\mathbb{T}}\Big{|}\leq\exp(-N)\cdot\|r\|_{L^{s}(\mu_{\mathbb{T}})},$

which tends to zero as $N$ grows to infinity, while for the second term
similarly we obtain

$\Big{|}\int_{F\setminus
A_{N}}g_{N}r\,d\mu_{\mathbb{T}}\Big{|}\leq\|\exp(f_{N})\|_{L^{t}(\mu_{\mathbb{T}})}\cdot\Big{(}\int_{F\setminus
A_{N}}|r|^{s}\,d\mu_{\mathbb{T}}\Big{)}^{1/s},$

which also tends to zero, by the absolute continuity of the finite measure
$|r|^{s}d\mu_{\mathbb{T}}$, assumption $(iii)$ in the statement of the lemma,
and the improved version of $(v)$ discussed above. Thus (3.1) vanishes for any
choice of $r$, and consequently $g\equiv 0$ on $F$.

The outer functions $g_{N}$ satisfy $g_{N}(0)=1$ by the assumption $(i)$, and
the estimate in (19) implies that

$\limsup_{N\to+\infty}|g_{N}(z)|\leq 1.$

The estimate in (19) also implies that the family $\\{g_{N}\\}_{N}$ is bounded
uniformly on compact subsets of $\mathbb{D}$. It follows now from Montel’s
theorem and the maximum modulus theorem for analytic functions that we have

$\lim_{N\to+\infty}g_{N}(z)=1,$

uniformly on compact subsets of the unit disk $\mathbb{D}$. By the norm bound
in (20) and again passing to a subsequence of the $N$, we can also assume that
the analytic functions $g_{N}$ converge weakly in the space
$\mathcal{P}^{t}(\mu_{\mathbb{D}})$ to the constant 1.

All in all, we see that we can ensure that some sequence of the functions
$g_{N}\in\mathcal{P}^{t}(\mu)$ converges weakly as $N\to+\infty$ to a function
$g\in\mathcal{P}^{t}(\mu)$ which is identically equal to $1$ in $\mathbb{D}$
and vanishes almost everywhere on the set $F$ on $\mathbb{T}$. Then the
function $\tilde{g}:=g-1\in\mathcal{P}^{t}(\mu)$ vanishes on $\mathbb{D}$ and
it is equal to the constant $1$ almost everywhere on the set $F$.

Recall that we have ensured that $\tilde{g}\in L^{t_{*}}(\mu_{\mathbb{T}})$
for some $t_{*}>t$. Using this, we re-write

$\displaystyle\log(|\tilde{g}|^{t}w)$
$\displaystyle=(t/t_{*})\log(|\tilde{g}|^{t_{*}}w^{t^{*}/t})$
$\displaystyle=(t/t_{*})\log(|\tilde{g}|^{t_{*}}w)+(t/t_{*})\log(w^{t_{*}/t-1})$
$\displaystyle=(t/t_{*})\log(|\tilde{g}|^{t_{*}}w)+(t/t_{*})(t_{*}/t-1)\log(w).$

In the last line we see a linear combination, with positive coefficients, of
the term $\log w$, which satisfies

$\int_{\mathbb{T}}\log w\,d\textit{m}=-\infty,$

and the term $\log(|\tilde{g}|^{t_{*}}w)$ which is the logarithm of an
$L^{1}(m)$ function. This term might or might not be absolutely integrable,
but certainly satisfies

$\int_{\mathbb{T}}\log(|\tilde{g}|^{t_{*}}w)\,d\textit{m}<+\infty$

as a consequence of the integrability of $|\tilde{g}|^{t_{*}}w$. It follows
that

(22) $\int_{\mathbb{T}}\log(|\tilde{g}|^{t}w)\,d\textit{m}=-\infty.$

Since $\tilde{g}$ vanishes on $\mathbb{D}$, the smallest subpace $\mathcal{S}$
of $\mathcal{P}^{t}(\mu)$ which is invariant for multiplication by analytic
polynomials and which contains $\tilde{g}$ is actually a subspace of
$L^{t}(\mu_{\mathbb{T}})$. Then beurlingwienerlemma applies to identify
$\mathcal{S}$ as $L^{t}(\mu_{\tilde{F}})$, where $\tilde{F}$ is a carrier for
$\tilde{g}$. Since $\tilde{g}\equiv 1$ on $F$, the proof is complete in the
case $t>1$.

An elementary argument extends the conclusion from finite $t>1$ to $t>0$. In
the case $t\in(0,1]$, let $h\in L^{\infty}(m)$ live only on the set $F$. Then
$h\in L^{2}(\mu_{\mathbb{T}})$, so by what has already been proved there
exists a sequence of analytic polynomials $\\{p_{n}\\}_{n}$ which converges to
$h$ in the norm of $\mathcal{P}^{2}(\mu)$ and also pointwise $\mu$-almost
everywhere. Letting

$A_{n}:=\\{x\in\overline{\mathbb{D}}:|p_{n}(x)-h(x)|>1\\}$

we note that the sequence $|p_{n}-h|^{t}1_{\overline{\mathbb{D}}\setminus
A_{n}}$ is bounded pointwise by 1 and converges $\mu$-almost everwhere to zero
on $\overline{\mathbb{D}}$, so we have that

$\displaystyle\lim_{n\to\infty}\int_{\overline{\mathbb{D}}}|p_{n}-h|^{t}d\mu$
$\displaystyle=\lim_{n\to\infty}\int_{A_{n}}|p_{n}-h|^{t}d\mu+\int_{\overline{\mathbb{D}}\setminus
A_{n}}|p_{n}-h|^{t}d\mu$
$\displaystyle\leq\limsup_{n\to\infty}\int_{A_{n}}|p_{n}-h|^{2}d\mu+\int_{\overline{\mathbb{D}}}|p_{n}-h|^{t}1_{\overline{\mathbb{D}}\setminus
A_{n}}d\mu$ $\displaystyle=0,$

where in the last step we used the dominated convergence theorem on the second
integral. This shows that $h\in\mathcal{P}^{t}(\mu)$. The set of $h\in
L^{\infty}(m)$ which live only on $F$ is clearly a dense subset of
$L^{t}(\mu_{F})$, and so by taking closures we obtain
$L^{t}(\mu_{F})\subset\mathcal{P}^{t}(\mu)$ also for $t\in(0,1]$. ∎

###  3.2. A Poisson integral estimate

Our primary tool for ensuring the critical property $(iv)$ in reductionLemma
will be the following estimate for Poisson integrals.

######  Lemma 3.2.

SuperExpPoissonEstimate Let $\\{I_{j}\\}_{j}$ be a finite family of disjoint
intervals on $\mathbb{T}$, and let $f=\sum_{j}f_{j}$ be a real-valued function
such that

  1. (i)

$f_{j}$ is supported on the interval $I_{j}$,

  2. (ii)

$\int_{I_{j}}f_{j}\,d\textit{m}=0$,

  3. (iii)

$\int_{I_{j}}|f_{j}|\,d\textit{m}\leq C,$ where $C$ is some positive constant
which is independent of $j$.

Then we have the Poisson integral growth estimate

$P_{f}(z)\leq\frac{4C}{1-|z|},$

where $C$ is the constant in $(iii)$ above.

For the sake of the proof of the lemma we introduce a notation. If a real-
valued function $g$ and an interval $I$ are given, then we define the
variation of $g$ over $I$ as the non-negative number

$\text{var}(g,I):=\sup_{x\in I}g(x)-\inf_{x\in I}g(x).$

###### Proof.

By symmetry, it will be sufficient to show that

$P_{f}(r)=\int_{\mathbb{T}}P_{r}\,d\textit{m}\leq\frac{4C}{1-r},$

where $P_{r}=P_{r}(x)=\frac{1-r^{2}}{|x-r|^{2}}$ is the Poisson kernel. Let

$f_{j}^{+}(x)=\max[f_{j}(x),0]$

and

$f_{j}^{-}(x)=-\min[f_{j}(x),0].$

It of course holds that

$\int_{I_{j}}P_{r}f_{j}^{+}\,d\textit{m}\leq\sup_{x\in
I_{j}}P_{r}(x)\cdot\int_{I_{j}}f_{j}^{+}\,d\textit{m}$

and

$\inf_{x\in
I_{j}}P_{r}(x)\cdot\int_{I_{j}}f_{j}^{-}\,d\textit{m}\leq\int_{I_{j}}P_{r}f_{j}^{-}\,d\textit{m},$

from which we deduce, using property $(ii)$ in the statement of the lemma

$\displaystyle\int_{I_{j}}P_{r}f_{j}\,d\textit{m}$
$\displaystyle\leq\sup_{x\in
I_{j}}P_{r}(x)\cdot\int_{I_{j}}f_{j}^{+}\,d\textit{m}-\inf_{x\in
I_{j}}P_{r}(x)\cdot\int_{I_{j}}f_{j}^{-}\,d\textit{m}$
$\displaystyle=\text{var}(P_{r},I_{j})\int_{I_{j}}f_{j}^{+}\,d\textit{m}.$

It follows now from $(i)$ and $(iii)$ that

$\displaystyle\int_{\mathbb{T}}P_{r}f\,d\textit{m}$
$\displaystyle\leq\sum_{j}\text{var}(P_{r},I_{j})\int_{I_{j}}f_{j}^{+}$
$\displaystyle\leq C\sum_{j}\text{var}(P_{r},I_{j}).$

But since the intervals in the family $\\{I_{j}\\}_{j}$ are disjoint, and the
Poisson kernel is unimodal, we can in fact deduce the estimate

(23) $\sum_{j}\text{var}(P_{r},I_{j})\leq\frac{2(1+r)}{1-r}\leq\frac{4}{1-r},$

see Figure 1 for a visual proof of this inequality. This establishes the
required growth estimate. ∎

VarPsnPlot

Figure 1. The height of each gray area equals $\text{var}(P_{r},I_{j})$

###  3.3. Main construction and proof of the Kriete-MacCluer conjecture

Our task on identification of $L^{t}$-summands has now been reduced to a
construction of real-valued functions $f_{N}$ on $\mathbb{T}$ which satisfy
the five assumptions appearing in reductionLemma.

We need one more tool before going into our main construction. We recall that
a collection of open intervals $\mathcal{C}=\\{I\\}$ is a Vitali covering of a
set $F\subset\mathbb{T}$ if for every $x\in F$ there exists an interval
$I\in\mathcal{C}$ which contains $x$ and is of arbitrarily short length. The
following formulation of Vitali’s well-known theorem will be used in our
construction. A proof can be found, for instance, in [11, pages 128-129].

######  Lemma 3.3.

(Vitali’s covering theorem) VitaliCoveringTheorem Let $F$ be a subset of
$\mathbb{T}$ which has positive Lebesgue measure, and let
$\mathcal{C}=\\{I\\}$ be a Vitali covering of $F$. Given any $\delta>0$, there
exists a finite collection $\\{I_{j}\\}_{i=j}^{n}\subset\mathcal{C}$
consisting of pairwise disjoint intervals for which we have

$|F\setminus\cup_{i=j}^{n}I_{j}|<\delta.$

The main result of the section is the following.

######  Proposition 3.4.

FwSplittingOffProp Let $G$ be a function which satisfies the condition
(ExpDec), $w\in L^{1}(m)$ be a non-negative measurable function, and $E$ and
$F$ be respectively the carrier and the residual set of $w$, as defined in (6)
and (7). Assume that $F$ has positive $m$-measure. If
$d\mu=G(1-|z|)dA(z)+w\,d\textit{m}$, then for any finite $t>0$ we have that
$L^{t}(\mu_{F})\subset P^{t}(\mu)$.

###### Proof.

To start, note that the set $F$ is equal, up to a set of Lebesgue measure
zero, to the countable union

$\cup_{n=1}^{\infty}F_{n}=\cup_{n=1}^{\infty}F\cap\\{x\in E:w(x)>1/n\\}.$

It is thus sufficient to show that for each $n$ we have the containment
$L^{t}(\mu_{F_{n}})\subset P^{t}(\mu)$. Consequently, we lose no generality
assuming that the set $F$ is contained in

(24) $\\{x\in E:w(x)>c\\}$

for some arbitrary number $c>0$.

Let $N>0$ be some large number. By definition of $F$, for each point $x\in F$
there exist intervals $I$ of arbitrary small length which contain $x$ and for
which we have $\int_{I}\log w\,d\textit{m}=-\infty.$ Moreover, it is easy to
see that the family $\mathcal{C}=\mathcal{C}_{N}$ of all such intervals which
additionally satisfy the condtition

(25) $N|I|\leq 1$

constitutes a Vitali covering of the set $F$. Extract from $\mathcal{C}$ a
finite collection $\\{I_{j}\\}_{j}$ of pairwise disjoint intervals such that

(26) $|F\setminus\cup_{j}I_{j}|<1/N.$

We proceed to show how to define

(27) $f_{N}=\sum_{j}f_{N,j}$

where $f_{N,j}$ will live only on the interval $I_{j}$, so that all the five
assumptions of reductionLemma will be satisfied. Fix one of the sets $I_{j}$,
and recall the definition of $c$ in (24). We decompose the logarithmic
integral of $w$ over $I_{j}$ into three pieces:

$\displaystyle+\infty$ $\displaystyle=\int_{I_{j}}\log(1/w)\,d\textit{m}$
$\displaystyle=\int_{I_{J}\setminus
E}\log(1/w)\,d\textit{m}+\int_{\\{w>c\\}\cap I_{j}}\log(1/w)\,d\textit{m}$
$\displaystyle+\lim_{d\to 0^{+}}\int_{\\{d\leq w\leq c\\}\cap
I_{j}}\log(1/w)\,d\textit{m}.$

The second piece satisfies

$-\infty<\int_{\\{w>c\\}\cap I_{j}}\log(1/w)\,d\textit{m}<+\infty.$

Indeed, the upper bound follows from the inequality $\log(1/w)<\log(1/c)$
which holds on the set $\\{w>c\\}$, and the lower bound is an easy consequence
of the integrability of $w$. We see from this that at least one of the two
remaining pieces must be equal to $+\infty$. Consequently, we deduce that at
least one of the following situations must occur: either we have that

(C1) $I_{j}\setminus E\text{ has positive Lebesgue measure}$

or we have that the family of sets

$E_{d}:=\\{x\in E:d\leq w(x)\leq c\\}$

satisfies

(C2) $\lim_{d\to 0^{+}}\int_{I_{j}\cap E_{d}}\log(1/w)\,d\textit{m}=\infty.$

Of course, it might be so that both situations occur.

The first alternative (C1) is simpler. In this case, we may set

(28) $f_{N,j}=N\frac{|I_{j}\cap F|}{|1_{I_{j}\setminus E}|}1_{I_{j}\setminus
E}-N1_{I_{j}\cap F}.$

If we are in the situation (C2), then there exists a positive number $d$ and
some measurable subset $I_{j}^{+}$ of $I_{j}\cap E_{d}$ for which we have

(29) $N|I_{j}|\leq\int_{I_{j}^{+}}\log(1/w)\,d\textit{m}\leq 2N|I_{j}|\leq 2.$

Indeed, by (C2) there exists some $d$ small enough so that
$|N|I_{j}\leq\int_{I_{j}\cap E_{d}}\log(1/w)\,d\textit{m}$, and then by
absolute continuity of the finite measure $\log(1/w)1_{I_{j}\cap
E_{d}}d\textit{m}$ we might pick a subset $I_{j}^{+}$ so that (29) holds. Let

(30) $\alpha_{j}:=\int_{I_{j}^{+}}\log(1/w)\,d\textit{m}\simeq N|I_{j}|$

and

$I_{j}^{-}:=I_{j}\setminus I_{j}^{+}$

If (C2) holds, then we may set

(31)
$f_{N,j}=\log(1/w)1_{I^{+}_{j}}-\frac{\alpha_{j}}{|I_{j}^{-}|}1_{I_{j}^{-}}.$

If both (C1) and (C2) hold, then we are free to construct $f_{N,j}$ in either
way.

We now verify that this way of constructing $f_{N,j}$ implies that the
functions $f_{N}$ in (27) satisfy all five conditions of reductionLemma. Both
alternative definitions of $f_{N,j}$ ensure that

$\int_{I_{j}}f_{N,j}\,d\textit{m}=0,$

which means of course that

$\int_{\mathbb{T}}f_{N}\,d\textit{m}=\sum_{j}\int_{I_{j}}f_{N,j}\,d\textit{m}=0.$

Hence $(i)$ of reductionLemma holds. By setting $A_{N}:=\cup_{j}I_{j}$, we see
from (26) that assumpion $(iii)$ of reductionLemma also holds. Next, we verify
$(ii)$. If $f_{N,j}$ was constructed according to (31), and $x\in I_{j}\cap
F$, then $w(x)>c$, and so $x\not\in I_{j}^{+}$. Thus from (30) we deduce that
in the second case

$-f_{N,j}(x)=\frac{\alpha_{j}}{|I_{j}^{-}|}\geq\frac{\alpha_{j}}{|I_{j}|}\geq
N,$

while clearly $-f_{N,j}(x)=N$ if $x\in I_{j}\cap F$ and we constructed
$f_{N,j}$ according to (28). We have hence verified that $(ii)$ of
reductionLemma is satisfied. The assumption $(iv)$ is satisfied as a
consequence of SuperExpPoissonEstimate and (25), since from
$\int_{I_{j}}f_{N,j}\,d\textit{m}=0$ we deduce

$\int_{I_{j}}|f_{N,j}|\,d\textit{m}=2\int_{I_{j}}\max[f_{N,j},0]\,d\textit{m}$

which, in the first case, can be estimated by

$2\int_{I_{j}}\max[f_{N,j},0]\,d\textit{m}=2N|I_{j}\cap F|\leq 2N|I_{j}|\leq
2,$

and similarly in the second case by

$2\int_{I_{j}}\max[f_{N,j},0]\leq 2\alpha_{j}\leq 4N|I_{j}|\leq 4.$

Finally, the fifth assumption in reductionLemma is satisfied since at any
$x\in\mathbb{T}$ the function $\exp(f_{N})$ will attain a value which is
either less than or equal to $1$, or equal to $\exp(f_{N,j}(x))$ for some
unique index $j$. If $f_{N,j}$ was constructed according to the first case,
and so is given by (28), then

$\exp(f_{N,j}(x))w(x)=0$

at any point $x$ where $f_{N,j}(x)>0$, and if it was constructed according to
the second case in (31), then

$\exp(f_{N,j}(x))w(x)=1$

whenever $f_{N,j}(x)>0$. So in any case,

$\int_{\mathbb{T}}\exp(f_{N})w\,d\textit{m}\leq\int_{\mathbb{T}}(1+w)\,d\textit{m}=1+\|w\|_{L^{1}(m)}.$

Since $N>0$ is arbitrary, we have now verified the five requirements to apply
reductionLemma, and so the proof is complete. ∎

As an immediate corollary, we see that we have proved the conjecture of Kriete
and MacCluer from [7, Section 9], in an even stronger form than asserted.

###### Proof of KrieteMaccluerConjectureProof.

The assumption that $\log w$ is not integrable on any interval clearly implies
that the set $F$ in FwSplittingOffProp coincides with the carrier set $E$ of
the weight $w$, defined in (6). Thus the result follows directly from
FwSplittingOffProp. ∎

##  4\. Identifying the analytic summand

We specialize now to the Hilbertian setting $t=2$ and show that if $I$ is any
interval on which the weight $w$ is $\log$-integrable, then this interval has
trivial intersection with any $L^{2}$-summand contained in
$\mathcal{P}^{2}(\mu)$ of the form (5). The result is more or less anticipated
from the work of Kriete and MacCluer, [7, Theorem D]. However, we shall need a
slightly more specialized statement than what appears there. Consequently,
some of the techniques in the proofs below are only a minor adaptation of the
ones used in their work.

###  4.1. A criterion for local irreducibility

The next lemma is the key to showing irreducibility of a space.

######  Lemma 4.1.

irreducibilityCriterionLemma Assume that there exists a function

$f=f_{\mathbb{D}}+f_{\mathbb{T}}\in\mathcal{P}^{2}(\mu_{\mathbb{D}})\oplus
L^{2}(\mu_{\mathbb{T}})$

which is orthogonal to $\mathcal{P}^{2}(\mu)$. Let $C$ be a carrier set of the
part of $f$ living on $\mathbb{T}$:

$C:=\\{x\in\mathbb{T}:|f_{\mathbb{T}}(x)|>0\\}.$

Then $\mathcal{P}^{2}(\mu)$ does not contain the characteristic function of
any measurable subset of $C$ which is of positive Lebesgue measure.

###### Proof.

Assume, seeking a contradiction, that the characteristic function $1_{B}$ of
the set $B\subset C$ is contained in $\mathcal{P}^{2}(\mu)$, and $|B|>0$. We
will show that $f\equiv 0$ on $B$, which is a contradiction to $C$ being a
carrier of $f_{\mathbb{T}}$.

We conclude from beurlingwienerlemma (or from cor22) that
$1_{B}\in\mathcal{P}^{2}(\mu)$ implies that $L^{2}(\mu_{B})$ is contained in
$\mathcal{P}^{2}(\mu)$. Since the function $f$ is orthogonal to
$\mathcal{P}^{2}(\mu)$, we have that

$\int_{B}f_{\mathbb{T}}\overline{g}w\,d\textit{m}=0$

for any function $g\in L^{2}(\mu_{B})$. This means that

$fw\equiv 0\text{ on }B.$

Since the carrier set $C$ of $f_{\mathbb{T}}$ must be (up to a subset of
Lebesgue measure zero) contained in the carrier set $E$ of $w$, this means
that $f=0$ almost everywhere on $B$. This is the desired contradiction. ∎

###  4.2. Beurling-Malliavin majorants

We now want to construct a large family of functions $f$ to which
irreducibilityCriterionLemma applies. To this end we will find useful the
following simple special case of the famous, and much more difficult to prove,
general form of the Beurling-Malliavin multiplier theorem.

######  Lemma 4.2.

babyBMtheorem Let $P$ be a positive function defined on $\mathbb{R}$ which is
even and decreasing for $x\geq 0$. If

$\int_{\mathbb{R}}\frac{\log P(x)}{1+x^{2}}\,dx>-\infty,$

then for any $a>0$ there exists a $C^{\infty}$ function $g$ with support
inside the interval $(-a,a)$ and with $g(0)\neq 0$, for which the Fourier
transform

$\hat{g}(x)=\int_{\mathbb{R}}e^{-ixt}g(t)dt$

satisfies the inequality

$|\hat{g}(x)|\leq P(x),\quad x\in\mathbb{R}.$

A proof of babyBMtheorem can be found in [3, pages 276-277]. An exposition and
proof of the general Beurling-Malliavin multiplier theorem is also contained
in [3].

###  4.3. An estimate for the moments of $G$

For every analytic polynomial $p(z)=\sum_{n}p_{n}z^{n}$ we have that

$\int_{\mathbb{D}}|p|^{2}d\mu_{\mathbb{D}}=\int_{\mathbb{D}}|p(z)|^{2}G(1-|z|)\,dA(z)=\sum_{n}|p_{n}|^{2}\alpha_{n}$

where

(32) $\alpha_{n}=2\int_{0}^{1}r^{2n+1}G(1-r)dr$

are the moments of the function $G(1-r)$. The rate of decay of $G(1-|z|)$ near
the boundary of the disk and the rate of decay of the moments $\alpha_{n}$ is
connected through the following lemma.

######  Lemma 4.3.

GmomentsLogSummability If $G$ is continuously differentible on $(0,1)$ and
satisfies (LogLogInt), then the function

$P(x):=\int_{0}^{1}r^{|x|}G(1-r)dr,\quad x\in\mathbb{R}$

satisfies

(33) $\int_{\mathbb{R}}\frac{\log P(x)}{1+x^{2}}\,dx>-\infty.$

Note that $P$ in the above statement is an even and decreasing function of
$x\geq 0$. We also have

$P(2n+1)=\alpha_{n}/2.$

The lemma follows from a combination of two arguments in [2] and [3].

###### Proof.

Following Beurling in [2], we set

$m(x):=\log(1/G(x)),\quad x\in(0,1]$

and

(34) $k(x):=\inf_{y\in(0,1]}m(y)+yx,\quad x>0.$

Then $m$ is a decreasing, non-negative and differentiable on $(0,1)$, and $k$
is positive. Beurling proves (see [2, Lemma 1]) that our assumption
(LogLogInt) on $G$ implies that

(35) $\int_{0}^{\infty}\frac{k(x)}{1+x^{2}}dx<\infty.$

Now, we follow [3, Page 229]. In the reference, the context is somewhat
different and some additional assumptions on $m$ are present, but they are not
needed for our purposes. For all sufficiently large positive $x$, we will now
estimate $P(x)$ from below in terms of $k(x)$. We have

(36) $\displaystyle P(x)$ $\displaystyle=\int_{0}^{1}r^{x}G(1-r)\,dr$
$\displaystyle=\int_{0}^{1}\exp\Big{(}x\log r-m(1-r)\Big{)}dr$
$\displaystyle\geq\int_{0}^{1}\exp\Big{(}2x(r-1)-m(1-r)\Big{)}dr$
$\displaystyle=\int_{0}^{1}\exp\Big{(}-2xt-m(t)\Big{)}dt.$

Since $\lim_{y\to 0^{+}}m(y)=+\infty$ the infimum on the right-hand side of
(34) is attained at least at one point $y\in(0,1]$. By comparing the values of
the expression in (34) for $y=1/2$ and $y=1$, we see that for sufficiently
large $x$, the the infimum will be actually be attained inside of the interval
$(0,1)$. At any such point $y$ we must have, by basic calculus, the equality
$m^{\prime}(y)=-x$. But $m^{\prime}(y)=-G^{\prime}(y)/G(y)$, and this
expression reveals that $m^{\prime}(y)$ is bounded uniformly on each interval
of form $[\delta,1],\delta>0$. It follows that

$\lim_{x\to+\infty}\sup_{m^{\prime}(y)=-x}y=0,$

so for sufficiently large $x$, there is always a point $\zeta=\zeta_{x}$ which
satisfies $\zeta_{x}<1/2$ and $k(x)=m(\zeta_{x})+\zeta_{x}x$. Returning to
(36), we use that $m$ is decreasing to estimate

$\displaystyle P(x)$
$\displaystyle\geq\int_{\zeta_{2x}}^{1}\exp\Big{(}-2xt-m(t)\Big{)}dt$
$\displaystyle\geq\exp\big{(}-m(\zeta_{2x})\big{)}\int_{\zeta_{2x}}^{1}\exp(-2xt)dt$
$\displaystyle=\exp\Big{(}-m(\zeta_{2x})\Big{)}\frac{\exp(-2x\zeta_{2x})-\exp(-2x)}{2x}$
$\displaystyle=\frac{\exp\big{(}-k(2x)\big{)}}{2x}\Big{(}1-\exp(-2x+2x\zeta_{2x})\Big{)}$
$\displaystyle\geq\frac{\exp\big{(}-k(2x)\big{)}}{4x}.$

In the last step we used that $\zeta_{2x}<1/2$ when $x$ is sufficiently large,
and consequently $1-\exp(-2x+2x\zeta_{2x})$ is larger than $1/2$ when $x$ is
sufficiently large. Now (33) follows easily from (35) and our last estimate. ∎

###  4.4. Finalizing the irreducibility proof

Recall the definition of the residual set $F$ in (7), which is a subset of the
carrier $E$ of $w$ defined in (6). The set $E\setminus F$ has the property
that for almost every $x$ in that set, there exists an interval $I$ containing
$x$ on which $w$ is $\log$-integrable. We will now show that the space
$\mathcal{P}^{2}(\mu_{\mathbb{D}}+\mu_{\mathbb{T}\setminus F})$ contains no
characteristic functions, i.e, that it is irreducible. Together with the
result of FwSplittingOffProp, this will essentially complete the proof of
maintheoremSuperExp.

In the proof of the next lemma we use the same core idea as Kriete and
MacCluer in their proof of [7, Theorem D].

######  Lemma 4.4.

irreducibilityLemmaLogInt Assume that $G$ is as in GmomentsLogSummability, and
$w$ has the property that its carrier set is, up to a set of Lebesgue measure
zero, a union of intervals $I$ on which we have

$\int_{I}\log w\,d\textit{m}>-\infty.$

If $\mu$ is given by (5), then $\mathcal{P}^{2}(\mu)$ is irreducible.

###### Proof.

Let $I$ be any interval on which $\log w$ is integrable, and let $x$ be an
arbitrary point in the interior of $I$. We will construct a tuple
$f_{\mathbb{D}}+f_{\mathbb{T}}\in\mathcal{P}^{2}(\mu_{\mathbb{D}})\oplus
L^{2}(\mu_{\mathbb{T}})$ which is orthogonal to $\mathcal{P}^{2}(\mu)$ and
such that the carrier set $C$ of $f_{\mathbb{T}}$ contains an interval
$J\subset I$ centered at $x$. An application of irreducibilityCriterionLemma
then shows that $\mathcal{P}^{2}(\mu)$ contains no characteristic functions of
subsets of $J$. Since $x$ is arbitrary, we deduce from cor22 by patching
together the interval $I$ with the subintervals $J$, that
$\mathcal{P}^{2}(\mu)$ contains no characteristic functions of subsets of $I$.
Finally, since up tu a null set, the entire carrier of $w$ can be covered by
intervals $I$ on which $\log w$ is integrable, it will follow that
$\mathcal{P}^{2}(\mu)$ is irreducible.

We proceed to construct such a tuple. Without loss of generality we can assume
that $x=1$ and that $I$ is some interval centered at $1$. Let $P$ be as in
GmomentsLogSummability and set

$\tilde{P}(x)=2\frac{P(2|x|+1)}{1+|x|}.$

Then it is not hard to see that $\tilde{P}$ satisfies the hypothesis of
babyBMtheorem, and so there exists a smooth function $g$ supported on an
interval $(-a,a)$ of $\mathbb{R}$ which satisfies $g(0)\neq 0$ and has a
Fourier transform obeying the estimate

$|\hat{g}(x)|\leq\tilde{P}(x).$

If $h$ is defined on the circle by the equation

$h(e^{it})=g(t),\quad t\in(-\pi,\pi),$

then $h$ does not vanish at $1\in\mathbb{T}$, is supported in an interval $J$
around $1$ which we might suppose is contained in $I$ (since support of $g$
can be chosen arbitrarily short), and it has a Fourier series
$h=\sum_{n}h_{n}e^{itn}$ which satisfies

(37) $|h_{n}|=|\hat{g}(n)|\leq\tilde{P}(n)=\frac{\alpha_{n}}{1+n},$

where $\alpha_{n}$ is the $n$:th moment of $G$ defined in (32). Let
$F(z)=\sum_{n=0}^{\infty}F_{n}z^{n}$, where

$F_{n}=-\frac{h_{n}}{\alpha_{n}}.$

Then the coefficients $F_{n}$ satisfy $|F_{n}|\leq\frac{1}{1+n}$, so that $F$
is an analytic function in $\mathbb{D}$. The computation

$\displaystyle\int_{\mathbb{D}}|F(z)|^{2}G(1-|z|)dA(z)$
$\displaystyle=\sum_{n\geq 0}|F_{n}|^{2}\alpha_{n}$
$\displaystyle\leq\sum_{n\geq 0}\frac{|\alpha_{n}|}{(1+n)^{2}}<\infty$

shows that $f$ is a member of $\mathcal{P}^{2}(\mu_{\mathbb{D}})$, since the
moments $\alpha_{n}$ are certainly bounded. The Taylor coefficients of $F$
have been chosen in such a way that

$\int_{\mathbb{D}}F(z)\overline{z}^{n}G(1-|z|)dA(z)=F_{n}\alpha_{n}=-h_{n}=-\int_{\mathbb{T}}h\overline{z}^{n}d\textit{m}$

for non-negative integers $n$. This means precisely that the monomials
$\\{z^{n}\\}_{n\geq 0}=\\{z^{n}_{\mathbb{D}}+z^{n}_{\mathbb{T}}\\}_{n\geq 0}$
are orthogonal to $F+h$, considered as an element of
$\mathcal{P}^{2}(\mu_{\mathbb{D}})\oplus L^{2}(\textit{m})$. Consequently the
closed linear span of these monomials, which of course is
$\mathcal{P}^{2}(\mu_{\mathbb{D}}+\textit{m})$, is orthogonal to $F+h$. The
assumption that $\int_{I}\log w\,d\textit{m}>-\infty$ implies that we can
construct a function $u\in
H^{\infty}\subset\mathcal{P}^{2}(\mu_{\mathbb{D}}+\textit{m})$ which satisfies
$|u|=\min(w,1)$ on the interval $I$. Then $z^{n}u$ is orthogonal to $F+h$ in
$\mathcal{P}^{2}(\mu_{\mathbb{D}})\oplus L^{2}(\textit{m})$. By boundedness of
$u$ we have that $F\overline{u}$ is a member of $L^{2}(\mu_{\mathbb{D}})$, and
we can orthogonally project it onto
$f_{\mathbb{D}}\in\mathcal{P}^{2}(\mu_{\mathbb{D}})$. We then compute

$\displaystyle 0$
$\displaystyle=\int_{\mathbb{D}}F\overline{u}\overline{z}^{n}G(1-|z|)dA+\int_{\mathbb{T}}h\overline{u}\overline{z}^{n}d\textit{m}$
$\displaystyle=\int_{\mathbb{D}}f_{\mathbb{D}}\overline{z}^{n}G(1-|z|)dA+\int_{I}h\frac{\overline{u}}{w}\overline{z}^{n}wd\textit{m}.$

This means that if we set $f_{\mathbb{T}}:=h\frac{\overline{u}}{w}1_{I}$, then
$f_{\mathbb{T}}$ is a bounded measurable function, and the tuple
$f_{\mathbb{D}}+f_{\mathbb{T}}\in\mathcal{P}^{2}(\mu_{\mathbb{D}})\oplus
L^{2}(\mu_{\mathbb{T}})$ is orthogonal to $\mathcal{P}^{2}(\mu)$. Moreover,
$f_{\mathbb{T}}$ is non-zero almost everywhere on $J$. The proof is complete,
by the remarks made in the first paragraph of the proof.

∎

###### Proof of maintheoremSuperExp.

By FwSplittingOffProp, the space $L^{2}(\mu_{F})$ is contained in
$\mathcal{P}^{2}(\mu)$. The orthogonal complement of this space in
$\mathcal{P}^{2}(\mu)$ is clearly
$\mathcal{P}^{2}(\mu_{\mathbb{D}}+\mu_{\mathbb{T}\setminus F})$. By
irreducibilityLemmaLogInt, the space
$\mathcal{P}^{2}(\mu_{\mathbb{D}}+\mu_{\mathbb{T}\setminus F})$ is
irreducible. ∎

## References

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# Witgenstein’s influence on artificial intelligence

Piero Molino and Jacopo Tagliabue111This is an English pre-print of the
chapter that appeared in Spanish in CENTENARIO DEL SILENCIO. At the time of
submitting our work (2021), Piero Molino was at Stanford University and Jacopo
Tagliabue at Coveo Labs: the current text has been only slightly edited for
content and citations.

###### Abstract

We examine how much of the contemporary progress in artificial intelligence
(and, specifically, in natural language processing), can be, more or less
directly, traced back to the seminal work and ideas of the Austrian-British
philosopher Ludwig Wittgenstein, with particular focus on his late views.
Discussing Wittgenstein’s original theses will give us the chance to survey
the state of artificial intelligence, and comment on both its strengths and
weaknesses. A similar text appeared first in Spanish as a chapter of
CENTENARIO DEL SILENCIO (2021), a book celebrating 100 years since the
publication of the Tractatus.

The thesis we are proposing here is one that is not actively debated in
artificial intelligence research circles to the best of our knowledge.
However, we believe it could help interpret recent lines of research that have
proved themselves effective. The thesis is that much of the progress in
artificial intelligence, and specifically in natural language processing and
machine learning in recent years can be, albeit indirectly, traced back to
Witgenstein, specifically to the “Philosophical Investigations” (henceforth,
PI). In some instances the connection is easier to trace, in some others there
is a leap to take in order to accept the thesis, but despite that, we believe
exploring it could be beneficial, even if only for the inspirations it could
trigger by analogy.

We suggest following the thread of research on natural language processing
that more directly follows from Witgenstein through the work of Margaret
Masterman as it will lead to an intersection point with machine learning later
on. Margaret Masterman was a student in Wittgenstein’s 1933-1934 course, the
lecture notes of which were collected in the “Blue Book”. The ideas in that
course were the foundational ones later contained in the “Philosophical
Investigations”: the specific insight that the meaning of words derives from
their use in language constituted the basis of Masterman’s computational
approach to linguistics - focused on meaning and thesauri rather than syntax
-, which she applied in particular to machine translation. She founded the
Cambridge Language Research Unit in 1955, and work coming from there
influenced researchers all around the world for the following decades, in a
time when Chomskian ideas [Cho57] were predominant (Sixties) and Montague
proposed his logic-based theories [Mon74] (early Seventies).

Around the same time as Masterman, Zellig Harris was publishing his most
influential structuralist works [Har51, Har54]. They followed in the American
linguistic tradition of Leonard Bloomfield, but in fact echoed the
structuralist theories coming from Cambridge. With “Papers in Linguistics”
[Fir57] and “Mathematical Structure of Language” [Har68] we have some of the
first appearances of the distributional hypothesis, which states that the
degree of semantic similarity between two linguistic expressions is a function
of the similarity of the linguistic contexts in which they can appear.

This hypothesis has been the foundation of countless empirical and
computational works in the sub-field of natural language processing called
distributional semantics (for a detailed historical account see [Len18]). In
short, many different algorithms have been proposed to represent words as
vectors in a multi-dimensional space, usually by statistical methods observing
the co-occurrence of words with other words or other representations of
contexts within a specific corpus. The vectors obtained this way have
interesting properties that follow the distributional hypothesis and have been
fruitfully applied in many applications, particularly in the context of
information retrieval and search engines starting in the 70s with the vector
space model up to the current day.

A very influential work in 2013 [MSC+13] proposed word2vec 222Technically
word2vec is a tool containing two algorithmic approaches: continuous bag of
words and skip gram., a novel algorithm to obtain vectors representing words.
The algorithms consist of a function that predicts a word given its context
(continuous bag of words) or the context given a word (skip gram), where words
and contexts are n-grams obtained from a large corpus. For instance, first a
text like “The quick brown fox jumps over the lazy dog” is transformed in
n-grams of consecutive words like “The quick brown fox jumps”, “quick brown
fox jumps over”, “brown fox jumps over the” and so on, then a function that
computes $P(word|context)$ is trained to maximize the probability of “brown”
given “The quick fox jumps”, and the process is repeated for all the other
n-grams in the corpus. The function is parametrized by a shallow neural
network made of only two linear layers and its parameters are learned through
optimization (stochastic gradient descent). Importantly, the learned
parameters of the first layer are then used as vector representations of the
words in the corpus. One of the interesting properties of the vector spaces
obtained with this method is that some relationships between words are
captured as linear directions in the space, making it possible to use linear
algebra for simple reasoning, like the famous example “king is to man what
queen is to …” that can be solved by calculating the closest vector to
“vec(king) - vec(man) + vec(queen)”.

Despite the simplicity and the fact that fundamentally its workings are no
more than a direct implementation of structuralist principles, this algorithm
changed the perception of the entire field, as it brought machine learning and
distributional semantics communities much closer. Its success is also due to
the timeliness of proposing a method to obtain distributed representations of
words well suited for use with neural networks, making it possible to adapt
deep learning approaches to textual data, in a moment when deep learning was
successfully applied mostly to other types of perceptual data. Word2vec was
not the first neural network approach to obtain vector representations though,
but it was the first one to popularize the idea, because it was practical,
fast and easy to use. [BDVJ03] proposed a neural network algorithm that
considered only the words preceding a certain word in order to predict the
following one, what is commonly referred to as language modeling, but they did
not use dense word representations, which made the approach impractical.
[MD88] very pionieristically proposed the FGREP algorithm, a kind of auto-
encoder method that obtained representations by means of compressing and
predicting back sentences and semantic frames, clearly ahead of its time.

After studies on strengths and limitations of learning-based algorithms as
opposed to classical distributional semantics ones [BDK14, LGD15, SL16], the
natural language processing community accepted machine learning as the de
facto standard way to obtain word representations. Since then many other
neural network based approaches have been proposed as alternatives, with one
of the most widely applied today being GloVe [PSM14], and those algorithms
have been applied for all sorts of natural language processing tasks, and have
influenced new developments in recommender systems, graph algorithms, speech
recognition and many more.

The reason they work is the basic structuralist assumption that meaning
emerges by use and is determined by context. The reason why structuralist
assumptions are so important is because of the cost of acquiring data. The
most widely adopted paradigm in machine learning is supervised learning, which
consists in having humans tag or categorize objects and then iteratively
improving an algorithm by means of how well its predictions match the ones
provided by humans. Having humans annotate all the objects of interest is very
expensive and error prone, and the most accurate supervised algorithms are
data hungry. The famous ImageNet competition [DDS+09] collected a million
images and that was a sufficiently large amount of data for the most effective
models to learn to recognize objects in images with high accuracy, but was
also really expensive to collect. The structuralist principle instead suggests
that meaning depends on use, and as such, you just need to observe use to
obtain representations. Word2vec and similar algorithms use huge amounts of
textual data, but no human labels are provided, the words to predict are
simply removed from the ngrams collected, and unlabeled data is obviously
extremely cheaper to collect, and it works well because applied well known and
very effective supervised learning techniques to huge amounts of inexpensive
unsupervised data. In recent years this idea of removing something from the
context it appears in and predicting it back has been named self-supervision
and has been one of the main focus of research and source of advancement in
both natural language processing and machine learning.

Specifically in the natural language processing space, algorithms that learn
to represent entire sentences by training on unsupervised corpora have been
shown to be very effective when then applied and adapted to supervised tasks.
Algorithms like ELMo [PNI+18], BERT [DCLT19] and their many derivatives work
exactly in this way. BERT for instance is trained with the so-called masked
language model objective, which consists in obtaining a sentence from a
corpus, masking a few random words and then asking the neural network to
predict the masked words from the unmasked ones. This allows training on huge
amounts of text crawled from the web and is very effective. At the same time,
generative language models, like the GPT [RN18] family of algorithms, in which
every word in a document is predicted from the words that preceded it, have
been shown to be really effective at learning how to generate text that is
fluent, usually grammatical, and usually thematically coherent with what came
before.

In the broader machine learning field, self-supervision is attracting more and
more the interest of the research community. In computer vision, autoencoding
and in-filling tasks (in which an area of an image is masked and the algorithm
is asked to predict it using the unmasked portion as input) are used and self-
supervision is quickly becoming the most effective way to obtain pre-trained
models [PKD+16]. In video analysis, prediction of the next frame is a widely
used self-supervision task [WG15], and in graph learning, representations of
nodes in graphs are obtained by removing a node from its neighborhood and
predicting it back [GL16]. Finally, in reinforcement learning, researchers are
learning models of the dynamics of the environment in which agents interact by
means of observing sequences of actions and states, removing the last one and
predicting it given the previous ones, in a similar way to language models
[HS18].

Notwithstanding the influence on contemporary distributional semantics that we
just outlined, we do believe that the “meaning is use”-thesis at the heart of
the “Philosophical Investigations” still has insights waiting to be fully
appreciated by mainstream Natural Language Processing. As parting thoughts, we
highlight here two broad areas of inquiry for which Wittgenstein’s
philosophical work may provide interesting intuitions.

The first topic is the relation between words and objects. While it is true
that word2vec models meaning as a function of linguistic context, it is
important to remember that Wittgenstein’s main foe in the PI was his younger
self, as well as the entire tradition of scholars analyzing human languages
with formal tools (i.e. Tarski-like semantics [Tar36, Dav67]). When
Wittgenstein stresses the importance of context is mostly to reject the idea
of a one-to-one correspondence between words and objects, but he certainly
does not imply that language is entirely self-referential and can be analyzed
independently from reality, like word2vec and BERT do. To the contrary, even
the simpler example of “language game” involves an inextricable mixture of
linguistic signs and extra-linguistic activity:

> The language is meant to serve for communication between a builder A and an
> assistant B. A is building with building-stones: there are blocks, pillars,
> slabs and beams. B has to pass the stones, in the order in which A needs
> them. For this purpose they use a language consisting of the words ”block”,
> ”pillar” ”slab”, ”beam”. A calls them out; — B brings the stone which he has
> learnt to bring at such-and-such a call. Conceive this as a complete
> primitive language. (PI 2.)

While most of the practical success of recent NLP is due to intra-language
models, the community is starting to recognize the importance of grounding
language in an extra-linguistic reality [BHT+20, BGT21]. Applying the PI
lesson to this growing field means not only recognizing the importance of the
outside world for the concept of meaning, but also trying to understand
communication in the context of human activities: formal studies tend to focus
on descriptive language (e.g. Wikipedia pages), but PI made clear that
possibly the majority of language use is non-descriptive [Aus62]: promises,
greetings, jokes, etc.. How to account for the variety of intents of human
language is an open research question that likely involves modelling social
and psychological traits.

The value of practical communication above descriptive sentences and the
importance of social interactions for meaning bring us to the second topic.
While Wittgenstein was not particularly fond of mental states, it is true that
the PI marked a discontinuity in philosophy of language: if formal accounts
mostly assumed that meaning was by and large explicitly coded in linguistic
expressions, “meaning is use” highlighted that in many cases what is said is
significantly less/different than what is meant - semantics is important, but
so is pragmatics. Some twenty years after PI, Paul Grice [Gri75] put forward a
theory of meaning based on “presuppositions”, that is, a framework in which
meaning is heavily dependent on context via the expectations of the
participants in a conversation. If Richard Montague is mostly worried about
logical implication [Mon73], Grice studies implicatures: if Jack writes a
supporting letter to a Ph.D. candidate, Bob, saying he is a very pleasant
person, the committee will most likely think that Jack does not believe Bob is
very talented, or otherwise he would have said that first. The extent to which
Grice’s rules are necessary to account for empirical phenomena is subject to
debate, but there is general consensus that successful communication involves
at least partially the theory of mind of the participants. To the best of our
knowledge, no practical NLP application has been so far grounded in formal
pragmatics, but in recent years the Rational Speech Act [GF16] theory has made
some progress (mostly in toy examples) in translating Grice’s insights into a
unified Bayesian framework for linguistic inference. Nested Bayesian inference
is not without practical challenges and it is too early to judge how
influential this thread of research will be on mainstream NLP; however, RSA is
a good example of how formal tooling from statistics and computer science can
be used to make philosophical intuitions precise, and, in turn, raise new
philosophical questions.

This historical excursus followed by a brief description of the current state
of artificial intelligence research in many of its sub-fields was aimed at
trying to draw a fil rouge that can trace back the underlying (and in most
cases unconscious and un-acknowledged) assumptions of current work in self-
supervision all the way back to structuralism and the theories of Witgenstein
and this hopefully also makes the case for researchers in artificial
intelligence to be curious about philosophy and its history and to be thought
provoking for philosophers with regards to the practical computational
implications of their theories.

It is unlikely that a single discipline, in isolation, will solve the mystery
of language; multi-disciplinary, bold thoughts are needed to make significant
progress. To quote Ludwig one last time:

> You could attach prices to thoughts. Some cost a lot, some a little. And how
> does one pay for thoughts? The answer, I think, is: with courage.

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