Orthogonal Polynomials and Weighted Polynomial Approximation

Chapter 2Orthogonal Polynomials and WeightedPolynomial Approximation

2.1 Orthogonal Systems and Polynomials

2.1.1 Inner Product Space and Orthogonal Systems

Suppose that X is a complex linear space of functions with an inner product(f, g):X2 → C such that

(a) (f + g,h) = (f,h) + (g,h) (Linearity),

(b) (αf,g) = α(f,g) (Homogeneity),

(c) (f, g) = (g, f ) (Hermitian Symmetry),

(d) (f,f ) ≥ 0, (f, f ) = 0 ⇐⇒ f = 0 (Positivity),

where f,g,h ∈ X and α is a complex scalar. The bar in the above line denotes thecomplex conjugate. The space X is called an inner product space.

If X is a real linear space, then the inner product (f, g):X2 → R is such that thecondition (c) is reduced to

(c′) (f, g) = (g, f ) (Symmetry).

An important inequality for the inner product is the Cauchy-Schwarz-Buniakow-sky inequality (cf. [328, p. 87])

|(f, g)| ≤ ‖f ‖‖g‖ (f, g ∈ X), (2.1.1)

where the norm of f is defined by ‖f ‖ = √(f,f ).

A system S of elements of an inner product space is called orthogonal if(f, g) = 0 for every f �= g (f,g ∈ S). An orthogonal system S is called ortho-normal if (f,f ) = 1 for all f ∈ S.

Suppose that U = {g0, g1, g2, . . .} is a system of linearly independent functionsin a complex inner product space X. Starting from this system of elements andusing the well-known Gram-Schmidt orthogonalizing process we can construct thecorresponding orthogonal (orthonormal) system S = {ϕ0, ϕ1, ϕ2, . . .}, where ϕn is,in fact, a linear combination of the functions g0, g1, . . . , gn, such that (ϕn,ϕk) = 0for n �= k.

G. Mastroianni, G.V. Milovanovic, Interpolation Processes,© Springer 2008

75

76 2 Orthogonal Polynomials and Weighted Polynomial Approximation

Using the functions gn and Gram matrix of order n + 1,

Gn+1 =

⎡⎢⎢⎢⎢⎣

(g0, g0) (g0, g1) · · · (g0, gn)

(g1, g0) (g1, g1) (g1, gn)...

(gn, g0) (gn, g1) (gn, gn)

⎤⎥⎥⎥⎥⎦

an explicit expression for the orthogonal functions ϕn can be obtained. Notice thatthis matrix is non-singular. Namely, it is well-known that Δn+1 = det Gn+1 �= 0if and only if the system of functions {g0, g1, g2, . . . , gn} is linearly independent.Moreover, we can prove that det Gn+1 > 0.

Firstly, the matrix Gn+1 is Hermitian, because of the property (c) of the innerproduct, i.e., (gi, gj ) = (gj , gi). Putting x = [x0 x1 · · · xn]T and ψn = ∑n

k=0 xkgk ,we can see that the Gram matrix is also positive definite. Namely, then

x∗Gn+1x =n∑

i=0

n∑j=0

(gi, gj )xixj

can be expressed in the form

x∗Gn+1x = (ψn,ψn) = ‖ψn‖2,

which is positive, except if ψn = 0 (i.e., x = 0). This means that Δn+1 =det Gn+1 > 0.

Theorem 2.1.1 The orthonormal functions ϕn are given by

ϕn(z) = 1√ΔnΔn+1

∣∣∣∣∣∣∣∣∣∣

(g0, g0) (g0, g1) · · · (g0, gn−1) g0(z)

(g1, g0) (g1, g1) (g1, gn−1) g1(z)...

(gn, g0) (gn, g1) (gn, gn−1) gn(z)

∣∣∣∣∣∣∣∣∣∣(2.1.2)

where Δn = det Gn and Δ0 = 1.

Proof For the proof of this statement it is enough to prove that ϕn given by (2.1.2)satisfies the orthogonality condition (ϕn, gk) = 0 for each k = 0,1, . . . , n − 1.

Since

(ϕn, gk) = 1√ΔnΔn+1

∣∣∣∣∣∣∣∣∣∣

(g0, g0) (g0, g1) · · · (g0, gn−1) (g0, gk)

(g1, g0) (g1, g1) (g1, gn−1) (g1, gk)...

(gn, g0) (gn, g1) (gn, gn−1) (gn, gk)

∣∣∣∣∣∣∣∣∣∣,

2.1 Orthogonal Systems and Polynomials 77

we see immediately that this determinant is equal to zero for each k = 0,1, . . . ,

n − 1, and for k = n we have (ϕn, gn) = √Δn+1/Δn. Expanding the determinant

from (2.1.2) along the last column, we obtain an expansion

ϕn(z) = 1√ΔnΔn+1

(c0g0(z) + c1g1(z) + · · · + cngn(z)

),

in terms of gk , where cn = Δn. Therefore,

(ϕn,ϕn) = cn√ΔnΔn+1

(gn,ϕn) = 1. �

2.1.2 Fourier Expansion and Best Approximation

Taking an orthonormal system of functions S = {ϕ0, ϕ1, ϕ2, . . .}, it is easy to con-struct the corresponding Fourier expansion for a given function f ∈ X,

f (z) ∼

+∞∑k=0

fkϕk(z). (2.1.3)

The Fourier coefficients fk are given by

fk = (f,ϕk) (k = 0,1, . . .), (2.1.4)

which follows directly from (2.1.3). This sequence of coefficients is bounded. In-deed, applying the Cauchy-Schwarz-Buniakowsky inequality (2.1.1) to (2.1.4), weobtain

|fk| = |(f,ϕk)| ≤ ‖f ‖‖ϕk‖ = ‖f ‖.The partial sums of (2.1.3), i.e.,

sn(z) =n∑

k=0

fkϕk(z), (2.1.5)

play a very important role in approximation theory.Let Xn be a subspace of X spanned by Sn = {ϕ0, ϕ1, . . . , ϕn} (dimXn = n + 1),

i.e., Xn = spanSn. The following theorem shows that sn is the closest element tof ∈ X among all elements of the subspace Xn with respect to the metric induced bythe given norm. Thus, the partial sum sn is the best approximation to f ∈ X in thesubspace Xn.

Theorem 2.1.2 Let f ∈ X and Xn be a subspace of X spanned by {ϕ0, ϕ1, . . .}.Then

minϕ∈Xn

‖f − ϕ‖2 = ‖f − sn‖2 = ‖f ‖2 −n∑

k=0

|fk|2, (2.1.6)

where sn is given by (2.1.5).


Proof Let f ∈ X and let sn be given by (2.1.5). An arbitrary element of Xn can beexpressed as a linear combination of the orthonormal functions ϕ0, ϕ1, . . . , ϕn, i.e.,ϕ = ∑n

k=0 akϕk . Then

‖f − ϕ‖2 = (f − ϕ,f − ϕ) = (f,f ) − (f,ϕ) − (ϕ,f ) + (ϕ,ϕ).

Since

(f,ϕ) =n∑

k=0

ak(f,ϕk) =n∑

k=0

akfk, (ϕ,f ) =n∑

k=0

akf k, (ϕ,ϕ) =n∑

k=0

akak,

we get

‖f − ϕ‖2 = ‖f ‖2 −n∑

k=0

|fk|2 +n∑

k=0

(fkf k − akfk − akf k + akak

)

= ‖f ‖2 −n∑

k=0

|fk|2 +n∑

k=0

|fk − ak|2.

This expression attains the minimal value for ak = fk (k = 0,1, . . . , n), i.e., whenϕ = sn, and the minimum is given by (2.1.6). �

Since (f, sn) = (sn, sn) = ∑nk=0 |fk|2, we see that the error en = f − sn in the

best approximation is orthogonal to sn, i.e.,

(f − sn, sn) = 0.

Also, for each k = 0,1, . . . , n,

(f − sn,ϕk) =(f −

n∑ν=0

fνϕν,ϕk

)= (f,ϕk) −

n∑ν=0

fν(ϕν,ϕk) = 0, (2.1.7)

In other words, the error en is orthogonal to each ϕk , i.e., en is orthogonal to thesubspace Xn.

Based on (2.1.6) we conclude that

‖f − s0‖ ≥ ‖f − s1‖ ≥ ‖f − s2‖ ≥ · · · ,

which also follows directly from the fact that X0 ⊂ X1 ⊂ X2 ⊂ · · ·.Notice also that (2.1.6) implies the Bessel inequality

n∑k=0

|fk|2 ≤ ‖f ‖2,


which holds for every n ∈ N. When n → +∞, this becomes

+∞∑k=0

|fk|2 ≤ ‖f ‖2.

Thus, the series on the left hand side in this limit inequality converges, which impliesthat

limk→+∞fk = lim

k→+∞(f,ϕk) = 0.

Therefore, we conclude that the Fourier coefficients of any function f ∈ X approachzero.

The limit Bessel inequality reduces to an equality (Parseval’s equality) in thecase when span{ϕ0, ϕ1, ϕ2, . . .} is dense in X.

2.1.3 Examples of Orthogonal Systems

In this section we mention several interesting orthogonal systems.

2.1.3.1 Trigonometric System

In Sect. 1.2.2 we have seen that the trigonometric system (1.2.18), i.e.,{1, cosx, sinx, cos 2x, sin 2x, . . . , cosnx, sinnx, . . .

}

is orthogonal with respect to the inner product defined by (1.2.19). According to(1.2.16) and (1.2.17), the corresponding orthonormal system is

{ 1√2π

,cosx√

π,

sinx√π

,cos 2x√

π,

sin 2x√π

, . . . ,cosnx√

π,

sinnx√π

, . . .}.

2.1.3.2 Chebyshev Polynomials

Let

(f, g) =∫ 1

−1f (x)g(x)(1 − x2)λ−1/2dx, λ > −1/2. (2.1.8)

The Chebyshev polynomials of the first kind {Tn}n∈N0 and of the second kind{Un}n∈N0 are orthogonal on [−1,1] with respect to the inner product (2.1.8) forλ = 0 and λ = 1, respectively (see Sect. 1.1.4). The corresponding orthonormal sys-tems are

{1√π

,

√2

πT1,

√2

πT2, . . .

}and

{√2

π,

√2

πU1,

√2

πU2, . . .

}, (2.1.9)

respectively.


2.1.3.3 Orthogonal Polynomials on the Unit Circle

The system of monomials {zn}n∈N0 is orthonormal with respect to the inner product

(f, g) = 1

2π

∫ π

−π

f (eiθ )g(eiθ ) v(θ) dθ, (2.1.10)

where v(θ) = 1. This is the simplest case of polynomials orthogonal on the unitcircle with respect to (2.1.10). Such polynomials were introduced and studied bySzego ([468, 469]) and Smirnov ([445, 446]). A more general case was consid-ered by Achieser and Kreın [7], Geronimus ([177, 178]), Nevai ([378, 379]), Simon[436], etc. (see also surveys [9] and [334], as well as in a very impressive new bookof Barry Simon in two volumes [437, 438]). These polynomials are linked to manyquestions in the theory of time series, digital filters, statistics, image processing,scattering theory, control theory, etc.

The general theory of orthogonality on a union of circular arcs was also con-sidered by several authors (cf. Peherstorfer and Steinbauer [394–396], Simon [437,438]). For the zeros of such polynomials see Lukashov and Peherstorfer [277], Si-mon [439, 440], etc.

2.1.3.4 Orthogonal Polynomials on the Unit Disk

The system of monomials pn(z) = √(n + 1)/π zn, n = 0,1, . . . , is orthonormal

with respect to the inner product defined by the following double integral

(f, g) =∫ ∫

|z|≤1

f (z)g(z) dx dy.

2.1.3.5 Orthogonal Polynomials on the Ellipse

Let Er (r > 1) denote the ellipse with its foci at ±1 and such that the sum of itssemi-axes is r .

The Chebyshev polynomials of the first kind {Tn}n∈N0 are orthogonal with re-spect to the inner product defined by the following contour integral

(f, g) =∮

Er

f (z)g(z)

|1 − z2|1/2ds (ds2 = dx2 + dy2).

Namely, we have

(Tn, Tn) =⎧⎨⎩

π

2

(r2n + r−2n

), n > 0,

2π, n = 0.


However, the polynomials

pn(z) = 2

√n + 1

π

(r2n+2 − r−2n−2

)Un(z), n = 0,1, . . . ,

where Un(z) are the Chebyshev polynomials of the second kind, are orthonormalwith respect to the inner product

(f, g) =∫ ∫

intEr

f (z)g(z) dx dy.

2.1.3.6 Malmquist-Takenaka System of Rational Functions

Let {aν}ν∈N0 be a sequence of complex numbers such that |aν | < 1 (ν = 0,1, . . .).The system of rational functions

wn(z) =

n−1∏ν=0

(z − aν)

n∏ν=0

(z − 1/aν)

, n ∈ N0, (2.1.11)

considered by Malmquist [281], Takenaka [472], Walsh [501, Sects. 9.1 and 10.7],Djrbashian [102], etc., is orthogonal with respect to the inner product defined by(2.1.10), where again v(θ) = 1. Thus, the orthogonal functions (2.1.11), which areknown as the Malmquist-Takenaka functions (basis), generalize Szego’s orthogonalpolynomials. Here,

‖wn‖2 = (wn,wn) = |a0a1 · · ·an|21 − |an|2

(cf. [345, 362]).

2.1.3.7 Polynomials Orthogonal on the Radial Rays

The following sequence of polynomials

1, z, z2, z3, z4 − 1

3, z5 − 5

11z, z6 − 7

13z2, z7 − 3

5z3, z8 − 14

17z4 + 21

221, . . .

(2.1.12)is orthogonal with respect to the inner product

(f, g) =∫ 1

0

[f (x)g(x) + f (ix)g(ix) + f (−x)g(−x) + f (−ix)g(−ix)

]ω(x)dx,


where ω(x) = (1 − x4)1/2x2. This is a special case of polynomials orthogonal onthe radial rays introduced by Milovanovic in [336]. For details and several prop-erties of such polynomials see [333, 337, 341, 360, 361]. It is interesting that thepolynomial sequence (2.1.12) contains a subsequence which was obtained from aphysical problem connected to a non-linear diffusion equation (cf. [448]).

2.1.3.8 Müntz Orthogonal Polynomials

Let Λ = {λ0, λ1, . . .

}be a given sequence of complex numbers such that Re(λk) >

−1/2, k ∈ N0, and let Λn = {λ0, λ1, . . . , λn}. As we mentioned in Chap. 1 (Re-mark 1.3.1), Müntz systems can be considered also for the complex sequences.According to [23, 473], and [48], we can introduce the so-called Müntz-Legendrepolynomials on (0,1] by

Pn(x) = Pn(x;Λn) = 1

2πi

∮

Γ

Wn(z)xz dz, n = 0,1, . . . , (2.1.13)

where the simple contour Γ surrounds all poles of the rational function

Wn(z) =n−1∏ν=0

z + λν + 1

z − λν

· 1

z − λn

(n ∈ N0).

An empty product for n = 0 should be taken to be equal to 1.The polynomials (2.1.13) are orthogonal with respect to the inner product

(f, g) = ∫ 10 f (x)g(x) dx.

The corresponding orthonormal polynomials are P ∗n (x) = (1+λn+λn)

1/2Pn(x).In the simplest case when λν �= λμ (ν �= μ) it is easy to show that the polynomialsPn(x) can be expressed in a power form Pn(x) = ∑n

k=0 cn,kxλk , where

cn,k =

n−1∏ν=0

(1 + λk + λν)

n∏ν=0ν �=k

(λk − λν)

, k = 0,1, . . . , n.

An important special case of the Müntz-Legendre polynomials when

λ2k = λ2k+1 = k, k = 0,1, . . . ,

was considered in [338]. Namely, we put λ2k = k and λ2k+1 = k + ε, k = 0,1, . . . ,where ε decreases to zero. The corresponding limit process leads to the orthogonalMüntz polynomials with logarithmic terms,

Pn(x) = Rn(x) + Sn(x) logx, n = 0,1, . . . , (2.1.14)


where Rn(x) and Sn(x) are algebraic polynomials of degree[

n2

]and

[n−1

2

], respec-

tively, i.e.,

Rn(x) =[n/2]∑ν=0

a(n)ν xν, Sn(x) =

[(n−1)/2]∑ν=0

b(n)ν xν. (2.1.15)

Notice that Pn(1) = Rn(1) = 1. The first few Müntz polynomials (2.1.14) are:

P0(x) = 1,

P1(x) = 1 + logx,

P2(x) = −3 + 4x − logx,

P3(x) = 9 − 8x + 2(1 + 6x) logx,

P4(x) = −11 − 24x + 36x2 − 2(1 + 18x) logx,

P5(x) = 19 + 276x − 294x2 + 3(1 + 48x + 60x2) logx,

P6(x) = −21 − 768x + 390x2 + 400x3 − 3(1 + 96x + 300x2) logx.

The explicit expressions for the coefficients of the polynomials (2.1.15) for ar-bitrary n are given in [338]. These Müntz polynomials can be used in the proof ofthe irrationality of ζ(3) and of other familiar numbers (see [47, pp. 372–381] and[486]).

Similarly, if we take

λ3k = λ3k+1 = λ3k+2 = k, k = 0,1, . . . ,

i.e., λ3k = k − ε, λ3k+1 = k, λ3k+2 = k + ε, k = 0,1, . . . , where ε tends to zero, weget the corresponding orthogonal Müntz polynomials:

P0(x) = 1,

P1(x) = 1 + logx,

P2(x) = 1 + 2 logx + 1

2log2 x,

P3(x) = −7 + 8x − 4 logx − 1

2log2 x,

P4(x) = 29 − 28x + (11 + 24x) logx + log2 x,

P5(x) = −97 + 98x − 4(7 + 15x) logx + (36x − 2) log2 x,

P6(x) = 127 − 342x + 216x2 + (32 − 108x) logx + (2 − 108x) log2 x.

These polynomials have the form

Pn(x) = Rn(x) + Sn(x) logx + Tn(x) log2 x,


where Rn(x), Sn(x), and Tn(x) are algebraic polynomials of degree[

n3

],[

n−13

], and[

n−23

], respectively. Notice that Pn(1) = Rn(1) = 1.

A direct evaluation of the Müntz polynomials Pn(x) in the power form can beproblematic in finite arithmetic, especially when n is a large number and x is closeto 1. A numerical method for a stable evaluation of Müntz polynomials and someapplications were given in [338] (see also [339]).

2.1.3.9 Müntz Orthogonal Polynomials of the Second Kind

In [72] and [362] an external operation for the Müntz polynomials from M(Λ) andthe corresponding inner product were defined. Namely, at first an operation � formonomials was introduced in the following way:

xα � xβ = xαβ (x ∈ (0,+∞), α,β ∈ C),

and then it was extended to the Müntz polynomials P ∈ Mn(Λ) and Q ∈ Mm(Λ),i.e.,

P(x) =n∑

i=0

pixλi and Q(x) =

m∑j=0

qjxλj ,

as

(P � Q)(x) =n∑

i=0

m∑j=0

piqjxλiλj .

Under the restrictions that for each i and j , |λi | > 1 and Re(λiλj − 1) > 0, thefollowing inner product can be defined ([362])

[P,Q] =∫ 1

0(P � Q)(x)

dx

x2, (2.1.16)

as well as the Müntz polynomials

Qn(x) = 1

2πi

∮

Γ

Wn(z)xz dz, n = 0,1, . . . , (2.1.17)

where

Wn(s) =

n−1∏ν=0

(s − 1/λν)

n∏ν=0

(s − λν)

, n = 0,1, . . . , (2.1.18)

and the simple contour Γ surrounds all the points λν , ν = 0,1, . . . , n.We note that the rational functions (2.1.18) form a Malmquist-Takenaka system.

Indeed, putting aν = 1/λν ν = 0,1, . . . , these functions reduce to (2.1.11).


Under the previous conditions on the sequence Λ, the Müntz polynomials Qn(x),n = 0,1, . . . , defined by (2.1.17), are orthogonal with respect to the inner product(2.1.16). Furthermore, this orthogonality is connected to the orthogonality of theMalmquist-Takenaka system (2.1.11) (see ([362]) and

[Qn,Qm] = δn,m

(|λn|2 − 1)|λ0λ1 · · ·λn−1|2 .

2.1.3.10 Generalized Exponential Polynomials

Let A = {α0, α1, α2, . . . } be a complex sequence such that Reαk > 0. For each k

(k ≥ 0) denote by mk ≥ 1 the multiplicity of the appearance of the numbers αk inthe set Ak = {α0, α1, α2, . . . , αk}. With the sequence A we associate the sequence offunctions

{tmk−1e−αkt

}k∈N0

, which can be orthogonalized with respect to the innerproduct

(f, g) =∫ +∞

0f (t)g(t) dt, (2.1.19)

for example, using the well-known Gram-Schmidt method. Such an orthonormalsystem {gk(t)}k∈N0

is unique up to a multiplicative constant of the form eiγk , withImγk = 0.

For example, if we take A = {1/2,1,1,2,5/2, . . . }, for which m0 = m1 = 1,m2 = 2, m3 = m4 = 1, . . . , using an orthogonalizing process we get the exponentialfunctions (generalized exponential polynomials)

g0(t) = e−t/2,

g1(t) = √2 e−t

(3 − 2et/2),

g2(t) = √2 e−t

(5 − 6et/2 + 6t

),

g3(t) = 2e−2t(15 − 8et − 6e3t/2 + 12tet

),

g4(t) = 12

√5 e−5t/2(147 − 240et/2 + 80e3t/2 + 15e2t − 48te3t/2), . . . ,

which are orthonormal with respect to the inner product (2.1.19) (cf. [339]).

2.1.3.11 Discrete Chebyshev Polynomials

Let

(f, g) = f (0)g(0) + f (1)g(1) + f (2)g(2) + f (3)g(3).

Starting from monomials and using the Gram-Schmidt orthogonalizing process weget orthogonal polynomials with respect to this inner product:

g0(x) = 1, g1(x) = x − 1

3, g2(x) = x2 − 3x + 1, g3(x) = x3 − 9

2x2 + 47

10x − 3

10.


The corresponding orthonormal polynomials are

1

2,

2x − 3

2√

5,

x2 − 3x + 1

2,

10x3 − 45x2 + 47x − 3

6√

5.

Continuing this process we find g4(x) = x(x − 1)(x − 2)(x − 3) and, as we can see,g4(x) = 0 for x = 0,1,2,3. Therefore, ‖g4‖ = 0.

This is a special case of the so-called discrete Chebyshev orthogonal polynomials(cf. Szego [470, pp. 33–34]) which can be expressed (except for constant factors)

in the form

tn(x) = n!Δn

{(x

n

)(x − N

n

)}, n = 0,1, . . . ,N − 1, (2.1.20)

where the inner product is given by

(f, g) =N−1∑k=0

f (k)g(k) (2.1.21)

and N is a given natural number. In (2.1.20) the symbol Δ is the forward differenceoperator with unit spacing acting on the variable x. The inner product (2.1.21) canbe rewritten in the integral form

(f, g) =∫

R

f (x)g(x)dμ(x),

with the distribution dμ(x) = ∑N−1k=0 δ(x −k)dx, where μ(x) is a step function with

jumps of one unit at the points x = k, k = 0,1, . . . ,N − 1. Here, δ is the Dirac deltafunction. Notice that

‖tn‖2 = (tn, tn) = N(N2 − 12)(N2 − 22) · · · (N2 − n2)

2n + 1.

Chebyshev also considered the case when the set of equidistant points {0,1, . . . ,

N − 1} is replaced by an arbitrary set of N distinct points. Finally, we mention thatthere are several other cases of discrete polynomials when the jumps in μ(x) aredifferent from one unit (see [29, pp. 221–227], [382], [470, pp. 34–37]).

2.1.3.12 Formal Orthogonal Polynomials with Respect to a MomentFunctional

Let a complex valued linear functional L be given on the linear space of all algebraicpolynomials P. The values of the linear functional L at the set of monomials arecalled moments and they are denoted by μk . Thus, L[xk] = μk , k ∈ N0. A sequenceof polynomials {Pn(x)}∞n=0 is called a formal orthogonal polynomial sequence withrespect to a moment functional L provided for all nonnegative integers k and n,


1◦ Pn(x) is a polynomial of degree n,2◦ L[Pn(x)Pk(x)] = 0 for k �= n,3◦ L[Pn(x)2] �= 0.

If a sequence of orthogonal polynomial exists for a given linear functional L,then L is called quasi-definite linear functional. Under the condition L[P 2

n (x)] > 0,the functional L is called positive definite. For details see Chihara [60, pp. 6–17].

The necessary and sufficient conditions for the existence of a sequence of orthog-onal polynomials with respect to the linear functional L are that for each n ∈ N theHankel determinants

Δn =

∣∣∣∣∣∣∣∣∣

μ0 μ1 . . . μn−1μ1 μ2 μn

...

μn−1 μn μ2n−2

∣∣∣∣∣∣∣∣∣�= 0. (2.1.22)

When L is positive definite, we can define (p, q) := L[p(x)q(x)] for all alge-braic polynomials p(x) and q(x), so that the orthogonality with respect to the mo-ment functional L is consistent with the standard definition of orthogonality withrespect to an inner product. On the other hand, there are several interesting quasi-definite cases when the moments are non-real. We mention here two cases:

(a) Orthogonality with respect to an oscillatory weight. Let w(x) = xeimπx , wherex ∈ [−1,1] and m ∈ Z. Putting

L[p] :=∫ 1

−1p(x)w(x)dx (p ∈ P), (2.1.23)

i.e.,

μk = (−1)m+k(k + 1)!(imπ)k+1

k∑ν=0

(1 + (−1)ν)(−imπ)ν

(ν + 1)! ,

in [346] it was proved that for every integer m(�= 0), the sequence of formal or-thogonal polynomials with respect to the functional L exists uniquely. Such or-thogonal polynomials have several interesting properties and they can be appliedin numerical integration of highly oscillatory functions (see [346]). The corre-sponding case with the Chebyshev weight, i.e., when w(x) = x(1 − x2)−1/2eiζx ,ζ ∈ R, was recently investigated in [347].

According to (2.1.23) we can define

(p, q) :=∫ 1

−1p(x)q(x)w(x)dx (p,q ∈ P),

but, as we can see, (p, q) is not Hermitian and not positive definite. Namely, theproperties (c) and (d) of the inner product do not hold!


(b) Orthogonality on the semicircle. Polynomials orthogonal on the semicircle Γ ={z ∈ C | z = eiθ , 0 ≤ θ ≤ π} with respect to the moment functional

L[zk] = μk =∫ π

0eikθdθ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

π, k = 0,

2i

k, k odd,

0, k �= 0 even,

have been introduced by Gautschi and Milovanovic (see [170] and [171]). Thecorresponding non-Hermitian inner product is given by

(p, q) =∫

Γ

p(z)q(z)(iz)−1 dz =∫ π

0p(eiθ )q(eiθ ) dθ.

A few first polynomials of this orthogonal system are

π0(x) = 1,

π1(x) = x − 2i

π,

π2(x) = x2 − πi

6x − 1

3,

π3(x) = x3 − 8i

5πx2 − 3

5x + 8i

15π,

π4(x) = x4 − 9πi

56x3 − 6

7x2 + 27πi

280x + 3

35,

π5(x) = x5 − 128i

81πx4 − 10

9x3 + 256i

189πx2 + 5

21x − 128i

945π.

The general case of complex polynomials orthogonal with respect to a complexweight function was considered by Gautschi, Landau and Milovanovic [175].For some properties and applications see [153, 326, 327, 329]. A generalizationof such polynomials on a circular arc was given by de Bruin [50], and furtherinvestigations were done by Milovanovic and Rajkovic [354].

In this chapter we mainly consider those polynomial orthogonal systems, whenan inner product is defined on some lines or on a curve in the complex plane C.Furthermore, starting from Sect. 2.2, we only consider polynomials orthogonal onthe real line.

By Pn we denote the set of all algebraic polynomials (with complex coefficients)of degree at most n. Further, let Pn be the set of all monic polynomials of degree n,i.e.,

Pn = {zn + q(z) | q(z) ∈ Pn−1

}.


A system of polynomials {pn}, where

pn(z) = γnzn + lower degree terms, γn > 0,

(pn,pm) = δnm, n,m ≥ 0,(2.1.24)

is called a system of orthonormal polynomials with respect to the inner product( · , · ). In many considerations and applications we use the monic orthogonal poly-nomials

πn(z) = pn(z)

γn

= zn + lower degree terms. (2.1.25)

Sometimes, we also use the notation pn(z) for monic orthogonal polynomials in-stead of πn(z).

2.1.4 Basic Facts on Orthogonal Polynomials and ExtremalProblems

Let dμ be a finite positive Borel measure in the complex plane C, with an infinite setas its support, and let L2(dμ) denote the Hilbert space of measurable functions f

for which∫ |f (z)|2 dμ(z) < +∞. Finally, let the inner product ( · , · ) be defined by

(f, g) =∫

f (z)g(z) dμ(z) (f, g ∈ L2(dμ)). (2.1.26)

Starting from the system of monomials U = {1, z, z2, . . . }, by the Gram-Schmidt orthogonalization process, we can obtain the unique orthonormal poly-nomials (2.1.24). In order to emphasize the orthogonality with respect to the givenmeasure dμ, we write

pn(z) = pn(dμ; z) = γnzn + lower degree terms, γn = γn(dμ) > 0.

Also, the monic orthogonal polynomials (2.1.25) are unique.The following extremal property characterizes orthogonal polynomials:

Theorem 2.1.3 The polynomial πn(z) = pn(z)/γn = zn + · · · is the unique monicpolynomial of degree n of minimal L2(dμ)-norm, i.e.,

minp∈Pn

∫|p(z)|2dμ(z) =

∫|πn(z)|2dμ(z) = 1

γ 2n

. (2.1.27)

Proof Using the polynomials {pk(z)}, orthonormal with respect to the measure dμ,an arbitrary monic polynomial p(z) ∈ Pn can be expressed in the form

p(z) =n−1∑k=0

ckpk(z) + 1

γn

pn(z).


Then, we have

‖p‖2 =n−1∑k=0

|ck|2 + 1

γ 2n

≥ 1

γ 2n

,

with equality if and only if c0 = c1 = · · · = cn−1 = 0, i.e.,

p(z) = p∗(z) = 1

γn

pn(z) = πn(z). �

This extremal property is completely equivalent to orthogonality. Namely, manyquestions regarding orthogonal polynomials can be answered by using only thisextremal property (cf. [477] and [452]).

Notice also that the previous theorem gives the polynomial of the best approxi-mation to the monomial zn in the class Pn−1. Indeed, according to Theorem 2.1.2,the best L2-approximation to f (z) = zn is given by sn−1(z) = ∑n−1

k=0 fkpk(z),where fk = (zn,pk), k = 0,1, . . . , n − 1. But, by Theorem 2.1.3, f (z) − sn−1(z) =πn(z), so that the polynomial of the best approximation in this case can be expressedin the form sn−1(z) = zn − πn(z).

Now, we define the function (z, t) �→ Kn(z, t) by

Kn(z, t) =n∑

k=0

pk(z)pk(t) (n ≥ 0), (2.1.28)

which plays a fundamental role in the integral representation of the partial sums ofthe orthogonal expressions.

For a function f ∈ L2(dμ) we can determine its Fourier coefficients fk withrespect to the inner product (2.1.26). Thus, using (2.1.4) and the orthonormal poly-nomials {pk(z)}, we have

fk = (f,pk) =∫

f (t)pk(t) dμ(t).

Then, the partial sums (2.1.5) can be expressed in an integral form

sn(z) =n∑

k=0

fkpk(z) =n∑

k=0

(f,pk)pk(z) =∫

f (t)Kn(z, t) dμ(t).

Suppose that f is an arbitrary polynomial of degree at most n, i.e., f (z) = P(z)

(P (z) ∈ Pn). Then, the corresponding partial sum sn coincides with f and we obtain

P(z) =∫

P(t)Kn(z, t) dμ(t) (P (z) ∈ Pn). (2.1.29)


Because of that, the function Kn is very often called the reproducing kernel. Noticethat Kn(z, t) = Kn(t, z) and

Kn(z, z) =n∑

k=0

|pk(z)|2 ≥ |p0(z)|2 = γ 20 > 0

for each z ∈ C and n ≥ 0.The reciprocal of this function is known as the Christoffel function,

λn(z) = λn(dμ; z) = 1

Kn−1(z, z)=

(n−1∑k=0

|pk(z)|2)−1

. (2.1.30)

The following extremal problem is related to the reproducing kernel (cf. [359]and [485]):

Theorem 2.1.4 For every P(z) ∈ Pn such that P(t) = 1, we have∫

|P(z)|2 dμ(z) ≥ λn+1(dμ; t), (2.1.31)

with equality only for

P(z) = P ∗(z) = Kn(z, t)

Kn(t, t).

Proof Let t be a fixed complex number and P(z) ∈ Pn. In order to find the minimumof the integral on the left hand side in (2.1.31) under the constraint P(t) = 1, werepresent P(z) as a linear combination of the orthonormal polynomials pk(z) =pk(dμ; z), i.e., P(z) = ∑n

k=0 ckpk(z). Then, we have

F(P ) =∫

|P(z)|2 dμ(z) =n∑

k=0

|ck|2.

Since P(t) = ∑nk=0 ckpk(t) = 1, using the Cauchy inequality for the complex se-

quences c = {ck}nk=0 and p = {pk(t)}nk=0 (see Mitrinovic [364, p. 32]), we have

1 =∣∣∣∣∣

n∑k=0

ckpk(t)

∣∣∣∣∣2

≤(

n∑k=0

|ck|2)(

n∑k=0

|pk(t)|2)

= F(P )Kn(t, t), (2.1.32)

which implies F(P ) ≥ 1/Kn(t, t) = λn+1(dμ; t), i.e., (2.1.31).In the case of equality in (2.1.32), which is attained only if the sequences c and

p are proportional, i.e., when ck = γ pk(t) (k = 0,1, . . . , n), with some complexconstant λ, we find that

P(t) = γ

n∑k=0

|pk(t)|2 = γKn(t, t) = 1.


Thus, γ = 1/Kn(t, t) and the extremal polynomial is given by

P(z) = P ∗(z) = γ

n∑k=0

pk(t)pk(z) = Kn(z, t)

Kn(t, t).

�

According to this theorem, the Christoffel function can also be expressed in theform

λn(dμ; t) = minP∈Pn−1P (t)=1

∫|P(z)|2 dμ(z). (2.1.33)

Using the previous theorem we can also prove:

Theorem 2.1.5 Let t be a fixed complex constant, and let P(z) be an arbitrarypolynomial of degree at most n, normalized by the condition

‖P ‖2 =∫

|P(z)|2 dμ(z) = 1.

The maximum of |P(t)|2 taken over all such polynomials is attained for

P(z) = γKn(z, t)√Kn(t, t)

(|γ | = 1).

The maximum itself is Kn(t, t).

Proof Taking Q(z) = P(z)/P (t) we see that Q(t) = 1 and, according to Theo-rem 2.1.4,

∫|Q(z)|2 dμ(z) = 1

|P(t)|2 ≥ λn+1(dμ; t),

with equality case for Q(z) = P(z)/P (t) = Kn(z, t)/Kn(t, t).Thus, |P(t)|2 ≤ Kn(t, t), with equality only for

P(z) = P(t)Q(z) = γ√

Kn(t, t)Kn(z, t)

Kn(t, t)= γ

Kn(z, t)√Kn(t, t)

,

where |γ | = 1. �

Under certain conditions, the extremal property from Theorem 2.1.3, can beextended to Lr(dμ)-norm (1 < r < +∞), so that the unique monic polynomialp∗

n(z) = zn + · · · of the minimal Lr(dμ)-norm exists, i.e.,

minp∈Pn

∫|p(z)|rdμ(z) =

∫|p∗

n(z)|rdμ(z). (2.1.34)


For measures with support on the real line, an interesting special case r = 2s +2,where s ∈ N0, leads to a case of the power orthogonality. Then, the extremal (monic)polynomials in (2.1.34), denoted by p∗

n(x) = πn,s(x) = πn,s(x;dμ), exist uniquelyand they are known as s-orthogonal polynomials (for more details see [146, 179,180, 340, 387]). These polynomials must satisfy the “orthogonality conditions” (cf.Ghizzetti and Ossicini [180], Milovanovic [340])

∫R

πn,s(x)2s+1xk dμ(x) = 0, k = 0,1, . . . , n − 1. (2.1.35)

In the case s = 0, the s-orthogonal polynomials reduce to the standard orthogonalpolynomials, πn,0 = πn.

Also, the generalized Christoffel function can be defined for 0 < r < +∞, by

λn(dμ, r; t) = minP∈Pn−1P (t)=1

∫|P(z)|r dμ(z). (2.1.36)

Notice that (2.1.36) for r = 2 reduces to (2.1.33), i.e., λn(dμ,2; t) = λn(dμ; t).Several properties of the generalized Christoffel functions for measures on R can

be found in Nevai [375, pp. 106–123].

2.1.5 Zeros of Orthogonal Polynomials

Now, we study some basic properties of the zeros of orthogonal polynomials. Ac-cording to the fundamental theorem of algebra, we know that any polynomial ofdegree n has exactly n zeros, counting multiplicities. The zeros of orthogonal poly-nomials play a very important role in interpolation theory, quadrature formulas, etc.

Using Theorem 2.1.3 it is easy to prove a general result on the location of zeros.This result is connected with the support of the measure supp (dμ), which is a closedset. Firstly, we need some definitions:

Definition 2.1.1 A set A ⊂ C is convex if for each pair of points z, t ∈ A the lineconnecting z and t is a subset of A.

Definition 2.1.2 The convex hull Co (B) of a set B ⊂ C is the smallest convex setcontaining B .

Definition 2.1.3 Let D∞ be the connected component of the complement of E thatcontains the point ∞, then D∞ is open and

Pc (E) = C \ D∞

is the polynomial convex hull of E.


It is clear that Co (supp (dμ)) is the intersection of all closed half-planes con-taining supp (dμ). Also,

supp (dμ) ⊂ Pc (supp (dμ)) ⊂ Co (supp (dμ)).

The following result is due to Fejér (see [422] and [485]).

Theorem 2.1.6 All the zeros of the (monic) polynomial πn(dμ; z) lie in the convexhull of the support E = supp (dμ).

Proof Suppose that ζ is a zero of πn(dμ; z) = πn(z) such that ζ �∈ Co (E), whereE = supp (dμ). Then

πn(z) = (z − ζ )q(z) (q(z) ∈ Pn).

Since ζ �∈ Co (E), there exists a line L separating E and ζ . Let ζ be the orthogonalprojection of ζ on L. Then, for every z ∈ E, we have |z − ζ | < |z − ζ |, i.e.,

|(z − ζ )q(z)| < |(z − ζ )q(z)| = |πn(z)|,

from which it follows that∫

|(z − ζ )q(z)|2 dμ(z) <

∫|πn(z)|2 dμ(z),

which is a contradiction to Theorem 2.1.3. Thus, we conclude that there are no zerosoutside Co (E). �

An improvement of this theorem was given by Saff [422]:

Theorem 2.1.7 If Co (supp (dμ)) is not a line segment, then all the zeros of thepolynomial πn(dμ; z) lie in the interior of Co (supp (dμ)).

For example, if C is the unit circle |z| = 1 and supp (dμ) ⊂ C, then Theo-rem 2.1.7 asserts that all the zeros of πn(dμ; z) must lie in the open unit disk |z| < 1.This is a classical result of Szego [470, p. 292] for orthogonal polynomials on theunit circle.

An interesting question is related with a number of zeros of πn(dμ; z) which areoutside E = supp (dμ) (i.e., in Co (E) \E). If the set E has holes, it is possible thatall the zeros are in the holes, as in the case of polynomials orthogonal on the unitcircle. Here, we mention a result of Widom [505] (see Saff [422] for the proof).

Theorem 2.1.8 Let E = supp (dμ) and A be a closed set such that A∩ Pc (E) = ∅.Then the number of zeros of πn(dμ; z) on A is uniformly bounded in n.

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