Anti-Gaussian quadrature rule for trigonometric polynomials

Abstract

In this paper, anti-Gaussian quadrature rules for trigonometric polynomials are introduced. Special attention is paid to an even weight function on [-?, ?). The main properties of such quadrature rules are proved and a numerical method for their construction is presented. That method is based on relations between nodes and weights of the quadrature rule for trigonometric polynomials and the quadrature rule for algebraic polynomials. Some numerical examples are included. Also, we compare our method with other available methods.

Full text

Filomat 36:3 (2022), 1005–1019
https://doi.org/10.2298/FIL2203005P
Published by Faculty of Sciences and Mathematics,
University of Niˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Anti-Gaussian Quadrature Rule for Trigonometric Polynomials
Nevena Z. Petrovi´ca, Marija P. Stani´ca, Tatjana V. Tomovi ´c Mladenovi´ca
aUniversity of Kragujevac, Faculty of Science, Department of Mathematics and Informatics, Radoja Domanovi´ca 12, 34000 Kragujevac, Serbia
Abstract. In this paper, anti-Gaussian quadrature rules for trigonometric polynomials are introduced.
Special attention is paid to an even weight function on [−π, π). The main properties of such quadrature
rules are proved and a numerical method for their construction is presented. That method is based
on relations between nodes and weights of the quadrature rule for trigonometric polynomials and the
quadrature rule for algebraic polynomials. Some numerical examples are included. Also, we compare our
method with other available methods.
1. Introduction
Let ube a given weight function over an interval [a,b]. By Pnand Tn,n∈N0, we denote the linear space
of all algebraic and trigonometric polynomials of degree less than or equal to n, respectively. Let Gnbe the
corresponding n-point Gaussian quadrature rule:
Gn(f)=
n
X
k=1
ωkf(xk)
of degree 2n−1 for the integral
I(f)=Zb
a
f(x)u(x) dx.
The formula Gnhas property Gn(p)=I(p), for all p∈ P2n−1. This famous method for numerical integration
was derived by C.F. Gauss in 1814 in [7]. During the period of more than 200 years, such rules were
considered by large number of mathematicians, and were generalized in different ways.
For an arbitrary function f, it may be very difficult to determine an accurate estimate of the error
I(f)−Gn(f). By using some quadrature rule Awith at least n+1 additional points and degree greater
than 2n−1, this error can be estimated by difference A(f)−Gn(f). In order to increase the accuracy of the
approximation value of a desired integral, Laurie [14] introduced an anti-Gaussian quadrature rule, with
2020 Mathematics Subject Classification. Primary 65D32
Keywords. anti-Gaussian quadrature rules, recurrence relation, averaged Gaussian formula
Received: 27 July 2021; Accepted: 08 October 2021
Communicated by Miodrag Spalevi´
c
This work was supported by the Serbian Ministry of Education, Science and Technological Development (Agreement No. 451-03-
9/2021-14/200122).
Email addresses: nevena.z.petrovic@pmf.kg.ac.rs (Nevena Z. Petrovi´
c), marija.stanic@pmf.kg.ac.rs (Marija P. Stani´
c),
tatjana.tomovic@pmf.kg.ac.rs (Tatjana V. Tomovi ´
c Mladenovi´
c)

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1006
algebraic degree of exactness 2n+1, that gives an error equal in magnitude but of opposite sign to that of
the corresponding Gaussian rule. A similar idea has been used in the numerical solution of initial value
problems in ordinary differential equations [5, 18, 20].
The anti-Gaussian quadrature rule
Hn+1(f)=
n+1
X
k=1
λkf(ξk)
is an (n+1)-point formula such that I(p)−Hn+1(p)=−(I(p)−Gn(p)), for all p∈ P2n+1. Here, we see that
Hn+1(p)=(2I−Gn)(p), for each p∈ P2n+1. Because of that, we introduce the following inner product:
f,1u=(2I−Gn)( f1).(1)
Let us denote by (pk) the sequence of monic polynomials orthogonal with respect to the inner product
f,1u=I(f1). It is well known that such polynomials satisfy the following three-term recurrence relation
of the following form (see, e.g., [8]):
pk+1(x)=(x−ak)pk(x)−bkpk−1(x),k=0,1,..., p0(x)=1,p−1(x)=0.
Let (πk) be the sequence of monic polynomials orthogonal with respect to the inner product (1). They also
satisfy the three-term recurrence relation (see [14]):
πk+1(x)=(x−αk)πk(x)−βkπk−1(x),k=0,1,...,n, π0(x)=1, π−1(x)=0.
Coefficients {ak|k=0,1, . . .},{bk|k=1,2, . . .},{αk|k=0,1,...,n}and {βk|k=1,...,n}are given by:
ak=I(xp2
k)
I(p2
k),bk=I(p2
k)
I(p2
k−1), αk=(2I−Gn)(xπ2
k)
(2I−Gn)(π2
k), βk=(2I−Gn)(π2
k)
(2I−Gn)(π2
k−1).
It is easy to see that these coefficients satisfy:
αk=ak,k=0,...,n;βk=bk,k=0,...,n−1; βn=2bn,
and, therefore πk=pk, for k=0,1,...,n(see [14]).
Modified and generalized anti-Gaussian quadrature rules for real-valued measures were investigated
in [2] and [17]. Other generalizations to matrix-valued measures were given in [6] and [1]. Applications
of anti-Gaussian quadrature rules for the estimates of the error I(f)−Gn(f) can be found in [23]. Based on
anti-Gaussian quadrature rule, Laurie in [14] introduced an averaged Gaussian rule. Different averaged
Gaussian quadrature rules relative to those introduced by Laurie in [14] can be found in [19, 22, 24, 25].
Sun-mi Kim and Reichel in [12] introduced anti-Szeg˝
o quadrature rules which were characterized by
the property that the quadrature error for Laurent polynomials of order at most nis a specified negative
multiple of the quadrature error obtained with the n-node Szeg˝
o rule. By using Szeg ˝
o and anti-Szeg˝
o
quadrature rules, Jages, Reichel, and Tang in [13] defined generalized averaged Szeg˝
o quadrature rules
which are exact for all Laurent polynomial in Λ−n+1,n−1.
In this paper, we combine two different types of generalizations of Gaussian rules: generalizations to
the rules which are exact on spaces of functions different from the space of algebraic polynomials as well
as generalizations to an anti-Gaussian quadrature rule. Our attention is restricted to the anti-Gaussian
quadrature rules with trigonometric degree of exactness for an even weight function on [−π, π). Also, we
introduce the averaged Gaussian quadrature formula with trigonometric degree of exactness.
The paper is organized as follows. In Section 2 we introduce and consider an anti-Gaussian quadrature
rule for trigonometric polynomials. In Section 3, we present method for numerical construction of the
anti-Gaussian quadrature rules with respect to symmetric weight function. Some numerical examples and
comparisons with other available methods are presented in Section 4.

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1007
2. Anti-Gaussian quadrature rule for trigonometric polynomials
Let wbe a weight function, integrable and nonnegative on the interval [−π, π). For every nonnegative
integer nand γ∈ {0,1/2}by Tn,γ we denote the linear span of the set {cos (k+γ)x,sin (k+γ)x|k=0,1,...,n}.
Thus, Tn,0=Tnis the linear space of all trigonometric polynomials of degree less than or equal to n, and
Tn,1/2is the linear space of all trigonometric polynomials of semi-integer degree less than or equal to n+1/2.
Let us consider the integral:
b
I(f)=Zπ
−π
f(x)w(x) dx.
By (Ak,γ), Ak,γ ∈ Tn,γ ,k=0,1, . . ., we denote the sequence of trigonometric polynomials orthogonal on
[−π, π) with respect to the inner product
f,1w=b
I(f1).(2)
The corresponding Gaussian quadrature rule
b
Ge
n+1(f)=e
n
X
k=0
ωkf(xk),e
n=2n−1+2γ, γ ∈ {0,1/2},(3)
is exact for every t∈ T2n−1+2γif and only if the nodes xk,k=0,1,...,e
nare the zeros of the trigonometric
polynomial An,γ ∈ Tn,γ (see [3, 4, 16, 26–28]).
For the fixed positive integer n, we introduce the inner product (∗,∗)ωas follows:
(f,1)w=(2
b
I−b
Ge
n+1)( f1).(4)
Remark 2.1. It is important to emphasize that the inner product (4) depends on the number n. Because of that, in
what follows when we write that positive integer n is fixed in advance, we assume that we consider the inner product
(4) with respect to that fixed number n.
Let positive integer nbe fixed in advance. By (Bk,γ ) we denote the sequence of trigonometric polynomials
orthogonal with respect to the inner product (4). The corresponding Gaussian rule has e
n+1 nodes, while
the corresponding anti-Gaussian rule (where n7→ n+1) has e
n+3 nodes (so we denote it by b
He
n+3, and it is
exact for all t∈ T2n+1+2γ), which are the zeros of the trigonometric polynomial
Bn+1,γ(x)=
n+1
X
k=0pk,γ cos k+γx+qk,γ sin k+γx,|pn+1,γ|+|qn+1,γ |,0.
Specially, for (pn+1,γ ,qn+1,γ)=(1,0) and (pn+1,γ ,qn+1,γ)=(0,1) we get
BC
n+1,γ(x)=cos n+1+γx+
n
X
k=0p(n+1)
k,γ cos k+γx+q(n+1)
k,γ sin k+γx
and
BS
n+1,γ(x)=sin n+1+γx+
n
X
k=0r(n+1)
k,γ cos k+γx+s(n+1)
k,γ sin k+γx,
respectively. For k, ` ∈N0we define:
b
IC,γ
k=BC
k,γ,BC
k,γw,b
IS,γ
k=BS
k,γ,BS
k,γw,b
Iγ
k=BC
k,γ,BS
k,γw,
b
JC,γ
k, ` =2 cos xBC
k,γ,BC
`,γw,b
JS,γ
k, ` =2 cos xBS
k,γ,BS
`,γw,b
Jγ
k, ` =2 cos xBC
k,γ,BS
`,γw.

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1008
By IC,γ
k,IS,γ
k,Iγ
k,JC,γ
k, ` ,JS,γ
k, ` ,Jγ
k, `,k, ` ∈N0, we denote the corresponding values related to the inner product (2)
(e.g., IC,γ
k=AC
k,γ,AC
k,γw, . . . ).
Theorem 2.2. Let positive integer n be fixed in advance. Trigonometric polynomials BC
k,γ(x)and BS
k,γ(x), k >1,
orthogonal with respect to the inner product (4), satisfy the following recurrence relations:
BC
k,γ(x)=2 cos x−a(1)
k,γBC
k−1,γ(x)−b(1)
k,γBS
k−1,γ(x)−a(2)
k,γBC
k−2,γ(x)−b(2)
k,γBS
k−2,γ(x),(5)
BS
k,γ(x)=2 cos x−d(1)
k,γBS
k−1,γ(x)−c(1)
k,γBC
k−1,γ(x)−d(2)
k,γBS
k−2,γ(x)−c(2)
k,γBC
k−2,γ(x),(6)
where the coefficients are the solutions of the following systems for j =1,2:
b
JC,γ
k−1,k−j=a(j)
k,γb
IC,γ
k−j+b(j)
k,γb
Iγ
k−j,b
Jγ
k−1,k−j=a(j)
k,γb
Iγ
k−j+b(j)
k,γb
IS,γ
k−j,(7)
b
Jγ
k−1,k−j=c(j)
k,γb
IC,γ
k−j+d(j)
k,γb
Iγ
k−j,b
JS,γ
k−1,k−j=c(j)
k,γb
Iγ
k−j+d(j)
k,γb
IS,γ
k−j,
with a(2)
1,γ =b(2)
1,γ =c(2)
1,γ =d(2)
1,γ =0.
Proof. The polynomials BC
j,γ(x) and BS
j,γ(x), for j=0,...,k, are linearly independent, so we have
2 cos xBC
k−1,γ(x)=BC
k,γ(x)+
k−1
X
j=0a(k−j)
k,γ BC
j,γ(x)+b(k−j)
k,γ BS
j,γ(x).(8)
Applying the inner product (4) for i=0,1,...,k−1 we get:
2 cos xBC
k−1,γ(x),BC
i,γ(x)w=BC
k,γ(x),BC
i,γ(x)w+
k−1
X
j=0
a(k−j)
k,γ BC
j,γ(x),BC
i,γ(x)w+
k−1
X
j=0
b(k−j)
k,γ BS
j,γ(x),BC
i,γ(x)w.
Now, for i=0,...,k−3, one has:
a(k−i)
k,γ b
IC,γ
i+b(k−i)
k,γ b
Iγ
i=0.(9)
Similarly, for i=0,1,...,k−1 we get:
2 cos xBC
k−1,γ(x),BS
i,γ(x)w=BC
k,γ(x),BS
i,γ(x)w+
k−1
X
j=0
a(k−j)
k,γ BC
j,γ(x),BS
i,γ(x)w+
k−1
X
j=0
b(k−j)
k,γ BS
j,γ(x),BS
i,γ(x)w.
Again, for i=0,...,k−3, one has
a(k−i)
k,γ b
Iγ
i+b(k−i)
k,γ b
IS,γ
i=0.(10)
Therefore, we get homogeneous systems of linear equations:
b
IC,γ
i·a(k−i)
k,γ +b
Iγ
i·b(k−i)
k,γ =0,
b
Iγ
i·a(k−i)
k,γ +b
IS,γ
i·b(k−i)
k,γ =0,i=0,...,k−3,(11)
and the corresponding determinants of these systems are given by
b
Dγ
i=b
IC,γ
ib
Iγ
i
b
Iγ
ib
IS,γ
i.

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1009
Sinceb
I(f)=b
Ge
n+1(f) for all f∈ Te
n, we have
b
He
n+3(f)=(2
b
I−b
Ge
n+1)( f)=b
I(f)+(
b
I−b
Ge
n+1)( f)=b
I(f) for each f∈ Te
n.
Further, BC
k,γ(x)=AC
k,γ(x) and BS
k,γ(x)=AS
k,γ(x), for every k=0,...,n. For 2k+1<2nit follows:
b
IC,γ
k=BC
k,γ,BC
k,γw=(2
b
I−b
Ge
n+1)BC
k,γ ·BC
k,γ=b
IBC
k,γ ·BC
k,γ=IC,γ
k.
Similarly, b
IS,γ
k=IS,γ
kand b
Iγ
k=Iγ
k. Now, using Cauchy-Schwarz-Bunyakovsky integral inequality, for
i=0,...,k−3, we have:
b
Dγ
i=
b
IC,γ
ib
IS,γ
i−(
b
Iγ
i)2=IC,γ
iIS,γ
i−(Iγ
i)2
= Zπ
−πAC
i,γ(x)2w(x) dx!· Zπ
−πAS
i,γ(x)2w(x) dx!− Zπ
−π
AC
i,γ(x)AS
i,γ(x)w(x) dx!2
> Zπ
−πAC
i,γ(x)2w(x) dx!· Zπ
−πAS
i,γ(x)2w(x) dx!− Zπ
−πAC
i,γ(x)2w(x) dx!· Zπ
−πAS
i,γ(x)2w(x) dx!
=0.
Therefore, for all i=0,...,k−3 the system (11) has only the trivial solutions a(k−i)
k,γ =b(k−i)
k,γ =0. Thus, equality
(8) gets the following form
2 cos xBC
k−1,γ(x)=BC
k,γ(x)+a(1)
k,γBC
k−1,γ(x)+b(1)
k,γBS
k−1,γ(x)+a(2)
k,γBC
k−2,γ(x)+b(2)
k,γBS
k−2,γ(x),(12)
i.e., form (5). Analogously, one can obtain the recurrence relation (6) for BS
k,γ(x).
Using recurrence relation (12), for j=1,2, we have:
2 cos xBC
k−1,γ(x),BC
k−j,γ(x)w=a(1)
k,γ BC
k−1,γ(x),BC
k−j,γ(x)w+b(1)
k,γ BS
k−1,γ(x),BC
k−j,γ(x)w
+a(2)
k,γ BC
k−2,γ(x),BC
k−j,γ(x)w+b(2)
k,γ BS
k−2,γ(x),BC
k−j,γ(x)w,
i.e., b
JC,γ
k−1,k−j=a(j)
k,γb
IC,γ
k−j+b(j)
k,γb
Iγ
k−j,j=1,2.
Analogously, one gets b
Jγ
k−1,k−j=a(j)
k,γb
Iγ
k−j+b(j)
k,γb
IS,γ
k−j,j=1,2.
Similarly, starting from (6) one can obtain b
Jγ
k−1,k−j=c(j)
k,γb
IC,γ
k−j+d(j)
k,γb
Iγ
k−j,b
JS,γ
k−1,k−j=c(j)
k,γb
Iγ
k−j+d(j)
k,γb
IS,γ
k−j,j=1,2.
Using the recurrence relations (5), (6) and orthogonality conditions, the following Lemma could be
easily proved.
Lemma 2.3. For k >1the following equations hold:
b
IC,γ
k=b
JC,γ
k,k−1,b
IS,γ
k=b
JS,γ
k,k−1,b
Iγ
k=b
Jγ
k,k−1.
Let us denote b
JC,γ
k=b
JC,γ
k,k,b
JS,γ
k=b
JS,γ
k,k,b
Jγ
k=b
Jγ
k,k.
Corollary 2.4. The recurrence coefficients in (5) and (6) are given as follows:
a(1)
k,γ =b
IS,γ
k−1b
JC,γ
k−1−b
Iγ
k−1b
Jγ
k−1
b
Dγ
k−1
,a(2)
k,γ =b
IC,γ
k−1b
IS,γ
k−2−b
Iγ
k−1b
Iγ
k−2
b
Dγ
k−2
,b(1)
k,γ =b
IC,γ
k−1b
Jγ
k−1−b
Iγ
k−1b
JC,γ
k−1
b
Dγ
k−1
,b(2)
k,γ =b
Iγ
k−1b
IC,γ
k−2−b
IC,γ
k−1b
Iγ
k−2
b
Dγ
k−2
,
c(1)
k,γ =b
IS,γ
k−1b
Jγ
k−1−b
Iγ
k−1b
JS,γ
k−1
b
Dγ
k−1
,c(2)
k,γ =b
Iγ
k−1b
IS,γ
k−2−b
IS,γ
k−1b
Iγ
k−2
b
Dγ
k−2
,d(1)
k,γ =b
IC,γ
k−1b
JS,γ
k−1−b
Iγ
k−1b
Jγ
k−1
b
Dγ
k−1
,d(2)
k,γ =b
IS,γ
k−1b
IC,γ
k−2−b
Iγ
k−1b
Iγ
k−2
b
Dγ
k−2
,
where b
Dγ
k−j=b
IC,γ
k−jb
IS,γ
k−j−(
b
Iγ
k−j)2, j =1,2, for k >1.

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1010
Proof. Formulas can be obtained solving systems (7).
Let positive integer nbe fixed in advance. Since BC
k,γ(x)=AC
k,γ(x) and BS
k,γ(x)=AS
k,γ(x), for every
k=0,...,n, we have:
a(j)
k,γ =α(j)
k,γ,b(j)
k,γ =β(j)
k,γ,c(j)
k,γ =γ(j)
k,γ,d(j)
k,γ =δ(j)
k,γ,j=1,2,(13)
where α(j)
k,γ,β(j)
k,γ,γ(j)
k,γ and δ(j)
k,γ are the coefficients in the corresponding recurrence relations for polynomials
AC
k,γ and AS
k,γ (see [16, 28]).
Lemma 2.5. Let positive integer n be fixed in advance. For k =0,...,n−1the following equations
b
IS,γ
k=IS,γ
k,b
JS,γ
k=JS,γ
k,b
IC,γ
k=IC,γ
k,b
JC,γ
k=JC,γ
k,b
Iγ
k=Iγ
k,b
Jγ
k=Jγ
k,
hold, while
b
IS,γ
n=2IS,γ
n,b
JS,γ
n=2JS,γ
n,b
IC,γ
n=2IC,γ
n,b
JC,γ
n=2JC,γ
n,b
Iγ
n=2Iγ
n,b
Jγ
n=2Jγ
n,
and for k =n+1coefficients in the five-term recurrence relations (5) and (6) satisfy
a(1)
n+1,γ =α(1)
n+1,γ,a(2)
n+1,γ =2α(2)
n+1,γ,b(1)
n+1,γ =β(1)
n+1,γ,b(2)
n+1,γ =2β(2)
n+1,γ,
c(1)
n+1,γ =γ(1)
n+1,γ,c(2)
n+1,γ =2γ(2)
n+1,γ,d(1)
n+1,γ =δ(1)
n+1,γ,d(2)
n+1,γ =2δ(2)
n+1,γ.(14)
Proof. For k=0,...,n−1, we have
b
Iγ
k=BC
k,γ ,BS
k,γw=2
b
I−b
Ge
n+1BC
k,γBS
k,γ=2
b
I−b
Ge
n+1AC
k,γAS
k,γ
=
b
IAC
k,γAS
k,γ+b
I−b
Ge
n+1AC
k,γAS
k,γ=DAC
k,γ ,AS
k,γEw=Iγ
k.
In a similar way, one can prove the other equalities for k6n−1.
Due to the fact that the nodes of the Gaussian quadrature rule are the zeros of trigonometric polynomial
An,γ(x), for k=n, we have
b
Iγ
n=BC
n,γ ,BS
n,γw=(2
b
I−b
Ge
n+1)BC
n,γBS
n,γ=(2
b
I−b
Ge
n+1)AC
n,γAS
n,γ=2Iγ
n.
Analogously, the other equalities can be proved.
Using the previous equalities, we deduce:
b
Dγ
k=b
IC,γ
kb
IS,γ
k−(
b
Iγ
k)2=IC,γ
kIS,γ
k−(Iγ
k)2=Dγ
k,k=0,...,n−1,
b
Dγ
n=b
IC,γ
nb
IS,γ
n−(
b
Iγ
n)2=2IC,γ
n·2IS,γ
n−(2Iγ
n)2=4Dγ
n,
where Dγ
k=IC,γ
kIS,γ
k−(Iγ
k)2,k=0,1,...,n.
Finally, for the coefficients a(1)
n+1,γ and a(2)
n+1,γ we have
a(1)
n+1,γ =b
IS,γ
nb
JC,γ
n−b
Iγ
nb
Jγ
n
b
Dγ
n
=2IS,γ
n·2JC,γ
n−2Iγ
n·2Jγ
n
4·Dγ
n
=α(1)
n+1,γ
and
a(2)
n+1,γ =b
IC,γ
nb
IS,γ
n−1−b
Iγ
nb
Iγ
n−1
b
Dγ
n−1
=2IC,γ
nIS,γ
n−1−2Iγ
nIγ
n−1
Dγ
n−1
=2α(2)
n+1,γ.
The equalities for other coefficients can be proved in a similar way.

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1011
3. Numerical construction of anti-Gaussian quadrature rule
In this section we consider only even weight functions won (−π, π). Let positive integer nbe fixed in
advance. Using equalities BC
k,γ(x)=AC
k,γ(x), BS
k,γ(x)=AS
k,γ(x), for k=0,...,n, and (14), we get
BC
n+1,γ(x)=2 cos x−a(1)
n+1,γBC
n,γ(x)−b(1)
n+1,γBS
n,γ(x)−a(2)
n+1,γBC
n−1,γ(x)−b(2)
n+1,γBS
n−1,γ(x)
=2 cos x−α(1)
n+1,γAC
n,γ(x)−β(1)
n+1,γAS
n,γ(x)−2α(2)
n+1,γAC
n−1,γ(x)−2β(2)
n+1,γAS
n−1,γ(x)
=AC
n+1,γ(x)−α(2)
n+1,γAC
n−1,γ(x)−β(2)
n+1,γAS
n−1,γ(x).
Due to [16, Lemma 4.1.] and [28, Lemma 2.5.] we know that β(j)
k,γ =0, j=1,2, for each k∈N, so β(2)
n+1,γ is
equal to zero. Now, because of the form AC
k,γ(x)=
k
P
j=0
u(k)
j,γ cos j+γx, with u(k)
k,γ =1, we have:
BC
n+1,γ(x)=AC
n+1,γ(x)−α(2)
n+1,γAC
n−1,γ(x)=
n+1
X
k=0
p(n+1)
k,γ cos k+γx,
with p(n+1)
n+1,γ =1. Similarly, using (14), [16, Lemma 4.1] and [28, Lemma 2.5.], we get:
BS
n+1,γ(x)=AS
n+1,γ(x)−δ(2)
n+1,γAS
n−1,γ(x)=
n+1
X
k=0
s(n+1)
k,γ sin k+γx,
with s(n+1)
n+1,γ =1.
Also, polynomials BC
n+1,γ and BS
n+1,γ satisfy the following three-term recurrence relations:
BC
n+1,γ(x)=2 cos x−a(1)
n+1,γBC
n,γ(x)−a(2)
n+1,γBC
n−1,γ(x),(15)
BS
n+1,γ(x)=2 cos x−d(1)
n+1,γBS
n,γ(x)−d(2)
n+1,γBS
n−1,γ(x).(16)
The following result is obvious.
Lemma 3.1. For every k ∈N0we have
BC
k+1,γ(π)=0,BS
k+1,γ(0) =0.
3.1. Quadrature rule with an even number of nodes
Let positive integer nbe fixed in advance. For γ=0 we get a quadrature formula with an even number
of nodes.
Let us introduce the following notation for k=0,1, . . .:
Ck,0(x)=
k
X
j=0
p(k)
j,0Tj(x),Sk,0(x)=
k
X
j=0
s(k)
j,0Uj−1(x),
u1(x)=w(arccos x)
√1−x2,u2(x)=√1−x2w(arccos x),
where Tk(x)=cos (k·arccos x) and Uk(x)=sin ((k+1) arccos x)/√1−x2,x∈(−1,1), are Chebyshev poly-
nomials of the first and second kind, respectively. Also, by τ(i)
kand σ(i)
k,k=1,...,n, we denote the nodes
and weights of the Gaussian quadrature rule constructed for the algebraic polynomials with respect to the
weight function ui(x), i=1,2.
The connections of quadrature rule (3) with certain Gaussian quadrature rule for algebraic polynomials
in the case γ=0 were considered in [27]. Using these connections we get the following results.

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1012
Theorem 3.2. Let positive integer n be fixed in advance. For an even weight function w(x)on (−π, π), the following
equalities
Cn,0(x),Ck,0(x)u1=0and Sn,0(x),Sk,0(x)u2=0 (17)
hold for all k =0,1,...,n−1.
Proof. Using [27, Lemma 2.4.], orthogonality of the polynomials BC
k,0(x) and substitution x=arccos t, for
each k=0,...,n−1, we have
0=BC
n,0(x),BC
k,0(x)w=2
b
I−b
G2nBC
n,0(x)BC
k,0(x)=4Zπ
0
BC
n,0(x)BC
k,0(x)w(x) dx−
2n−1
X
j=0
ωjBC
n,0(xj)BC
k,0(xj)
=4Z1
−1
BC
n,0(arccos t)BC
k,0(arccos t)w(arccos t)
√1−t2dt−2
n
X
j=1
σ(1)
jBC
n,0arccos τ(1)
jBC
k,0arccos τ(1)
j
=2(2I−Gn)BC
n,0(arccos x)BC
k,0(arccos x)=2BC
n,0(arccos x),BC
k,0(arccos x)u1.
Now, using BC
k,0(arccos x)=
k
P
j=0
p(k)
j,0cos (jarccos x)=
k
P
j=0
p(k)
j,0Tj(x), for all k=0,...,n−1 we get:
BC
n,0(arccos x),BC
k,0(arccos x)u1
=







n
X
j=0
p(n)
j,0Tj(x),
k
X
j=0
p(k)
j,0Tj(x)






u1
=Cn,0(x),Ck,0(x)u1,
i.e., (Cn(x),Ck(x))u1=0.
The second equality can be proved in the similar way, using [27, Lemma 2.5.]:
0=BS
n,0(x),BS
k,0(x)w=2
b
I−b
G2nBS
n,0(x)BS
k,0(x)=4Zπ
0
BS
n,0(x)BS
k,0(x)w(x) dx−
2n−1
X
j=0
ωjBS
n,0(xj)BS
k,0(xj)
=2·
2Z1
−1
BS
n,0(arccos t)
√1−t2·BS
k,0(arccos t)
√1−t2u2(t) dt−
n
X
j=1
σ(2)
j
BS
n,0arccos τ(2)
j
r1−τ(2)
j2·
BS
k,0arccos τ(2)
j
r1−τ(2)
j2

=2·(2I−Gn)






BS
n,0(arccos x)
√1−x2·BS
k,0(arccos x)
√1−x2





=2






BS
n,0(arccos x)
√1−x2,BS
k,0(arccos x)
√1−x2





u2
.
Now, using
BS
k,0(arccos x)=
k
X
j=0
s(k)
j,0sin (jarccos x)=
k
X
j=0
s(k)
j,0√1−x2Uk−1(x),
i.e.,
BS
k,0(arccos x)
√1−x2
=
k
X
j=0
s(k)
j,0Uj−1(x),
for all k=0,...,n−1 we get:







BS
n,0(arccos x)
√1−x2,BS
k,0(arccos x)
√1−x2





u2
=







n
X
j=0
s(n)
j,0Uj−1(x),
k
X
j=0
s(k)
j,0Uj−1(x)






u2
=Sn,0(x),Sk,0(x)u2,
i.e., (Sn(x),Sk(x))u2=0.

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1013
If we put x:=arccos xin (15), using equality BC
n,0(arccos x)=Cn,0(x), we get:
Cn,0(x)=2x−a(1)
n,0Cn−1,0(x)−a(2)
n,0Cn−2,0(x),a(2)
1,0=0,C0,0(x)=1.
Therefore, the zeros of the polynomial Cn+1,0(x) (and hence the zeros of the trigonometric polynomial
BC
n+1,0(x)) can be calculated using QR-algorithm.
Lemma 3.3. Let w be the even weight function on (−π, π)and let τkand σk, k =1,...,n+1, be the nodes and
weights of the (n+1)-point anti-Gaussian quadrature rule constructed for algebraic polynomials with respect to the
weight function u1(x)=w(arccos x)/√1−x2on (−1,1). Then the weights ωkand the nodes xk, k =0,1,...,2n+1,
of the (2n+2)-point anti-Gaussian quadrature rule with respect to w are given as follows:
ωk=ω2n+1−k=σk+1,k=0,...,n,
xk=−x2n+1−k=−arccos τk+1,k=0,...,n.
Proof. Due to the equality BC
n+1,0(arccos x)=Cn+1,0(x) it is easy to see that the nodes of the anti-Gaussian
quadrature rule for trigonometric polynomials are given by
xk=−x2n+1−k=−arccos τk+1,k=0,...,n.
Weights can be constructed using Shohat formula (see [16, 21]):
σk=µ0
n
X
j=0















Cj,0(τk)
j
Q
i=2
a(2)
i,0















2
−1
=µ0
n
P
j=0








BC
j,0(arccos τk)
j
Q
i=2
a(2)
i,0









2,k=1,...,n+1,
where
µ0=Z1
−1
w(arccos x)
√1−x2dx.
Due to the fact that the function w(x) is even on (−π, π), we have
ω2n+1−k=b
µ0
2·
n
P
j=0








BC
j,0(x2n+1−k)
j
Q
i=2
a(2)
i,0









2,k=0,...,n,
where
b
µ0=Zπ
−π
w(x) dx=2Z1
−1
w(arccos t)dt
√1−t2
=2µ0.
Therefore we get ω2n+1−k=ωk=σk+1, for k=0,...,n.
The following result could be proved by using the similar arguments.
Lemma 3.4. Let w be the even weight function on (−π, π)and let τkand σk, k =1,...,n+1, be the nodes and
weights of the (n+1)-point anti-Gaussian quadrature rule constructed for algebraic polynomials with respect to the
weight function u2(x)=√1−x2w(arccos x)on (−1,1). Then the weights ωkand the nodes xk, k =0,1,...,2n+1,
of the (2n+2)-point anti-Gaussian quadrature rule with respect to w are given as follows:
ωk=ω2n+1−k=σk+1
1−τ2
k+1
,k=0,...,n,
xk=−x2n+1−k=−arccos τk+1,k=0,...,n.

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1014
3.2. Quadrature rule with an odd number of nodes
In the case γ=1/2, we get quadrature rule with an odd number of nodes.
Let us introduce the following notation for k=0,1, . . .:
Ck,1/2(x)=
k
P
j=0
p(k)
j,1/2Tj(x)−(1 −x)Uj−1(x),u3(x)=r1+x
1−xw(arccos x),
Sk,1/2(x)=
k
P
j=0
s(k)
j,1/2Tj(x)+(1 +x)Uj−1(x),u4(x)=r1−x
1+xw(arccos x).
Also, by τ(i)
kand σ(i)
k,k=1,...,n, we denote the nodes and weights of the Gaussian quadrature rule
constructed for algebraic polynomials with respect to the weight function ui(x), i=3,4.
Theorem 3.5. Let positive integer n be fixed in advance. For an even weight function w(x)on (−π, π), the following
equalities
Cn,1/2(x),Ck,1/2(x)u3=0and Sn,1/2(x),Sk,1/2(x)u4=0
hold for all k =0,1,...,n−1.
Proof. Using the following simple trigonometric equality:
cos k+1
2arccos x=Tk(x)r1+x
2−√1−x2·Uk−1(x)r1−x
2,
we have
BC
k,1/2(arccos x)=
k
X
j=0
p(k)
j,1/2cos j+1
2arccos x=r1+x
2Ck,1/2(x),(18)
i.e., BC
k,1/2(arccos x)/√1+x=Ck,1/2(x)/√2. Using these equalities, orthogonality of the polynomials BC
k,1/2(x),
Lemma 3.1, and [16, Lemma 5.2], for every k=0,...,n−1, we get:
0=BC
n,1/2(x),BC
k,1/2(x)w=2
b
I−b
G2n+1BC
n,1/2(x)·BC
k,1/2(x)
=4Zπ
0
BC
n,1/2(x)BC
k,1/2(x)w(x) dx−
2n
X
j=0
ωjBC
n,1/2(xj)BC
k,1/2(xj)
=4Z1
−1
BC
n,1/2(arccos t)
√1+t·BC
k,1/2(arccos t)
√1+tr1+t
1−tw(arccos t) dt
−2
n
X
j=1
σ(3)
j
BC
n,1/2arccos τ(3)
j
q1+τ(3)
j
·
BC
k,1/2arccos τ(3)
j
q1+τ(3)
j
=2Z1
−1
Cn,1/2(t)Ck,1/2(t)u3(t) dt−
n
X
j=1
σ(3)
jCn,1/2(τ(3)
j)Ck,1/2(τ(3)
j)
=(2I−Gn)Cn,1/2(x)Ck,1/2(x)=Cn,1/2(x),Ck,1/2(x)u3,
i.e., Cn,1/2(x),Ck,1/2(x)u3=0.
For the second part, we use the following trigonometric equality
sin k+1
2arccos x=√1−x2Uk−1(x)r1+x
2+Tk(x)r1−x
2,

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1015
and get
BS
k,1/2(arccos x)=
k
X
j=0
s(k)
j,1/2sin j+1
2arccos x=r1−x
2Sn,1/2(x),(19)
i.e., BS
k,1/2(arccos x)/√1−x=Sk,1/2(x)/√2. Now, using these equalities, the orthogonality of the polynomials
BS
k,1/2(x), Lemma 3.1, and [16, Lemma 5.3], for every k=0,...,n−1, we get:
0=BS
n,1/2(x),BS
k,1/2(x)w=Sn,1/2(x),Sk,1/2(x)u4.
If we put x:=arccos xin (15), using equality (18), and dividing both sides by p(1 +x)/2, we get
Cn+1,1/2(x)=2x−a(1)
n+1,1/2Cn,1/2(x)−a(2)
n+1,1/2Cn−1,1/2(x),a(2)
1,1/2=0,C0,1/2(x)=1.(20)
Lemma 3.6. Let w be the even weight function on (−π, π)and let τkand σk, k =1,...,n+1, be the nodes and weights
of the (n+1)-point anti-Gaussian quadrature rule constructed for algebraic polynomials with respect to the weight
function u3(x)=w(arccos x)√1+x/√1−x on (−1,1). Then the weights ωkand the nodes xk, k =0,1,...,2n+2,
of the (2n+3)-point anti-Gaussian quadrature rule with respect to w are given as follows:
ωk=ω2n+2−k−1=σk+1
1+τk+1
,k=0,...,n, ω2n+2=Zπ
−π
w(x) dx−
2n+1
X
k=0
ωk,
xk=−x2n+2−k−1=−arccos τk+1,k=0,...,n,x2n+2=π.
Proof. Using the three-term recurrence relation (20), we can construct the sequence of the polynomials
(Ck,1/2(x)), and using QR-algorithm we can obtain the zeros of the polynomial Cn+1,1/2(x), i.e., nodes τk,
k=1,...,n+1. According to (18) it is easy to see that xk=−x2n+2−k−1=−arccos τk+1,k=0,...,n. Due to
Lemma 3.1 we have x2n+2=π.
The weights can be constructed using Shohat formula (see [16, 21]):
σk=µ0
n
X
j=0















Cj,1/2(τk)
j
Q
i=2
a(2)
i,1/2















2
−1
=µ0(1 +τk)
2
n
P
j=0








BC
j,1/2(x2n+2−k)
j
Q
i=2
a(2)
i,1/2









2,k=1,...,n+1,
where
µ0=Z1
−1r1+x
1−xw(arccos x) dx.
Since the function w(x) is even on (−π, π), we have
ω2n+2−k−1=b
µ0
2
n
P
j=0








BC
j,1/2(x2n+2−k−1)
j
Q
i=2
a(2)
i,1/2









2,k=0,...,n,
where
b
µ0=Zπ
−π
cos2x
2w(x) dx=2Z1
−1
1+t
2w(arccos t)dt
√1−t2
=µ0,

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1016
so we obtain:
ω2n+2−k−1=ωk=σk+1
1+τk+1
,k=0,...,n.
Finally, using equality
2n+2
P
k=0
ωk=
π
R
−π
w(x) dx, we get the last weight
ω2n+2=Zπ
−π
w(x) dx−
2n+1
X
k=0
ωk.
Lemma 3.7. Let w be the even weight function on (−π, π)and let τkand σk, k =1,...,n+1, be the nodes and weights
of the (n+1)-point anti-Gaussian quadrature rule constructed for algebraic polynomials with respect to the weight
function u4(x)=w(arccos x)√1−x/√1+x on (−1,1). Then the weights ωkand the nodes xk, k =0,1,...,2n+2,
of the (2n+3)-point anti-Gaussian quadrature rule with respect to w are given as follows:
ωk=ω2n+2−k=σk+1
1−τk+1
,k=0,...,n, ωn+1=Zπ
−π
w(x) dx−
2n+2
X
k=0
k,n+1
ωk,
xk=−x2n+2−k=−arccos τk+1,k=0,...,n,xn+1=0.
Proof. This lemma can be proved in the same way as the previous one, using polynomials BS
j,1/2(x), relation
(19) and b
µ0=
π
R
−π
sin2x
2w(x) dx.
Finally, using Gaussian ( b
Ge
n+1(f)) and anti-Gaussian (b
He
n+3(f)) quadrature rule, we can introduce the
so-called averaged Gaussian rule for trigonometric polynomials:
b
A2e
n+4(f)=1
2(b
Ge
n+1+b
He
n+3)( f).
4. Numerical examples
4.1. Quadrature formulas with an even number of nodes
The errors in the Gaussian (b
Ge
n+1), anti-Gaussian (b
He
n+3) and averaged (b
A2e
n+4) quadrature rules in the
case
w(x)=1−cos2x,for x∈(−π, π) and
u1=w(arccos x)
√1−x2
=√1−x2,for x∈(−1,1)
for the integrand f(x)=(1 +cos x)(e−x+4/3) are given in Table 1.

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1017
Table 1: Errors in the Gaussian (b
Ge
n+1), anti-Gaussian (b
He
n+3) and averaged ( b
A2e
n+4) formula for w(x)=1−cos2x,x∈(−π, π) and
f(x)=(1 +cos x)(e−x+4/3)
e
n+1b
I(f)−b
Ge
n+1(f)b
I(f)−b
He
n+3(f)b
I(f)−b
A2e
n+4(f)
80 −9.30463 ·10−98.99386 ·10−9−1.55389 ·10−10
60 −4.97942 ·10−84.82213 ·10−8−7.86464 ·10−10
40 −5.16734 ·10−75.00653 ·10−7−8.04024 ·10−9
20 −0.0000254069 0.0000246255 −3.90685 ·10−7
4.2. Quadrature formulas with odd number of nodes
Errors in the Gaussian (b
Ge
n+1), anti-Gaussian (b
He
n+3) and averaged ( b
A2e
n+4) quadrature rules in the case
w(x)=1+cos x,for x∈(−π, π) and
u4(x)=w(arccos x)r1−x
1+x=√1−x2,for x∈(−1,1)
for the integrand f(x)=(1 +cos x)(e−x+4/3) are given in Table 2.
Table 2: Errors in the Gaussian (b
Ge
n+1), anti-Gaussian (b
He
n+3) and averaged ( b
A2e
n+4) formula for w(x)=1+cos x,x∈(−π, π) and
f(x)=(1 +cos x)(e−x+4/3)
e
n+1b
I(f)−b
Ge
n+1(f)b
I(f)−b
He
n+3(f)b
I(f)−b
A2e
n+4(f)
81 −4.63804 ·10−94.49229 ·10−9−7.28786 ·10−11
61 −2.48222 ·10−82.40457 ·10−8−3.88281 ·10−10
41 −2.56852 ·10−72.48826 ·10−7−4.01318 ·10−9
21 −0.0000124339 0.0000120453 −1.94297 ·10−7
Finally, we compare our method with the other methods.
The nodes of the generalized averaged Szeg˝
o quadrature rule are the eigenvalues of (2n−2) ×(2n−2)
unitary upper Hessenberg matrix ˘
H2n−2(τ), determined by parameter τfrom unit circle and the so-called
Schur parameters γ1, . . . , γn−1. Weights are determined by the square of the first component of associated
unit eigenvectors (see [12, 13]). Matrix ˘
H2n−2(τ) is given by ˘
H2n−2(τ)=b
D−1/2
2n−2b
H2n−2(τ)b
D1/2
2n−2, where
b
H2n−2(τ)=

−¯
γ0γ1−¯
γ0γ2··· − ¯
γ0γn−1−¯
γ0γn−2··· − ¯
γ0γ1−¯
γ0τ
1− |γ1|2−¯
γ1γ2··· − ¯
γ1γn−1−¯
γ1γn−2··· − ¯
γ1γ1−¯
γ1τ
0 1 − |γ2|2··· − ¯
γ2γn−1−¯
γ2γn−2··· − ¯
γ2γ1−¯
γ2τ
.
.
.
0 0 ··· 0 1 − |γ1|2−¯
γ1τ

,
γ0=1, b
D2n−2=diag[b
δ0,b
δ1,...,b
δ2n−3], b
δ0=1, b
δj=b
δj−11− |b
γj|2,j=1,...,2n−3, b
γj=γj,j=1,...,n−1
and b
γj=γ2n−2−j,j=n,...,2n−3. There are several algorithms for the eigen decomposition of such kind
of matrices (see [9–11]). Computation of eigensystem can be performed very efficiently by using compact
representation of Hessenberg matrices (see e.g., [9, 11]).
In our method we use recurrence relations to obtain wanted orthogonal systems in order to escape nu-
merical non-stability which is characteristic for Gram-Schmidt method. Also, recurrence relations provide
a stable way for computation of values of trigonometric polynomials in contrast to using expanded forms.
We demonstrated how in the case of symmetric weight function the anti-Gaussian quadrature rules can be
constructed using orthogonal polynomials on the real line.
Using averaged Gaussian quadrature rules for trigonometric polynomials introduced in this article,
we were able to achieve much greater accuracy in comparison with averaged Szeg˝
o quadrature rules on

N. Z. Petrovi´c et al. /Filomat 36:3 (2022), 1005–1019 1018
the class of symmetric weight functions, which we will demonstrate with the following example. Also,
introducing the anti-Gaussian quadrature rules with an odd number of nodes (the case γ=1/2) we achieved
accuracy for the trigonometric polynomials of higher degree, for all t∈ T2n+2. We give one example from
[12] supplemented by our results.
Consider the weight function w(x)=2 sin2(x/2) on the interval (−π, π) and the integrand f(x)=
1
2log (5 +4 cos x). In Table 3 we give errors in Szeg ˝
o (Sn
1(f)), anti-Szeg˝
o (An
1(f)), generalized averaged
Szeg˝
o (b
S(2n−2)
1(f)); Gaussian (b
Ge
n+1(f)), anti-Gaussian (b
He
n+3(f)) and averaged Gaussian ( b
A2e
n+4) formula, with
e
n+1=n. One can see that errors obtained by using Gaussian and Szeg˝
o are similar with those obtained
by using anti-Gaussian and anti-Szeg˝
o quadrature rules, respectively. However, it is obvious that averaged
Gaussian quadrature rules for trigonometric polynomials give a significant improvement over the gener-
alized averaged Szeg˝
o quadrature rules (which gave the best results in [12]). That improvement is more
significant with increasing number of nodes. Also, we can see that in this example quadrature rules with
an even number of nodes lead to a greater improvement of results than rules with an odd number of nodes.
Table 3: Errors for several quadrature rules for w(x)=2 sin2(x/2), x∈(−π, π) and f(x)=1
2log (5 +4 cos x)
Rule n=12 n=15 n=18
Sn
1(f)−2.2·10−52.2·10−6−2.3·10−7
An
1(f) 2.3·10−5−2.3·10−62.4·10−7
b
S(2n−2)
1(f)−1.5·10−79.2·10−9−6.7·10−10
b
Ge
n+1(f)−1.98 ·10−51.38 ·10−5−2·10−7
b
He
n+3(f) 1.98 ·10−5−1.38 ·10−52·10−7
b
A2e
n+4−5.31 ·10−10 1.04 ·10−10 −8.75 ·10−14
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