An exponential-trigonometric spline minimizing a seminorm in a Hilbert space

In the present paper, using the discrete analogue of the operator d6/dx6 – 1, we construct an interpolation spline that minimizes the quantity ∫ 1 0 (φ ′′′(x) + φ(x))2 dx in the Hilbert spaceW (3,0) 2 . We obtain explicit formulas for the coefficients of the interpolation spline. The obtained interpolation spline is exact for the exponential-trigonometric functions e–x , e x 2 cos( √ 3 2 x), and e x 2 sin( √ 3 2 x).

By K m (a, b) we denote the class of all functions f (x) defined on [a, b] that possess absolutely continuous (n -1)th derivatives on [a, b] and have the nth derivatives in L 2 (a, b); it is a Hilbert space with the inner product (1) if functions that differ by a solution of the equation Lf = 0 are identified. If : a = x 0 < x 1 < · · · < x N = b is a mesh on [a, b], then a generalized spline of deficiency k (0 ≤ k ≤ m) with respect to is a function S (x) from K 2m-k (a, b) that satisfies the differential equation on each open mesh interval (x i-1 , x i ) (i = 1, 2, . . . , N ) of . The ordinary spline of deficiency 1 allows discontinuities in the (2m -1)th derivative, but only at mesh points.
From the results of Ahlberg, Nilson, and Walsh [1] ti is known that for generalized spline of deficiency 1, the following is true: Let : a = x 0 < x 1 < · · · < x N = b and Y = {y i , i = 0, 1, . . . , N} be given. Then among all functions f (x) in K m (a, b) such that f (x i ) = y i (i = 0, 1, . . . , N), the generalized spline S (Y ; x), when it exists, minimizes the quantity b a (Lf (x)) 2 dx. Further, we give some results obtained from the theory of L-splines. First, contributions to the theory of splines include the works of Greville, Ahlberg, Nilson, and Walsh and of Schultz and Varga (see [22, p. 459]). These works are concentrated on natural L-splines, which appear as solutions to the corresponding best interpolation problems, and the order of approximation of generalized splines was first studied.
Trigonometric splines were first considered by Schoenberg [20]. Construction of local B-splines for general spaces of L-splines was studied in [9,14]. Further results on the of approximation of trigonometric splines were obtained in [16] based on construction of quasi-interpolation operators. In [12], control curves of trigonometric splines are constructed, and it is proved that they have similar properties as polynomial splines. Then trigonometric B-splines were studied in [8,10,19].
The space of natural hyperbolic splines was introduced in [22, p. 407]. In [21], it is shown that hyperbolic splines can be treated as an example of L-splines.
In the present work, we solve the problem of construction of a natural L-spline in the case m = 3, L ≡ d 3 /dx 3 + 1 on the interval [0, 1], and we denote by W (3,0) 2 (0, 1) the corresponding space, that is, is a Hilbert space if we identify functions that differ by a solution of the equation find a function S(x) that gives the minimum of the norm (2), where x β ∈ [0, 1] are the nodes of interpolation, and ϕ(x β ) = y β are given values.
The solution S(x) of Problem 1 is a ordinary generalized spline and is uniquely defined with respect to mesh : 0 = x 0 < x 1 < · · · < x N = 1 on the interval [0, 1] as follows: (v) S(x) satisfies the following boundary conditions: We consider the fundamental solution of the differential operator d 6 dx 6 -1, that is, the solution of the equation where δ(x) is the Dirac delta-function.

Remark
The following rule for finding a fundamental solution of a linear differential operator where a j are real numbers, is given in [29, pp. 88-89]: replacing d dx by p, we get a polynomial P(p). Then we expand the expression 1 P(p) to partial fractions: and with every partial fraction (pλ) -k , we associate x k-1 sgn x 2(k-1)! · e λx . Using this rule, we find the function G(x) that is the fundamental solution of the operator d 6 dx 6 -1 and has the form (4).
It is easy to check that the fifth derivative of the function has a discontinuity equal to 1 at the point x γ , and the third and the fours derivatives of Then the spline S(x) can be written in the form where , and d 0 , d 1 , and d 2 are real numbers. It is known that (see, e.g., [28]) the solution S(x) of the form (4) of Problem 1 exists and is unique when N ≥ 2, and the coefficients C γ , d 0 , d 1 , and d 2 of S(x) are determined by the following system of N + 4 linear equations: It is easy to show that the spline S(x) defined by equation (8) with coefficients C γ , d 0 , d 1 , and d 2 satisfies conditions (i)-(v).
It should be noted that using Sobolev method, interpolation splines minimizing the seminorms in different Hilbert spaces were constructed in [2,4,6,7,23,24]. Furthermore, connection between interpolation splines and optimal quadrature formulas in the sense of Sard in L (m) 2 (0, 1) and K 2 (P 2 ) spaces were shown in [4] and [6]. The rest of the paper is organized as follows. In Sect. 2, we give some definitions and known results. In Sect. 3, we give an algorithm for solving system (9)-(12) when the nodes x β are equally spaced. Using this algorithm, the coefficients of the interpolation spline S(x) are computed in Sect. 4.

Preliminaries
In this section, we give some definitions and known results that we need to prove the main results.
We mainly use the concept of discrete argument functions and operations on them. The theory of discrete argument functions is given in [25,27]. For completeness, we give some definitions about functions of discrete argument.

Definition 1 The function ϕ[β]
is a function of discrete argument if it is given on some set of integer values of β.

Definition 2 The inner product of two discrete-argument functions ϕ[β] and ψ[β] is given by
if the series on the right-hand side of the last equality converges absolutely.

Definition 3 The convolution of two functions ϕ[β] and ψ[β] is the inner product
In our computations we need the discrete analogue D[β] of the differential operator d 6 dx 6 -1, which satisfies the equality where G[β] is the discrete-argument function corresponding to G(x) defined by (4), δ[β] is equal to 0 when β = 0 and to 1 when β = 0, that is, δ[β] is the discrete delta-function. Equation (13) is a discrete analogue of equation (5). In [2,3] the discrete analogue D[β] of the differential operator d 6 dx 6 -1, which satisfies equation (13), was constructed, and the following theorem was proved.

Theorem 1
The discrete analogue of the differential operator d 6 dx 6 -1 satisfying equation (13) has the form where , h is a small positive parameter, and λ k are the roots of the polynomial A 4 (λ) such that |λ k | < 1.
Furthermore, several properties of the discrete-argument function D[β] were given in [2,3]. Here we give the following its properties, which we need in our computations.

Theorem 2
The discrete analogue D[β] of the differential operator d 6 dx 6 -1 satisfies the following equalities:

The algorithm for computation of coefficients of interpolation spline
In the present section, we give an algorithm for solving system (9)-(12) when the nodes x β are equally spaced, that is, x β = hβ, h = 1 N , N = 1, 2, . . . . Here we use a similar method suggested by Sobolev [26,27] for finding the coefficients of optimal quadrature formulas in the Sobolev space L (m) 2 (0, 1). We note that here [β] means (hβ). Suppose that C[β] = 0 when β < 0 and β > N . Using Definition 3, we rewrite system (9)-(12) in the convolution form: Thus we have the following problem.
Further, we investigate Problem 2, which is equivalent to Problem 1. Instead of C[β], we introduce the following discrete-argument functions:

Now we express the coefficients C[β] by the function u[β].
Taking into account (14), (20), and Theorems 1 and 2, for the coefficients, we have Thus, if we find the function u[β], then the coefficients C[β] will be found from equality (21).
To calculate convolution (21), it is required to find the representation of the function u[β] for all integer values of β. From equality (15) (8) and (16) Thus, when β ≤ 0, we get We denote Taking into account (20), (22), and (24), we get the following problem.
Thus Problem 3 and, respectively, Problems 2 and 1 will be solved.

Computation of coefficients of interpolation spline (8)
In this section, using the presented algorithm, we obtain explicit formulas for the coefficients of the interpolation spline (8), which, as we have proved in the previous section, is the solution of Problem 1.
Proof First, we find the expressions for d -0 and d + 0 . When β = 0 and β = N , from (28) for d -0 and d + 0 we get where β < 0 and β > N . First, we consider the cases β = -1, -2. From (34) we obtain the following system of two linear equations where 2 ) λ ke h , Applying S(x) with N = 10 and N = 100 and using Theorem 3 for the functions f 1 (x) and f 2 (x), we obtain the absolute errors. The graphs of the corresponding absolute errors are displayed in Figs. 1 and 2. We can see that by increasing the value of N the absolute errors between interpolation splines and the given functions decrease and the order of convergence of the interpolation formula (8) in these examples is O(h 3 ).