PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 95 (109) (2014), 29–47 DOI: 10.2298/PIM1409029H

OPTIMAL QUADRATURE FORMULA IN THE SENSE OF SARD IN K_{2}(P_{3}) SPACE

Abdullo R. Hayotov, Gradimir V. Milovanović, and Kholmat M. Shadimetov

Communicated by Stevan Pilipović

Abstract. We construct an optimal quadrature formula in the sense of Sard in the Hilbert space K_{2}(P_{3}). Using Sobolev’s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L_{2}^{(3)} (0, 1). The obtained optimal quadrature formula is exact for the trigonometric functions sin x, cos x and for constants. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.

1. Introduction and preliminaries

We consider the following quadrature formula

(1.1)  ∫_{0}^{1} φ(x) dx ≅ ∑_{ν=0}^{N} C_{ν}φ(x_{ν}),

with an error functional given by

(1.2)  ℓ(x) = χ_{[0,1]}(x) − ∑_{ν=0}^{N} C_{ν}δ(x − x_{ν}),

2010 Mathematics Subject Classification: Primary 65D32. Key words and phrases: optimal quadrature formulas, error functional, extremal function,

Hilbert space, optimal coefficients.

The work of the first author (A. R. Hayotov) was supported by the program Erasmus Mundus Action 2, Stand 1, Lot 10, Marco XXI (project number: 204513-EM-1-2011-1-DE-ERA MUNDUS-EMA21). He thanks to Professor A. Cabada for his hospitality and useful discussion during his visit at the University of Santiago de Compostela, Spain. The second author (G. V. Milovanović) was supported in part by the Serbian Ministry of Education, Science and Technological Development (Project: Approximation of integral and differential operators and applications, grant number #174015).

where C_{ν} and x_{ν} (∈ [0, 1]) are coefficients and nodes of formula (1.1), respectively, χ_{[0,1]}(x) is the characteristic function of the interval [0, 1], and δ(x) is Dirac’s deltafunction. We suppose that the functions φ(x) belong to the Hilbert space

K_{2}(P_{3}) = {φ : [0, 1] → ℝ | φ′′is absolutely continuous and φ′′′∈ L_{2}(0, 1)},

equipped with the norm

(1.3)  ∥φ |K_{2}(P_{3})∥ = \{ ∫_{0}^{1} ( P_{3}( \frac{d}{dx}) φ(x))^{2}dx\}^{1/2} ,

where

P_{3}( \frac{d}{dx}) = \frac{d^{3}}{dx^{3}} + \frac{d}{dx}  and  ∫_{0}^{1}( P_{3}( \frac{d}{dx}) φ(x))^{2}dx < ∞.

Equality (1.3) is semi-norm and ∥φ∥ = 0 if and only if φ(x) = c_{1} sinx + c_{2} cosx + c_{3}. The case without the constant term in φ(x) was considered in our previous work [12].

It should be noted that for a linear differential operator of order n, L ≡ P_{n}(d/dx), Ahlberg, Nilson, and Walsh in the book [1, Chapter 6] investigated the Hilbert spaces in the context of generalized splines. Namely, with the inner product

⟨φ, ψ⟩ = ∫_{0}^{1} Lφ(x) · Lψ(x) dx,

K_{2}(P_{n}) is a Hilbert space if we identify functions that differ by a solution of Lφ = 0. Also, such a type of spaces of periodic functions and optimal quadrature formulae were discussed in [6].

The corresponding error of the quadrature formula (1.1) can be expressed in the form

(1.4)  R_{N} (φ) = ∫_{0}^{1} φ(x) dx − ∑_{ν=0}^{N} C_{ν}φ(x_{ν}) = (ℓ, φ) = ∫_{ℝ} ℓ(x)φ(x) dx

and it is a linear functional in the conjugate space K_{2}^{∗}(P_{3}) to the space K_{2}(P_{3}). By the Cauchy–Schwarz inequality

|(ℓ, φ)| ⩽ ∥φ |K_{2}(P_{3})∥ · ∥ℓ |K_{2}^{∗}(P_{3})∥

error (1.4) can be estimated by the norm of error functional (1.2), i.e.,

∥ℓ |K_{2}^{∗}(P_{3})∥ = sup_{∥φ |K_{2} (P_{3} )∥=1} |(ℓ, φ)|.

In this way, the error estimate of quadrature formula (1.1) on the space K_{2}(P_{3}) can be reduced to finding a norm of the error functional ℓ(x) in the conjugate space K_{2}^{∗}(P_{3}).

OPTIMAL QUADRATURE FORMULA IN THE SENSE OF SARD IN K_{2} (P_{3} ) SPACE 31

Obviously, this norm of the error functional ℓ(x) depends on the coefficients C_{ν} and the nodes x_{ν}, ν = 0, 1, …, N. The problem of finding the minimal norm of the error functional ℓ(x) with respect to the coefficients C_{ν} and the nodes x_{ν} is called Nikol’skiĭ problem, and the obtained formula is called optimal quadrature formula in the sense of Nikol’skiĭ. This problem was first considered by Nikol’skiĭ [16], and continued by many authors (cf. [3]– [6], [17, 34] and references therein). A minimization of the norm of the error functional ℓ(x) with respect only to the coefficients C_{ν}, when the nodes are fixed, is called Sard’s problem. The obtained formula is called the optimal quadrature formula in the sense of Sard. This problem was first investigated by Sard [19].

There are several methods of construction of optimal quadrature formulas in the sense of Sard (see e.g., [3,28]). In the space L_{2}^{(m)} (a, b), based on these methods, Sard’s problem was investigated by many authors (see, for example, [2,3,5], [7]– [9], [13]– [15], [20, 21], [23]– [30], [32, 33] and references therein). Here, L_{2}^{(m)} (a, b) is the Sobolev space of functions, with a square integrable m-th generalized derivative.

It should be noted that a construction of optimal quadrature formulas in the sense of Sard, which are exact for solutions of linear differential equations, was given in [9, 14], using the Peano kernel method, including several examples for some number of nodes.

An optimal quadrature formula in the sense of Sard was constructed in [22], using Sobolev’s method in the space W_{2}^{(m,m−1)} (0, 1), with the norm defined by

∥φ |W_{2}^{(m,m−1)} (0, 1)∥ = { ∫_{0}^{1} ( φ^{(m)} (x) + φ^{(m−1)} (x)) ^{2} dx}^{1/2} .

In this paper we give the solution of Sard’s problem in the space K_{2}(P_{3}), using Sobolev’s method for an arbitrary number of nodes N + 1. Namely, we find the coefficients C_{ν} (and the error functional ℓ̊) such that

(1.5)  ∥ℓ̊|K_{2}^{∗}(P_{3})∥ = inf_{C_{ν}} ∥ℓ |K_{2}^{∗}(P_{3})∥.

Thus, in order to construct an optimal quadrature formula in the sense of Sard in K_{2}(P_{3}), we need consequently to solve the following two problems:

Problem 1. Calculate the norm of the error functional ℓ(x) for the given quadrature formula (1.1).

Problem 2. Find such values of the coefficients C_{ν} such that equality (1.5) is satisfied with fixed nodes x_{ν}.

The paper is organized as follows. In Section 2 we determine the extremal function which corresponds to the error functional ℓ(x) and give a representation of the norm of the error functional (1.2). Section 3 is devoted to a minimization of ∥ℓ∥^{2} with respect to the coefficients C_{ν}. We obtain a system of linear equations for the coefficients of the optimal quadrature formula in the sense of Sard in the space K_{2}(P_{3}). Moreover, the existence and uniqueness of the corresponding solution is proved. Explicit formulas for coefficients of the optimal quadrature formula of the form (1.1) are found in Section 4. In Section 5 we calculate the norm of the error functional (1.2) of the optimal quadrature formula (1.1). Furthermore, we give an asymptotic analysis of this norm. Finally, in Section 6 some numerical results are presented. It should be noted that the results of this paper is a continuation of the results of [12].

2. The extremal function and representation of the error functional ℓ(x)

In order to solve Problem 1, i.e., to calculate the norm of the error functional (1.2) in the space K_{2}^{∗}(P_{3}), we use a concept of the extremal function for a given functional. The function ψ_{ℓ} is called the extremal for the functional ℓ (cf. [29]) if the following equality is fulfilled

(ℓ, ψ_{ℓ}) = ∥ℓ |K_{2}^{∗}(P_{3})∥ · ∥ψ_{ℓ} |K_{2}(P_{3})∥.

Since K_{2}(P_{3}) is a Hilbert space, the extremal function ψ_{ℓ} in this space can be found using the Riesz theorem about general form of a linear continuous functional on Hilbert spaces. Then, for the functional ℓ and for any φ ∈ K_{2}(P_{3}) there exists such a function ψ_{ℓ} ∈ K_{2}(P_{3}), for which the following equality

(2.1)  (ℓ, φ) = ⟨ψ_{ℓ}, φ⟩

holds, where

(2.2)  ⟨ψ_{ℓ}, φ⟩ = ∫_{0}^{1}(ψ′′′_{ℓ}(x) + ψ′_{ℓ}(x))(φ′′′(x) + φ′(x))dx

is an inner product defined on the space K_{2}(P_{3}).

Further, we will investigate the solution of Equation (2.1). Let first φ ∈ C̊^{(∞)} (0, 1), where C̊^{(∞)} (0, 1) is a space of infinity-differentiable and finite functions in the interval (0, 1). Then from (2.2), an integration by parts gives

(2.3)  ⟨ψ_{ℓ}, φ⟩ = − ∫_{0}^{1}(ψ_{ℓ}^{(6)}(x) + 2ψ_{ℓ}^{(4)}(x) + ψ′′_{ℓ}(x))φ(x) dx.

According to (2.1) and (2.3) we conclude that

(2.4)  ψ_{ℓ}^{(6)}(x) + 2ψ_{ℓ}^{(4)}(x) + ψ′′_{ℓ}(x) = −ℓ(x).

Thus, when φ ∈ C̊^{(∞)} (0, 1), the extremal function ψ_{ℓ} is a solution of the equation (2.4). But, we have to find the solution of (2.1) when φ ∈ K_{2}(P_{3}).

Since the space C̊^{(∞)} (0, 1) is dense in K_{2}(P_{3}), then functions from K_{2}(P_{3}) can be uniformly approximated as closely as desired by functions from the space C̊^{(∞)} (0, 1). For φ ∈ K_{2}(P_{3}) we consider the inner product ⟨ψ_{ℓ}, φ⟩. An integration by parts gives

⟨ψ_{ℓ}, φ⟩ = (ψ′′′_{ℓ}(x) + ψ′_{ℓ}(x))(φ′′(x) + φ(x))|_{0}^{1} − (ψ_{ℓ}^{(4)}(x) + ψ′′_{ℓ}(x))φ′(x)|_{0}^{1}

+ (ψ_{ℓ}^{(5)}(x) + ψ′′′_{ℓ}(x))φ(x)|_{0}^{1} − ∫_{0}^{1}(ψ_{ℓ}^{(6)}(x) + 2ψ_{ℓ}^{(4)}(x) + ψ′′_{ℓ}(x))φ(x) dx.

Hence, taking into account the arbitrariness φ(x) and uniqueness of the function ψ_{ℓ}(x) (up to functions sinx, cosx and 1), taking into account (2.4), the following equation must be fulfilled

(2.5)  ψ_{ℓ}^{(6)}(x) + 2ψ_{ℓ}^{(4)}(x) + ψ′′_{ℓ}(x) = −ℓ(x),

with boundary conditions

(2.6)  ψ′′′_{ℓ}(0) + ψ′_{ℓ}(0) = 0,  ψ′′′_{ℓ}(1) + ψ′_{ℓ}(1) = 0,
(2.7)  ψ_{ℓ}^{(4)}(0) + ψ′′_{ℓ}(0) = 0, ψ_{ℓ}^{(4)}(1) + ψ′′_{ℓ}(1) = 0,
(2.8)  ψ_{ℓ}^{(5)}(0) + ψ′′′_{ℓ}(0) = 0, ψ_{ℓ}^{(5)}(1) + ψ′′′_{ℓ}(1) = 0.

Thus, we conclude, that the extremal function ψ_{ℓ}(x) is a solution of boundary value problem (2.5)–(2.8).

Taking the convolution of two functions f and g, i.e.,

(2.9)  (f ∗ g)(x) = ∫_{ℝ} f(x − y) g(y)dy = ∫_{ℝ} f(y) g(x − y)dy,

we can state the following result:

Theorem 2.1. The solution of boundary value problem (2.5)–(2.8) is the extremal function ψ_{ℓ}(x) of the error functional ℓ(x) and it has the following form

ψ_{ℓ}(x) = (G ∗ ℓ)(x) + d_{1} sinx + d_{2} cosx + d_{3},

where d_{1}, d_{2} and d_{3} are arbitrary real numbers, and

(2.10)  G(x) = \frac{1}{4} sgnx · (x cosx − 3 sinx + 2x)

is the solution of the equation

ψ_{ℓ}^{(6)}(x) + 2ψ_{ℓ}^{(4)}(x) + ψ′′_{ℓ}(x) = δ(x).

Proof. It is known that the general solution of a non-homogeneous differential equation can be represented as a sum of its particular solution and the general solution of the corresponding homogeneous equation. In our case, the general solution of the corresponding homogeneous equation for (2.5) is given by

ψ_{ℓ}^{h}(x) = d_{1} sinx + d_{2} cosx + d_{3} + d_{4}x sinx + d_{5}x cosx + d_{6}x,

where d_{k}, k = \overline{1, 6}, are arbitrary constants. It is not difficult to verify that a particular solution of the differential equation (2.5) can be expressed as a convolution of the functions ℓ(x) and G(x) defined by (2.9). The function G(x) is the fundamental solution of the equation (2.5), and it is determined by (2.10).

According to the general rule for finding fundamental solutions of a linear differential operators (cf. [31, p. 88]), in our case for the operator \frac{d^{6}}{dx^{6}} + 2\frac{d^{4}}{dx^{4}} + \frac{d^{2}}{dx^{2}} we get (2.10).

Thus, we have the following general solution of equation (2.5)

(2.11) ψ_{ℓ}(x) = (ℓ ∗ G)(x) + d_{1} sinx + d_{2} cosx + d_{3} + d_{4}x sinx + d_{5}x cosx + d_{6}x.

In order that in the space K_{2}(P_{3}) the function ψ_{ℓ}(x) will be unique (up to functions sinx, cosx and 1), it has to satisfy conditions (2.6)–(2.8), where derivatives are taken in a generalized sense. In computations we need the first five derivatives of the function G(x):

G′(x) = \frac{sgn x}{4} (2 − 2 cosx − x sinx),
G′′(x) = \frac{sgn x}{4} (sinx − x cosx),  G′′′(x) = \frac{sgn x}{4} x sinx,

G^{(4)}(x) = \frac{sgn x}{4} (sinx + x cosx),  G^{(5)}(x) = \frac{sgn x}{4} (2 cosx − x sinx),

where we used the following formulas from the theory of generalized functions [31], (sgnx)′ = 2δ(x), δ(x)f(x) = δ(x)f(0). Further, using the well-known formula

\frac{d}{dx}(f ∗ g)(x) = (f′∗ g)(x) = (f ∗ g′)(x),

we can get expressions for ψ_{ℓ}^{(ν)}(x), ν = 1, …, 5, and then, using these expressions and (2.11), as well as expressions for G^{(k)}(x), k = \overline{0, 5}, boundary conditions (2.6)– (2.8) reduce to

(ℓ(y), 1) − (ℓ(y), cosy) + 4d_{5} − 2d_{6} = 0,

(ℓ(y), 1) − cos1(ℓ(y), cosy) − sin1(ℓ(y), siny) − 4d_{4} sin1 − 4d_{5} cos1 + 2d_{6} = 0,

(ℓ(y), siny) − 4d_{4} = 0,
sin1 (ℓ(y), cosy) − cos1 (ℓ(y), siny) − 4d_{4} cos1 + 4d_{5} sin1 = 0,
(ℓ(y), cosy) − 4d_{5} = 0,
cos1 (ℓ(y), cosy) + sin1 (ℓ(y), siny) + 4d_{4} sin1 + 4d_{5} cos1 = 0.

Hence we have d_{4} = 0, d_{5} = 0, d_{6} = 0 and

(2.12)  (ℓ(y), siny) = 0, (ℓ(y), cosy) = 0, (ℓ(y), 1) = 0.

Substituting these values into (2.11), we get the assertion of the theorem. □

The equalities (2.12) mean that our quadrature formula will be exact for constants and for the functions sinx and cosx.

Now, using Theorem 2.1, we immediately obtain a representation of the norm of the error functional

(2.13) ∥ℓ|K_{2}^{∗}(P_{3})∥^{2} = (ℓ(x), ψ_{ℓ}(x)) = − ∑_{ν=0}^{N} ∑_{γ=0}^{N} C_{ν}C_{γ} G(x_{ν} − x_{γ})

+ 2 ∑_{ν=0}^{N} C_{ν} ∫_{0}^{1} G(x − x_{ν}) dx − ∫_{0}^{1}∫_{0}^{1} G(x − y) dx dy.

Thus, Problem 1 is solved. Further in Sections 3 and 4 we deal with Problem 2.

3. Existence and uniqueness of optimal coefficients

Let the nodes x_{ν} of quadrature formula (1.1) be fixed. The error functional (1.2) satisfies conditions (2.12). The norm of error functional ℓ(x) is a multidimensional function of the coefficients C_{ν} (ν = 0, 1, …, N). For finding its minimum under conditions (2.12), we apply the Lagrange method, i.e., we consider the function

Ψ(C_{0}, …, C_{N} , d_{1}, d_{2}, d_{3}) = ∥ℓ∥^{2} + 2d_{1}(ℓ(x), sinx) + 2d_{2}(ℓ(x), cosx) + 2d_{3}(ℓ(x), 1).

It leads to the following system of linear equations

∑_{γ=0}^{N} C_{γ}G(x_{ν} − x_{γ}) + d_{1} sinx_{ν} + d_{2} cosx_{ν} + d_{3} = f(x_{ν}), ν = 0, 1, …, N, (3.1)

(3.2)  ∑_{γ=0}^{N} C_{γ} sinx_{γ} = 1 − cos1, ∑_{γ=0}^{N} C_{γ} cosx_{γ} = sin1, ∑_{γ=0}^{N} C_{γ} = 1,

where G(x) is determined by (2.10) and f(x_{ν}) = ∫_{0}^{1} G(x − x_{ν}) dx .

First, we put C = (C_{0}, C_{1}, …, C_{N}) and d = (d_{1}, d_{2}, d_{3}) for the solution of the system of equations (3.1)–(3.2), which represents a stationary point of the function Ψ(C, d). Setting C_{ν} = C̄_{ν} + C_{1ν}, ν = 0, 1, …, N, (2.13) and system (3.1)–(3.2) becomes

(3.3) ∥ℓ∥^{2} = − ∑_{ν=0}^{N} ∑_{γ=0}^{N} C̄_{ν} C̄_{γ}G(x_{ν} − x_{γ}) + 2 ∑_{ν=0}^{N} (C̄_{ν} + C_{1ν}) ∫_{0}^{1} G(x − x_{ν}) dx

− ∑_{ν=0}^{N} ∑_{γ=0}^{N} (2C̄_{ν}C_{1γ} + C_{1ν}C_{1γ})G(x_{ν} − x_{γ}) − ∫_{0}^{1}∫_{0}^{1} G(x − y) dx dy

and

∑_{γ=0}^{N} C̄_{γ}G(x_{ν} − x_{γ}) + d_{1} sinx_{ν} + d_{2} cosx_{ν} + d_{3} = F (x_{ν}), ν = 0, 1, …, N, (3.4)

∑_{γ=0}^{N} C̄_{γ} sinx_{γ} = 0, ∑_{γ=0}^{N} C̄_{γ} cosx_{γ} = 0, ∑_{γ=0}^{N} C̄_{γ} = 0, (3.5)

respectively, where F (x_{ν}) = f(x_{ν})−∑ _{γ=0}^{N} C_{1γ} G(x_{ν} −x_{γ}) and C_{1γ}, γ = 0, 1, …, N, are partial solutions of system (3.2).

System (3.1)–(3.2) has a unique solution and it gives the minimum to ∥ℓ∥^{2} under conditions (3.2). The uniqueness of this solution was proved in [30, Chapter I]. However, we directly get that the minimization of (2.13) under conditions (2.12) by C_{ν} is equivalent to the minimization of expression (3.3) by C̄_{ν} under conditions (3.5). Therefore, it is sufficient to prove that system (3.4)–(3.5) has the unique solution with respect to C̄ = (C̄_{0}, C̄_{1}, …, C̄_{N}) and d = (d_{1}, d_{2}, d_{3}) and this solution gives the conditional minimum for ∥ℓ∥^{2}. From the theory of conditional extrema, we need the positivity of the quadratic form

(3.6)  Φ(C̄) = ∑_{ν=0}^{N} ∑_{γ=0}^{N} \frac{∂^{2}Ψ}{∂C̄_{ν}∂C̄_{γ}} C̄_{ν} C̄_{γ}

on the set of vectors C̄ = (C̄_{0}, C̄_{1}, …, C̄_{N}), under the condition SC̄ = 0, where S is the matrix of the system of equations (3.5)

S = ( \begin{matrix}sinx_{0} & sinx_{1} & ⋯ & sinx_{N} \\ cosx_{0} & cosx_{1} & ⋯ & cosx_{N} \\ 1 & 1 & ⋯ & 1 \end{matrix} )

Now we show that in this case the condition is satisfied.

Theorem 3.1. For any nonzero vector C̄ ∈ ℝ^{N+1} lying in the subspace SC̄ = 0 , the function Φ(C̄) is strictly positive.

Proof. Using the definition of the function Ψ(C̄, d) and the previous equations, (3.6) reduces to

(3.7)  Φ(C̄) = −2 ∑_{ν=0}^{N} ∑_{γ=0}^{N} G(x_{ν} − x_{γ})C̄_{ν} C̄_{γ}.

Consider now a linear combination of delta functions

(3.8)  δC̄(x) = \sqrt{2} ∑_{ν=0}^{N} C̄_{ν}δ(x − x_{ν}).

By virtue of the condition SC̄ = 0, this functional belongs to the space K_{2}^{∗}(P_{3}). So, it has an extremal function uC̄(x) ∈ K_{2}(P_{3}), which is a solution of the equation

(3.9)  ( \frac{d^{6}}{dx^{6}} + 2\frac{d^{4}}{dx^{4}} + \frac{d^{2}}{dx^{2}} ) uC̄(x) = −δC̄(x).

As uC̄(x), we can take a linear combination of shifts of the fundamental solution G(x), uC̄(x) = −\sqrt{2}∑ _{ν=0}^{N} C̄_{ν}G(x − x_{ν}) , and we can see that

∥uC̄|K_{2}(P_{3})∥^{2} = (δC̄, uC̄) = −2 ∑_{ν=0}^{N} ∑_{γ=0}^{N} C̄_{ν} C̄_{γ}G(x_{ν} − x_{γ}) = Φ(C̄).

Thus, it is clear that for a nonzero C̄, the function Φ(C̄) is strictly positive and Theorem 3.1 is proved. □

If the nodes x_{0}, x_{1}, …, x_{N} are selected such that the matrix S has the right inverse, then the system of equations (3.4)–(3.5) has the unique solution, as well as the system of equations (3.1)–(3.2).

Theorem 3.2. If the matrix S has the right inverse matrix, then the main matrix Q of the system of equations (3.4)–(3.5) is nonsingular.

Proof. We denote by M the matrix of the quadratic form −\frac{1}{2}Φ(C̄), given in (3.7). It is enough to consider the homogenous system of linear equations

(3.10)  Q ( C̄ d) = ( \begin{matrix}M & S^{∗} \\ S & 0 \end{matrix}) ( C̄ d) = 0

and prove that it has only the trivial solution.

Let C̄, d be a solution of (3.10). Consider the function δC̄(x), defined before by (3.8). As an extremal function for δC̄(x) we take the following function

uC̄(x) = −\sqrt{2}( ∑_{ν=0}^{N} C̄_{ν}G(x − x_{ν}) + d_{1} sinx + d_{2} cosx + d_{3}) .

This is possible, because uC̄ belongs to the space K_{2}(P_{3}) and it is a solution of equation (3.9). The first N + 1 equations of system (3.10) mean that uC̄(x) takes the value zero at all the nodes x_{ν}, ν = 0, 1, …, N. Then, for the norm of the functional δC̄(x) in K_{2}^{∗}(P_{3}), we have

∥δC̄|K_{2}^{∗}(P_{3})∥^{2} = (δC̄, uC̄) = −2 ∑_{ν=0}^{N} C̄_{ν} uC̄(x_{ν}) = 0,

which is possible only when C̄ = 0. Taking into account this, from the first N + 1 equations of system (3.10) we obtain S^{∗}d = 0. Since the matrix S is right-inversive (by the hypotheses of this theorem), we conclude that S^{∗} has the left inverse matrix, and therefore d = 0, i.e., d_{1} = d_{2} = d_{3} = 0, which completes the proof. □

According to (2.13) and Theorems 3.1 and 3.2, it follows that at fixed values of the nodes x_{ν}, ν = 0, 1, …, N, the norm of the error functional ℓ(x) has the unique minimum for some concrete values of C_{ν} = C̊_{ν}, ν = 0, 1, …, N. As we mentioned in the first section, the quadrature formula with such coefficients C̊_{ν} is called optimal quadrature formula in the sense of Sard, and C̊_{ν}, ν = 0, 1, …, N, are the optimal coefficients. In the sequel, for convenience the optimal coefficients C̊_{ν} will be denoted only as C_{ν}.

4. Coefficients of the optimal quadrature formula

In this section we solve system (3.1)–(3.2) and find an explicit formula for the coefficients C_{ν}. We use a similar method, offered by Sobolev [28] for finding optimal coefficients in the space L_{2}^{(m)} (0, 1). Here, we mainly use a concept of functions of a discrete argument and the corresponding operations (see [29, 30]). For completeness we give some of definitions.

Let nodes x_{ν} be equaly spaced, i.e., x_{ν} = νh, h = 1/N. Assume that φ(x) and ψ(x) are real-valued functions defined on the real line ℝ.

Definition 4.1. The function φ(hν) is a function of a discrete argument if it is given on some set of integer values of ν.

Definition 4.2. The inner product of two discrete argument functions φ(hν) and ψ(hν) is given by [φ, ψ] = ∑ _{ν=−∞}^{∞} φ(hν)·ψ(hν), if the series on the right hand side converges absolutely.

Definition 4.3. The convolution of two functions φ(hν) and ψ(hν) is the inner product

φ(hν) ∗ ψ(hν) = [φ(hγ), ψ(hν − hγ)] = ∑_{γ=−∞}^{∞} φ(hγ) · ψ(hν − hγ).

Suppose that C_{ν} = 0 when ν < 0 and ν > N. Using these definitions, the system (3.1)–(3.2) can be rewritten in the convolution form

G(hν) ∗ C_{ν} + d_{1} sin(hν) + d_{2} cos(hν) + d_{3} = f(hν), ν = 0, 1, …, N, (4.1)

(4.2)  ∑_{ν=0}^{N} C_{ν} sin(hν) = 1 − cos1, ∑_{ν=0}^{N} C_{ν} cos(hν) = sin1, ∑_{γ=0}^{N} C_{γ} = 1,

where

(4.3) f(hν) = cos(1 − hν) + \frac{1}{4} sin(1 − hν) − \frac{1}{4}hν sin(1 − hν)

+ cos(hν) + \frac{1}{4}hν sin(hν) + \frac{1}{2}(hν)^{2} − \frac{1}{2}hν − \frac{7}{4} .

Now, we consider the following problem.

Problem 1. For a given f(hν) find a discrete function C_{ν} and unknown coefficients d_{1}, d_{2}, d_{3}, which satisfy system (4.1)–(4.2).

Further, instead of C_{ν} we introduce the functions v(hν) and u(hν) by

v(hν) = G(hν) ∗ C_{ν} and u(hν) = v(hν) + d_{1} sin(hν) + d_{2} cos(hν) + d_{3}.

In such a statement it is necessary to express C_{ν} by the function u(hν). For this we have to construct such an operator D(hν), which satisfies the equation

(4.4)  D(hν) ∗ G(hν) = δ(hν),

where δ(hν) is equal to 0 when ν ≠ 0 and is equal to 1 when ν = 0, i.e., δ(hν) is a discrete delta-function.

In connection with this, a discrete analogue D(hν) of the differential operator

D_{6} = \frac{d^{6}}{dx^{6}} + 2\frac{d^{4}}{dx^{4}} + \frac{d^{2}}{dx^{2}} ,

which satisfies (4.4) was constructed in [11] and some properties were investigated. Following [11] we have:

Theorem 4.1. The discrete analogue of the differential operator D_{6} satisfying the equation (4.4) has the form

D(hν) = \frac{2}{2h + h cosh − 3 sinh} \{ \begin{matrix}∑ _{k=1}^{2} A_{k} λ_{1}^{|ν|−1} , & |ν| ⩾ 2, \\ 1 + ∑ _{k=1}^{2} A_{k}, & |ν| = 1, \\ \frac{3 sin2h − 4h cos^{2} h − 2h}{2h + h cosh − 3 sinh} + ∑ _{k=1}^{2} \frac{A_{k}}{λ_{k}} , & ν = 0, \end{matrix}

where

λ_{1} = \frac{−p_{1} − \sqrt{p_{1}^{2} − 4p_{2} + 8} + \sqrt{2p_{1}^{2} − 4p_{2} − 8 + 2p_{1} \sqrt{p_{1}^{2} − 4p_{2} + 8}}}{4}  ,

λ_{2} = \frac{−p_{1} + \sqrt{p_{1}^{2} − 4p_{2} + 8} + \sqrt{2p_{1}^{2} − 4p_{2} − 8 − 2p_{1} \sqrt{p_{1}^{2} − 4p_{2} + 8}}}{4}

are zeros of the polynomial P_{4}(λ) = λ^{4} + p_{3}λ^{3} + p_{2}λ^{2} + p_{1}λ + 1, with coefficients

p_{3} = p_{1} = \frac{3 sin2h + 6 sinh − 10h cosh − 2h}{2h + h cosh − 3 sinh} = 26 + O(h^{2}),

p_{2} = \frac{8h cos^{2} h + 8h + 2h cosh − 6 sinh − 6 sin2h}{2h + h cosh − 3 sinh} = 66 + O(h^{2}),

h is a small parameter, |λ_{k}| < 1, and A_{k} = \frac{Q_{6}(λ_{k})}{λ_{k}P′_{4}(λ_{k})} , where

Q_{6}(λ) = λ^{6} − (2 + 4 cosh)λ^{5} + (4 cos^{2} h + 8 cosh + 3)λ^{4}
− (4 cosh + 8 cos^{2} h + 4 + cosh)λ^{3} + (4 cos^{2} h + 8 cosh + 3)λ^{2}
− (2 + 4 cosh)λ + 1.

Theorem 4.2. The discrete analogue D(hν) of the differential operator D_{6} satisfies the following equalities:

1) D(hν) ∗ sin(hν) = 0,  5) D(hν) ∗ 1 = 0,
2) D(hν) ∗ cos(hν) = 0,  6) D(hν) ∗ (hν) = 0,
3) D(hν) ∗ (hν) sin(hν) = 0,  7) D(hν) ∗ G(hν) = δ(hν).
4) D(hν) ∗ (hν) cos(hν) = 0,

Here G(hν) is the function of discrete argument, corresponding to the function G(x) defined by (2.10), and δ(hν) is the discrete delta-function.

Then, taking into account (4.4) and Theorems 4.1 and 4.2, for optimal coefficients we have

(4.5)  C_{ν} = D(hν) ∗ u(hν).

Thus, if we find the function u(hν), then the optimal coefficients can be obtained from (4.5). In order to calculate convolution (4.5), we need a representation of the function u(hν) for all integer values of ν. According to (4.1) we get that u(hν) = f(hν) when hν ∈ [0, 1]. Now, we need a representation of the function u(hν) when ν < 0 and ν > N.

Since C_{ν} = 0 for hν ∉ [0, 1], then C_{ν} = D(hν) ∗ u(hν) = 0, hν ∉ [0, 1]. Now, we calculate the convolution v(hν) = G(hν) ∗ C_{ν} when hν ∉ [0, 1].

Let ν < 0. Taking into account the equalities (2.10) and (4.2) and denoting

b_{1} = \frac{1}{4} ∑_{γ=0}^{N} C_{γ} hγ sin(hγ), b_{2} = \frac{1}{4} ∑_{γ=0}^{N} C_{γ} hγ cos(hγ), b_{3} = \frac{1}{2} ∑_{γ=0}^{N} C_{γ} hγ,

we get

v(hν) = −\frac{1}{4}[(hν cos(hν) − 3 sin(hν)) sin1 + (hν sin(hν) + 3 cos(hν))(1 − cos1)

+ 2hν − 4b_{1} sin(hν) − 4b_{2} cos(hν) − 4b_{3} ]

Similarly, for ν > N we obtain

v(hν) = \frac{1}{4}[(hν cos(hν) − 3 sin(hν)) sin1 + (hν sin(hν) + 3 cos(hν))(1 − cos1)

+ 2hν − 4b_{1} sin(hν) − 4b_{2} cos(hν) − 4b_{3} ].

Now, setting

d_{1}^{−} = d_{1} + b_{1}, d_{2}^{−} = d_{2} + b_{2}, d_{3}^{−} = d_{3} + b_{3},
d_{1}^{+} = d_{1} − b_{1}, d_{2}^{+} = d_{2} − b_{2}, d_{3}^{+} = d_{3} − b_{3}

we formulate the following problem:

Problem 2. Find the solution of the equation

(4.6)  D(hν) ∗ u(hν) = 0, hν ∉ [0, 1],

in the form

u(hν) = \{ \begin{matrix}−\frac{sin 1}{4}(hν cos(hν) − 3 sin(hν)) + \frac{1−cos 1}{4} [hν sin(hν) + 3hν cos(hν)] \\ −\frac{1}{2}hν + d_{1}^{−} sin(hν) + d_{2}^{−} cos(hν) + d_{3}^{−} ,  ν < 0, \\ f(hν),  0 ⩽ ν ⩽ N, \\ \frac{sin 1}{4}(hν cos(hν) − 3 sin(hν)) + \frac{1−cos 1}{4} [hν sin(hν) + 3hν cos(hν)] \\ +\frac{1}{2}hν + d_{1}^{+} sin(hν) + d_{2}^{+} cos(hν) + d_{3}^{+} ,  ν > N, \end{matrix}

where d_{1}^{−} , d_{2}^{−} , d_{3}^{−} , d_{1}^{+} , d_{2}^{+} , d_{3}^{+} are unknown coefficients.

It is clear that

d_{1} = \frac{1}{2}(d_{1}^{−} + d_{1}^{+} ), d_{2} = \frac{1}{2}(d_{2}^{−} + d_{2}^{+} ), d_{3} = \frac{1}{2}(d_{3}^{−} + d_{3}^{+} ),
b_{1} = \frac{1}{2}(d_{1}^{−} − d_{1}^{+} ), b_{2} = \frac{1}{2}(d_{2}^{−} − d_{2}^{+} ), b_{3} = \frac{1}{2}(d_{3}^{−} − d_{3}^{+} ).

These unknowns d_{1}^{−} , d_{2}^{−} , d_{3}^{−} , d_{1}^{+} , d_{2}^{+} , d_{3}^{+} can be found from equation (4.6), using the function D(hν). Then, the explicit form of the function u(hν) and optimal coefficients C_{ν} can be obtained. Thus, in this way, Problem 2, as well as Problem 1, can be solved.

However, instead of this, using D(hν) and u(hν) and taking into account (4.5), we find here expressions for the optimal coefficients C_{ν}, ν = 1, …, N − 1. For this purpose we introduce the following notations

p = \frac{2}{2h + h cosh − 3 sinh} ,
m_{k} = \frac{pA_{k}}{λ_{k}} ∑_{γ=1}^{∞} λ_{k}^{γ}[−f(−hγ) − \frac{1}{4}(3 sin(hγ) − hγ cos(hγ)) sin1 − \frac{1}{4}(3 cos(hγ)
+ hγ sin(hγ))(1 − cos1) + \frac{1}{2}hγ − d_{1}^{−} sin(hγ) + d_{2}^{−} cos(hγ) + d_{3}^{−} ],
n_{k} = \frac{A_{k}p}{λ_{k}} ∑_{γ=1}^{∞} λ_{k}^{γ}[ \frac{1}{4}(h(N + γ) cos((N + γ)h) − 3 sin((N + γ)h)) sin1
+ \frac{1}{4}(3 cos((N + γ)h) + h(N + γ) sin((N + γ)h))(1 − cos1) + \frac{1}{2}h(N + γ)
+ d_{1}^{+} sin((N + γ)h) + d_{2}^{+} cos((N + γ)h) + d_{3}^{+} − f((N + γ)h)].

for k = 1, 2. The series in the previous expressions are convergent, because |λ_{k}| < 1. Using the previous results we can get the following relation for the coefficients

C_{ν} = D(hν) ∗ f(hν) + ∑_{k=1}^{2} (m_{k}λ_{k}^{ν} + n_{k}λ_{k}^{N−ν} ) ,

and then, using Theorems 4.1 and 4.2 and equality (4.3), we can calculate the convolution D(hν) ∗ f(hν) = D(hν) ∗ \frac{1}{2}(hν)^{2} = h, so that we obtain the following statement:

Theorem 4.3. The coefficients of optimal quadrature formulas in the sense of Sard of form (1.1) in the space K_{2}(P_{3}) have the following representation

(4.7)  C_{ν} = h + ∑_{k=1}^{2} (m_{k}λ_{k}^{ν} + n_{k}λ_{k}^{N−ν} ) , ν = 1, …, N − 1,

where m_{k} and n_{k} are defined above, and λ_{k} are given in Theorem 4.1.

According to Theorem 4.3, it is clear that in order to obtain the exact expressions of the optimal coefficients C_{ν} we need only m_{k} and n_{k}. They can be found from an identity with respect to (hν), which can be obtained by substituting equality (4.7) into (4.1). Namely, equating the corresponding coefficients on the left and the right hand sides of equation (4.1) we find m_{k} and n_{k}. The coefficients C_{0} and C_{N} follow directly from (4.2).

Finally, we can formulate the following result:

Theorem 4.4. The coefficients of the optimal quadrature formulas in the sense of Sard of the form (1.1) in the space K_{2}(P_{3}) are

C_{ν} = \{ \frac{2(1 − cos1)(1 − cosh) − h(sin(1 − h) + sinh − sin1)}{2(1 − cosh) sin1} − ∑ _{k=1}^{2} m_{k} \frac{(λ_{k} + λ_{k}^{N+1} )(sin(1 − h) + sinh) − (λ_{k}^{N} + λ_{k}^{2}) sin1}{(λ_{k}^{2} + 1 − 2λ_{k} cosh) sin1} , ν = 0, N, h + ∑ _{k=1}^{2} m_{k} (λ_{k}^{ν} + λ_{k}^{N−ν} ) , ν = \overline{1, N − 1} ,

where

(4.8)  m_{1} = \frac{A_{22}B_{1} − A_{12}B_{2}}{A_{11}A_{22} − A_{12}A_{21}} ,  m_{2} = \frac{A_{11}B_{2} − A_{21}B_{1}}{A_{11}A_{22} − A_{12}A_{21}} ,

and

A_{11} = \frac{λ_{1} + λ_{1}^{N+1}}{λ_{1}^{2} + 1 − 2λ_{1} cosh} ,  A_{12} = \frac{λ_{2} + λ_{2}^{N+1}}{λ_{2}^{2} + 1 − 2λ_{2} cosh} ,
A_{21} = \frac{(λ_{1} + λ_{1}^{N+1} )(sin(1 − h) + sinh) − (λ_{1}^{2} + λ_{1}^{N}) sin1}{λ_{1}^{2} + 1 − 2λ_{1} cosh}  − \frac{(λ_{1} − λ_{1}^{N}) sin1}{1 − λ_{1}} ,
A_{22} = \frac{(λ_{2} + λ_{2}^{N+1} )(sin(1 − h) + sinh) − (λ_{2}^{2} + λ_{2}^{N}) sin1}{λ_{2}^{2} + 1 − 2λ_{2} cosh}  − \frac{(λ_{2} − λ_{2}^{N}) sin1}{1 − λ_{2}} ,
B_{1} = \frac{2(1 − cosh) − h sinh}{2(1 − cosh) sinh} ,  B_{2} = \frac{(1 − cos1)(2(1 − cosh) − h sinh)}{2(1 − cosh)} ,

λ_{k} are given in Theorem 4.1 and |λ_{k}| < 1.

Because of complexity, the proof of this theorem will be omitted. Only we mention here that the convolution G(hν) ∗ C_{ν} in equation (4.1), i.e.,

G(hν) ∗ C_{ν} = ∑_{γ=0}^{N} C_{γ}G(hν − hγ) = S_{1} − S_{2},

where

S_{1} = \frac{1}{2} ∑_{γ=0}^{ν} C_{γ} [(hν − hγ) cos(hν − hγ) − 3 sin(hν − hγ) + 2(hν − hγ)] ,

S_{2} = \frac{1}{4} ∑_{γ=0}^{N} C_{γ} [(hν − hγ) cos(hν − hγ) − 3 sin(hν − hγ) + 2(hν − hγ)] ,

provides the following identity with respect to (hν),

(4.9)  S_{1} − S_{2} + d_{1} sin(hν) + d_{2} cos(hν) + d_{3} = f(hν),

where f(hν) is defined by (4.3). Unknowns in (4.9) are m_{k}, n_{k} (k = 1, 2), d_{1}, d_{2} and d_{3}. Equating the corresponding coefficients of (hν) sin(hν), (hν) cos(hν) and (hν) of both sides of this identity, for unknowns m_{k} and n_{k} (k = 1, 2) we obtain a system of linear equations. After some simplifications we conclude that m_{k} = n_{k}, k = 1, 2, and then, using (4.7), we can find the optimal coefficients C_{ν}, ν = 0, 1, …, N.

Theorem 4.4 gives the solution of Problem 1, which is equivalent to Problem 2. Thus, Problem 2 is solved, i.e., the coefficients of the optimal quadrature formula (1.1) in the sense of Sard in the space K_{2}(P_{3}) for equal spaced nodes are found.

5. The norm of the error functional of the optimal quadrature formula

This section is devoted to the calculation of the square of the norm of error functional (1.2) for the optimal quadrature formula (1.1).

Theorem 5.1. The square of the norm of error functional (1.2) of the optimal quadrature formula (1.1) on the space K_{2}(P_{3}) has the form

∥ℓ̊∥^{2} = \frac{h^{2} − 18}{12} + \frac{5(1 − cos1)}{2 sin1} + \frac{h[5 sinh(1 − cos1) − sin1(h + 2 sinh)]}{4 sin1(cosh − 1)}

+∑_{k=1}^{2} m_{k}[ \frac{λ_{k}(1+λ_{k}^{N})[5(cos1−1) sinh + (cosh−sinh) sin1] − (λ_{k}^{N} +λ_{k}^{2}) sin1}{2(λ_{k}^{2} + 1 − 2λ_{k} cosh) sin1}

+ \frac{λ_{k}(λ_{k}^{2} − 1)(λ_{k}^{N} − 1)h sinh}{(λ_{k}^{2} + 1 − 2λ_{k} cosh)^{2}} + \frac{λ_{k}(λ_{k} + 1)(λ_{k}^{N} − 1)h^{2}}{(λ_{k} − 1)^{3}} − \frac{λ_{k}^{N} − λ_{k}}{2(λ_{k} − 1)} ],

where λ_{k} are given in Theorem 4.1 and |λ_{k}| < 1, m_{k} are defined in Theorem 4.4.

Proof. In the equal spaced case of the nodes, the expression (2.13), using (2.10), we can rewrite in the following form

∥ℓ∥^{2} = − ∑_{ν=0}^{N} C_{ν}( ∑_{γ=0}^{N} C_{γ}G(hν −hγ)−f(hν)) +∑_{ν=0}^{N} C_{ν}f(hν)− \frac{5}{2} sin1+ \frac{1}{2} cos1+ \frac{11}{6},

where f(hν) is defined by (4.3).

Then taking into account equality (3.1) we get

∥ℓ∥^{2} = − ∑_{ν=0}^{N} C_{ν}[−d_{1} sin(hν)−d_{2} cos(hν)−d_{3}]+∑_{ν=0}^{N} C_{ν}f(hν)− \frac{5}{2} sin1+ \frac{1}{2} cos1+ \frac{11}{6}.

Using (4.3), (3.2), and (4.9), after some simplifications, we obtain

d_{1} = \frac{sin1 − cos1}{4} + \frac{3(1 − cos1)}{2 sin1} − \frac{3(1 − cos1)h sinh}{4 sin1 · (1 − cosh)} − \frac{1}{4} ∑_{γ=0}^{N} C_{γ}(hγ) sin(hγ)

+ \frac{1}{2} ∑_{k=1}^{2} \frac{m_{k}λ_{k} sinh}{λ_{k}^{2} + 1 − 2λ_{k} cosh} ( \frac{3(λ_{k}^{N} + 1)(cos1 − 1)}{sin1} + \frac{h(λ_{k}^{2} − 1)(λ_{k}^{N} − 1)}{λ_{k}^{2} + 1 − 2λ_{k} cosh} ) ,

d_{2} = \frac{sin1 + cos1 + 7}{4} − \frac{h(h + 3 sinh)}{4(1 − cosh)} − \frac{1}{4} ∑_{γ=0}^{N} C_{γ}(hγ) cos(hγ)

− \frac{1}{2} ∑_{k=1}^{2} \frac{m_{k}λ_{k}(1 + λ_{k}^{N})}{λ_{k}^{2} + 1 − 2λ_{k} cosh} ( 3 sinh − \frac{h(λ_{k}^{2} cosh − 2λ_{k} + cosh)}{λ_{k}^{2} + 1 − 2λ_{k} cosh} ) ,

d_{3} = −\frac{7}{4} − \frac{h(h + 3 sinh)}{4(cosh − 1)} − \frac{1}{2} ∑_{γ=0}^{N} C_{γ}(hγ) + ∑_{k=1}^{2} \frac{m_{k}λ_{k}h(1 + λ_{k}^{N})}{(λ_{k} − 1)^{2}} .

as well as the corresponding value of ∥ℓ∥^{2},

∥ℓ∥^{2} = \frac{18(1 − cos1) − 7 sin1}{6 sin1} + \frac{3h(1 − cos1)^{2} sinh}{4(cosh − 1) sin1} + \frac{h(sin1 − 1)(h + 3 sinh)}{4(cosh − 1)}

+ ∑_{k=1}^{2} m_{k}[ \frac{3λ_{k}(1 + λ_{k}^{N})(cos1 − 1) sinh}{(λ_{k}^{2} + 1 − 2λ_{k} cosh) sin1}

+ \frac{λ_{k}h}{2(λ_{k}^{2} + 1 − 2λ_{k} cosh)^{2}} ((1 − cos1)(λ_{k}^{2} − 1)(λ_{k}^{N} − 1) sinh

+ (1 + λ_{k}^{N})(λ_{k}^{2} cosh − 2λ_{k} + cosh) sin1) + \frac{λ_{k}h(1 + λ_{k}^{N})}{(λ_{k} − 1)^{2}} ]

+ ∑_{γ=0}^{N} C_{γ} ( \frac{cos1}{2}(hγ) sin(hγ) − \frac{sin1}{2}(hγ) cos(hγ) − hγ + \frac{1}{2}(hγ)^{2}) .

Finally, using the expression for optimal coefficients C_{γ} from Theorem 4.4, after some calculations and simplifications, we get the assertion of Theorem 5.1. □

Theorem 5.2. For the norm of error functional (1.2) of the optimal quadrature formula of form (1.1) on the space K_{2}(P_{3}) we have

(5.1)  ∥ℓ̊|K_{2}^{∗}(P_{3})∥^{2} = \frac{1}{30240}h^{6} + O(h^{7}) as N → ∞.

Proof. Since

λ_{1} = \frac{−p_{1} − \sqrt{p_{1}^{2} − 4p_{2} + 8} + \sqrt{2p_{1}^{2} − 4p_{2} − 8 + 2p_{1} \sqrt{p_{1}^{2} − 4p_{2} + 8}}}{4}

= \frac{−13 − \sqrt{105} + \sqrt{270 + 26\sqrt{105}}}{2} + O(h^{2}),

λ_{2} = \frac{−p_{1} + \sqrt{p_{1}^{2} − 4p_{2} + 8} + \sqrt{2p_{1}^{2} − 4p_{2} − 8 − 2p_{1} \sqrt{p_{1}^{2} − 4p_{2} + 8}}}{4}

= \frac{−13 + \sqrt{105} + \sqrt{270 − 26\sqrt{105}}}{2} + O(h^{2}),

that is |λ_{k}| < 1, k = 1, 2 and then λ_{k}^{N} → 0 as N → ∞. Thus, when N → ∞ from Theorem 5.1 for ∥ℓ∥^{2} we get the following asymptotic expression

∥ℓ̊∥^{2} = \frac{h^{2} − 18}{12} + \frac{5(1 − cos1)}{2 sin1} + \frac{h[5 sinh · (1 − cos1) − sin1 · (h + 2 sinh)]}{4 sin1 · (cosh − 1)}

+ ∑_{k=1}^{2} m_{k}[ \frac{λ_{k}[5(cosh − 1) sinh + (cosh − sinh) sin1] − λ_{k}^{2} sin1}{2(λ_{k}^{2} + 1 − 2λ_{k} cosh) sin1}

− \frac{h sinh(λ_{k}^{2} − 1)λ_{k}}{(λ_{k}^{2} + 1 − 2λ_{k} cosh)^{2}} − \frac{h^{2}(λ_{k} + 1)λ_{k}}{(λ_{k} − 1)^{3}} + \frac{λ_{k}}{2(λ_{k} − 1)} ],

where m_{k} are defined in Theorem 4.4.

The expansion of the last expression in a series in powers of h gives the assertion of Theorem 5.2. □

The following theorem gives an asymptotic optimality for our optimal quadrature formula.

Theorem 5.3. The optimal quadrature formula of form (1.1) with the error functional (1.2) in the space K_{2}(P_{3}) is asymptotic optimal in the Sobolev space L_{2}^{(3)}(0, 1), i.e.,

(5.2)  lim_{N→∞} \frac{∥ℓ̊|K_{2}^{∗}(P_{3})∥^{2}}{∥ℓ̊|L_{2}^{(3)∗} (0, 1)∥^{2}} = 1.

Proof. Indeed, using Corollary 5.2 from [23] (for m = 2 and η_{0} = 0), for square of the norm of error functional (1.2) of the optimal quadrature formula of form (1.1) on the Sobolev space L_{2}^{(3)}(0, 1) we get the following expression

∥ℓ̊|L_{2}^{(3)∗} (0, 1)∥^{2} = \frac{h^{6}}{30240} + \frac{2h^{7}}{6!} ∑_{k=1}^{2} d_{k} ∑_{i=1}^{6} \frac{−q_{k}^{N+i} + (−1)^{i}q_{k}}{(1 − q_{k})^{i+1}} Δ^{i}0^{6},

i.e.,

(5.3)  ∥ℓ̊|L_{2}^{(3)∗} (0, 1)∥^{2} = \frac{h^{6}}{30240} + O(h^{7}),

where d_{k}, k = 1, 2 are known,

q_{1} = \frac{−13 − \sqrt{105} + \sqrt{270 + 26\sqrt{105}}}{2} , q_{2} = \frac{−13 + \sqrt{105} + \sqrt{270 − 26\sqrt{105}}}{2} ,

Δ^{i}γ^{6} is the finite difference of order i of γ^{6}, Δ^{i}0^{6} = Δ^{i}γ^{6}|_{γ=0} .

Using (5.1) and (5.3) we obtain (5.2). □

As we mentioned in Section 1, error (1.4) of the optimal quadrature formula of form (1.1) in the space K_{2}(P_{3}) is estimated by the Cauchy–Schwarz inequality

(5.4)  |R_{N} (φ)| ⩽ ∥φ|K_{2}(P_{3})∥ · ∥ℓ̊|K_{2}^{∗}(P_{3})∥.

Hence taking into account Theorem 5.2 we get

|R_{N} (φ)| ⩽ ∥φ|K_{2}(P_{3})∥( \frac{\sqrt{210}}{2520}h^{3} + O(h^{7/2} )) ,

from which we conclude that order of the convergence of our optimal quadrature formula is O(h^{3}).

In the next section we give some numerical examples which confirm our theoretical results.

6. Numerical results

As in [12] we considered numerical results for the following functions

f_{1}(x) = e^{x}, f_{2}(x) = tanx, f_{3}(x) = \frac{313 x^{4} − 6900 x^{2} + 15120}{13 x^{4} + 660 x^{2} + 15120} .

The corresponding integrals are

I_{1} = ∫_{0}^{1} e^{x}dx = e − 1 = 1.718281828459045 …,
I_{2} = ∫_{0}^{1} tanxdx = − log(cos1) = 0.6156264703860142 …,

I_{3} = ∫_{0}^{1} \frac{313x^{4} − 6900x^{2} + 15120}{13x^{4} + 660x^{2} + 15120} dx = 0.84147101789394123457 ….

Applying the optimal quadrature formula (1.1) with coefficients C_{ν}, ν = \overline{0, N} which are given in Theorem 4.4, for N = 10, 100, 1000, to the previous integrals we obtain their approximate values denoted by I_{k}^{(N)} , k = 1, 2, 3, respectively. Using Theorem 5.1 and taking into account inequality (5.4), we get upper bounds for absolute errors of these integrals. The corresponding absolute errors and upper bounds are displayed in Table 1. Numbers in parentheses indicate decimal exponents.

Table 1. Absolute errors of quadrature approximations I_{1}^{(N)} , I_{2}^{(N)} , and I_{3}^{(N)} and corresponding upper bounds for N = 10^{k}, k = 1, 2, 3

\begin{matrix}N & |I_{1}^{(N)} − I_{1}| & ∥f_{1}∥ ∥ℓ̊∥ & |I_{2}^{(N)} − I_{2}| & ∥f_{2}∥ ∥ℓ̊∥ & |I_{3}^{(N)} − I_{3}| & ∥f_{3}∥ ∥ℓ̊∥ \\ 10^{1} & 4.15(−6) & 3.86(−5) & 5.61(−5) & 1.92(−4) & 2.44(−10) & 6.99(−10) \\ 10^{2} & 4.49(−10) & 2.30(−8) & 7.32(−9) & 1.15(−7) & 3.17(−14) & 4.17(−13) \\ 10^{3} & 4.52(−14) & 2.08(−11) & 7.50(−13) & 1.04(−10) & 3.25(−18) & 3.77(−16) \end{matrix}

We can see that optimal quadrature formula (1.1) gives the best results for the last integral, because its integrand f_{3}(x) is a rational approximation for the function cosx (cf. [10, p. 66]).

All calculations were performed in Mathematica with 34 decimal digits mantissa. The same results can be obtained using Fortran in quadruple precision arithmetic (with machine precision m.p. ≈ 1.93 × 10^{−34} ).

References

1. J. H. Ahlberg, E. N. Nilson, J. L. Walsh, The Theory of Splines and Their Applications, Academic Press, New York–London, 1967.

2. I. Babuška, Optimal quadrature formulas Dokladi Akad. Nauk SSSR 149 (1963), 227–229 (Russian).

3. P. Blaga, Gh. Coman, Some problems on optimal quadrature, Stud. Univ. Babeş-Bolyai Math. 52, no. 4 (2007), 21–44.

4. B. Bojanov, Optimal quadrature formulas, Uspekhi Mat. Nauk 60, no. 6(366) (2005), 33–52 (Russian).

5. T. Catinaş,Gh. Coman, Optimal quadrature formulas based on the ϕ-function method, Stud. Univ. Babeş-Bolyai Math. 51, no. 1 (2006), 49–64.

6. M. A. Chakhkiev, Linear differential operators with real spectrum, and optimal quadrature formulas, Izv. Akad. Nauk SSSR Ser. Mat. 48, no. 5 (1984), 1078–1108 (Russian).

7. Gh. Coman, Quadrature formulas of Sard type, Studia Univ. Babeş-Bolyai Ser. Math.-Mech. 17, no. 2 (1972), 73–77 (Romanian).

8. , Monosplines and optimal quadrature formulae in L_{p}, Rend. Mat. (6) 5 (1972), 567– 577.

9. A. Ghizzetti, A. Ossicini, Quadrature Formulae, Akademie Verlag, Berlin, 1970.

10. J. F. Hart, at al., Computer Approximations, John Wiley & Sons, Inc., New York-London-Sydney, 1968.

11. A. R. Hayotov, Discrete analogues of some differential operators, Uzbek. Math. Zh. 2012, no. 1 (2012), 151–155 (Russian).

12. A. R.Hayotov, G. V. Milovanović, Kh. M. Shadimetov, On an optimal quadrature formula in the sense of Sard, Numer. Algorithms 57 (2011), 487–510.

13. P. Köhler, On the weights of Sard’s quadrature formulas, Calcolo 25 (1988), 169–186.

14. F. Lanzara, On optimal quadrature formulae, J. Ineq. Appl. 5 (2000), 201–225.

15. A. A. Maljukov, I. I. Orlov, Construction of coefficients of the best quadrature formula for the class W_{L_{2}}^{(2)} (M; ON) with equally spaced nodes, Optimization methods and operations research, applied mathematics (Russian), pp. 174–177, 191. Akad. Nauk SSSR Sibirsk. Otdel. Sibirsk. Ènerget. Inst., Irkutsk, 1976.

16. S. M. Nikol’skiĭ, To question about estimation of approximation by quadrature formulas, Uspekhi Matem. Nauk 5:2 (36) (1950), 165–177 (Russian).

17. , Quadrature Formulas, Nauka, Moscow, 1988 (Russian).

18. L. F. Meyers, A. Sard, Best approximate integration formulas, J. Math. Physics 29 (1950), 118–123.

19. A. Sard, Best approximate integration formulas; best approximation formulas, Amer. J. Math. 71 (1949), 80–91.

20. Kh. M. Shadimetov, Optimal quadrature formulas in L_{2}^{m} (Ω) and L_{2}^{m} (R^{1}), Dokl. Akad. Nauk UzSSR 1983, no. 3 (1983), 5–8 (Russian).

21. , Construction of weight optimal quadrature formulas in the space L_{2}^{(m)} (0, N), Siberian J. Comput. Math. 5, no. 3 (2002), 275–293 (Russian).

22. Kh. M. Shadimetov, A. R. Hayotov, Computation of coefficients of optimal quadrature formulas in the space W_{2}^{(m,m−1)} (0, 1), Uzbek. Math. Zh., 2004, no. 3 (2004), 67–82 (Russian).

23. , Optimal quadrature formulas with positive coefficients in L_{2}^{(m)} (0, 1) space, J. Comput. Appl. Math. 235 (2011), 1114–1128.

24. Kh. M. Shadimetov, A. R. Hayotov, S. S. Azamov, Optimal quadrature formula in K_{2}(P_{2}) space, Appl. Numer. Math. 62 (2012), 1893–1909.

25. I. J. Schoenberg, On monosplines of least deviation and best quadrature formulae, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (1965), 144–170.

26. , On monosplines of least square deviation and best quadrature formulae II, SIAM J. Numer. Anal. 3 (1966), 321–328.

27. I. J. Schoenberg, S. D. Silliman, On semicardinal quadrature formulae, Math. Comp. 28 (1974), 483–497.

28. S. L. Sobolev, The coefficients of optimal quadrature formulas, Selected Works of S. L. Sobolev, pp. 561–566, Springer, 2006.

29. , Introduction to the Theory of Cubature Formulas, Nauka, Moscow, 1974 (Russian).

30. S. L. Sobolev, V. L. Vaskevich, The Theory of Cubature Formulas, Kluwer Academic Publishers Group, Dordrecht, 1997.

31. V. S. Vladimirov, Generalized Functions in Mathematical Physics, Nauka, Moscow, 1979 (Russian).

32. F. Ya. Zagirova, On construction of optimal quadrature formulas with equal spaced nodes, Novosibirsk, 1982, 28 p. [Preprint No. 25, Institute of Mathematics SD of AS of USSR] (Russian).

33. Z. Zh. Zhamalov, Kh. M. Shadimetov, About optimal quadrature formulas, Dokl. Akademii Nauk UzSSR 7 (1980), 3–5 (Russian).

34. A. A. Zhensikbaev, Monosplines of minimal norm and the best quadrature formulas, Uspekhi Matem. Nauk. 36 (1981), 107–159 (Russian).

Institute of Mathematics, National University of Uzbekistan (Received 27 12 2013) Do‘rmon yo‘li str., 29, 100125 Tashkent Uzbekistan

    hayotov@mail.ru

Mathematical Institute of Serbian Academy of Sciences and Arts

Kneza Mihaila 36, 11000 Beograd

Serbia

    gvm@mi.sanu.ac.rs

Institute of Mathematics, National University of Uzbekistan

Do‘rmon yo‘li str., 29, 100125 Tashkent

Uzbekistan

    kholmatshadimetov@mail.ru