# A Pilot Survey of an M Dwarf Flare Star with Swift’s UV Grism

Shashank Chavali Laboratory for Atmospheric and Space Physics, University of
Colorado, 600 UCB, Boulder, CO 80309, USA  Allison Youngblood Exoplanets and
Stellar Astrophysics Laboratory, NASA Goddard Space Flight Center, Greenbelt,
MD 20771, USA  Rishi R. Paudel University of Maryland, Baltimore County,
Baltimore, MD 21250, USA  Exoplanets and Stellar Astrophysics Laboratory, NASA
Goddard Space Flight Center, Greenbelt, MD 20771, USA  Center for Research and
Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771,
USA  R. O. Parke Loyd Eureka Scientific, Oakland, CA 94602, USA  Karan
Molaverdikhani Universitäts-Sternwarte, Ludwig-Maximilians-Universität
München, Scheinerstrasse 1, 81679 München, Germany  Exzellenzcluster Origins,
Boltzmannstrasse 2, 85748 Garching, Germany  Max-Planck-Institut für
Astronomie, Königstuhl 17, 69117 Heidelberg, Germany  J. Sebastian Pineda
Laboratory for Atmospheric and Space Physics, University of Colorado, 600 UCB,
Boulder, CO 80309, USA  Thomas Barclay University of Maryland, Baltimore
County, Baltimore, MD 21250, USA  Exoplanets and Stellar Astrophysics
Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA  Center
for Research and Exploration in Space Science and Technology, NASA/GSFC,
Greenbelt, MD 20771, USA  Laura D. Vega University of Maryland, College Park,
MD 20742, USA  Exoplanets and Stellar Astrophysics Laboratory, NASA Goddard
Space Flight Center, Greenbelt, MD 20771, USA  Center for Research and
Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771,
USA

###### Abstract

The near-ultraviolet (NUV) spectral region is a useful diagnostic for stellar
flare physics and assessing the energy environment of young exoplanets,
especially as relates to prebiotic chemistry. We conducted a pilot NUV
spectroscopic flare survey of the young M dwarf AU Mic with the Neil Gehrels
Swift Observatory’s UltraViolet and Optical Telescope. We detected four flares
and three other epochs of significantly elevated count rates during the 9.6
hours of total exposure time, consistent with a NUV flare rate of $\sim$0.5
hour-1. The largest flare we observed released a minimum energy of
6$\times$1033 erg between 1730-5000 Å. All flares had durations longer than
the $\sim$14-17 minute duration of each Swift visit, making measuring total
flare energy and duration infeasible.

##  1 Introduction

The near-ultraviolet (NUV; $\sim$1700-3200 Å) is a historically under-studied
region for stellar flares, but has an enormous potential for diagnosing flare
physics. In flare models such as Kowalski et al. (2013), the broad-band
visible light from a flare comes from a sunspot-sized patch of hot,
$\sim$10,000 K plasma on the stellar photosphere, making NUV measurements
potentially more efficient at constraining this flaring mechanism than any
other wavelength range. The NUV is also thought to be an essential energy
source for triggering abiogenesis, the origin of life, and open questions
remain about whether the cool photospheres of M dwarfs could provide enough
NUV photons to initiate chemical reactions that ultimately lead to the
production of RNA precursors and whether flares could make up for the deficit
in NUV energy (Ranjan et al., 2017; Rimmer et al., 2018).

Figure 1: Swift grism data from the brightest flaring epoch show that AU Mic
regularly exceeds the abiogenesis threshold and that the flaring emission is
comprised of chromospheric emission lines, Balmer continuum, and possibly
blackbody continuum. Left: the average stellar flux from 2000-2800 Å is shown
as a function of time from the start of Visit 18. The red shaded area shows
the abiogenesis threshold level from Rimmer et al. (2018) at 0.31 AU (AU Mic’s
conservative inner habitable zone edge), and the dashed grey line shows the
quiescent stellar flux level. Right: a flare-only spectrum (i.e., the
quiescent stellar spectrum is subtracted from the in-flare spectrum) of the
brightest data point from the left panel is shown as a black line with the
gray shading representing the 1-$\sigma$ uncertainty. Various emission lines
and the Balmer continuum are labeled. The colored lines show blackbody curves
with temperatures ranging from 5000-18,000 K normalized to match the spectrum
at 4200 Å.

##  2 Observations & Reductions

We designed a pilot program to determine the viability of NUV spectroscopic
monitoring of flare stars with the Neil Gehrels Swift Observatory (Swift)
UltraViolet Optical Telescope (UVOT) grism. The UVOT UV grism is optimized for
1700-2900 Å first-order spectra at spectral resolving power $R\sim$150\. We
targeted AU Mic, a pre-main sequence M0Ve star at 9.7 pc, for its brightness
(V=8.6 mag), high flare rate ($\sim$0.5 hour-1 based on archival UVOT UVM2
photometry; E. Gilbert et al., 2022 in preparation), and sky location away
from the Galactic center where source confusion and oversubscription of Swift
observing time are less of a concern. We observed AU Mic for 52 ks (14.4
hours) over 42 individual spacecraft visits between 2020 April 5 and 2021
April 2 using the UVOT 0x0384 mode, which alternates between the UV grism
(UGRISM, clocked mode, 120 s exposures) and UVM2 images (30 s exposures, used
for aspect correction, 1997-2495 Å), resulting in $\sim$14-18 minutes elapsed
time per visit. After overhead, the total target exposure time was 34.5 ks
(9.6 hours). Event mode is generally not used with the grism because of the
burden on telemetry.

We reduced the UVOT data using standard procedures from HEASoft (Nasa High
Energy Astrophysics Science Archive Research Center (2014), Heasarc) and
uvotpy (Kuin, 2014). We corrected for offsets in the uvotpy wavelength
solutions by fitting the blended Mg II h&k emission lines with a Gaussian and
shifting the centroid to 2800 Å (Kuin et al., 2015). We corrected for
contamination by background stars by visually inspecting every 2D spectrum and
using linear interpolation to splice out parts of the spectrum contaminated by
background stars. These corrections were different for every spectrum because
the roll angle of the telescope varies over the course of a year.

##  3 Results

We established median flux levels for the ensemble UVM2 (11.4 counts/s) and
grism data and searched for data that lay $>$3$\sigma$ above the median in
order to identify flares. In the total 34.5 ks spent on target, we detected
four flares (one shown in Figure 1) and three additional epochs where the
count rate was significantly elevated above the median, but the light curve
morphology was inconsistent with flaring. Note, however, that the cadence and
total duration of our visits do not allow us to rule out flaring for these
three elevated-flux epochs. Without event mode, 30-120 s cadence combined with
14-18 minute visit durations is a challenging setup for fully resolving flare
light curves. We could not determine the start and stop times of any of our
flares, so we adopt the visit durations (16.6-18.2 minutes) as the minimum
flare durations. For the three elevated epochs, the visit durations were only
14.2 minutes.

The dates corresponding to the four flaring epochs are: 2020 May 10, 2020 July
05, 2020 July 31 (Figure 1), and 2020 October 18. The three periods of
elevated flux that could not confidently be identified as flares occurred on
2020 April 26, 2020 June 28, and 2020 July 26 . The observed flare rate of
0.42-0.73 hour-1 is consistent with that expected from archival UVM2
photometry of AU Mic (E. Gilbert et al., 2022 in preparation).

Sufficient NUV photons may be a necessary ingredient for triggering
photoreactions that kick off prebiotic chemical pathways. We compared all
spectra to the 2000-2800 Å abiogenesis threshold from Rimmer et al. (2018) to
determine how flares could impact the ability of M dwarf stars to meet this
flux threshold. Unlike older, later type M dwarfs, AU Mic provides $\sim
4\times$ the abiogenesis threshold energy during quiescence to its
conservative inner-habitable zone edge (0.31 AU; Kane et al. (2022)). Our
results indicate that AU Mic spends 98% of the time $>$5$\times$ above the
threshold, 26% $>10\times$ above the threshold, 4% $>25\times$ above the
threshold, and 0.3% $>50\times$ above the threshold. Note that AU Mic’s two
known Neptune-sized exoplanets at 0.06 and 0.11 AU are closer to the star than
the habitable zone (Cale et al., 2021).

We subtracted the median spectrum from each of the flare spectra in order to
analyze the type of emission processes responsible for the flux increases we
observe during flaring and/or elevated epochs. We obtained useful flare
spectra between $\sim$1730-5000 Å for our largest flare (Figure 1), which
lasted $>$18.2 minutes and released a minimum of 6$\times$1033 erg in the
1730-5000 Å bandpass, with roughly equal energy contribution from emission
above and below the Balmer series limit at 3646 Å. We find Balmer emission
from H$\beta$ through the Balmer series limit at 3646 Å, and Ca II H&K
emission. Continuum in the 4000-5000 Å region is roughly flat and consistent
with a $\sim$6,000-15,000 K blackbody. Shortward of 3646 Å, we observe bright
Balmer continuum emission that dominates over any hot blackbody component. The
emission lines shortward of the Balmer series limit are blended multiplets
from Mg II and Fe II. This flare spectral energy distribution is consistent
with that found for GJ 1243 (M4Ve) by Kowalski et al. (2019).

##  4 Conclusion

The goal of this Swift pilot survey of AU Mic was to explore the utility of
Swift’s UV grism mode for stellar flare monitoring. Aside from the Hubble
Space Telescope, which has detector protection restrictions in place against
flaring M dwarfs, Swift’s grism uniquely provides NUV spectroscopy for
studying stellar flares. We have shown that although the 0x0384 mode’s lack of
event mode and short visit duration make determining flare morphology and
total duration challenging, flux-calibrated flare spectra are readily
measurable. Future Swift flare surveys should explore the feasibility of
consecutive executions of the 0x0384 mode so that individual visits could
extend longer.

The authors thank Paul Kuin for advice in setting up this observing program
and using uvotpy. This work was supported by NASA’s Swift Guest Investigator
Program under award 80NSSC20K1112.

## References

  * Cale et al. (2021) Cale, B. L., Reefe, M., Plavchan, P., et al. 2021, AJ, 162, 295, doi: 10.3847/1538-3881/ac2c80
  * Kane et al. (2022) Kane, S. R., Foley, B. J., Hill, M. L., et al. 2022, AJ, 163, 20, doi: 10.3847/1538-3881/ac366b
  * Kowalski et al. (2013) Kowalski, A. F., Hawley, S. L., Wisniewski, J. P., et al. 2013, ApJS, 207, 15, doi: 10.1088/0067-0049/207/1/15
  * Kowalski et al. (2019) Kowalski, A. F., Wisniewski, J. P., Hawley, S. L., et al. 2019, ApJ, 871, 167, doi: 10.3847/1538-4357/aaf058
  * Kuin et al. (2015) Kuin, N. P. M., Landsman, W., Breeveld, A. A., et al. 2015, MNRAS, 449, 2514, doi: 10.1093/mnras/stv408
  * Kuin (2014) Kuin, P. 2014, UVOTPY: Swift UVOT grism data reduction, Astrophysics Source Code Library, record ascl:1410.004.  http://ascl.net/1410.004
  * Nasa High Energy Astrophysics Science Archive Research Center (2014) (Heasarc) Nasa High Energy Astrophysics Science Archive Research Center (Heasarc). 2014, HEAsoft: Unified Release of FTOOLS and XANADU, Astrophysics Source Code Library, record ascl:1408.004.  http://ascl.net/1408.004
  * Ranjan et al. (2017) Ranjan, S., Wordsworth, R., & Sasselov, D. D. 2017, ApJ, 843, 110, doi: 10.3847/1538-4357/aa773e
  * Rimmer et al. (2018) Rimmer, P. B., Xu, J., Thompson, S. J., et al. 2018, Science Advances, 4, eaar3302, doi: 10.1126/sciadv.aar3302


# PROPERTIES OF CHEBYSHEV POLYNOMIALS

Natanael Karjanto
Department of Applied Mathematics, University of Twente
P.O. Box 217, 7500 AE Enschede, The Netherlands
<EMAIL_ADDRESS>
###### Abstract

Ordinary differential equations and boundary value problems arise in many
aspects of mathematical physics. Chebyshev differential equation is one
special case of the Sturm-Liouville boundary value problem. Generating
function, recursive formula, orthogonality, and Parseval’s identity are some
important properties of Chebyshev polynomials. Compared with a Fourier series,
an interpolation function using Chebyshev polynomials is more accurate in
approximating polynomial functions.

Keywords: Sturm-Liouville boundary value problem, Chebyshev differential
equation, Chebyshev polynomials, generating function, recursive formula,
orthogonality, Parseval’s identity.

###### Résumé

_Des équations différentielles ordinaires et des problèmes de valeurs limites
se posent dans de nombreux aspects de la physique mathématique. L’équation
différentielle de Chebychev est un cas particulier du problème de la valeur
limite de Sturm-Liouville. La fonction génératrice, la formule récursive,
l’orthogonalité et l’identité de Parseval sont quelques propriétés importantes
du polynôme de Chebyshev. Par rapport à une série de Fourier, une fonction
d’interpolation utilisant des polynômes de Chebyshev est plus précise dans
l’approximation des fonctions polynomiales._

_Mots-clés : problème de valeur limite de Sturm-Liouville, équation
différentielle de Chebyshev, polynôme de Chebyshev, fonction génératrice,
formule récursive, orthogonalité, identité de Parseval._

##  INTRODUCTION

Based on its name, these polynomials were first investigated by a Russian
mathematician Pafnuty Lvovich Chebyshev (1821-1894). In addition to orthogonal
functions, Chebyshev also studied inequalities, prime numbers, probability
theory, quadratic forms, integral theory, geographical map, geometrical volume
formulas, and mechanics as well as problems in changing circular motion to
straight motion with mechanical coupling [1]. In this paper, we will observe
the appearance of Chebyshev polynomials as solutions of one particular case
from a Sturm-Liouville boundary value problem. We discuss and prove several
essential properties, such as the generating formula, the recursive relation,
and Parseval’s identity. We cover but do not prove Rodrigues’ formula, even
and odd functions, as well as orthogonality. Depending on the type of
functions, an interpolation function using Chebyshev polynomials has a faster
rate of convergence and better accuracy than a Fourier series when
approximating a polynomial. On the other hand, a Fourier series converges
faster and better when interpolating non-polynomial functions.

##  CHEBYSHEV DIFFERENTIAL EQUATION AND ITS POLYNOMIALS

######  Definition 1 (Sturm-Liouville boundary value problem).

An ordinary differential equation (ODE) defined on an interval $a\leq x\leq b$
with a general form

$\frac{d}{dx}\left(p(x)\,\frac{dy}{dx}\right)+\left[q(x)+\lambda\,r(x)\right]\,y=0$
(1)

and boundary conditions

$\left\\{\begin{aligned} a_{1}\,y(a)+a_{2}\,y^{\prime}(a)&=0\\\
b_{1}\,y(b)+b_{2}\,y^{\prime}(b)&=0\end{aligned}\right.$ (2)

is called a Sturm-Liouville boundary value problem or a Sturm-Liouville system
with $p(x)>0$, $q(x)$, and a weighted function $r(x)>0$ are given functions;
$a_{1}$, $a_{2}$, $b_{1}$, and $b_{2}$ are given constants; and the
characteristic value $\lambda$ is an undetermined parameter.

The boundary value problem (1) admits several special cases that depend on the
defined interval, given functions, and characteristic values [2]. Each of
these special cases leads to a particular ODE with orthogonal functions as its
solutions. Among them are Bessel’s functions, Legendre polynomials, Hermite
polynomials, Laguerre polynomials, and Chebyshev polynomials. This paper
discusses the latter in particular.

For the values of an endpoint interval $a=-1$ and $b=1$, the given functions
$p(x)=\sqrt{1-x^{2}}$, $q(x)=0$, $r(x)=1/\sqrt{1-x^{2}}$, and the
characteristic value $\lambda=n^{2}$, the ODE (1) becomes

$\frac{d}{dx}\left(\sqrt{1-x^{2}}\,\frac{dy}{dx}\right)+\frac{n^{2}}{\sqrt{1-x^{2}}}\,y=0.$
(3)

######  Definition 2 (Chebyshev differential equation).

The ODE (3), or expressed in another form as

$(1-x^{2})\frac{d^{2}y}{dx^{2}}-x\frac{dy}{dx}+n^{2}\,y=0$ (4)

with $n\in\mathbb{N}_{0}$ for $|x|<1$ is called the Chebyshev differential
equation. The solution for this ODE is called the Chebyshev function with
singularities at $x=-1,1$, and $\infty$.

$x$$y$$1$$-1$1$-1$$n=0$$n=1$$n=2$$n=3$$n=4$$n=5$ Figure 1: Plot of Chebyshev
polynomials $T_{n}(x)$ for $n=0,1,2,\dots,5$.

######  Definition 3 (Chebyshev polynomials).

The functions

$T_{n}(x)=\cos\,(n\,\cos^{-1}x)\qquad\text{or}\qquad
T_{n}\,(\cos\theta)=\cos\,(n\,\theta)\qquad\text{for}\qquad x=\cos\theta$ (5)

are called the Chebyshev polynomials or the Chebyshev polynomials of the first
kind [3, 4, 5]. These are solutions of the ODE (4) for non-negative integers
$n$, $n\in\mathbb{N}_{0}$. See Figure 1.

Using a series expansion ${\displaystyle
y(x)=\sum_{n=0}^{\infty}\,a_{n}\,x^{n}}$, a general solution for the ODE (4)
is givey by

$y(x)=b_{1}\,T_{n}(x)+b_{2}\,\sqrt{1-x^{2}}\,U_{n-1}(x)$ (6)

where $U_{n}(x)=\sin\left[(n+1)\cos^{-1}x\right]/\sin(\cos^{-1}x)$ are the
Chebyshev polynomials of the second kind.

##  PROPERTIES OF THE CHEBYSHEV POLYNOMIALS

######  Property 4 (Rodrigues’ formula).

The Chebyshev polynomials $T_{n}(x)$ can be expressed as Rodrigues’ formula in
terms of derivatives

$T_{n}(x)=\frac{\sqrt{1-x^{2}}}{(-1)^{n}\,(2n-1)(2n-3)\dots
1}\frac{d^{n}}{dx^{n}}(1-x^{2})^{n-\frac{1}{2}}$ (7)

with $n=0,1,2,3,\dots$.

######  Property 5 (Generating function).

The generating function for the Chebyshev polynomials $T_{n}(x)$ reads

$\frac{1-z\,x}{1-2\,z\,x+z^{2}}=\sum_{n=0}^{\infty}\,T_{n}(x)\,z^{n}.$ (8)

###### Proof.

By letting $x=\cos\theta$, we have

$\displaystyle\sum_{n=0}^{\infty}\,T_{n}(x)\,z^{n}$
$\displaystyle=\sum_{n=0}^{\infty}\,T_{n}(\cos\theta)\,z^{n}=\sum_{n=0}^{\infty}\,\cos(n\,\theta)\,z^{n}=1+\frac{1}{2}\,\sum_{n=-\infty,\,n\neq
0}^{\infty}\,z^{|n|}\,e^{\,i\,n\,\theta}$
$\displaystyle=1+\frac{1}{2}\,\left(\sum_{n=1}^{\infty}\,z^{n}\,e^{\,i\,n\,\theta}+\sum_{n=1}^{\infty}\,z^{n}\,e^{-i\,n\,\theta}\right)=1+\frac{1}{2}\,\left(\frac{z\,e^{\,i\,\theta}}{1-z\,e^{\,i\,\theta}}+\frac{z\,e^{-i\,\theta}}{1-z\,e^{-i\,\theta}}\right)$
$\displaystyle=1+\frac{1}{2}\,\left(\frac{z\,e^{\,i\,\theta}-z^{2}+z\,e^{-i\,\theta}-z^{2}}{1-z\,e^{\,i\,\theta}-z\,e^{-i\,\theta}+z^{2}}\right)=1+\frac{z\,\cos\theta-z^{2}}{1-2\,z\,\cos\theta+z^{2}}$
$\displaystyle=1+\frac{x\,z}{1-2\,x\,z+z^{2}}=\frac{1-x\,z}{1-2\,x\,z+z^{2}}.$

Hence, we have shown the generating function formula for the Chebyshev
polynomials.

######  Property 6 (Even and odd functions).

Based on their degrees, the Chebyshev polynomials $T_{n}(x)$ admit two
possibilities as either even or odd functions.

  * •

For $n$ even, then $T_{n}(x)$ are even functions.

  * •

For $n$ odd, then $T_{n}(x)$ are odd functions.

######  Property 7 (Recursive relationship).

One Chebyshev polynomial $T_{n}(x)$ at a particular point can be expressed in
terms of its neighboring Chebyshev polynomials at the identical point

$T_{n+1}(x)=2\,x\,T_{n}(x)-T_{n-1}(x).$ (9)

We refer (9) as a three-term recursive relationship since the formula forms a
relationship among three terms of successive Chebyshev polynomial.

###### Proof.

From the definition of the Chebyshev polynomials, we have
$T_{n}(\cos\theta)=\cos(n\theta)$. We also have

$T_{n+1}(x)=T_{n+1}(\cos\theta)=\cos(n+1)\,\theta=\cos
n\theta\,\cos\theta-\sin n\theta\,\sin\theta$

and

$T_{n-1}(x)=T_{n-1}(\cos\theta)=\cos(n-1)\,\theta=\cos
n\theta\,\cos\theta+\sin n\theta\,\sin\theta.$

Next, by adding these two expressions, we obtain

$T_{n+1}(x)+T_{n-1}(x)=2\,\cos n\theta\,\cos\theta=2\,T_{n}(x)\,x.$

Finally, we attain the recursive formula
$T_{n+1}(x)=2\,x\,T_{n}(x)-T_{n-1}(x)$.

This recursive relationship can also be derived from the generating function
for the Chebyshev polynomials (Property 5) by differentiating it twice with
respect to $z$ and equating the coefficients for $z^{n}$ on both sides of the
equation. We will obtain the desired result accordingly.

######  Property 8 (Orthogonality).

The Chebyshev polynomials $T_{n}(x)$ form a complete orthogonal set on the
interval $-1\leq x\leq 1$ with a weighted function $1/\sqrt{1-x^{2}}$.

This means

$\int_{-1}^{1}\,\frac{1}{\sqrt{1-x^{2}}}\,T_{m}(x)\,T_{n}(x)\,dx=\left\\{{\begin{array}[]{*{20}c}0,\\\
\pi,\\\ \frac{\pi}{2},\\\
\end{array}}\right.\qquad\left.{\begin{array}[]{l}m\neq n\\\ m=n=0\\\
m=n=1,2,3,\dots\end{array}}\right.$ (10)

Using this orthogonality property, a piecewise continuous function on interval
$-1\leq x\leq 1$ can be expressed in terms of the Chebyshev polynomials

$\sum_{n=0}^{\infty}\,C_{n}\,T_{n}(x)=\left\\{{\begin{array}[]{*{20}c}f(x),\\\
\frac{1}{2}\,[f(x^{-})+f(x^{+})],\\\
\end{array}}\right.\qquad\left.{\begin{array}[]{l}\textmd{for continuous
$f(x)$ }\\\ \textmd{at discontinuous points}\end{array}}\right.$ (11)

where

$C_{0}=\frac{1}{\pi}\,\int_{-1}^{1}\,\frac{f(x)}{\sqrt{1-x^{2}}}\,dx\qquad\textmd{and}\qquad
C_{n}=\frac{2}{\pi}\,\int_{-1}^{1}\,\frac{1}{\sqrt{1-x^{2}}}\,f(x)\,T_{n}(x)\,dx,\qquad\textmd{for}\quad
n\in\mathbb{N}.$ (12)

######  Property 9 (Parseval’s identity).

Parseval’s identity related to the series expansion ${\displaystyle
f(x)=\sum_{n=0}^{\infty}\,C_{n}\,T_{n}(x)}$ with the Chebyshev polynomials
$T_{n}(x)$ is given by

$\int_{-\infty}^{\infty}\,\frac{1}{\sqrt{1-x^{2}}}\,[f(x)]^{2}\,dx=C_{0}^{2}\,\pi+\frac{\pi}{2}\,\sum_{n=1}^{\infty}\,C_{n}^{2}.$
(13)

###### Proof.

The first step is by taking the square of both sides of the series expansion
${\displaystyle f(x)=\sum_{n=0}^{\infty}\,C_{n}\,T_{n}(x)}$ to yield

$[f(x)]^{2}=\sum_{n=0}^{\infty}\,\sum_{m=0}^{\infty}\,C_{n}\,C_{m}\,T_{n}(x)\,T_{m}(x).$
(14)

We then multiply (14) with the weighted function $1/\sqrt{1-x^{2}}$ and
integrate it with respect to the variable $x$ from $-\infty$ to $\infty$ to
obtain

$\int_{-\infty}^{\infty}\,\frac{1}{\sqrt{1-x^{2}}}\,[f(x)]^{2}\,dx=\sum_{n=0}^{\infty}\,\sum_{m=0}^{\infty}\,C_{n}\,C_{m}\,\int_{-\infty}^{\infty}\,\frac{1}{\sqrt{1-x^{2}}}T_{n}(x)\,T_{m}(x)\,dx.$
(15)

Using Property 8, we can write (15) as follows:

$\int_{-\infty}^{\infty}\,\frac{1}{\sqrt{1-x^{2}}}\,[f(x)]^{2}\,dx=C_{0}^{2}\,\pi+\frac{\pi}{2}\,\sum_{n=1}^{\infty}\,C_{n}^{2}$

which gives Parseval’s identity for the Chebyshev polynomials.

##  APPLICATION

The Chebyshev polynomials play an important role in function approximation,
integral approximation, and polynomial interpolation problem. For a periodic
function $f(x)$ on a particular interval, a Fourier series is more superior in
approximating it with a relatively tiny error. However, in applications, we
often encounter non-periodic functions which are only defined in a limited
interval. The Chebyshev polynomials have a substantial contribution to these
cases [5, 6]. By approximating the function $f(x)$ with the Chebyshev
polynomials $T_{n}(x)$, we obtain the following interpolation function
$P_{N}(x)$:

$f(x)\approx P_{N}(x)=\sum_{n=0}^{N}\,C_{n}\,T_{n}(x).$ (16)

We must select the coefficients $C_{n}$ such that the norm
$\|f(x)-P_{n}(x)\|_{2}$ is minimum. By seeking extreme values (in this case
minimum values) of the $L^{2}$-norm, we obtain the coefficients $C_{n}$ as in
(12).

Figure 2: Plot of a seventh-order polynomial
$y=f(x)=x^{7}-14x^{5}+49x^{3}-36x$ (blue curve) approximated by interpolation
functions corresponding to the Chebyshev polynomials with $N=7$ (red curve on
the left panel) and a Fourier series with $N=20$ (red curve on the right
panel). We observe that for an $N^{\text{th}}$-order polynomial, a Chebyshev
approximation with $N$ terms provides an accurate interpolation while more
terms might be needed in a Fourier series approximation for a comparable
qualitative accuracy.

An $N^{\text{th}}$-order interpolation function using the Chebyshev
polynomials provides an accurate approximation for any $N^{\text{th}}$-order
polynomial with zero error. Approximating a polynomial using the Fourier
series requires more terms to reach a qualitative comparable accuracy. Figure
2 illustrates an example for a seventh-order polynomial
$y=f(x)=x^{7}-14x^{5}+49x^{3}-36x$ when it is approximated by the Chebyshev
polynomials and Fourier series. For the former, taking $N=7$ in the
interpolation function $P_{N}(x)$ provides an accurate interpolation. For the
latter, the polynomial could be interpolated with a relatively tiny but
visible error by taking at least $N=10$.

For non-polynomial functions, we observe that an interpolation function using
Fourier series converges faster than the one employing the Chebyshev
polynomials. Figure 3 illustrates another example of a step (Heaviside)
function approximated by the Chebyshev polynomials and Fourier series on the
interval $-1\leq x\leq 1$. We need up to $N=20$ in the partial sum for the
former while it is sufficient to include up to $N=10$ for the latter to reach
a qualitative comparable accuracy. Note that for this particular case, the
chosen step function is odd, and so is the integrands in (12) for $n$ is even.
Thus, the coefficients $C_{n}=0$ for $n$ is even. This means that although we
take $N=20$ for the interpolation function, only 10 nonzero terms contribute
to the approximation, i.e., only the odd terms of the Chebyshev polynomial.
For both cases, we observe the occurrence of the Gibbs phenomenon.111The Gibbs
phenomenon is an event where a jump or a deviation occurs near discontinous
points of a piecewise continuous function when the function is approximated by
Fourier series or other function series.

Figure 3: A step function (blue curve) approximated by interpolation functions
corresponding to the Chebyshev polynomials with $N=20$ (red curve on the left
panel) and a Fourier series with $N=10$ (red curve on the right panel). For
non-polynomial functions, more terms are required in the interpolation
function for the Chebyshev polynomials to reach a similar qualitative accuracy
with the one using the Fourier series.

##  CONCLUSION

In this paper, we have observed that the Chebyshev differential equation is a
special case of the Sturm-Liouville boundary value problem. Its solutions, the
Chebyshev polynomials, have properties which are also found in other
polynomials. One of these properties is orthogonoality, which plays an
important role in approximating functions, similar to the well-known Fourier
series. Similar to the case in the Fourier series, approximating piecewise
continuous functions using the Chebyshev polynomials also produces the Gibbs
phenomenon. When approximating polynomials, the Chebyshev polynomials exhibit
a superiority in comparison to the Fourier series due to its accuracy.
However, the Fourier series seems to converge faster in interpolating non-
polynomial functions than the Chebyshev polynomials.

### ACKNOWLEDGMENT

The author thanked Hendra Gunawan (Bandung Institute of Technology), Intan
Detiena Muchtadi-Alamsyah (The University of Picardy Jules Verne, Amiens), and
Natanael Dewobroto (Paul Verlaine University, The University of Lorraine,
Metz) for fruitful discussion, constructive advice, and valuable suggestion.

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