# Some properties of a modified Hilbert transform

Matteo Ferrari
Faculty of Mathematics
University of Vienna
1090 Vienna, Austria
<EMAIL_ADDRESS>
###### Abstract

Recently, Steinbach et al. introduced a novel operator
$\mathcal{H}_{T}:L^{2}(0,T)\to L^{2}(0,T)$, known as the modified Hilbert
transform. This operator has shown its significance in space-time formulations
related to the heat and wave equations. In this paper, we establish a direct
connection between the modified Hilbert transform $\mathcal{H}_{T}$ and the
canonical Hilbert transform $\mathcal{H}$. Specifically, we prove the
relationship $\mathcal{H}_{T}\varphi=-\mathcal{H}\widetilde{\varphi}$, where
$\varphi\in L^{2}(0,T)$ and $\widetilde{\varphi}$ is a suitable extension of
$\varphi$ over the entire $\mathbb{R}$. By leveraging this crucial result, we
derive some properties of $\mathcal{H}_{T}$, including a new inversion
formula, that emerge as immediate consequences of well-established findings on
$\mathcal{H}$.

##  1 Introduction and main result

In [10], a modified Hilbert transform $\mathcal{H}_{T}$, associated with the
bounded interval $(0,T)$, has been defined. Given the Fourier series of
$\varphi\in L^{2}(0,T)$

$\varphi(t)=\sum_{k=0}^{\infty}\varphi_{k}\sin\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right)\quad\text{with}\quad\varphi_{k}=\frac{2}{T}\int_{0}^{T}\varphi(s)\sin\left(\left(\frac{\pi}{2}+k\pi\right)\frac{s}{T}\right)\differential
s,$

the modified Hilbert transform is defined as

$\mathcal{H}_{T}\varphi(t)=\sum_{k=0}^{\infty}\varphi_{k}\cos\left(\left(\frac{\pi}{2}+k\pi\right)\frac{t}{T}\right)\quad
t\in(0,T).$ (1)

This operator has been employed in the context of space-time discretizations
of PDEs using both finite element [2, 7, 10] and boundary element [9] methods.
It is particularly well-suited because, by defining the Sobolev space

$H_{0,}^{\nicefrac{{1}}{{2}}}(0,T)=\\{\varphi\in
H^{\nicefrac{{1}}{{2}}}(0,T):\varphi(0)=0\\},$

it can be shown (see [10]) that
$-\partial_{t}\mathcal{H}_{T}:H_{0,}^{\nicefrac{{1}}{{2}}}(0,T)\to[H_{0,}^{\nicefrac{{1}}{{2}}}(0,T)]^{\prime}$
induces an equivalent norm in $H_{0,}^{\nicefrac{{1}}{{2}}}(0,T)$.
Furthermore, it holds
$\langle\varphi,\mathcal{H}_{T}\varphi\rangle_{L^{2}(0,T)}\geq 0$ for all
$\varphi\in L^{2}(0,T)$.

An alternative integral representation of $\mathcal{H}_{T}$ suited for
numerical schemes has been presented in [11, Lemma 2.1].

######  Lemma 1.

For $\varphi\in L^{2}(0,T)$, the operator $\mathcal{H}_{T}$ allows the
integral representation

$\mathcal{H}_{T}\varphi(t)=\frac{1}{2T}\operatorname{p.v.}\int_{0}^{T}\varphi(s)\left[\csc\left(\frac{\pi(s+t)}{2T}\right)+\csc\left(\frac{\pi(s-t)}{2T}\right)\right]\differential
s,\quad t\in(0,T)$ (2)

as Cauchy principal value integral. Moreover, if $v\in H^{1}(0,T)$ it holds

$\displaystyle\mathcal{H}_{T}\varphi(t)=$
$\displaystyle-\frac{2}{\pi}\varphi(0)\log\left(\tan\left(\frac{\pi
t}{4T}\right)\right)-\frac{1}{\pi}\int_{0}^{T}\partial_{t}\varphi(s)\log\left(\tan\left(\frac{\pi(s+t)}{4T}\right)\tan\left(\frac{\pi|s-t|}{4T}\right)\right)\differential
s,$ (3)

for $t\in(0,T)$ as a weakly singular integral.

The Hilbert transform $\mathcal{H}$ of a function $\varphi$ is defined as
Cauchy principal value integral

$\mathcal{H}\varphi(t)=\frac{1}{\pi}\operatorname{p.v.}\int_{\mathbb{R}}\frac{\varphi(s)}{t-s}\,ds,$

whenever it exists (see [1, Chapter 9], [5] and references therein).

The following relationship between the Hilbert transform $\mathcal{H}$ and its
modified version $\mathcal{H}_{T}$ has been established in [8, Theorem 4.3].

######  Theorem 1.

For $\varphi\in L^{2}(0,T)$, it holds

$\mathcal{H}_{T}\varphi=-\mathcal{H}\bar{\varphi}+B\varphi\quad\quad\text{in~{}}L^{2}(0,T),$

where $B:L^{2}(0,T)\to L^{2}(0,T)$ is a compact operator, and $\bar{\varphi}$
is the reflection (see Figure 1)

$\bar{\varphi}(s)=\begin{cases}-\varphi(s+2T)&s\in(-2T,-T),\\\
-\varphi(-s)&s\in(-T,0),\\\ \varphi(s)&s\in(0,T),\\\
\varphi(2T-s)&s\in(T,2T),\\\ 0&\text{else}.\end{cases}$ (4)

Figure 1: For $\varphi(x)=x^{2}$, $\varphi:[0,T]\to\mathbb{R}$, it is plotted
$\bar{\varphi}:\mathbb{R}\to\mathbb{R}$ defined in (4).

In this paper, we prove that $\mathcal{H}_{T}$ is, in fact, precisely the
Hilbert transform $\mathcal{H}$ applied to a specific odd periodic extension
with alternating signs.

######  Theorem 2.

For $\varphi\in L^{2}(0,T)$, it holds

$\mathcal{H}_{T}\varphi=-\mathcal{H}\widetilde{\varphi}\quad\quad\text{in~{}}L^{2}(0,T),$

where $\widetilde{\varphi}$ is the periodic reflection (see Figure 2)

$\widetilde{\varphi}(s)=\begin{cases}-\varphi(s+2T-4kT)&s\in((4k-2)T,(4k-1)T),\,\,k\in\mathbb{Z},\\\
-\varphi(4kT-s)&s\in((4k-1)T,4kT),\,\,k\in\mathbb{Z},\\\
\varphi(s-4kT)&s\in(4kT,(4k+1)T),\,\,k\in\mathbb{Z},\\\
\varphi(4kT+2T-s)&s\in((4k+1)T,(4k+2)T),\,\,k\in\mathbb{Z}.\\\ \end{cases}$
(5)

Figure 2: For $\varphi(x)=x^{2}$, $\varphi:[0,T]\to\mathbb{R}$, it is plotted
$\widetilde{\varphi}:\mathbb{R}\to\mathbb{R}$ defined in (5).

######  Example 1.

From (3), we obtain that for $\varphi(s)=1$ with $s\in(0,T)$, we have
$\mathcal{H}_{T}\varphi(t)=-\frac{2}{\pi}\log\left|\tan\left(\frac{\pi
t}{4T}\right)\right|$. It can be readily seen that, for $s\in\mathbb{R}$,
$\widetilde{\varphi}(s)=\operatorname{sgn}(\sin(as))$ with $a=\frac{\pi}{2T}$,
and for this function, it is well-known that
$\mathcal{H}\widetilde{\varphi}(t)=\frac{2}{\pi}\log\left|\tan\left(\frac{at}{2}\right)\right|$
(see [5, Equation 6.9]).

######  Example 2.

A function that satisfies a periodicity of the form (5) is
$\varphi(s)=\sin(\pi s)$ with $T=\frac{1}{2}$. In this case, on $\mathbb{R}$,
we have the trivial extension $\widetilde{\varphi}(s)=\sin(\pi s)$. It is
well-known that $\mathcal{H}\widetilde{\varphi}(t)=-\cos(\pi t)$ (see [5,
Equation 1.8]). From (2), we can calculate

$\displaystyle\mathcal{H}_{T}\varphi(t)$
$\displaystyle=\operatorname{p.v.}\int_{0}^{\frac{1}{2}}\sin(\pi
s)\left[\csc\left(\pi(s+t)\right)+\csc\left(\pi(s-t)\right)\right]\differential
s$
$\displaystyle=\operatorname{p.v.}\int_{t}^{t+\frac{1}{2}}\frac{\sin(\pi(s-t))}{\sin(\pi
s)}\differential
s+\operatorname{p.v.}\int_{-t}^{\frac{1}{2}-t}\frac{\sin(\pi(s-t))}{\sin(\pi
s)}\differential s=\cos(\pi t).$

##  2 Proofs of the main result

In this section, we present three distinct proofs of Theorem 2. The first
proof relies on the integral representation 2, while the second proof utilizes
an alternative definition of the Hilbert transform $\mathcal{H}$ specifically
designed for periodic functions. Finally, the third proof employs Fourier
series.

###  2.1 Proof based on the integral representation

The first proof is established by leveraging the Laurent series expansion of
the cosecant function, as presented in [4, Formula (9.3.30)]

$\csc(z)=\frac{1}{z}+2z\sum_{k=1}^{\infty}\frac{(-1)^{k}}{z^{2}-k^{2}\pi^{2}}.$
(6)

This series converges absolutely and uniformly for $|z|<\pi$, in fact for each
$N\in\mathbb{N}$ it holds

$\left|\csc(z)-\frac{1}{z}\right|\leq\frac{2|z|}{\pi^{2}}\sum_{k=1}^{N}\frac{1}{\left|\left(\frac{z}{k\pi}\right)^{2}-1\right|}\frac{1}{k^{2}}\leq\frac{|z|}{3}.$

Starting from the definitions of Hilbert transform $\mathcal{H}$ and
$\widetilde{\varphi}$ in (5), we can write

$\displaystyle\mathcal{H}\widetilde{\varphi}(t)$
$\displaystyle=\frac{1}{\pi}\left[\ldots+\operatorname{p.v.}\int_{-3T}^{-T}\frac{\widetilde{\varphi}(s)}{t-s}\differential
s+\operatorname{p.v.}\int_{-T}^{T}\frac{\widetilde{\varphi}(s)}{t-s}\differential
s+\operatorname{p.v.}\int_{T}^{3T}\frac{\widetilde{\varphi}(s)}{t-s}\differential
s+\ldots\right]$
$\displaystyle=\frac{1}{\pi}\left[\ldots-\operatorname{p.v.}\int_{-T}^{T}\frac{\widetilde{\varphi}(s)}{t-s-2T}\differential
s+\operatorname{p.v.}\int_{-T}^{T}\frac{\widetilde{\varphi}(s)}{t-s}\differential
s-\operatorname{p.v.}\int_{-T}^{T}\frac{\widetilde{\varphi}(s)}{t-s+2T}\differential
s+\ldots\right]$

where we have used the alternating signs periodicity of $\widetilde{\varphi}$.
Therefore, we can compactly write

$\displaystyle\mathcal{H}\widetilde{\varphi}(t)$
$\displaystyle=\frac{1}{\pi}\sum_{k=-\infty}^{+\infty}\operatorname{p.v.}\int_{-T}^{T}\widetilde{\varphi}(s)\frac{(-1)^{k}}{t-s+2kT}\differential
s.$

By interchanging the integral and the summation (which is possible due to the
uniform convergence of the series), and utilizing (6), we obtain

$\displaystyle\mathcal{H}\widetilde{\varphi}(t)$
$\displaystyle=\frac{1}{\pi}\operatorname{p.v.}\int_{-T}^{T}\widetilde{\varphi}(s)\left[\sum_{k=-\infty}^{+\infty}\frac{(-1)^{k}}{t-s+2kT}\right]\differential
s$
$\displaystyle=\frac{1}{\pi}\operatorname{p.v.}\int_{-T}^{T}\widetilde{\varphi}(s)\left[\frac{1}{t-s}+2(t-s)\sum_{k=1}^{+\infty}\frac{(-1)^{k}}{(t-s)^{2}-4k^{2}T^{2}}\right]\differential
s$
$\displaystyle=\frac{1}{2T}\operatorname{p.v.}\int_{-T}^{T}\widetilde{\varphi}(s)\csc\left(\frac{\pi(t-s)}{2T}\right)\differential
s.$

From the latter, we obtain the modified Hilbert transform $\mathcal{H}_{T}$ in
integral form (2) by recalling the definition (5), and writing

$\displaystyle\mathcal{H}\widetilde{\varphi}(t)$
$\displaystyle=-\frac{1}{2T}\operatorname{p.v.}\int_{-T}^{0}\varphi(-s)\csc\left(\frac{\pi(t-s)}{2T}\right)\differential
s+\frac{1}{2T}\operatorname{p.v.}\int_{0}^{T}\varphi(s)\csc\left(\frac{\pi(t-s)}{2T}\right)\differential
s$
$\displaystyle=-\frac{1}{2T}\operatorname{p.v.}\int_{0}^{T}\varphi(s)\csc\left(\frac{\pi(t+s)}{2T}\right)\differential
s-\frac{1}{2T}\operatorname{p.v.}\int_{0}^{T}\varphi(s)\csc\left(\frac{\pi(t-s)}{2T}\right)\differential
s$ $\displaystyle=-\mathcal{H}_{T}\varphi(t).$

###  2.2 Proof based on the Hilbert transform for periodic functions

Given $\varphi\in L^{2}(0,T)$, we observe that $\widetilde{\varphi}$ as in (5)
is actually periodic with a period of $4T$, and it is known that the Hilbert
transform $\mathcal{H}$ for a periodic function can be calculated using (see
[6]):

$\mathcal{H}\widetilde{\varphi}(t)=\frac{1}{4T}\operatorname{p.v.}\int_{-2T}^{2T}\widetilde{\varphi}(s)\cot\left(\frac{\pi(t-s)}{4T}\right)\differential
s.$ (7)

Let us expand the four contributions of the integral above to recover
$\mathcal{H}_{T}\varphi$

$\displaystyle\mathcal{H}\widetilde{\varphi}(t)$
$\displaystyle=-\frac{1}{4T}\left[\operatorname{p.v.}\int_{-2T}^{-T}\varphi(s+2T)\cot\left(\frac{\pi(t-s)}{4T}\right)\differential
s-\operatorname{p.v.}\int_{-T}^{0}\varphi(-s)\cot\left(\frac{\pi(t-s)}{4T}\right)\differential
s\right.$ $\displaystyle\hskip
12.80365pt\left.+\operatorname{p.v.}\int_{0}^{T}\varphi(s)\cot\left(\frac{\pi(t-s)}{4T}\right)\differential
s+\operatorname{p.v.}\int_{T}^{2T}\varphi(2T-s)\cot\left(\frac{\pi(t-s)}{4T}\right)\differential
s\right]$
$\displaystyle=-\frac{1}{4T}\operatorname{p.v.}\int_{0}^{T}\varphi(s)\cot\left(\frac{\pi(t-s+2T)}{4T}\right)\differential
s-\frac{1}{4T}\operatorname{p.v.}\int_{0}^{T}\varphi(s)\cot\left(\frac{\pi(t+s)}{4T}\right)\differential
s$ $\displaystyle\hskip
12.80365pt+\frac{1}{4T}\operatorname{p.v.}\int_{0}^{T}\varphi(s)\cot\left(\frac{\pi(t-s)}{4T}\right)\differential
s+\frac{1}{4T}\operatorname{p.v.}\int_{0}^{T}\varphi(s)\cot\left(\frac{\pi(t+s-2T)}{4T}\right)\differential
s.$

Finally, using the trigonometric formula

$\displaystyle
2\csc(2x)=-\cot\left(x\pm\frac{\pi}{2}\right)+\cot(x),\quad|x|<\frac{\pi}{2},$
(8)

we again conclude that
$\mathcal{H}_{T}\varphi=-\mathcal{H}\widetilde{\varphi}$ in $L^{2}(0,T)$.

###  2.3 Proof based on Fourier series

For $f$ with period $2T$ in $L^{2}(-T,T)$, the circular Hilbert transform is
defined as (see [5, Section 6.4])

$\mathcal{H}f(t)=\mathrm{i}\sum_{k=1}^{\infty}\left[f_{-k}e^{-\mathrm{i}k\pi\frac{t}{T}}-f_{k}e^{\mathrm{i}k\pi\frac{t}{T}}\right]\quad\text{with}\quad
f_{k}=\frac{1}{2T}\int_{-T}^{T}f(s)e^{-\mathrm{i}k\pi\frac{s}{T}}\differential
s.$

Let us verify that this definition is consistent with the original definition
of the modified Hilbert transform $\mathcal{H}_{T}$ by Steinbach and Zank (1).
That is, we will demonstrate that for a given function $\varphi\in
L^{2}(0,T)$, we have $\mathcal{H}\widetilde{\varphi}=-\mathcal{H}_{T}\varphi$
in $L^{2}(0,T)$, with $\widetilde{\varphi}$ as in (5). Recalling that
$\widetilde{\varphi}$ is a periodic function with period $4T$, let us begin by
writing

$\mathcal{H}\widetilde{\varphi}(t)=\mathrm{i}\sum_{k=1}^{\infty}\left[\widetilde{\varphi}_{-k}e^{-\mathrm{i}k\pi\frac{t}{2T}}-\widetilde{\varphi}_{k}e^{\mathrm{i}k\pi\frac{t}{2T}}\right],$

and readily one can verify

$\displaystyle\widetilde{\varphi}_{k}=\frac{1}{4T}\int_{-2T}^{2T}\widetilde{\varphi}(s)e^{-\mathrm{i}k\pi\frac{s}{2T}}\differential
s$
$\displaystyle=\frac{1}{4T}\int_{0}^{2T}\widetilde{\varphi}(s)\left[e^{-\mathrm{i}k\pi\frac{s}{2T}}-e^{\mathrm{i}k\pi\frac{s}{2T}}\right]\differential
s$
$\displaystyle=-\frac{\mathrm{i}}{2T}\int_{0}^{2T}\widetilde{\varphi}(s)\sin\left(k\pi\frac{s}{2T}\right)\differential
s=-\widetilde{\varphi}_{-k}.$

Hence, we continue

$\displaystyle\mathcal{H}\widetilde{\varphi}(t)$
$\displaystyle=-\sum_{k=1}^{\infty}\left[e^{-\mathrm{i}k\pi\frac{t}{2T}}+e^{\mathrm{i}k\pi\frac{t}{2T}}\right]\frac{1}{2T}\int_{0}^{2T}\widetilde{\varphi}(s)\sin\left(k\pi\frac{s}{2T}\right)\differential
s$
$\displaystyle=-\sum_{k=1}^{\infty}2\cos\left(k\pi\frac{t}{2T}\right)\frac{1}{2T}\int_{0}^{2T}\widetilde{\varphi}(s)\sin\left(k\pi\frac{s}{2T}\right)\differential
s.$

Moreover, we can now write

$\displaystyle\frac{1}{2T}\int_{0}^{2T}\widetilde{\varphi}(s)\sin\left(k\pi\frac{s}{2T}\right)\differential
s$
$\displaystyle=\frac{1}{2T}\int_{0}^{T}\varphi(s)\sin\left(k\pi\frac{s}{2T}\right)\differential
s+\frac{1}{2T}\int_{T}^{2T}\varphi(2T-s)\sin\left(k\pi\frac{s}{2T}\right)\differential
s$
$\displaystyle=\frac{1}{2T}\int_{0}^{T}\varphi(s)\left[\sin\left(k\pi\frac{s}{2T}\right)-\sin(k\pi)\sin\left(k\pi\frac{s}{2T}\right)\right]\differential
s$ $\displaystyle=\begin{cases}0&k\text{~{} even},\\\
\displaystyle{\frac{1}{T}\int_{0}^{T}\varphi(s)\sin\left(k\pi\frac{s}{2T}\right)\differential
s}&k\text{~{} odd}.\end{cases}$

Finally, we conclude

$\mathcal{H}\widetilde{\varphi}(t)=-\sum_{k=0}^{\infty}\cos\left((2k+1)\pi\frac{t}{2T}\right)\frac{2}{T}\int_{0}^{T}\widetilde{\varphi}(s)\sin\left((2k+1)\pi\frac{s}{2T}\right)\differential
s=-\mathcal{H}_{T}\varphi(t).$

##  3 Consequences of the main result

In this section we show simple consequences of Theorem 2.

###  3.1 Inversion formula

For $f\in L^{2}(-T,T)$ and periodic with period $2T$, the inversion formula
holds (see [5, Formula (6.35)])

$\mathcal{H}^{2}f(t)=-f(t)+\frac{1}{2T}\int_{-T}^{T}f(s)\differential
s,\quad\quad\text{in~{}}L^{2}(-T,T).$ (9)

For $\varphi\in L^{2}(0,T)$, we can calculate

$\mathcal{H}(\mathcal{H}_{T}\varphi)(t)=-\mathcal{H}^{2}\widetilde{\varphi}(t)=\widetilde{\varphi}(t)-\frac{1}{4T}\int_{-2T}^{2T}\widetilde{\varphi}(s)\differential
s=\varphi(t),\quad\quad\text{in~{}}L^{2}(0,T),$

since $\widetilde{\varphi}$ is an odd function.

###  3.2 Alternative formula

For an odd function $f\in L^{2}(-T,T)$ with period $2T$, it can be shown that

$\mathcal{H}f(t)=\frac{1}{T}\operatorname{p.v.}\int_{0}^{T}f(s)\frac{\sin\left(\pi\frac{s}{T}\right)}{\cos\left(\pi\frac{s}{T}\right)-\cos\left(\pi\frac{t}{T}\right)}\differential
s,\quad\quad\text{in~{}}L^{2}(-T,T).$

Therefore, we obtain the alternative formula in $L^{2}(0,T)$

$\displaystyle\mathcal{H}_{T}\varphi(t)=-\mathcal{H}\widetilde{\varphi}(t)$
$\displaystyle=-\frac{1}{2T}\operatorname{p.v.}\int_{0}^{2T}\widetilde{\varphi}(s)\frac{\sin\left(\pi\frac{s}{2T}\right)}{\cos\left(\pi\frac{s}{2T}\right)-\cos\left(\pi\frac{t}{2T}\right)}\differential
s$
$\displaystyle=-\frac{\cos\left(\pi\frac{t}{2T}\right)}{T}\operatorname{p.v.}\int_{0}^{T}\varphi(s)\frac{\sin\left(\pi\frac{s}{2T}\right)}{\cos^{2}\left(\pi\frac{s}{2T}\right)-\cos^{2}\left(\pi\frac{t}{2T}\right)}\differential
s.$

This formula can also be deduced from equation (2) using trigonometric
identities.

###  3.3 Integral representation

Let suppose that $\varphi\in H^{1}(0,T)$, then the derivative of
$\widetilde{\varphi}$ is

$\partial_{t}\widetilde{\varphi}(s)_{|_{(-2T,2T)}}=\begin{cases}-\partial_{t}\varphi(s+2T)&s\in(-2T,-T),\\\
\partial_{t}\varphi(-s)&s\in(-T,0),\\\ \partial_{t}\varphi(s)&s\in(0,T),\\\
-\partial_{t}\varphi(2T-s)&s\in(T,2T).\end{cases}$ (10)

We note that if $\varphi\in H^{1}(0,T)$, the only possible discontinuity of
$\widetilde{\varphi}_{|_{(-2T,2T)}}$ is in the point $s=0$.

Starting from (7), integrating by parts in the intervals of continuity
$(-2T,0)$ and $(0,2T)$ of $\widetilde{\varphi}$, we obtain

$\displaystyle\mathcal{H}_{T}\varphi(t)=-\mathcal{H}\widetilde{\varphi}(t)$
$\displaystyle=-\frac{1}{\pi}\int_{-2T}^{2T}\partial_{t}\widetilde{\varphi}(s)\log\left(\sin\left(\frac{\pi|t-s|}{4T}\right)\right)\differential
s$ $\displaystyle\hskip
9.95863pt+\frac{1}{\pi}\left[(\widetilde{\varphi}(0^{-})-\widetilde{\varphi}(0^{+}))\log\left(\sin\left(\frac{\pi
t}{4T}\right)\right)+(\widetilde{\varphi}(2T^{-})-\widetilde{\varphi}(-2T^{+}))\log\left(\cos\left(\frac{\pi
t}{4T}\right)\right)\right]$
$\displaystyle=-\frac{1}{\pi}\int_{-2T}^{2T}\partial_{t}\widetilde{\varphi}(s)\log\left(\sin\left(\frac{\pi|t-s|}{4T}\right)\right)\differential
s-\frac{2}{\pi}\varphi(0)\log\left(\tan\left(\frac{\pi t}{4T}\right)\right).$

This formula, in the case $\varphi(0)=0$ is the original form in which Hilbert
wrote the Hilbert transform in [3]. Let split the integral in the four usual
intervals, and use (10),

$\displaystyle\mathcal{H}_{T}\varphi(t)$
$\displaystyle=\frac{1}{\pi}\int_{0}^{T}\partial_{t}\varphi(s)\log\left(\cos\left(\frac{\pi(t+s)}{4T}\right)\right)\differential
s-\frac{1}{\pi}\int_{0}^{T}\partial_{t}\varphi(s)\log\left(\sin\left(\frac{\pi(t+s)}{4T}\right)\right)\differential
s$ $\displaystyle\hskip
8.5359pt-\frac{1}{\pi}\int_{0}^{T}\partial_{t}\varphi(s)\log\left(\sin\left(\frac{\pi|t-s|}{4T}\right)\right)\differential
s+\frac{1}{\pi}\int_{0}^{T}\partial_{t}\varphi(s)\log\left(\cos\left(\frac{\pi(t-s)}{4T}\right)\right)\differential
s$ $\displaystyle\hskip
8.5359pt-\frac{2}{\pi}\varphi(0)\log\left(\tan\left(\frac{\pi
t}{4T}\right)\right)$
$\displaystyle=-\frac{1}{\pi}\int_{0}^{T}\partial_{t}\varphi(s)\log\left(\tan\left(\frac{\pi(t+s)}{4T}\right)\tan\left(\frac{\pi|t-s|}{4T}\right)\right)\differential
s-\frac{2}{\pi}\varphi(0)\log\left(\tan\left(\frac{\pi t}{4T}\right)\right).$

We have obtained exactly the result of Lemma 3 in an alternative way.

##  4 Acknowledgements

This work is partially supported by the INdAM-GNCS project “Metodi numerici
per lo studio di strutture geometriche parametriche complesse” (CUP
E53C22001930001) and by the MIUR project “Dipartimenti di Eccellenza
2018-2022” (CUP E11G18000350001). Moreover, this research was funded in part
by the Austrian Science Fund (FWF) project 10.55776/F65

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