OPTIMAL QUADRATURE FORMULAS FOR FOURIER COEFFICIENTS IN W ( m , m − 1 ) 2 SPACE

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the W (m,m−1) 2 [0, 1] space for calculating Fourier coefficients. Using S. L. Sobolev’s method we obtain new optimal quadrature formulas of such type for N + 1 ≥ m, where N + 1 is the number of the nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We investigate the order of convergence of the optimal formula for m = 1. The obtained optimal quadrature formula in the W (m,m−1) 2 [0, 1] space is exact for exp(−x) and Pm−2(x), where Pm−2(x) is a polynomial of degree m − 2. Furthermore, we present some numerical results, which confirm the obtained theoretical results.


Introduction
Numerical calculation of integrals of highly oscillating functions is one of the more important problems of numerical analysis, because such integrals are encountered in applications in many branches of mathematics as well as in other science such as quantum physics, flow mechanics and electromagnetism.Main examples of strongly oscillating integrands are encountered in different transformation, for example, the Fourier transformation and Fourier-Bessel transformation.Standard methods of numerical integration frequently require more computational works and they cannot be successfully applied.The earliest formulas for numerical integration of highly oscillatory functions were given by Filon [11] in 1928.Filon's approach for Fourier integrals is based on piecewise approximation of f (x) by arcs of the parabola on the integration interval.Then finite integrals on the subintervals are exactly integrated.
In [22] the authors studied approximate computation of univariate oscillatory integrals (Fourier coefficients) for the standard Sobolev spaces H s of periodic and non-periodic functions with an arbitrary integer s ≥ 1.They found matching lower and upper bounds on the minimal worst case error of algorithms that use n function or derivative values.They also found sharp bounds on the information complexity which is the minimal n for which the absolute or normalized error is at most ε.
In the work [29] the weight lattice optimal cubature formulas in the periodic Sobolev's space L(m) 2 (Ω) were constructed.In particular, from the result of the work [29], in univariate case when the weight is the function exp(iσx) (where x ∈ [0, 2π] and σ is an integer), the Babuška optimal quadrature formula for Fourier coefficients was obtained [3].
Recently, some optimal quadrature formulas for Fourier coefficients in the Sobolev space L (m) 2 (0, 1) of non-periodic functions have been constructed in [6].This paper is devoted to construction of optimal quadrature formulas for approximate calculation of Fourier integrals in a Hilbert space of non-periodic functions.Precisely, we study the problem of construction such optimal formulas in the sense of Sard in the W (m,m−1) 2 [0, 1] space.We consider the following quadrature formula with the error functional where C β are the coefficients of formula (1.1), h = 1/N , N ∈ N, i 2 = −1, ω ∈ Z, ε [0,1] (x) is the indicator of the interval [0, 1] and δ(x) is the Dirac delta-function.
Functions ϕ belong to the space W (m,m−1) 2 [0, 1], where is the Hilbert space of complex valued functions and in this space the inner product is defined by the equality where ψ is the conjugate function to the function ψ and the norm of the function ϕ is correspondingly defined by the formula ϕ|W (m,m−1) and 1 0 ϕ (m) (x) + ϕ (m−1) (x) ϕ (m) (x) + ϕ (m−1) (x) dx < ∞.
We note that the coefficients C β depend on ω, N and m, i.e., C β = C β (ω, N, m).
It should be noted that for a linear differential operator of order m, L ≡ P m ( d/ dx), Ahlberg, Nilson, and Walsh in the book [1,Chapter 6] investigated the Hilbert spaces K 2 (P m ) in the context of generalized splines.Namely, with the inner product ϕ, ψ = 1 0 Lϕ(x) • Lψ(x) dx, K 2 (P m ) 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 formulas were discussed in [8].
The difference is called the error of the quadrature formula (1.1).The error of the formula (1.1) is a linear functional in W . Clearly the norm of the error functional depends on the coefficients C β .The problem of finding the minimum of the norm of the error functional by coefficients C β when the nodes are fixed (in our case distances between neighbor nodes of formula (1.1) are equal, i.e., x β = hβ, β = 0, 1, . . ., N , h = 1/N ) is called Sard's problem.And the obtained formula is called the optimal quadrature formula in the sense of Sard.This problem was first investigated by A. Sard [24] in the space L (m) 2 for some m.Here L (m) 2 is the Sobolev space of functions which (m − 1)-st derivative is absolutely continuous and m-th derivative is square integrable.
The main aim of the present paper is to solve the Sard problem for quadrature formulas (1.1) in the space W (m,m−1) 2 [0, 1] using S. L. Sobolev's method with N +1 ≥ m, i.e., to look for the coefficients C β that satisfy the following equality (1.5) Thus, to construct Sard's optimal quadrature formula of the form (1.1) in the space W (m,m−1) 2 [0, 1], we need to solve the following problems.It should be noted that Problems 1 and 2 were solved in [34] for the case ω = 0, i.e., in the work [34] the optimal quadrature formulas of the form in the sense of Sard were constructed.In the sequel we will solve Problems 1 and 2 in the cases when ω ∈ Z and ω = 0.
The paper is organized as follows.In the second section the extremal function, which corresponds to the error functional , is given and, with its help, a representation of the norm of the error functional (1.2) is calculated, i.e., Problem 1 is solved.In Section 3 we obtain the system of linear equations for coefficients of the optimal quadrature formulas in the space W (m,m−1) 2 [0, 1].Moreover, the existence and uniqueness of the solution of this system are discussed.In Section 4, in the cases m ≥ 2, the explicit formulas for the coefficients of the optimal quadrature formulas of the form (1.1) are found, i.e., Problem 2 is solved in the cases m ≥ 2. The obtained optimal quadrature formulas are exact for any polynomial of order ≤ m − 2 and for the exponential function exp(−x).In Section 5 we solve Problem 2 in the case m = 1 and we calculate the norm of the error functional of the optimal quadrature formula in the W (1,0) 2 [0, 1] space.The obtained explicit formula for the norm of the error functional shows dependence on ω and h of the error of the optimal quadrature formula of the form (1.1) in W (1,0) 2 [0, 1] space.Finally, in Section 6 we present some numerical results which are confirm the obtained theoretical results of the present work.

Extremal function and norm of the error functional
To solve Problem 1, i.e., to get the explicit expression for the norm of the error functional (1.2) in the space W (m,m−1) * 2 [0, 1], we use the concept of the extremal function.The function ψ is called the extremal function for the functional (see, [37]), if the following equality holds Since W (m,m−1) 2 [0, 1] is a Hilbert space, then the extremal function ψ in this space, is found with the help of the general form of a linear continuous functional on Hilbert spaces given by the Riesz theorem.Then for the functional and for any ϕ ∈ W (m,m−1) 2 [0, 1] there exists the function ψ ∈ W (m,m−1) 2 [0, 1] for which the following equation holds where is the inner product defined in the space W (m,m−1) 2 [0, 1].From (2.2) taking into account (2.3) for the extremal function ψ we get the following boundary value problem where ¯ is the conjugate to . where is a solution of the equation d is any complex number and P m−2 (x) is a polynomial of degree m − 2 with complex coefficients, and * is the operation of convolution.
Theorem 2.1 can be proved as Theorem 2.1 in [34].
For the error functional (1.2) to be defined on the space W (m,m−1) 2 (0, 1) it is necessary to impose the following conditions Hence, it is clear that for existence of the quadrature formulas of the form (1.1) the condition N + 1 ≥ m has to be met.The equalities (2.9) mean that our quadrature formula is exact for the function e −x and for any polynomial of degree ≤ m − 2. Now, using Theorem 2.1 we will get the representation of the square of the norm of the error functional (1.2).
We recall that a convolution of two functions is defined by the formula Taking into account the definition of convolution and equality (1.2) we calculate the convolution ¯ (x) * G m (x), i.e., where ¯ and Cβ are conjugates to and C β , respectively.Then keeping in mind (2.2), (2.3) and Theorem 2.1, we have Hence we obtain (2.10) Now we show that the right hand side of (2.10) is real.Really, let β and C I β are real.Using Euler's formula e 2πiωx = cos 2πωx + i sin 2πωx, we get the following equalities Keeping in mind the last three equalities, from (2.10) for the norm of the error functional we have and from (2.9), we have the following equalities x α cos 2πωx dx, α = 0, 1, 2, . . ., m − 2, (2.12) 3. The system for coefficients of optimal quadrature formulas (1.1) in the space W (m,m−1) 2
From (2.11) and these theorems from [34], it follows that the square of the norm of the error functional , being a quadratic functions of the coefficients C β has a unique minimum in some concrete value of C β = Cβ .
As it was said in the first section, the quadrature formula with the coefficients Cβ (β = 0, N ), corresponding to this minimum, is called the optimal quadrature formula in the sense of Sard, and Cβ (β = 0, N ) are called the optimal coefficients.
Below, for the purposes of convenience, the optimal coefficients Cβ will be denoted as C β .

Coefficients of optimal quadrature formulas (1.1)
In the present section we solve the system (3.7)-(3.9)and we find the explicit formulas for the optimal coefficients C β .Here we use a similar method to the one suggested by S. L. Sobolev [38] for finding the coefficients of optimal quadrature formulas in the space L (m) 2 (0, 1).Here the main concept used is that of functions of discrete argument and operations on them.Theory of discrete argument functions is given in [37,39].For the purposes of completeness we give some definitions about functions of discrete argument.
Suppose that ϕ(x) and ψ(x) are real-valued functions of real variable and are defined in real line R.
Now, we return to our problem.Suppose that C β = 0 when β < 0 and β > N .Using the above mentioned definitions, we rewrite the system (3.7)-(3.9) in the following convolution form where ) Further we investigate Problem 3 and instead of C β we introduce the functions In this statement it is necessary to express the coefficients C β by the function u(hβ).
For this, we need such an operator D m (hβ) which satisfies the equality where δ d (hβ) is equal to 0 when β = 0 and is equal to 1 when β = 0, i.e., δ d (hβ) is the discrete delta-function.
Theorem 4.2.The discrete analogue D m (hβ) of the differential operator satisfies the following equalities Thus, if we find the function u(hβ), then the optimal coefficients can be obtained from equality (4.10).
To calculate this convolution, it is required to find the representation of the function u(hβ) for all integer values of β.From equality (4.1), we get that u(hβ) = f m (hβ) when hβ ∈ [0, 1].Now we need to find the representation of the function u(hβ) when β < 0 and β > N . Since where is a polynomial of degree 2m − 3 in (hβ), is an unknown polynomial of degree m − 2 also in (hβ), and Similarly, in the case β > N , for the convolution We denote having the form Here, Q m−2 (hβ) are unknown polynomials of degree m − 2 with respect to (hβ), a − and a + are unknown constants.
If we find Q m−2 (hβ), a − and a + , then from (4.16), (4.17) we have m−2 (hβ), a − and a + can be found from the equation (4.18), using the function D m (hβ).Then we can obtain the explicit form of the function u(hβ) and find the optimal coefficients C β .Thus, Problem 4 and, respectively, Problem 3 can be solved.
But here we will not find Q m−2 (hβ), a − and a + .Instead of them, using D m (hβ) and u(hβ), taking into account (4.10), we find now the expressions for the optimal coefficients C β when β = 1, . . ., N − 1.
We denote where λ k are roots and p is the leading coefficient of the polynomial P 2m−2 (λ) of degree 2m − 2 defined by (4.9) and |λ k | < 1.The series in the notations (4.19), (4.20) are convergent.
The following statement holds: Theorem 4.3 (Theorem 3, [31]).The coefficients of optimal quadrature formulas in the sense of Sard of the form (1.1) in the space W (m,m−1) 2 [0, 1] have the following form where a k and b k are unknowns and have the form (4. 19) and (4.20) respectively, λ k are the roots of the polynomial P 2m−2 (λ) which is defined by equality (4.9) and From Theorem 4.3, it is clear that to obtain the explicit forms of the optimal coefficients C β in the space W . But here we will not calculate series (4.19) and (4.20).Instead of that substituting equality (4.21) into (4.1)we obtain the identity with respect to (hβ).Whence, equating the corresponding coefficients in the left and the right hand sides of equation (4.1) and using (4.2) when α = 1, 2, . . ., m − 2, we find a k and b k .The coefficients C 0 and C N can be found from (4.2) when α = 0 and (4.3), respectively.Below we do it.
In the present section we solve the system (4.1)-(4.3)for any m ≥ 2 and for natural N that N + 1 ≥ m.As it was mentioned above, it is sufficient to find a k and b k (k = 1, m − 1) in (4.21).
The case m = 1 we consider in the next section.In the case m ≥ 2 the following results hold: where a k and b k (k = 1, m − 1) are defined by the following system of 2m − 2 linear equations where a k and b k , k = 1, m − 1, are defined by the following system of 2m − 2 linear equations Here, λ k are the roots of the polynomial (4.9) and |λ k | < 1, A k and C are defined in Theorem 4.1.
In order to prove Theorem 4.4 we use the following formulas (cf.[13], [12]) where is the finite difference of order i of γ k , and B k+1−j are the Bernoulli numbers, as well as where K is given in Theorem 4.4.
Therefore, from Theorem 4.3, taking into account the last equality, for coefficients C β , β = 1, N − 1, we have For the convolution G m (hβ) * C β of equality (4.1) we have where Then, using (4.25), (4.22), (4.23) and taking into account that λ k are roots of the polynomial (4.9), after some simplifications, we get Now, using the binomial formula and equalities (4.2) and (4.3), we obtain Unknown polynomial P m−2 (hβ) and the coefficient d can be found from (4.29) by equating the corresponding coefficients of (hβ) α when α = 0, 1, . . ., m − 2 and e −hβ , respectively.Now, from equations (4.2) when α = 0 and (4.3), taking into account (4.25), using identities (4.22) and (4.23), after some simplifications for the coefficients C 0 and C N , we get the following expressions and respectively.Then, from (4.31), using (4.32), grouping the coefficients of same degrees of (hβ) and equating to zero, for a k and b k we obtain the following m − 1 linear equations: and Further, from (4.2) when α = 1, . . ., m − 2, using equalities (4.25), (4.33) and identities (4.22) and (4.23) for a k and b k we have the following m−2 linear equations: Finally, after some simplifications in (4.30) and the previous systems of equations for a k and b k , we get the system which is given in the assertion of this theorem.(2,1) 2

The proof of Theorem
[0, 1], are expressed by formulas where Remark 4.1.For λ 1 in Corollary 4.1 the following expansion Corollary 4.2.The coefficients of optimal quadrature formulas of the form (1.1), with the error functional (1.2), and with equal spaced nodes when ωh ∈ Z in the space W (3,2) 2 [0, 1], are expressed by formulas where a k and b k (k = 1, 2) are defined by the following system of linear equations Here λ k , k = 1, 2, are roots of the polynomial where a k and b k (k = 1, 3) are defined by the following system of linear equations Here λ k , k = 1, 2, 3 are the roots of the polynomial where , and 1 − e 2h + 2he h .
Corollary 4.5.The coefficients of optimal quadrature formulas of the form (1.1), with the error functional (1.2), and with equal spaced nodes when ωh ∈ Z and ω = 0 in the space W (3,2) 2 [0, 1], are expressed by formulas where a k and b k (k = 1, 2) are defined by the following system of linear equations Here λ k , k = 1, 2, are roots of the polynomial for which |λ k | < 1.
Corollary 4.6.The coefficients of optimal quadrature formulas of the form (1.1), with the error functional (1.2), and with equal spaced nodes when ωh ∈ Z and ω = 0 in the space W (4,3) 2 [0, 1], are expressed by formulas where a k and b k (k = 1, 3) are defined by the following system of linear equations Here λ k , k = 1, 2, 3, are roots of the polynomial for which |λ k | < 1.

5.
Coefficients and norm of the error functional of optimal quadrature formulas (1.1) in W (1,0) 2 Here we get the explicit expressions for coefficients and calculate the square of the norm of the error functional (1.2), of the optimal quadrature formula (1.1), on the space W (1,0) 2 [0, 1].For m = 1 the system (4.1)-(4.3)takes the form where and C β (β = 0, 1, . . ., N ) and d are unknowns.In this case Problem 4 is expressed as follows: Problem 5. Find the solution of the equation having the form where f 1 (hβ) is defined by (5.3), a − and a + are unknowns.
Note that, in Theorem 5.1, the formulas for the optimal coefficients C β are decomposed into two parts -real and imaginary parts.Therefore, from Theorem 5.1 we get the following results: Corollary 5.1.Coefficients for the optimal quadrature formulas of the form in the sense of Sard in the space W (1,0) 2 [0, 1] have the form

5.2.
Coefficients for the optimal quadrature formulas of the form in the sense of Sard in the space W (1,0) 2 [0, 1] have the form Theorem 5.2.The square of the norm of the error functional (1.2), of the optimal quadrature formula (1.1), on the space W (1,0) 2 [0, 1], has the form h(e 2h − 1) . (5.10) Proof.For m = 1 we rewrite the equality (2.11) in the following form where G 1 (x) is defined by (5.3).Taking into account (5.8), from (5.1) we get Then, using the last two equalities, for 2 we obtain Finally, calculating these integrals and using Corollaries 5.1 and 5.2, after some simplifications, we get (5.10).

Numerical results
In this section we give some numerical results of the upper bounds for the errors in the optimal quadrature formulas of the form (1.1), as well their analysis in the cases m = 1 and m = 2.
According to the Cauchy-Schwarz inequality, in the space W (m,m−1) 2 [0, 1] for the absolute value of the difference (1.4) we get where ˚ is the norm of the optimal error functional which corresponds to the optimal quadrature formulas (1.1).  1. Numbers in parenthesis indicate the decimal exponents.From the first column of this table we see that order of convergence of our optimal quadrature formula is O(N −1 ) and from the first row of Table 1 it is clear that the quantity ˚ converges as O(|ω| −1 ).From other columns and rows of Table 1 we conclude that order of convergence of our optimal quadrature formula in the case m = 1 is O((N + |ω|) −1 ).For the same values of N and ω, using formulas for the optimal coefficients C β from Theorem 5.1 and formula (5.10), we get the numerical values for the actual error R N (ω) and for the bound B N (ω) on the right hand side in the previous inequality.These results are presented in Table 2.These numerical results confirm our theoretical results obtained in the previous sections.
2 • Now we consider the case m = 2. From (2.11), taking into account (2.7), after some calculations for the norm of the error functional of the optimal quadrature formula (1.1) we get the following expression error (1.4) of formula (1.1) is estimated by the norm |W functional (1.2).Thus, the estimation of the error of the quadrature formula (1.1) over functions of the space W (m,m−1) 2 is reduced to finding the norm of the error functional in the conjugate space W (m,m−1) * 2

Problem 1 .Problem 2 .
Find the norm of the error functional of quadrature formulas (1.1) in the space W Find the coefficients C β that satisfy equality (1.5).

Theorem 2 . 1 .
The solution of the boundary value problem (2.4)-(2.6) is the extremal function ψ of the error functional and has the following form
1 is solved.Further in Sections 3 and 4 we solve Problem 2.

Theorem 4 . 1 .
The discrete analogues to the differential operator d 2m dx 2m − d 2m−2 dx 2m−2 satisfying the equation (4.7) has the form Here, G m (hβ) is the function of discrete argument corresponding to the function G m (x), defined by equality (2.7) and δ d (hβ) is the discrete delta function.Then taking into account (4.6), (4.7) and Theorems 4.1 and 4.2, for the optimal coefficients we have C β = D m (hβ) * u(hβ).(4.10) ) and, taking into account (4.11), (4.15), (4.6), we get the following problem.Problem 4. Find the solution of the equation

Theorem 4 . 4 .
The coefficients of optimal quadrature formulas of the form (1.1) with the error functional (1.2) and with equal spaced nodes in the space W (m,m−1) 2[0, 1] when m ≥ 2, N + 1 ≥ m and ωh ∈ Z are expressed by formulas

Corollary 4 . 1 .
4.5 is similar to one of Theorem 4.4.Only one difference is that D m (hβ) * f m (hβ) = K = 0.For m = 2, m = 3 and m = 4, from Theorem 4.4 we have the following results: The coefficients of optimal quadrature formulas of the form (1.1), with the error functional (1.2), and with equal spaced nodes when ωh ∈ Z in the space W 2k , and A k and C are defined in Theorem 4.1.

Corollary 4 . 3 .
The coefficients of optimal quadrature formulas of the form (1.1), with the error functional (1.2), and with equal spaced nodes when ωh ∈ Z in the space W 1], are expressed by formulas

) 6 Corollary 4 . 4 .
is the leading coefficient of the polynomial P 6 (λ), and A k and C are defined in Theorem 4.1 for m = 4. Now from Theorem 4.5 for m = 2, m = 3 and m = 4, we have the following corollaries: The coefficients of optimal quadrature formulas of the form (1.1), with the error functional (1.2), and with equal spaced nodes when ωh ∈ Z and ω = 0 in the space W (2,1) 2 [0, 1], are expressed by formulas

Theorem 5 . 1 . 2 [ 0 , 1 ]
Coefficients of the optimal quadrature formulas of the form (1.1) in the sense of Sard in the space W (1,0) have the form