Georgian Mathematical Journal Volume 9 (2002), Number 2, 331–338

CHARACTERISTIC FUNCTIONS AND s-ORTHOGONALITY PROPERTIES OF CHEBYSHEV POLYNOMIALS OF THIRD

AND FOURTH KIND

MARIA RENATA MARTINELLI

Abstract. The properties of two families of s-orthogonal polynomials, which are connected with Chebyshev polynomials of third and fourth kind, are studied. Evaluations of the remainders are given and asymptotic formulae are calculated for the corresponding hyper-Gaussian formulae used for an approximate estimation of integrals.

2000 Mathematics Subject Classification: 33C45, 65D32.

Key words and phrases: Orthogonal polynomials.

Introduction

In their various works A. Ossicini and F. Rosati dealt with the problem of construction of families of orthogonal polynomials which are at the same time s-orthogonal with respect to predefined weights.

Recently, in collaboration with the above-mentioned authors we have carried out a study of two families of s-orthogonal polynomials, connected with Chebyshev polynomials of first and second kind [1].

This paper is concerned with the properties of two families of s-orthogonal polynomials “connected” with Chebyshev polynomials of third and fourth kind.^{1} Using proper formulae of an upper bound, hyper-Gaussian functionals are studied and used for an approximate estimation of “integrals with weight”; evaluations of the remainders are given and asymptotic formulae are derived. The above results allow one to go beyond those obtained by A. Ossicini and F. Rosati in [3] and [4].

1. s-Orthogonal Polynomials And Fundamental Formulae

Let [a, b], a < b, be a finite interval on the x-axis, and p(x) a fixed measurable function which is almost everywhere positive and summable in [a, b] (p(x) ∈ L[a, b]).

Under such a hypothesis, having fixed an integer s ≥ 0, it was proved ([5], [6]) that it is possible to determine a sequence {P_{s,m}(x)} of polynomials of degree m (each polynomial being determined up to a multiplicative constant factor c_{s,m}) s-orthogonal in [a, b] with respect to the weight p(x), i.e., such that for each integer m ≥ 1

1For the definition of such polynomials, which are of course Jacobi polynomials, see [2].

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

∫_{a}^{b} p(x)Π_{m−1}(x)[P_{s,m}(x)]^{2s+1}dx = 0 ,

where Π_{m−1}(x) denotes an arbitrary polynomial of degree ≤ m − 1. Moreover, for m ≥ 1 it follows that the m zeros of the polynomial P_{s,m}(x) are real and distinct and located in the interior of [a, b].

For s = 0, P_{s,m}(x) are the classical orthogonal polynomials.

Such systems of s-orthogonal polynomials are of particular importance in studying of hyper-Gaussian quadrature formulae (see [7]) of the following type:

∫_{a}^{b} p(x)f(x)dx = ∑_{j=1}^{m} ∑_{h=0}^{2s} A_{hj}f^{(h)}(x_{m,j}) + R_{s,m}[f] for each f ∈ AC^{2s}[a, b] ,

where the coefficients A_{hj} (dependent on s and m) are independent of f and are uniquely determined by means of the condition: R_{s,m}[f] = 0 if f is an arbitrary polynomial of degree ≤ 2m(s + 1) − 1. Moreover, it follows that the nodes x_{m,1}, x_{m,2}, …, x_{m,m} (dependent in general on s) are necessarily m zeros of P_{s,m}(x).

With these preliminary remarks, let us consider two families of s-orthogonal polynomials connected to Chebyshev polynomials of 3rd and 4th kind of degree m = 0, 1, 2, . . .. We write such families as

{c_{m}^{∗}V_{m}(x)}, {c_{m}^{∗}W_{m}(x)}, x ∈ [−1, 1] ,  (1)

where c_{m}^{∗} is an appropriate normalization factor to be discussed later (see (18)). Let us specify the property of s-orthogonality of the above-mentioned systems of polynomials.

Theorem 1.1. Polynomials (1) orthogonal with respect to the weights (1 − x)^{\frac{1}{2}} (1 + x)^{− \frac{1}{2}} and (1 + x)^{\frac{1}{2}} (1 − x)^{− \frac{1}{2}} over the interval [−1, 1] are s-orthogonal over the interval [−1, 1] with respect to the weights

p^{[1]}(x) = (1 − x)^{\frac{1}{2} +s}(1 + x)^{− \frac{1}{2}} and p^{[2]}(x) = (1 − x)^{− \frac{1}{2}} (1 + x)^{\frac{1}{2} +s} (2)

for each integer s ≥ 0 (see, e.g., [3]).

Proof. For Chebyshev polynomials of 3rd and 4th kind given in (1) the formulae

(1 − x)^{s}[V_{m}(x)]^{2s+1} = 2^{−s} ∑_{k=0}^{s}(−1)^{k} (\begin{matrix}2s + 1 \\ s − k \end{matrix}) V_{m(2k+1)+k}(x)  (3)

and

(1 + x)^{s}[W_{m}(x)]^{2s+1} = 2^{−s} ∑_{k=0}^{s} (\begin{matrix}2s + 1 \\ s − k \end{matrix}) W_{m(2k+1)+k}(x)  (4)

hold, if we set x = cos θ and take into consideration the relations

(sin θ)^{2s+1} = 2^{−2s} ∑_{k=0}^{s}(−1)^{k} (\begin{matrix}2s + 1 \\ s − k \end{matrix}) sin(2k + 1)θ

and

(cos θ)^{2s+1} = 2^{−2s} ∑_{k=0}^{s} (\begin{matrix}2s + 1 \\ s − k \end{matrix}) cos(2k + 1)θ,

as well as the relations

cos^{2} \frac{θ}{2} = \frac{1 + cos θ}{2},  sin^{2} \frac{θ}{2} = \frac{1 − cos θ}{2} .  (5)

The proof follows from (3), (4) and the well known relations of orthogonality of the systems V_{m}(x) and W_{m}(x) (see also [3]).

Let us recall now some general formulae which we will use for the so-called “characteristic functions” [8]. Such functions prove to be particularly useful to determine the convergence of quadrature formulae; it is convenient to operate in the complex plane (see [3], [4]; for an easier reference see the formulae in [1]).

Let us begin now to discuss a question of calculating the integral I(f) = ∫_{−1}^{1} p(x)f(x)dx, where f(x) is a trace in the interval [−1, 1] of a function f(z), holomorphic in an open set A ⊃ [−1, 1]. Having defined a regular domain D ⊂ A such that D \ ∂D ⊃ [−1, 1], both the hyper-Gaussian quadrature formula and the remainder of the same formula can be formulated by means of integrals on +∂D; the following relations hold:

I(f) = J_{s,m}[f] + R_{s,m}[f] ,

with

J_{s,m}[f] = \frac{1}{2πi} ∫_{+∂D} f(z)ψ_{A}(z)dz,

having set

ψ_{A}(z) = ∑_{j=1}^{m} ∑_{h=0}^{2s} A_{hj} \frac{h!}{(z − x_{m,j})^{h+1}} ∀ z ∉ ⋃_{j=1}^{m}\{x_{m,j}\},

and with

R_{s,m}[f] = \frac{1}{2πi} ∫_{+∂D} f(z)Φ_{s,m}(z)dz ,  (6)

where Φ_{s,m}(z) is the characteristic function which is defined by

Φ_{s,m}(z) = \frac{Q_{s,m}(z)}{[P_{s,m}(z)]^{2s+1}} ∀ z ∉ [−1, 1],  (7)

where

Q_{s,m} = ∫_{−1}^{1} \frac{p(x)[P_{s,m}(x)]^{2s+1}}{z − x} dx, m = 1, 2, …,  (8)

and P_{s,m}(z) denotes the complex expression of P_{s,m}(x).^{2}

We assume that ∂D to be one of the confocal ellipse, E_{ρ} (ρ > 1) which have focuses at the ends of the segment [−1, 1], and are identified by the equations

z = \frac{1}{2} (ρe^{iθ} + ρ^{−1}e^{−iθ}) = \frac{1}{2} (ρ + ρ^{−1}) cos θ + \frac{i}{2} (ρ − ρ^{−1}) sin θ ,  (9)

and have the semiaxes

a_{ρ} = \frac{1}{2} (ρ + ρ^{−1}), b_{ρ} = \frac{1}{2} (ρ − ρ^{−1}) ,  (10)

an eccentric angle θ, a focal semidistance c = \sqrt{a_{ρ}^{2} − b_{ρ}^{2}} = 1.

The equations of E_{ρ} can also be put in the complex form

|z ± \sqrt{z^{2} − 1}| = ρ^{±1},  (11)

where the principal value is taken as a root.

For z ∈ E_{ρ}, we have some basic inequalities

|\sqrt{z^{2} − 1}| = \frac{1}{2} |(z + \sqrt{z^{2} − 1}) − (z − \sqrt{z^{2} − 1})| ≤ \frac{1}{2} (ρ + ρ^{−1}), (12)
|z| = \frac{1}{2} |(z + \sqrt{z^{2} − 1}) + (z − \sqrt{z^{2} − 1})| ≤ \frac{1}{2} (ρ + ρ^{−1}). (13)

We also have:

| \sqrt{\frac{z ± 1}{2}} | ≤ \frac{1}{2} (ρ^{\frac{1}{2}} + ρ^{− \frac{1}{2}} )  (14)

which can be immediately obtained by raising to square and considering (13).

Finally let us add that |z − x|, z ∈ E_{ρ}, x ∈ [−1, 1] with E_{ρ} ∩ [−1, 1] = ∅, has an absolute minimum which is obtained when z coincides with a vertex of E_{ρ} on the major axis and x with the focus near this vertex. Hence it follows that

|z − x| ≥ a_{ρ} − 1 .  (15)
2. Case in which P_{s,m}(x) = c_{m}^{∗}V_{m}(x)

Now, putting x = cos θ, we can calculate the Chebyshev polynomials of third and fourth kind (1):

V_{m}(cos θ) = \frac{sin(2m + 1)\frac{θ}{2}}{sin \frac{θ}{2}} = U_{2m} (cos \frac{θ}{2}) ,  (16)

W_{m}(cos θ) = \frac{cos(2m + 1)\frac{θ}{2}}{cos \frac{θ}{2}} = \frac{1}{cos \frac{θ}{2}} T_{2m+1} (cos \frac{θ}{2}) ,  (17)

where U_{2m}(·) and T_{2m+1}(·) denote respectively Chebyshev polynomials of first and second kind.

2(6), (7), (8) correspond to (2.5), (2.8), (2.9) of [1].

(16) and (17) allow us to construct the requested Chebyshev polynomials, sorthogonal with respect to the weights p(x) (see (2)), having used a “particular normalization”

c_{m}^{∗} = 2^{−m}  (18)

(it should be noted that c_{m}^{∗}V_{m} = x^{m} + … and, likewise, c_{m}^{∗}W_{m} = x^{m} + ⋯).

Taking into account the first equality (5), we obtain the s-orthogonal Chebyshev polynomials of third and fourth kind:

P_{s,m}^{[1]}(x) = 2^{−m}V_{2m}( \sqrt{\frac{1 + x}{2}} ),  (19)

P_{s,m}^{[2]}(x) = 2^{−m}\sqrt{\frac{2}{1 + x}} T_{2m+1}( \sqrt{\frac{1 + x}{2}} ).  (20)

Complex expressions of polynomials which appear in (19)and (20) can be given at once if we replace x by z, whenever necessary.

In the case of polynomials P_{s,m}^{[1]}(x) we will give (Theorem 2.1) an estimate of the remainder R_{s,m}[f] of (6). First we will establish the upper bounds of |Q_{s,m}(z)| of (8), and of |Φ_{s,m}(z)| of (7).

Lemma 2.1. For the function Φ_{s,m}(z) given by (7) the following inequality holds on the family of ellipses E_{ρ} (ρ > 1) given by (9) and (10):

|Φ_{s,m}^{[1]}(z)| ≤ 2^{s+2} \frac{1}{a_{ρ} − 1} \frac{(2s)!!}{(2s + 1)!!} (\frac{ρ^{\frac{1}{2}} + ρ^{− \frac{1}{2}}}{ρ^{m+ \frac{1}{2}} − ρ^{−m− \frac{1}{2}}} )^{2s+1} . ^{3}  (21)

Proof. Having taken care of (15), from (8) it follows that

|Q_{s,m}^{[1]}(z)| ≤ \frac{1}{a_{ρ} − 1} ∫_{−1}^{1}(1 − x)^{\frac{1}{2} +s}(1 + x)^{− \frac{1}{2}} |c_{m}^{∗}V_{m}(x)|^{2s+1}dx.

Putting x = cos θ and keeping in mind (16) and (18), we obtain

|Q_{s,m}^{[1]}(z)| ≤ \frac{1}{a_{ρ} − 1} ∫_{0}^{π}(1 − cos θ)^{\frac{1}{2} +s}(1 + cos θ)^{− \frac{1}{2}} sin θ |2^{−m} \frac{sin \frac{2m+1}{2}θ}{sin \frac{θ}{2}} |^{2s+1} dθ .

The above expression is reduced by the use of (5) to

|Q_{s,m}^{[1]}(z)| ≤ \frac{2^{s+1}}{2^{m(2s+1)}} \frac{1}{a_{ρ} − 1} ∫_{0}^{π} sin \frac{θ}{2} |sin \frac{2m + 1}{2} θ|^{2s+1} dθ ,

and further to

|Q_{s,m}^{[1]}(z)| ≤ \frac{2^{s+1}}{2^{m(2s+1)}} \frac{1}{a_{ρ} − 1} ∫_{0}^{π} |sin \frac{2m + 1}{2} θ|^{2s+1} dθ .  (22)

3Analogously to what it has been done in (2), we supply with superscripts [1] and [2] the entities relevant respectively to P_{s,m}^{[1]} and to P_{s,m}^{[2]}.

In order to evaluate the integral in (22) let us carry out the substitution φ = \frac{2m+1}{2}θ. Then the considered integral becomes

\frac{2}{2m + 1} ∫_{0}^{(2m+1)π/2}| sin φ|^{2s+1}dφ .  (23)

Having divided the integration interval [0, (2m + 1)π/2] into 2m + 1 intervals of length π/2 and taken into account periodicity of the integrand function, we obtain ∫_{0}^{(2m+1)π/2} | sin φ|^{2s+1}dφ = (2m + 1) ∫_{0}^{π/2} | sin φ|^{2s+1}dφ. Hence after substituting (23) into (22), we have the following upper bound of |Q_{s,m}^{[1]}(z)|:

|Q_{s,m}^{[1]}(z)| ≤ \frac{2^{s+2}}{2^{m(2s+1)}} \frac{1}{a_{ρ} − 1} ∫_{0}^{π/2}(sin φ)^{2s+1}dφ= \frac{2^{s+2}}{2^{m(2s+1)} (a_{ρ} − 1)} \frac{(2s)!!}{(2s + 1)!!} . (24)

Let us now proceed to proving (21) using (7) and (24). We obtain

|Φ_{s,m}^{[1]}(z)|_{z∈E_{ρ}} ≤ \frac{2^{s+2}}{2^{m(2s+1)}} \frac{1}{a_{ρ} − 1} \frac{(2s)!!}{(2s + 1)!!} \frac{1}{min |P_{s,m}(z)|_{z∈E_{ρ}}^{2s+1}} .  (25)

Consider now the polynomial P_{s,m}^{[1]} from (19), after using its complex expression and its modulus, i.e., 2^{−m}|V_{2m}(\sqrt{\frac{1+z}{2}})|.

Recalling the well-known formula

V_{2m}(ζ) = \frac{1}{2\sqrt{ζ^{2} − 1}} [(ζ + \sqrt{ζ^{2} − 1})^{2m+1} − (ζ − \sqrt{ζ^{2} − 1})^{2m+1}]  (26)

with the complex variable ζ and putting ζ^{2} = \frac{z+1}{2} and ζ^{2} − 1 = \frac{z−1}{2}, we have

|P_{s,m}^{[1]}(z)|_{z∈E_{ρ}} = 2^{−m−1}| \sqrt{\frac{2}{z − 1}} | |(\sqrt{\frac{z + 1}{2}} + \sqrt{\frac{z − 1}{2}})^{2m+1}

− ( \sqrt{\frac{z + 1}{2}} − \sqrt{\frac{z − 1}{2}} )^{2m+1}| .  (27)

Now, due to (14), from (27), on account of the following relation for z ∈ E_{ρ}

|\sqrt{\frac{z + 1}{2}} ± \sqrt{\frac{z − 1}{2}}| = ρ^{± \frac{1}{2}} , ^{4}  (28)

we have:

|P_{s,m}^{[1]}(z)|_{z∈E_{ρ}} ≥ 2^{−m} \frac{ρ^{m+ \frac{1}{2}} − ρ^{−m− \frac{1}{2}}}{ρ^{\frac{1}{2}} + ρ^{− \frac{1}{2}}} .  (29)

Having stated that, from (25) the assertion follows.

4See [4]. On the other hand, (28) can be immediately verified by raising to square and taking into account (8).

Theorem 2.1. Having fixed an integer s, the following asymptotic property holds with respect to (6) :

lim_{m→∞} R_{s,m}[f] = 0

and, more precisely, R_{s,m}[f] = O(ρ^{−m(2s+1)}), m → ∞ .

Proof. Let us denote by L_{ρ} the length of the ellipse E_{ρ} = ∂D and by M_{ρ} the maximum of |f(z)| on E_{ρ}. Then (6) and Lemma 2.1 imply

|R_{s,m}[f]| ≤ \frac{1}{π} L_{ρ}M_{ρ} \frac{2^{s+1}}{a_{ρ} − 1} \frac{(2s)!!}{(2s + 1)!!} (\frac{ρ^{\frac{1}{2}} + ρ^{− \frac{1}{2}}}{ρ^{m+ \frac{1}{2}} − ρ^{−m− \frac{1}{2}}} )^{2s+1} .

3. Case in which P_{s,m}(x) = c_{m}^{∗}P_{m}W_{m}(x)

In this case Lemma 2.1 and Theorem 2.1 are formulated in the same way, with proofs similar to those in Section 2, related however to (20); we will provide the details only concerning those steps which are different in the two cases.

From (2) it follows that

|Q_{s,m}^{[2]}(z)| ≤ \frac{1}{a_{ρ} − 1} ∫_{−1}^{1}(1 − x)^{− \frac{1}{2}} (1 + x)^{\frac{1}{2} +s}|c_{m}^{∗}W_{m}(x)|^{2s+1}dx .

Putting x = cos θ in the integral, (5) gives

|Q_{s,m}^{[2]}(z)| ≤ \frac{2^{s+1}}{2^{m(2s+1)}} \frac{1}{a_{ρ} − 1} ∫_{0}^{π} cos \frac{θ}{2} |cos \frac{2m + 1}{2} θ|^{2s+1} dθ .

Then, repeating the procedure used to find (23), (24), it follows that

|Q_{s,m}^{[2]}(z)| ≤ \frac{2^{s+2}}{2^{m(2s+1)}} \frac{1}{a_{ρ} − 1} ∫_{0}^{π/2}(cos φ)^{2s+1}dφ = \frac{2^{s+2}}{2^{m(2s+1)}} \frac{1}{a_{ρ} − 1} \frac{(2s)!!}{(2s + 1)!!} .

In the case which we are now handling, |P_{s,m}(z)| has to be considered as given by (20), subject to the complex expression and modulus, i.e.,

2^{−m}| \sqrt{\frac{2}{1 + z}} T_{2m+1}( \sqrt{\frac{1 + z}{2}} )| .

We apply the analogous formula of (26), i.e., the well-known formula:

T_{2m+1}(ζ) = \frac{1}{2} [(ζ + \sqrt{ζ^{2} − 1})^{2m+1} + (ζ − \sqrt{ζ^{2} − 1})^{2m+1}] ,

which for ζ = \sqrt{\frac{z+1}{2}}, due to (20) and with transformations analogous to those of Section 2, gives

|P_{s,m}^{[2]}(z)|_{z∈E_{ρ}} = 2^{−m−1}| \frac{2}{\sqrt{z + 1}} | |[(\sqrt{\frac{z + 1}{2}} + \sqrt{\frac{z − 1}{2}})^{2m+1}

+ (\sqrt{\frac{z + 1}{2}} − \sqrt{\frac{z − 1}{2}} )^{2m+1}]| .

Then we obtain a formula analogous to (29)

|P_{s,m}^{[2]}(z)|_{z∈E_{ρ}} ≥ 2^{−m} \frac{ρ^{m+ \frac{1}{2}} − ρ^{−m− \frac{1}{2}}}{ρ^{\frac{1}{2}} + ρ^{− \frac{1}{2}}} .

Acknowledgments

The paper was suggested by Prof. Alessandro Ossicini to whose memory it is dedicated.

Prof. Ossicini had been teaching and carried out research for about 30 years at the University of Rome “La Sapienza”. His remarkable personality, scientific achievements and merits will long be remembered by his colleagues and students.

The work was partially (40%) supported by Ministero Università Ricerca Scientifica Tecnologica.

References

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(Received 22.02.2001)

Author’s address:

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Facoltà di Ingegneria Università di Roma “La Sapienza” Via A. Scarpa 16, 00161 Roma Italy E-mail: martinelli@dmmm.uniroma1.it