A Sgeegö quadrature formula for a trigonometric polynomial modification of the Lebesgue measure

              Rcv.Acad.Canar.Cicnc .• XI (Núms. 1-2). 183-191 (1999)
A Szego quadrature formula for a trigonometric
polynomial modification of the Lebesgue me~ure *
J.C. Santos-León
Dcpartment of Mathematical Analysis, La Laguna University,
38271-La Laguna, Tenerife, Canary Islands, Spain
E-mail address: jcsantos@ull.es
Abstract
Szegi:i quadrature formulas are used for the computation of integrals over the unit
circle. They share sorne properties with the classical Gauss quadrature formulas
for integrals on the real line. Indeed, Szegi:i quadrature formulas have maximum
domain of validity. Furthermore, as Gauss quadrature formulas, they have positive
coefficients, and nodes located in the region of integration. Nevertheless, unlike
classical Gauss quadrature formulas, Szegi:i quadrature formulas are para-orthogonal
rather than orthogonal.
There are only a few known examples of Szegi:i quadrature formulas. In this note
a new Szegi:i quadrature formula for a trigonometric polynomial modification of the
Lebesgue measure on the. unit circle is constructed.
AMS Classification: 41, 65D.
Keywords: Construction of Szego quadrature formulas, modifications of the Lebesgue
measure, orthogonal polynomials on the unit circle, quadrature formulas on the unit circle
*This work was supported by the ministry of education and culture of Spain under contract PB96-1029.
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1 Introduction
We write T = { z E C : !z! = 1} for the unit circle.
Jones, Njastad and Thron studied in [8] the so-called Szego quadrature formulas for
the computation of integrals over the unit circle T, that is, integrals of the forro
where 7J¡ is a distribution function (real valued, bounded and non-decreasing ) on ( -7r, n}
The construction of Szego formulas is described below.
Let (p, q) be a pair of integers where p $ q. We denote by Ílp,q the linear space of ali
q
functions of the forro L c1zí, c1 E C. The functions of Ap,q are called Laurent polynomials
j=p
or briefly L-polynomials. We write A for the linear space of all L-polynomials. Consider
the inner product on A x A given by
(2) (!, g) = ¡: f (ei9) g(é9)d7J¡(8).
Let fon}O' be the sequence of polynomials obtained by orthogonalization of {zn}O' with
respect to the inner product (2). The sequence fon}O' is the sequence of Szego polynomials
with respect to the distribution function -¡p. As it is well known, see, e.g., [9], {}n has its
zeros in the region !zl < l. Thus they are not adequate as nodes for a general purpose
quadrature formula to approximate integrals over the unit circle. Quadrature formulas
with nodes not in T are of interest for functions with poles near but not in T. Taking the
poles as nodes is the underlying idea in the method of subtract out singularities [13].
Theorem 1 {8} Let fon}O' be the sequence of Szego polynomials with respect to the distri­butionfunction
7J¡. Let {Kn}o be a sequence of complex numbers satisfying IKnl = 1, n? O.
Let Bn(z, Kn) = {}n(z) + KnU~(z) where e~(z) = znen(l/z) . Then Bn(z, Kn) has n distinct
zeros c;J::> ( Kn) located on T. Let
>.~>(Kn) = r Bn(z, Kn) d7J¡(8), 1 $ m $ n.
}_1f (z - c;J::>(Kn)) B~ ( c;J::>(Kn), Kn)
Then
far all f E A-(n-1),n-1 · lt holds >.~)(Kn) > O, 1 $ m $ n, n 2: 1, and the quadrature
formula (3) gives the largest domain of validity, that is, there cannot exist an n-point
n
quadrature formulaμ(!)= L Amf(am), O:m ET which correctly integrates any fun ction
m=l
f E A-(n-1),n ar any function f E A-n,n- 1·
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Thf' polynomials Bn (z, "" ), n ~ O arP t.hP para-orthogonal polynomials with rt'SpPct.
to thP dist.rihntion fnnction ~· ·
Tlms Szf'go qnadrat.urf' formulas sharf' somt' propt'rt.i('S with th(' classical Gauss qnadra­t.
urt' formulas for iut('grals on t.hP rPal lü1('. Iud('Pd, Sz('gi:i quadrat.urn formulas hav(' max­imum
domain of validit.y, now in t.11P sapc(' of t11P Laurf'nt. polynomials. Furt.11('rmorc, as
Gauss quad.rat.urP formulas, thPy haw posit.iv(' coefficient.s, aud nodes located in the region
of int.('grat.iou. NPv('1t heless, unlike classical Gauss quadrat.ure formulas, Szego quadratnre
formulas arP para-orthogonal rathfi· thau orthogonal. One should take into account that
Ga.uss qnadrat.ure formulas ( maximum domain of exactness) for certain rational spaces of
fnnctions ar(' not. ort.hogonal [5] with respect to a fixPd distribution function.
DuP to thP difficultiPs in the construction of Szegi:i quadrature formulas, interpolatory
quadraturp formulas on the unit circle arise as alternative. They were introduced in
[2] for int.Pgrals on thP unit circl('. The interpolatory quadrature formulas with uniformly
distributed nodffi on the unit circle becom(' the most popular. Numerical experiments and
r('sults [7, 11, 12] show that these interpolatory quadrature formulas are competitive with
SzPgi:i formulas. In addition, the nodp,s for this quadrature formula are easily computable,
uniformly distributed on T, and the coefficients can be efficiently computable by means of
thP Fast Fourier Transform. algorithm, [11]. This facts make this interpolatory quadrature
formulas snitablP for practica! computations.
At the beginning [8], Szegi:i quadrature formulas were constructed as a too! for the
solution of the trigonometric moment problem. In [13], both interpolatory and Szegi:i
quadrature formulas were used as part of effi.cient quadrature formulas for the computa­tion
of integrals with Poisson type kernel that appear in the solution of boundary value
problPms for a circle.
Szegi:i quadrature formulas have been included in the more general topic of rational
Szegi:i quadrature formulas [1].
There are only a few known examples of Szegi:i quadrature formulas. Among them, for
the Lebesgue measure [3], for the Poisson integral [14], for rational modifications of the
Lebesgue measure on the unit circle [7], for a certain meruiure connected with q-starlike
functions [10], and for Jacobi type weight function on the unit circle [4]. Next we construct
a one parametric family of Szegi:i quadrature formulas for the trigonometric polynomial
modification of the Lebesgue measure on the unit circle given by
d'ljJ(8) = jei8 - ,Bj 2 d8, ,BE C, -7r:::; 8 < 7r.
The corresponding orthogonal polynomials were constructed in [ 6] . The associated mo­ments
are given by
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2 Construction of the quadrature formula
First we <leal with the case that /3 lies on the unit circle, that is, /3 E C, l/31 = l. Without
loss of generality, and for simplicity, we will take f3 = l. Indeed, if /3 E C, l/31 = 1 and
/3 ":11 then we can make an angle rotation on the complex unit circle.
The corresponding orthogonal polynomials l?n(z) are given by [6] ,
n k + 1 k
l?n(z) = L n + l z , n ~O.
k=O
Hence the para-orthogonal polynomials Bn(z, Kn) = l?n(z) + Knl?~(z), n ~ 1 where as
usual, Kn E C, lx:nl = 1, and l?~(z) = zngn(ljz) are given, for fixed Kn = 1 by
(5)
2 n 21 n+l
B ( ) _n+ '"°' k_n+ -z
n z, 1 - L..J Z - . n+l n+l 1-z
k=O
The nodes (~)(1), 1 ~ m ~ n, n ~ 1 of the n point Szegéi quadrature formula (x:n =
1, n ~ 1) are the n roots of Bn(z, 1), and its coeflicients >.~)(1), 1 ~ m ~ n, n ~ 1 are
given l:zy
where
From (4), and since /3 = 1, we get that m0 = 47r, m_1 = m1 = -27r and mk =O, lkl ~ 2.
Thus
Since
it holds
I n+2
Bn(O, 1) = Bn(O, 1) = -,
n+l
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From (5) and taking into account that 1 - ( (~)(1) ) =O we get
n+l
(n + 2) ((~)(l))n
B' (((nl(1) 1) = ------
n m ' 1 - (~)(1) '
and hence
27r ( 1 - (~\l) )2
(n + 1)(~)(1)
One has that Bn ( (~\l), 1) =O, or equivalently
(7)
From (6), (7), and taking into account that ( (~)(1) ) = 1 we deduce
n+l
,\~l(l) n~l (1-((!,:'l(1)f) (1-(,t,:'l(l))
n ~ 1 (2 - ( (,t,:'l(l) + ((,t,:'l(l)f))
n~l (2-((,t,:'l(1)+(~)(1))).
For the last equality take into account that ((~l(1))n = -(-~- = (~)(1) since 1 (~\l) 1 =
(,;: (1)
l. Then
Thus we have obtained that the n point Szego quadrature formula for the distribution
function
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aud Kn = 1 has uodes (.\:') ( 1) aud coefficieuts >.~) (1) giveu by
e27rrni/(n + 1)
1 :S m :S n, n 2'. l.
-47-r ( 1-cos ( -27r-m ) )
n+l n+l
This n poiut Szego quadrature formula satisfies
Next we consider the remain case
d1fJ(O) = le;8 - .Bl2 d(), ,BE IC, l.BI "l 1.
Without loss of generality, and for simplicity, we will take ,B E JR, ,B 2'. O, and ,B -1 l.
Otherwise we can make an angle rotation on the complex unit circle. The corresponding
orthogonal polynomials are given by [6],
{} (z) = 1 ~ ,Bk (,B2(n-k+l) _ l) zn-k.
n ,B2(n+l) - 1 L.,
k=O
After several elementary calculations is deduced that the para-orthogonal polynomials,
for fixed Kn = 1, n 2'. 1 are given by
- ,sn+2_1 (1-(,Bzr+1 ,sn+l_zn+l)
Bn(z, l) - ,B2(n+l) - 1 1 - ,Bz + ,B - Z •
The nodes (,~~.¡(1), 1 :S m :S n, n 2'. 1 of the n point Szego quadrature formula (Kn =
1, n 2'. 1) are the n roots of Bn(z, 1), and its coefficients ;.~>(1), 1 :S m :S n, n 2'.'l are
given by
where
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Thus
A~\1) L~l(O)mo + (L~l)' (O)m1
_ 27r (1 + ,a2) Bn(O, 1) + 27r,B ( B;,(o, l)(~l(l) + Bn(O, 1))
(~l(l)B~ ( (~l(l), 1) ( (~l(l)) 2 B~ ( (~l(l), 1)
27r (.B(~l(l)B~(O, 1) + Bn(O, 1) (.8 - (1 + ,82 )(~\1)))
(c;~l(1))2 B~ (c;~l(l),1)
One encounter that
and
where
Thus
where
B (O 1) = (,an+2 -1)(1 + .Bn) B' (O 1) = (,Bn+2 -1)(,B + ,an-1)
n ' ,82(n+I) _ 1 ' n ' ,82(n+I) _ 1 '
1 ( (n)( ) ) _ ,Bn+2 - 1 C¡
Bn (m 1 ' 1 - ,82(n+I) - 1 C2
C1 (n + 2)(,an+i + ,B) ((~l(1)t+1
-(n + 1)(,an+2+1) ((~l(1)t - c,an+2 + 1),
C2 (1- ,B(~l(l)) (.8-(~l(1)).
Taking into account that Bn (c;~l(l),1) =O, and Jc;~l(1)J 2 =(~)(1)(~)(1)=1 one can
deduce that
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Multiply numerator and denominator by -(IJ"+1 + /3) + (fJ"+2 + l)(t>(l). Aftcr sorne
elementary calculations is deduced that the coefficients >.~) (1), 1 ::; m ::; n, n ;::: 1 are
given by
where
C4 -1+2/3Re ((!,:i>(l)) - /32,
Cs (/3n+l + /3)2 - (pn+l + /3) (pn+2 + 1) 2Re ((!,:'l(l)) + (pn+2 + 1)2,
c6 -(n + 2) (,an+i + /3)2 - n (pn+2 + 1)2
+(n + 1) (/3n+l + /3) (13n+2 + 1) 2Re ((!,:'l(l)).
Thus the n point Szego quadrature formula for the distribution function
d'lf;(8) = lei9 - /31 2 d8, f3 E lR, f3 :'.:: O, f3 i' 1,
and for Kn = 1, n ? 1 is given as follows. lts nodes (~>(1), 1 ::; m ::; n, n ? 1 are the
roots of the polynomial
(pn+l + /3) Zn + (pn + 132) zn-1 + .. . + (/32 + pn) z + (/3 + 13n+l) = O,
and the coefficients >.~>(1), 1::; m::; n, n? 1 are given by (8). This quadrature formula
satisfies
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