PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 211–226 DOI: 10.2298/PIM1410211M

TRIGONOMETRIC MULTIPLE ORTHOGONAL POLYNOMIALS OF SEMI-INTEGER DEGREE AND THE CORRESPONDING QUADRATURE FORMULAS

Gradimir V. Milovanović, Marija P. Stanić, and Tatjana V. Tomović

Abstract. An optimal set of quadrature formulas with an odd number of nodes for trigonometric polynomials in Borges’ sense [Numer. Math. 67 (1994), 271–288], as well as trigonometric multiple orthogonal polynomials of semi-integer degree are defined and studied. The main properties of such a kind of orthogonality are proved. Also, an optimal set of quadrature rules is characterized by trigonometric multiple orthogonal polynomials of semiinteger degree. Finally, theoretical results are illustrated by some numerical examples.

1. Introduction

Motivated by a problem that arises in the evaluation of computer graphics illumination models, Carlos Borges [4] examined a more abstract problem of numerically evaluating a set of p definite integrals taken with respect to p distinct weight functions, but related to a common integrand and the same interval of integration. For such a problem it is not efficient to use a set of p Gauss–Christoffel quadrature rules, because valuable information is wasted.

Borges in [4] introduced a performance ratio in the following way:

R = \frac{Overall algebraic degree of precision + 1}{Number of integrand evaluation} .

Taking the set of p Gauss–Christoffel quadrature rules, one has R = 2/p and, hence, R < 1 for all p > 2.

If we select a set of n distinct nodes, common for all quadrature rules, then weight coefficients for each of p quadrature rules can be chosen in such a way that R = 1. Of course, an arbitrary selection of nodes does not lead us to the best possible quadrature rules. The optimal quadrature rules could be obtained by simulating the development of the Gauss–Christoffel quadrature rules. It was proved that the common nodes of an optimal set of quadrature rules are zeros of the corresponding type II multiple orthogonal polynomials, defined using orthogonality conditions spread out over p different measures [12,13]. A stable numerical method for construction of type II multiple orthogonal polynomials, as well as of optimal set of quadrature rules was given in [12].

2010 Mathematics Subject Classification: Primary 42C05; Secondary 65D32.

The authors were partly supported by the Serbian Ministry of Education, Science and Technological Development (grants #174015 and #44006).

Let us notice that multiple orthogonal polynomials arise naturally in the theory of simultaneous rational approximation, in particular in the Hermite–Padé approximation of a system of p Markov functions. When the Hermite–Padé approximation and multiple orthogonal polynomials are in question, we refer readers to the book by Nikishin and Sorokin [14, Chapter 4], the surveys by Aptekarev [1], de Bruin [6], and Milovanović and Stanić [13], as well as the papers by Piñeiro [16], Sorokin [17–19], Van Assche [22], Van Assche and Coussement [23], and Chapter 23 of Ismail’s book [7].

Type II multiple orthogonal polynomials are also connected with the generalized Birkhoff–Young quadrature rules [8, 11, 13].

Here we investigate the optimal set of quadrature rules in Borges’ sense for trigonometric polynomials. From the theory of Gaussian type quadrature rules, it is known that it is necessary to consider two different trigonometric orthogonal systems – standard trigonometric orthogonal polynomials and trigonometric orthogonal polynomials of semi-integer degree. As a matter of fact, for a quadrature rule with an even number of nodes and the maximal trigonometric degree of exactness one must consider orthogonality in the subspace of trigonometric polynomials, but in the case of an odd number of nodes and the maximal trigonometric degree of exactness orthogonality in subspace of trigonometric polynomials of semiinteger degree must be considered [2, 3, 5, 9, 10, 15, 20]. We restrict our attention to an optimal set of quadrature rules with an odd number of nodes. Therefore, here we introduce a concept of multiple orthogonality in the space of trigonometric polynomials of semi-integer degree. The paper is organized as follows. Trigonometric multiple orthogonal polynomials of semi-integer degree are defined in Section 2, where their main properties are proved. Section 3 is devoted to definition and characterization of an optimal sets of quadrature rules for trigonometric polynomials. Finally, numerical examples are presented in Section 4.

2. Trigonometric multiple orthogonal polynomials of semi-integer degree

For a nonnegative integer m and γ ∈ {0, 1/2}, we by 𝒯_{m}^{γ} denote the linear span of the set {cos(k + γ)x, sin(k + γ)x : k = 0, 1, …, m}. Obviously, 𝒯_{m}^{0} = 𝒯_{m} is the linear space of all trigonometric polynomials of degree less than or equal to m, while 𝒯_{m}^{1/2} is the linear space of all trigonometric polynomials of semi-integer degree less than or equal to m + 1/2. Of course, dim(𝒯_{m}) = 2m + 1 and dim(𝒯_{m}^{1/2} ) = 2(m + 1).

The trigonometric polynomial of semi-integer degree m + 1/2 has the following form

(2.1)  A_{m}^{1/2} (x) = ∑_{ν=0}^{m} [c_{ν} cos( ν + \frac{1}{2}) x + d_{ν} sin( ν + \frac{1}{2}) x],

where c_{ν}, d_{ν} ∈ ℝ, |c_{m}| + |d_{m}| ≠ 0. Every trigonometric polynomial of semi-integer degree m+1/2 of the form (2.1) can be represented in the form (cf. [21, Lemma 1])

(2.2)  A_{m}^{1/2} (x) = A ∏_{k=0}^{2m} sin \frac{x − x_{k}}{2} (A is a nonzero constant).

It is obvious that every A_{m}^{1/2} (x) given by (2.2) is a trigonometric polynomial of semi-integer degree m + 1/2.

Remark 2.1. For the rest of this paper an important fact is that a trigonometric polynomial of semi-integer degree m+1/2 has exactly 2m+1 zeros counting with their multiplicities and restricting to the strip −π ⩽ Rez < π, as well as that the nonreal zeros appear in conjugate pairs [3, Theorem 1.1]. This means that the number of sign changes of trigonometric polynomials of semi-integer degree on [−π, π) can not be even. The same is true also for any interval [L, L + 2π), L ∈ ℝ, of length 2π.

Let p be a positive integer and n = (n_{1}, n_{2}, …, n_{p}) a vector of p nonnegative integers, which is called a multi-index with length |n| = n_{1} + n_{2} + ⋯+ n_{p}. We introduce a partial order on multi-indices by

(2.3)  m ⪯ n ⇔ m_{ν} ⩽ n_{ν} for every ν = 1, 2, …, p.

Let W = {w_{1}, w_{2}, …, w_{p}} be a set of p weight functions, integrable and nonnegative on some interval E of length 2π, vanishing there only on a set of measure zero. In what follows, we always assume that the interval E is closed on the left and open on the right, i.e., that interval E is of the form [L, 2π + L), for L ∈ ℝ.

Analogously with the multiple algebraic orthogonal polynomials, we define two types of trigonometric multiple orthogonal polynomials of semi-integer degree.

Definition 2.1. Let n be a multi-index. Type I multiple trigonometric orthogonal polynomials of semi-integer degree with respect to W are collected in a vector (A_{n,1}^{1/2} , A_{n,2}^{1/2} , …, A_{n,p}^{1/2} ) of trigonometric polynomials of semi-integer degree, where A_{n,ν}^{1/2} has semi-integer degree n_{ν} −1/2, ν = 1, 2, …, p, such that the following orthogonality conditions hold

∑_{ν=1}^{p} ∫_{E} A_{n,ν}^{1/2} cos( k + \frac{1}{2}) x w_{ν}(x) dx = 0, k = 0, 1, 2, …, |n| − 2,

∑_{ν=1}^{p} ∫_{E} A_{n,ν}^{1/2} sin( k + \frac{1}{2}) x w_{ν}(x) dx = 0, k = 0, 1, 2, …, |n| − 2,

(2.4)

with the normalizations

∑_{ν=1}^{p} ∫_{E} A_{n,ν}^{1/2} cos( |n| − \frac{1}{2}) x w_{ν}(x) dx = 1,

∑_{ν=1}^{p} ∫_{E} A_{n,ν}^{1/2} sin( |n| − \frac{1}{2}) x w_{ν}(x) dx = 1.

(2.5)

Conditions (2.4)–(2.5) give a linear system of 2|n| equations for the 2|n| unknown coefficients of the trigonometric polynomials of semi-integer degree A_{n,ν}^{1/2} , ν = 1, 2, …, p. The multi-index n is normal for type I if system of equations (2.4)–(2.5) has a unique solution. For the type I multiple trigonometric orthogonal polynomials we define the following function

(2.6)  A_{n}(x) = ∑_{ν=1}^{p} A_{n,ν}^{1/2} w_{ν}(x).

Then, the orthogonality conditions (2.4) and the normalizations (2.5) become

∫_{E} A_{n}(x) cos( k + \frac{1}{2}) xdx = 0, k = 0, 1, 2, …, |n| − 2,

∫_{E} A_{n}(x) sin( k + \frac{1}{2}) xdx = 0, k = 0, 1, 2, …, |n| − 2,

(2.7)

and

(2.8)  ∫_{E} A_{n}(x) cos( |n| − \frac{1}{2}) xdx = 1, ∫_{E} A_{n}(x) sin( |n| − \frac{1}{2}) xdx = 1,

respectively.

Definition 2.2. Let n be a multi-index. Trigonometric polynomial of semiinteger degree T_{n}^{1/2} is a type II multiple trigonometric orthogonal polynomial of semi-integer degree with respect to W if it is of semi-integer degree |n| + 1/2 and satisfies the following orthogonality conditions

∫_{E} T_{n}^{1/2} (x) cos( k_{ν} + \frac{1}{2}) x w_{ν}(x) dx = 0, k_{ν} = 0, 1, …, n_{ν} − 1,

∫_{E} T_{n}^{1/2} (x) sin( k_{ν} + \frac{1}{2}) x w_{ν}(x) dx = 0, k_{ν} = 0, 1, …, n_{ν} − 1,

(2.9)

for ν = 1, 2, …, p.

Remark 2.2. Let us notice that if some n_{ν} = 0, we do not have orthogonality conditions (2.9) for the corresponding weight w_{ν}.

For p = 1 we have the case of ordinary trigonometric orthogonal polynomials of semi-integer degree.

Orthogonality conditions (2.9) give a system of linear equations for the unknown coefficients of the trigonometric polynomial T_{n}^{1/2} . Since

T_{n}^{1/2} (x) = ∑_{k=0}^{|n|} ( a_{k} cos( k + \frac{1}{2}) x + b_{k} sin( k + \frac{1}{2}) x) ∈ 𝒯_{|n|}^{1/2} ,

we have 2|n|+2 unknown coefficients a_{k}, b_{k}, k = 0, 1, …, |n|. Conditions (2.9) give 2(n_{1}+n_{2}+⋯ +n_{p}) = 2|n| equations for the 2|n|+2 unknown coefficients of T_{n}^{1/2} , so we have to fix two coefficients. We choose to fix in advance the leading coefficients a_{|n|} and b_{|n|} (of course, a_{|n|}^{2} + b_{|n|}^{2} ≠ 0). For the special choices of the leading coefficients, (a_{|n|} , b_{|n|}) ∈ {(1, 0), (0, 1)}, we introduce the following notations

T_{n}^{C,1/2} (x)= cos( |n| + \frac{1}{2}) x + ∑_{k=0}^{|n|−1} ( a_{k} cos( k + \frac{1}{2}) x + b_{k} sin( k + \frac{1}{2}) x) ,

T_{n}^{S,1/2} (x)= sin( |n| + \frac{1}{2}) x + ∑_{k=0}^{|n|−1} ( a_{k} cos( k + \frac{1}{2}) x + b_{k} sin( k + \frac{1}{2}) x) .

We call T_{n}^{C,1/2} (x) and T_{n}^{S,1/2} (x) the monic cosine and the monic sine multiple orthogonal polynomial of semi-integer degree, respectively.

If system (2.9) has a unique solution, then the multi-index n is normal for type II.

Lemma 2.1. A multi-index n is normal for type I if and only if it is normal for type II.

Proof. For ν = 1, …, p we introduce the following notations

I_{i,j}^{C,ν} = ∫_{E} cos( i + \frac{1}{2}) x cos( j + \frac{1}{2}) x w_{ν}(x) dx,

I_{i,j}^{S,ν} = ∫_{E} sin( i + \frac{1}{2}) x sin( j + \frac{1}{2}) x w_{ν}(x) dx,

I_{i,j}^{ν} = ∫_{E} cos( i + \frac{1}{2}) x sin( j + \frac{1}{2}) x w_{ν}(x) dx,

m_{i,j}^{(ν)} = [ \begin{matrix}I_{i,j}^{C,ν} & I_{i,j}^{ν} \\ I_{j,i}^{ν} & I_{i,j}^{S,ν} \end{matrix} ] ,

for i, j = 0, 1, …, and

M_{ν} = [ \begin{matrix}m_{0,0}^{(ν)} & m_{0,1}^{(ν)} & ⋯ & m_{0,|n|−1}^{(ν)} \\ m_{1,0}^{(ν)} & m_{1,1}^{(ν)} & ⋯ & m_{1,|n|−1}^{(ν)} \\ ⋮ & ⋮ & ... & ⋮ \\ m_{n_{ν} −1,0}^{(ν)} & m_{n_{ν} −1,1}^{(ν)} & ⋯ & m_{n_{ν} −1,|n|−1}^{(ν)} \end{matrix} ] .

Then, the matrix of system (2.4)–(2.5) is given by

M^{I} = [M_{1}^{T} M_{2}^{T} ⋯M_{p}^{T}]_{2|n|×2|n|} ,

and the matrix of system (2.9) (the leading coefficients of T_{n}^{1/2} are fixed) is given by

M^{II} = [ M_{1} ; M_{2} ; ⋮ ; M_{p} ]_{2|n|×2|n|} .

Obviously, M^{II} is the transpose of M^{I}, which means that determinants of systems (2.9) and (2.4)–(2.5) are equal, i.e., system (2.4)–(2.5) has a unique solution if and only if system (2.9) has a unique solution. □

According to Lemma 2.1, we could just talk about normal multi-indices. If all multi-indices are normal, then we have a perfect system.

Since the matrix of coefficients of systems (2.9) can be singular, we need some additional conditions on the p weight functions to provide the uniqueness of multiple trigonometric orthogonal polynomials of semi-integer degree. It is easy to see that the uniqueness is guaranteed if the following set of functions

\{w_{ν} cos( k_{ν} + \frac{1}{2}) x, w_{ν} sin( k_{ν} + \frac{1}{2}) x : ν = 1, 2, …, p, k_{ν} = 0, 1, …, n_{ν} − 1\},

form a Chebyshev system on E for the multi-index n. We called such set W = {w_{1}, w_{2}, …, w_{p}} trigonometric AT system (TAT system) of weight functions for multi-index n.

The properties of zeros of type I and II multiple trigonometric orthogonal polynomials are given in the following two theorems.

Theorem 2.1. Suppose that n is a multi-index such that W = {w_{1}, w_{2}, …, w_{p}} is a TAT system of weight functions for all multi-indices m such that m ⪯ n. For the type I multiple trigonometric orthogonal polynomials of semi-integer degree with respect to W the function A_{n}(x), given by (2.6), has exactly 2|n| − 1 simple zeros on E.

Proof. It is easy to see that the type I function A_{n}(x) has at least one sign change on E, since if we assume the contrary, for |n| ⩾ 1 the orthogonality

∫_{E} A_{n}(x) sin \frac{x − L}{2} dx = 0

would be impossible since sin(x − L)/2 does not change its sign on [L, 2π + L).

The function A_{n}(x) has at most 2|n| − 1 zeros on E since we are dealing with the TAT system. The number of its sign changes on E is odd (see Remark 2.1). Suppose it has 2m − 1, m < |n|, sign changes at the points x_{1}, x_{2}, …, x_{2m−1} ∈ E, and set

Q(x) = ∏_{i=1}^{2m−1} sin \frac{x − x_{i}}{2} .

Then, A_{n}(x)Q(x) does not change sign on E and ∫_{E} A_{n}(x)Q(x) dx ≠ 0, which is a contradiction with orthogonality conditions (2.4). Therefore, m = |n|, which means that A_{n}(x) has exactly 2|n| − 1 simple zeros on E. □

Theorem 2.2. Suppose that n is a multi-index such that W = {w_{1}, w_{2}, …, w_{p}} is a TAT system of weight functions for all multi-indices m such that m ⪯ n. Type II multiple trigonometric orthogonal polynomial of semi-integer degree T_{n}^{1/2} (x) with respect to W has exactly 2|n| + 1 simple zeros on E.

Proof. In the similar way as in proof of Theorem 2.1 we conclude that T_{n}^{1/2} (x) has an odd number of sign changes on E for |n| ⩾ 1.

Let us assume that the polynomial T_{n}^{1/2} (x) has 2m + 1 changes of sign on E at the points x_{0}, x_{1}, …, x_{2m} and that m < |n|. Let m = (m_{1}, m_{2}, …, m_{p}) be a multi-index such that m = |m|, m ⪯ n, and m_{j} < n_{j} for at least one j. Now, we construct

Q(x) = ∑_{i=1}^{p} Q_{m_{i}}^{1/2} (x) w_{i}(x),

where each Q_{m_{i}}^{1/2} is a trigonometric polynomial of semi-integer degree m_{i} − 1/2, for i ≠ j, Q_{m_{j}}^{1/2} is a trigonometric polynomial of semi-integer degree m_{j} +1/2, satisfying the interpolation conditions Q(x_{k}) = 0, k = 0, 1, …, 2m, and Q(x_{2m+1} ) = 1, for an additional point x_{2m+1} ∈ E. Since we have a Chebyshev system of 2m + 2 functions, this interpolation problem has a unique solution, and since the function Q already has 2m + 1 zeros it can have no additional sign changes. Of course, the function Q does not vanish identically since Q(x_{2m+1} ) ≠ 0. Obviously T_{n}^{1/2} (x)Q(x) does not change sign on E, so that

∫_{E} T_{n}^{1/2} (x)Q(x) dx ≠ 0,

but this is in contrast with the orthogonality relations (2.9). Hence, T_{n}^{1/2} (x) has exactly 2|n| + 1 simple zeros on E. □

We finish this section proving the following biorthogonality between the type II multiple trigonometric orthogonal polynomials of semi-integer degree T_{n}^{1/2} (x) and the type I functions A_{m} (x), given by (2.6).

Theorem 2.3. Suppose that n and m are two multi-indices and that W = {w_{1}, w_{2}, …, w_{p}} is a TAT system of weight functions for the both multi-indices n and m. Then the following relation of biorthogonality

∫_{E} T_{n}^{1/2} (x)A_{m} (x) dx = \{ \begin{matrix}0, & if m ⪯ n, \\ 0, & if |n| ⩽ |m| − 2, \\ a_{|n|} + b_{|n|} , & if |n| = |m| − 1, \end{matrix}

holds, where T_{n}^{1/2} (x) is the corresponding type II multiple trigonometric orthogonal polynomial of semi-integer degree with the leading coefficients a_{|n|} and b_{|n|}, and A_{m} (x) the corresponding type I function, given by (2.6).

Proof. Since A_{m} (x) is given by (2.6), we have

∫_{E} T_{n}^{1/2} (x)A_{m} (x) dx = ∫_{E} T_{n}^{1/2} (x)( ∑_{ν=1}^{p} A_{m,ν}^{1/2} w_{ν}(x)) dx

= ∑_{ν=1}^{p} ∫_{E} T_{n}^{1/2} (x)A_{m,ν}^{1/2} w_{ν}(x) dx.

First we assume that m ⪯ n. According to (2.3) and orthogonality conditions (2.9) for the T_{n}^{1/2} (x), we conclude that all integrals on the right hand side of the previous equality are equal to zero, i.e., ∫_{E} T_{n}^{1/2} (x)A_{m} (x) dx = 0. Since a_{|n|} and b_{|n|} are the leading coefficients of T_{n}^{1/2} , we have

(2.10)  ∫_{E} T_{n}^{1/2} (x)A_{m} (x) dx = a_{|n|} ∫_{E} A_{m} (x) cos( |n| + \frac{1}{2}) xdx

+ b_{|n|} ∫_{E} A_{m} (x) sin( |n| + \frac{1}{2}) xdx

+ ∑_{k=0}^{|n|−1} a_{k} ∫_{E} A_{m} (x) cos( k + \frac{1}{2}) xdx

+ ∑_{k=0}^{|n|−1} b_{k} ∫_{E} A_{m} (x) sin( k + \frac{1}{2}) xdx.

If |n| ⩽ |m| − 2, by using orthogonality conditions (2.7) for the A_{m} (x), we obtain that all integrals on the right hand side of the previous equality are equal to zero and hence ∫_{E} T_{n}^{1/2} (x)A_{m} (x) dx = 0.

To finish the proof we consider the case |n| = |m| − 1. By using conditions (2.8) and (2.7), from (2.10) we get what is stated. □

3. Optimal set of quadrature rules for trigonometric polynomials

Let n be a multi-index and let W = {w_{1}, w_{2}, …, w_{p}} be a TAT system for n on interval E. In this section we consider evaluation of a set of p definite integrals over the same interval E, taken with respect to the weight functions from W and related to a common integrand, i.e., the set of integrals of the form

∫_{E} f(x)w_{ν}(x) dx, ν = 1, 2, …, p.

As it has been already said, we are interested only in quadrature rules with an odd number of nodes. We require that such a set of quadrature rules is optimal for trigonometric polynomials in Borges’ sense. For that purpose, we start with definition of trigonometric degree of exactness.

Definition 3.1. For a weight function w, integrable and nonnegative on the interval E, vanishing there only on a set of a measure zero, a quadrature rule of the form

(3.1)  ∫_{E} f(x)w(x) dx = ∑_{ν=0}^{2n} σ_{ν}f(τ_{ν}) + R_{n}(f)

has a trigonometric degree of exactness equal to d if R_{n}(f) = 0 for all f ∈ 𝒯_{d} and there exists g ∈ 𝒯_{d+1} such that R_{n}(g) ≠ 0.

If τ_{ν} ∈ E, ν = 0, 1, …, 2n, are distinct points fixed in advance, then corresponding interpolatory quadrature rule (3.1) has a trigonometric degree of exactness equal to n. One can increase that trigonometric degree of exactness up to the maximal trigonometric degree of exactness, which is equal to 2n, with appropriate choice of nodes. The maximal trigonometric degree of exactness is achieved when nodes are zeros of the corresponding trigonometric orthogonal polynomials of semiinteger degree n + 1/2 [5,9,21]. In such a way we obtain the Gaussian quadrature rules for trigonometric polynomials.

Analogously with [4], we introduce the performance ratio with respect to trigonometric degree of exactness as follows

R_{T} = \frac{Overall trigonometric degree of exactness + 1}{Number of integrand evaluation} .

Taking distinct nodes for each of p quadrature rule, it is obvious that R_{T} will be maximal when we have the set of p Gaussian quadrature rules. In that case

R_{T} = \frac{2n + 1}{p(2n + 1)} = \frac{1}{p},

so that, R_{T} < 1/2 for all p > 2.

If we select a set of 2n + 1 distinct nodes, common for all quadrature rules, the weight coefficients for each of p quadrature rules can be chosen in such a way that R_{T} > 1/2, which will be shown in what follows.

Definition 3.2. Let n be a multi-index and let W = {w_{1}, w_{2}, …, w_{p}} be a TAT system for n on the interval E. A set of quadrature rules of the form

(3.2)  ∫_{E} f(x)w_{ν}(x) dx ≈ ∑_{k=0}^{2|n|} A_{ν,k} f(x_{k}), ν = 1, 2, …, p,

is an optimal set with respect to (W, n) if and only if the weight coefficients A_{ν,k}, ν = 1, 2, …, p, k = 0, 1, …, 2|n|, and the nodes x_{k}, k = 0, 1, …, 2|n|, satisfy the following system of equations

∑_{k=0}^{2|n|} A_{ν,k} = ∫_{E} w_{ν}(x) dx,

∑_{k=0}^{2|n|} A_{ν,k} cosm_{ν}x_{k} = ∫_{E} cosm_{ν}x w_{ν}(x) dx, m_{ν} = 1, 2, …, |n| + n_{ν},

∑_{k=0}^{2|n|} A_{ν,k} sinm_{ν}x_{k} = ∫_{E} sinm_{ν}x w_{ν}(x) dx, m_{ν} = 1, 2, …, |n| + n_{ν},

for ν = 1, 2, …, p.

Remark 3.1. Let us notice that an optimal set of quadrature rules have 2|n|+1 distinct nodes, common for all quadrature rules, and overall trigonometric degree of exactness is equal to |n| + m, where m = min{n_{1}, n_{2}, …, n_{p}}. Therefore,

R_{T} = \frac{|n| + m + 1}{2|n| + 1} = \frac{1}{2} + \frac{m + 1/2}{2|n| + 1} > 1/2.

In order to characterize the optimal set of quadrature rules, we prove the following representation of trigonometric polynomials. A special case of that representation for m = n was proved by Turetzkii [21].

Lemma 3.1. Every trigonometric polynomial of degree n + m, m ⩽ n,

B_{n+m} (x) = a_{0} + ∑_{k=1}^{n+m} (a_{k} coskx + b_{k} sinkx)

can be uniquely represented in the form

(3.3)  B_{n+m} (x) = A_{n}^{1/2} (x)S_{m−1}^{1/2} (x) + T_{n}(x),

where

A_{n}^{1/2} (x) = ∑_{k=0}^{n} ( c_{k} cos( k + \frac{1}{2}) x + d_{k} sin( k + \frac{1}{2}) x)

is a certain trigonometric polynomial of semi-integer degree n+ \frac{1}{2}, with c_{n} +id_{n} ≠ 0, and

S_{m−1}^{1/2} = ∑_{ν=0}^{m−1} ( γ_{ν} cos( ν + \frac{1}{2}) x + δ_{ν} sin( ν + \frac{1}{2}) x) ,

T_{n}(x)= u_{0} + ∑_{k=1}^{n} (u_{k} coskx + v_{k} sinkx)

are the required trigonometric polynomials of semi-integer degree m − \frac{1}{2} and of degree n, respectively.

Proof. Comparing the coefficients of cosℓx and sinℓx, ℓ= n+1, n+2, …, n+m , on both sides in (3.3), we obtain the following system of equations for determining the unknown coefficients γ_{ν}, δ_{ν}, ν = 0, 1, …, m − 1

(3.4) \frac{1}{2} ∑_{k=ℓ−m}^{n} (c_{k}γ_{ℓ−k−1} − d_{k}δ_{ℓ−k−1} ) = a_{ℓ}, \frac{1}{2} ∑_{k=ℓ−m}^{n} (c_{k}δ_{ℓ−k−1} + d_{k}γ_{ℓ−k−1} ) = b_{ℓ},

for ℓ = n + 1, n + 2, …, n + m. Since that system can be rewritten in the form

∑_{k=ℓ−m}^{n} (c_{k} + id_{k})(γ_{ℓ−k−1} + iδ_{ℓ−k−1} ) = 2(a_{ℓ} + ib_{ℓ}), ℓ = n + 1, …, n + m,

its determinant is equal to (c_{n}+id_{n})^{m} ≠ 0. Therefore, γ_{ν} and δ_{ν}, ν = 0, 1, …, m−1, can be uniquely determined from (3.4).

Further, from the equation

\frac{1}{2} ∑_{k=0}^{m−1} (c_{k}γ_{k} + d_{k}δ_{k}) + u_{0} = a_{0},

which is obtained by comparing free coefficients on both sides in (3.3), one can determine u_{0}.

Similarly, if we compare coefficients of cosℓx and sinℓx, ℓ = 1, 2, …, n, on both sides in (3.3), we obtain the following system of equations

\frac{1}{2} ∑_{k=ℓ}^{j} (c_{k}γ_{k−ℓ} + d_{k}δ_{k−ℓ} ) + \frac{1}{2} ∑_{k=0}^{m−ℓ−1} (c_{k}γ_{k+ℓ} + d_{k}δ_{k+ℓ} )

+ \frac{1}{2} ∑_{k=0}^{ℓ−1} (c_{k}γ_{ℓ−k−1} − d_{k}δ_{ℓ−k−1} ) + u_{ℓ} = a_{ℓ},

\frac{1}{2} ∑_{k=ℓ}^{j} (d_{k}γ_{k−ℓ} − c_{k}δ_{k−ℓ} ) + \frac{1}{2} ∑_{k=0}^{m−ℓ−1} (c_{k}δ_{k+ℓ} − d_{k}γ_{k+ℓ} )

+ \frac{1}{2} ∑_{k=0}^{ℓ−1} (c_{k}δ_{ℓ−k−1} + d_{k}γ_{ℓ−k−1} ) + v_{ℓ} = b_{ℓ},

where ℓ = 1, 2, …, m − 1, j = min{n, ℓ + m − 1}, and

\frac{1}{2} ∑_{k=ℓ}^{j} (c_{k}γ_{k−ℓ} + d_{k}δ_{k−ℓ} ) + \frac{1}{2} ∑_{k=ℓ−m}^{ℓ−1} (c_{k}γ_{ℓ−k−1} − d_{k}δ_{ℓ−k−1} ) + u_{ℓ} = a_{ℓ},

\frac{1}{2} ∑_{k=ℓ}^{j} (d_{k}γ_{k−ℓ} − c_{k}δ_{k−ℓ} ) + \frac{1}{2} ∑_{k=ℓ−m}^{ℓ−1} (c_{k}δ_{ℓ−k−1} + d_{k}γ_{ℓ−k−1} ) + v_{ℓ} = b_{ℓ},

for ℓ = m, m + 1, …, n and j = min{n, ℓ + m − 1}, from which one can uniquely obtain u_{ℓ}, v_{ℓ}, ℓ = 1, 2, …, n. □

The characterization of an optimal set of quadrature rules for trigonometric polynomials is given by the following statement, which is the counterpart to the fundamental theorem of Gaussian quadratures.

Theorem 3.1. Let n be a multi-index and let W = {w_{1}, w_{2}, …, w_{p}} be a TAT system for n on interval E. A set of quadrature rules (3.2) is an optimal set with respect to (W, n) if and only if

1^{∘} all rules are exact for all trigonometric polynomials from 𝒯_{|n|}; 2^{∘} T_{n}^{1/2} (x) = ∏ _{k=0}^{2|n|} sin \frac{x−x_{k}}{2} is the type II multiple trigonometric orthogonal polynomial of semi-integer degree |n| + 1/2 with respect to (W, n).

Proof. Suppose first that quadrature rules (3.2) form an optimal set with respect to (W, n).

For each ν = 1, 2, …, p, the corresponding quadrature rule with respect to the weight function w_{ν} is exact for all trigonometric polynomials of degree less than or equal to |n| + n_{ν}, hence, it is exact for those of degree less than or equal to |n|. Thus, 1^{∘} is proved.

In order to prove 2^{∘} assume that S_{m_{ν} −1}^{1/2} (x), ν = 1, 2, …, p, is a trigonometric polynomial of semi-integer degree m_{ν} − 1/2, where m_{ν} ⩽ n_{ν}, ν = 1, 2, …, p. Then T_{n}^{1/2} (x)S_{m_{ν} −1}^{1/2} (x) is a trigonometric polynomial of degree less than or equal to |n|+ n_{ν}, ν = 1, 2, …, p. Since the corresponding quadrature rule, with respect to the weight function w_{ν}, ν = 1, 2, …, p, is exact for all such trigonometric polynomials and T_{n}^{1/2} (x_{k}) = 0, k = 0, 1, …, 2|n|, it follows that

∫_{E} T_{n}^{1/2} (x)S_{m_{ν} −1}^{1/2} (x) w_{ν}(x) dx = ∑_{k=0}^{2|n|} A_{ν,k} T_{n}^{1/2} (x_{k})S_{m_{ν} −1}^{1/2} (x_{k}) = 0,

for ν = 1, 2, …, p, i.e., T_{n}^{1/2} (x) is the type II multiple trigonometric orthogonal polynomial of semi-integer degree |n| + 1/2 with respect to W.

Let us now suppose that 1^{∘} and 2^{∘} hold for (3.2). Let B_{|n|+n_{ν}} (x) ∈ 𝒯_{|n|+n_{ν}} , ν = 1, 2, …, p. According to Lemma 3.1, B_{|n|+n_{ν}} (x) can be represented in the form

B_{|n|+n_{ν}} (x) = T_{n}^{1/2} (x)S_{n_{ν} −1}^{1/2} (x) + P_{|n|} (x),

where S_{n_{ν} −1}^{1/2} ∈ 𝒯_{n_{ν} −1}^{1/2} and P_{|n|} ∈ 𝒯_{|n|}. Now we have that

∫_{E} B_{|n|+n_{ν}} (x) w_{ν}(x) dx = ∫_{E}[T_{n}^{1/2} (x)S_{n_{ν} −1}^{1/2} (x) + P_{|n|} (x)] w_{ν}(x) dx
= ∫_{E} T_{n}^{1/2} (x)S_{n_{ν} −1}^{1/2} (x) w_{ν}(x) dx + ∫_{E} P_{|n|} (x) w_{ν}(x) dx,

ν = 1, 2 …, p. From 2^{∘} it follows that

∫_{E} T_{n}^{1/2} (x)S_{n_{ν} −1}^{1/2} (x) w_{ν}(x) dx = 0, ν = 1, 2 …, p,

and, since P_{|n|} ∈ 𝒯_{|n|}, from 1^{∘} we have

∫_{E} P_{|n|} (x) w_{ν}(x) dx = ∑_{k=0}^{2|n|} A_{ν,k} P_{|n|} (x_{k}), ν = 1, 2 …, p,

and hence

∫_{E} B_{|n|+n_{ν}} (x) w_{ν}(x) dx = ∑_{k=0}^{2|n|} A_{ν,k} P_{|n|} (x_{k}), ν = 1, 2, …, p.

Finally, since T_{n}^{1/2} (x_{k}) = 0, k = 0, 1, …, 2|n|, it follows that B_{|n|+n_{ν}} (x_{k}) = P_{|n|} (x_{k}), k = 0, 1, …, 2|n|, ν = 1, 2, …, p, and we have

∫_{E} B_{|n|+n_{ν}} (x) w_{ν}(x) dx = ∑_{k=0}^{2|n|} A_{ν,k} B_{|n|+n_{ν}} (x_{k}), ν = 1, 2, …, p,

i.e., the quadrature rule for the weight w_{ν} is exact for all trigonometric polynomials of degree less that or equal to |n|+n_{ν}, ν = 1, 2, …, p. Therefore, a set of quadrature rules (3.2) is an optimal set with respect to (W, n). □

Remark 3.2. When p = 1 the optimal set of quadrature rules reduces to the Gaussian quadrature rule for trigonometric polynomials.

4. Numerical examples

In this section we illustrate characterization of an optimal set of quadrature rules (3.2) by using Theorem 3.1.

Example 4.1. Let us find parameters for the optimal set of quadrature rules on E = [−π, π), for p = 2, n = (2, 1), with respect to the weight functions w_{1}(x) = 1 and w_{2}(x) = 1 + sin2x. First we check whether the following set of functions

\{ cos \frac{x}{2}, sin \frac{x}{2}, cos \frac{3x}{2}, sin \frac{3x}{2}, cos \frac{x}{2}(1 + sin2x), sin \frac{x}{2}(1 + sin2x)\},

form a Chebyshev system on the interval [−π, π). Let −π ⩽ y_{1} < y_{2} ⋯< y_{6} < π be arbitrary distinct points. By using an elementary transformation and properties of determinants one can easily obtain that the determinant

||||||||||||||||| cos \frac{y_{1}}{2} sin \frac{y_{1}}{2} cos \frac{3y_{1}}{2} sin \frac{3y_{1}}{2} cos \frac{y_{1}}{2} (1 + sin2y_{1}) sin \frac{y_{1}}{2} (1 + sin2y_{1})

cos \frac{y_{2}}{2} sin \frac{y_{2}}{2} cos \frac{3y_{2}}{2} sin \frac{3y_{2}}{2} cos \frac{y_{2}}{2} (1 + sin2y_{2}) sin \frac{y_{2}}{2} (1 + sin2y_{2})
cos \frac{y_{3}}{2} sin \frac{y_{3}}{2} cos \frac{3y_{3}}{2} sin \frac{3y_{3}}{2} cos \frac{y_{3}}{2} (1 + sin2y_{3}) sin \frac{y_{3}}{2} (1 + sin2y_{3})
cos \frac{y_{4}}{2} sin \frac{y_{4}}{2} cos \frac{3y_{4}}{2} sin \frac{3y_{4}}{2} cos \frac{y_{4}}{2} (1 + sin2y_{4}) sin \frac{y_{4}}{2} (1 + sin2y_{4})
cos \frac{y_{5}}{2} sin \frac{y_{5}}{2} cos \frac{3y_{5}}{2} sin \frac{3y_{5}}{2} cos \frac{y_{5}}{2} (1 + sin2y_{5}) sin \frac{y_{5}}{2} (1 + sin2y_{5})
cos \frac{y_{6}}{2} sin \frac{y_{6}}{2} cos \frac{3y_{6}}{2} sin \frac{3y_{6}}{2} cos \frac{y_{6}}{2} (1 + sin2y_{6}) sin \frac{y_{6}}{2} (1 + sin2y_{6})

|||||||||||||||||

is equal to

−1024 ∏_{\begin{matrix}i,j=1 \\ i<j \end{matrix}}^{6} sin \frac{y_{i} − y_{j}}{2} ≠ 0.

Therefore, we have the Chebyshev set of functions, i.e., W = {w_{1}, w_{2}} is a TAT system for the multi-index n = (2, 1).

Now, the type II multiple trigonometric orthogonal polynomial of semi-integer degree |n|+1/2 = 3+1/2 can be obtained from orthogonality conditions (2.9). We choose the leading coefficients a_{3} = b_{3} = 1, so that

T_{n}^{1/2} (x) = cos \frac{7x}{2} + sin \frac{7x}{2} + ∑_{k=0}^{2} [a_{k} cos( k + \frac{1}{2}) x + b_{k} sin( k + \frac{1}{2}) x].

The solution of the corresponding system (2.9) is a_{0} = b_{0} = a_{1} = b_{1} = a_{2} = b_{2} = 0, i.e., T_{n}^{1/2} (x) = cos \frac{7x}{2} + sin \frac{7x}{2} . The zeros of T_{n}^{1/2} (x), i.e., nodes of the quadrature rules x_{i}, i = 0, 1, …, 6, are

x_{0} = −\frac{13π}{14} , x_{1} = −\frac{9π}{14}, x_{2} = −\frac{5π}{14}, x_{3} = −\frac{π}{14}, x_{4} = \frac{3π}{14}, x_{5} = \frac{π}{2}, x_{6} = \frac{11π}{14} .

We obtain the weight coefficients A_{ν,k}, ν = 1, 2, k = 0, 1, …, 6, by using the condition 1^{∘} of Theorem 3.1, i.e., from the conditions that the quadrature rules (3.2) are exact for all trigonometric polynomials from the space 𝒯_{3}. The results are given in Table 1. The numbers in parentheses indicate decimal exponents.

Table 1. The weight coefficients A_{ν,k}, ν = 1, 2, k = 0, 1, …, 6, of the optimal set of quadrature rules with respect to the set of functions W = {1, 1 + sin2x} and n = (2, 1)

\begin{matrix}k & A_{1,k} & A_{2,k} \\ 0 & 0.897597901025655 & 1.28705103454674 \\ 1 & 0.897597901025655 & 1.59936819864474 \\ 2 & 0.897597901025655 & 1.95827603406575(−1) \\ 3 & 0.897597901025655 & 5.08144767504572(−1) \\ 4 & 0.897597901025655 & 1.77269114865139 \\ 5 & 0.897597901025655 & 8.97597901025655(−1) \\ 6 & 0.897597901025655 & 2.25046533999260(−2) \end{matrix}

Example 4.2. Let us now construct the optimal set of quadrature rules on E = [0, 2π), for p = 3, n = (1, 1, 1), with respect to the weight functions w_{1}(x) = 3 − cos2x, w_{2}(x) = 1 + 2 sinx, and w_{3}(x) = 2 + cosx.

In this case we have to verify that the set of functions

\{cos \frac{x}{2}w_{1}(x), sin \frac{x}{2}w_{1}(x), cos \frac{x}{2}w_{2}(x), sin \frac{x}{2}w_{2}(x), cos \frac{x}{2}w_{3}(x), sin \frac{x}{2}w_{3}(x)\}

is a Chebyshev system on the interval [0, 2π), which can be proved in the same way as in Example 4.1.

We consider the monic sine type II multiple orthogonal polynomial of semiinteger degree T_{n}^{S,1/2} ∈ 𝒯_{3}^{1/2} . From orthogonality conditions (2.9) we get T_{n}^{S,1/2} = sin7x/2. Then the nodes x_{k}, k = 0, 1, …, 6, of the optimal set of quadrature rules are x_{k} = 2kπ/7, k = 0, 1, …, 6.

Table 2. The weight coefficients A_{ν,k}, ν = 1, 2, 3, k = 0, 1, …, 6, of the optimal set of quadrature rules with respect to the set of functions W = {3 − cos2x, 1 + 2 sinx, 2 + cosx} and n = (1, 1, 1)

\begin{matrix}k & A_{1,k} & A_{2,k} & A_{3,k} \\ 0 & 1.79519580205131 & 8.97597901025655(−1) & 2.69279370307697 \\ 1 & 2.89252802633042 & 2.30113849626382 & 2.35483893951061 \\ 2 & 3.50150146779564 & 2.64778439627711 & 1.59546147879785 \\ 3 & 2.13315056561766 & 1.67650416806782 & 9.86488037332638(−1) \\ 4 & 2.13315056561766 & 1.186916339834889(−1) & 9.86488037332638(−1) \\ 5 & 3.50150146779564 & −8.52588594225803(−1) & 1.59546147879785 \\ 6 & 2.89252802633042 & −5.05942694212505(−1) & 2.35483893951061 \end{matrix}

For each ν = 1, 2, 3, the corresponding weight coefficients A_{ν,k}, k = 0, 1, …, 6, given in Table 2, can be obtained by using condition 1^{∘} of Theorem 3.1.

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Mathematical Institute of the Serbian Academy of Sciences and Arts Belgrade Serbia

gvm@mi.sanu.ac.rs

Department of Mathematics and Informatics Faculty of Science University of Kragujevac Kragujevac Serbia

stanicm@kg.ac.rs
tomovict@kg.ac.rs