Volume 3, Issue 6 e1123

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Uniform algebraic hyperbolic spline quasi-interpolant based on mean integral values

Domingo Barrera Rosillo,

Corresponding Author

Domingo Barrera Rosillo

[email protected]

orcid.org/0000-0003-0045-8831

Department of Applied Mathematics, Faculty of Sciences, University of Granada, Granada, Spain

Correspondence Domingo Barrera Rosillo, Department of Applied Mathematics, Faculty of Sciences, University of Granada, Campus de Fuentenueva s/n, 18071 Granada, Spain.

Email: [email protected]

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Salah Eddargani,

Salah Eddargani

Department of Applied Mathematics, Faculty of Sciences, University of Granada, Granada, Spain

MISI Laboratory, Hassan First University, Settat, Morocco

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Abdellah Lamnii,

Abdellah Lamnii

MISI Laboratory, Hassan First University, Settat, Morocco

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Domingo Barrera Rosillo,

Corresponding Author

Domingo Barrera Rosillo

[email protected]

orcid.org/0000-0003-0045-8831

Department of Applied Mathematics, Faculty of Sciences, University of Granada, Granada, Spain

Correspondence Domingo Barrera Rosillo, Department of Applied Mathematics, Faculty of Sciences, University of Granada, Campus de Fuentenueva s/n, 18071 Granada, Spain.

Email: [email protected]

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Salah Eddargani,

Salah Eddargani

Department of Applied Mathematics, Faculty of Sciences, University of Granada, Granada, Spain

MISI Laboratory, Hassan First University, Settat, Morocco

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Abdellah Lamnii,

Abdellah Lamnii

MISI Laboratory, Hassan First University, Settat, Morocco

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First published: 31 July 2020

https://doi.org/10.1002/cmm4.1123

Citations: 1

Funding information: Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía, Research Group FQM 191 Matemática Aplicada, PAIDI; Universidad de Granada

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Summary

In the present work, a novel spline quasi-interpolation operator reproducing both constant polynomials and algebraic hyperbolic functions is presented. The quasi-interpolant to a given function is defined from the integrals on every interval of the function to be approximated. Compared to the other existing methods, this operator does not need any additional end conditions and it is easy to be implemented without solving any system of equations. The approximation properties of the operator are theoretically analyzed and some numerical tests are presented to illustrate its performance.

1 Introduction

Interpolation plays an essential role in defining approximants to functions or data. But, this kind of approximation has a strong inconvenient, because usually it requires the solution of linear systems. Spline quasi-interpolation avoids this limitation, so it is very convenient in practice. Since some properties, as stability and local control, are important in practice, we have to construct the quasi-interpolant (QI for short) to a given function f as a combination of the elements of a suitable set of functions, which are required to be positive and have a small local support. The coefficients of the linear combination are the values of linear functionals at f and (or) its derivatives and (or) integrals (see References 1-4 and references therein). In this paper, we deal with the case which only involves integral values of f, so our purpose is the construction of the approximating function using only this information. This kind of approximation arises in many different fields, as mechanics, mathematical statistics, electricity, environmental science, climatology, oceanography and so on (for more details, see References 1, 5-10 and references therein).

The interpolation of this kind of values has been considered in the literature. In References 5 and 6, Behforooz defines cubic and quintic integro spline interpolants. The associated operators need some end conditions and solving a tridiagonal system of linear equations. Then, for avoiding this last requirement, the authors in Reference 11 constructed cubic integro spline quasi-interpolants without solving any system of equations. In Reference 12, the authors constructed an integro quartic spline quasi-interpolation operator and analyzed its approximation properties. They showed that the integro quartic spline possesses superconvergence orders regarding point values and second-order derivative values at the knots, but a phenomenon of superconvergence remains without an answer. To this end, in Reference 13 other local integro quartic spline interpolants were constructed. Some other authors have devoted much attention to construct this kind of operators, as shown for instance in References 3, 14-18

In this paper, we propose an integro spline quasi-interpolant based on second order Uniform Algebraic Hyperbolic functions. The main tool of this approach is Marsden's identity.¹⁹ The advantage of this method is that it does not need any additional data and does not require the solution of any system of equations. This construction can be extended to derive fourth order approximating splines, but the expressions provided for the coefficients of the quasi-interpolants are very complex.

The remaining of the paper is described as follow: In Section 2, the approximation problem is stated as well as the spline space to deal with it. In Section 3, we derive linear combinations of the data to get good approximations to the values at the knots of the function to be approximated. The integral quasi-interpolant is constructed in Section 4, and the quasi-interpolation error is given in Section 5. Finally, the performance of the numerical scheme proposed to approximate the integral values is tested in Section 6.

2 The approximation problem and the spline space

Let f be a real function defined on an interval

I : = [a, b]

and

𝒳_{n} : = {t_{i} = a + i α, i = 0, \dots, n}

be a uniform partition of I with step size

α : = \frac{b - a}{n}

. Let us suppose that the values

𝒜_{i} : = \int_{t_{i}}^{t_{i + 1}} f (t) d t, i = 0, \dots, n - 1,

()

are known.

As described in the introduction, the construction of a polynomial spline s of low degree such that

\int_{t_{i}}^{t_{i + 1}} s (t) d t = 𝒜_{i}, i = 0, \dots, n - 1,

is a problem that has been widely studied in the literature and whose solution usually requires to solve a system of linear equations with a banded matrix after imposing some boundary conditions.

We propose to define an enough regular nonpolynomial spline q fulfilling the conditions

\int_{t_{i}}^{t_{i + 1}} q (t) d t ≃ 𝒜_{i}, i = 0, \dots, n - 1,

()

in such a way that the error committed is as small as possible. More precisely, we will use integral quasi-interpolation instead of interpolation to construct the spline without imposing any boundary conditions, and in such a way that the associated quasi-interpolation operator is exact on the space spanned by the functions 1,

\sinh t

, and

\cosh t

The function q will belong to the space of Uniform Algebraic Hyperbolic (UAH) splines defined by

Ω_{α, I} : = {s \in 𝒞^{1} (I) : s_{|_{[t_{i}, t_{i + 1}]}} \in span {1, \sinh \cdot, \cosh \cdot}, 0 \leq i \leq n - 1} .

()

In order to get a basis of the space, additional knots t₋₂ = t₋₁ = a and t_n + 2 = t_n + 1 = b are added and the Algebraic Hyperbolic (AH) B-splines

{N_{i}, 1 \leq i \leq n + 2}

are defined as follows (see References 19, 20, for instance, and References 21-23 for the construction and properties of AH B-splines on a uniform partition of an compact interval):

\begin{align} N_{1} (t) & : = \{\begin{matrix} \frac{\cosh (t_{1} - t) - 1}{\cosh α - 1}, & 0 \leq t < t_{1}, \\ 0, & otherwise, \end{matrix} \\ N_{2} (t) & : = \{\begin{matrix} - \frac{2 \cosh (t_{1} - t) - 2 \cosh α + \cosh t - 1}{2 (\cosh α - 1)}, & 0 \leq t < t_{1}, \\ \frac{1}{2} {csch}^{2} (\frac{α}{2}) \sinh^{2} (t_{1} - \frac{t}{2}), & t_{1} \leq t < t_{2}, \\ 0, & otherwise, \end{matrix} \\ N_{n + 1} (t) & : = \{\begin{matrix} - \frac{\cosh (t - t_{n - 2} - 1)}{2 (\cosh α - 1)}, & t_{n - 2} \leq t < t_{n - 1}, \\ \frac{1}{2} + \frac{1}{4} {csch}^{2} (\frac{α}{2}) (\cosh (t_{n} - t) - \cosh α) & t_{n - 1} \leq t < t_{n}, \\ + \coth (\frac{α}{2}) csch α (\cosh (t - t_{n - 1}) - 1), \\ 0, & otherwise, \end{matrix} \\ N_{n + 2} (t) & : = \{\begin{matrix} (1 - \cosh (t - t_{n - 1})) csch α \coth (\frac{α}{2}), & t_{n - 1} \leq t < t_{n}, \\ 0, & otherwise. \end{matrix} \end{align}

and

N_{i} (t) : = N_{3} (t - t_{i - 2}), 3 \leq i \leq n,

where

N_{3} (t) = \{\begin{matrix} \frac{\cosh (t - t_{0}) - 1}{2 (\cosh α - 1)}, & t_{0} \leq t < t_{1}, \\ \frac{2 (\cosh α - \cosh \frac{α}{2} \cosh (t - \frac{3}{2} α - t_{0}))}{2 (\cosh α - 1)}, & t_{1} \leq t < t_{2}, \\ \frac{\cosh (t - t_{3}) - 1}{2 (\cosh α - 1)}, & t_{2} \leq t < t_{3}, \\ 0, & otherwise. \end{matrix}

These compactly supported non-negative functions form a partition of unity and provide a basis of the space

Ω_{α, I}

. Figure 1 shows the plots of some UAH B-splines.

Details are in the caption following the image — **FIGURE 1**
Open in figure viewer PowerPoint

Uniform Algebraic Hyperbolic B-splines associated with the partition obtained by dividing the interval into 10 equal parts

3 Linking point and integral values

The main idea to solve the approximation problem stated in Section 2 is to use a discrete quasi-interpolant defined on the spline space but the point values of the function whose values $𝒜_{i}$ are going to be approximated are unknown. Then, good approximations to those values $f_{i} : = f (t_{i})$ will be provided.

Lemma 1.For a given regular function f, let

\tilde{f_{i}} : = δ_{2} (α) 𝒜_{i - 2} + δ_{1} (α) 𝒜_{i - 1} + δ_{0} (a) 𝒜_{i} + δ_{- 1} (a) 𝒜_{i + 1} + δ_{- 2} (α) 𝒜_{i + 2}, 2 \leq i \leq n - 3,

()

where

\begin{align} δ_{2} (α) & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 1 - 2 \cosh α + 3 α \coth α), \\ δ_{1} (α) & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (\sinh (3 α) csch α - 3 (- 4 + \cosh α + 3 α \coth α + csch α)), \\ δ_{0} (α) & : = \frac{1}{48 α} (40 - 6 {csch}^{2} (\frac{α}{2}) + 9 (α \coth (\frac{α}{2}) - 2) {csch}^{4} (\frac{α}{2})), \\ δ_{- 1} (α) & : = - \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 3 \cosh α + \cosh (2 α) + 3 α (3 + \cosh α) csch α - 10), \\ δ_{- 2} (α) & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 4 + \cosh α + 3 α csch α) . \end{align}

()

Then, $\tilde{f_{i}} - f_{i} = 𝒪 (α^{6})$ .

Proof.Let L_k be the differential operator defined as $L_{k} : = D^{k - 2} (D^{2} - 1)$ , and let $Γ_{k}$ be its null space, that is, the space of functions f such that L_k f = 0. The Green's function associated with L_k can be written as⁴

G_{k} (x; y) : = \{\begin{matrix} - \sum_{\underset{j odd}{j = 1}}^{k - 2} \frac{1}{j!} {(x - y)}_{+}^{j} + \sinh {(x - y)}_{+}, & k even, \\ - 1 - \sum_{\underset{j even}{j = 2}}^{k - 2} \frac{1}{j!} {(x - y)}_{+}^{j} + \cosh {(x - y)}_{+}, & k odd . \end{matrix}

Let 2 ≤ i ≤ n − 3 be fix. The algebraic hyperbolic Taylor expansion of f around t_i provides the expression (eg, Reference 19 and the references therein)

f (x) = f (t_{i}) + G_{2} (x, t_{i}) f^{'} (t_{i}) + G_{3} (x, t_{i}) f^{'^{'}} (t_{i}) + \sum_{j = 4}^{\infty} G_{j} (x, t_{i}) L_{j - 1} f (t_{i}) .

Then, we get

\begin{align} 𝒜_{i} & = \int_{t_{i}}^{t_{i + 1}} f (x) d x \\ = α f_{i} + (\cosh α - 1) f_{i}^{'} + (\sinh α - α) f_{i}^{'^{'}} + L_{3} f (t_{i}) (- \frac{1}{2} α^{2} + \cosh α - 1) \\ + L_{4} f (t_{i}) (\sinh α - \frac{1}{6} α (α^{2} + 6)) + 𝒪 (α^{6}), \end{align}

where f_i,

f_{i}^{'}

, and

f_{i}^{'^{'}}

stand for

f (t_{i})

f^{'} (t_{i})

and

f^{'^{'}} (t_{i})

, respectively. Similarly, we obtain the following results:

\begin{align} 𝒜_{i + 1} & = \int_{t_{i + 1}}^{t_{i + 2}} f (x) d x \\ = α f_{i} + (\cosh (2 α) - \cosh α) f_{i}^{'} + (- α - \sinh α + \sinh (2 α)) f_{i}^{'^{'}} \\ + L_{3} f (t_{i}) (- \frac{3}{2} α^{2} - \cosh α + \cosh (2 α)) + L_{4} f (t_{i}) (- \frac{7}{6} α^{3} - α - \sinh α + \sinh (2 α)) + 𝒪 (α^{6}), \\ 𝒜_{i + 2} & = \int_{t_{i + 2}}^{t_{i + 3}} f (x) d x \\ = α f_{i} + (\cosh (3 α) - \cosh (2 α)) f_{i}^{'} + (- α + \sinh (3 α) - 2 \sinh α \cosh α) f_{i}^{'^{'}} \\ + L_{3} f (t_{i}) (- \frac{5}{2} α^{2} - \cosh (2 α) + \cosh (3 α)) + L_{4} f (t_{i}) (- \frac{19}{6} α^{3} - α + \sinh (3 α) - 2 \sinh α \cosh α) \\ + 𝒪 (α^{6}), \\ 𝒜_{i - 1} & = \int_{t_{i - 1}}^{t_{i}} f (x) d x \\ = α f_{i} + (1 - \cosh α) f_{i}^{'} + (\sinh α - α) f_{i}^{'^{'}} \\ + L_{3} f (t_{i}) (\frac{1}{2} α^{2} - \cosh α + 1) + L_{4} f (t_{i}) (\sinh α - \frac{1}{6} α (α^{2} + 6)) + 𝒪 (α^{6}), \\ 𝒜_{i - 2} & = \int_{t_{i - 2}}^{t_{i - 1}} f (x) d x \\ = α f_{i} + (\cosh α - \cosh (2 α)) f_{i}^{'} + (- α - \sinh α + \sinh (2 α)) f_{i}^{'^{'}} \\ + L_{3} f (t_{i}) (\frac{3}{2} α^{2} + \cosh α - \cosh (2 α)) + L_{4} f (t_{i}) (- \frac{7}{6} α^{3} - α - \sinh α + \sinh (2 α)) + 𝒪 (α^{6}) . \end{align}

This linear system can be written in the form

(\begin{matrix} f_{i} + 𝒪 (α^{6}) \\ f_{i}^{'} + 𝒪 (α^{6}) \\ f_{i}^{'^{'}} + 𝒪 (α^{6}) \\ L_{3} f (t_{i}) + 𝒪 (α^{6}) \\ L_{4} f (t_{i}) + 𝒪 (α^{6}) \end{matrix}) = M (\begin{matrix} 𝒜_{i - 2} \\ 𝒜_{i - 1} \\ 𝒜_{i} \\ 𝒜_{i + 1} \\ 𝒜_{i + 2} \end{matrix}),

where

M

is the 5 × 5 matrix having the following columns:

\begin{align} M [1, .] & = (\begin{array}{c} \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 2 \cosh α + 3 α \coth α - 1) \\ - \frac{1}{16 α^{2}} {csch}^{4} (\frac{α}{2}) (α^{2} - 2 \cosh α + 2) \\ \frac{1}{16 α^{3}} {csch}^{4} (\frac{α}{2}) (α^{3} \coth α - 2 \cosh α + 2) \\ - \frac{1}{4 α^{2}} {csch}^{2} (\frac{α}{2}) \\ \frac{1}{4 α^{3}} {csch}^{2} (\frac{α}{2}) \end{array}), \\ M [2, .] & = (\begin{array}{c} \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (csch α \sinh (3 α) - 3 (\cosh α + 3 α \coth α + α csch α - 4)) \\ \frac{1}{16 α^{2}} {csch}^{4} (\frac{α}{2}) (3 α^{2} + 2 \cosh α - 2 \cosh (2 α)) \\ - \frac{1}{16 α^{3}} {csch}^{4} (\frac{α}{2}) (α^{3} (3 \coth α + csch α) - csch α \sinh (3 α) + 3) \\ \frac{1}{4 α^{2}} (3 {csch}^{2} (\frac{α}{2}) + 4) \\ - \frac{1}{α^{3}} \coth^{2} (\frac{α}{2}) \end{array}), \\ M [3, .] & = (\begin{array}{c} \frac{1}{48 α} (9 (α \coth (\frac{α}{2}) - 2) {csch}^{4} (\frac{α}{2}) - 6 {csch}^{2} (\frac{α}{2}) + 40) \\ \frac{1}{16 α^{2}} {csch}^{4} (\frac{α}{2}) (- 3 α^{2} - 2 \cosh α + 2 \cosh (2 α)) \\ \frac{1}{16 α^{3}} \coth (\frac{α}{2}) (3 α^{3} {csch}^{4} (\frac{α}{2}) - 32 \coth α - 16 csch α) \\ - \frac{1}{4 α^{2}} (3 {csch}^{2} (\frac{α}{2}) + 4) \\ \frac{1}{2 α^{3}} (3 {csch}^{2} (\frac{α}{2}) + 4) \end{array}), \\ M [4, .] & = (\begin{array}{c} - \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 3 \cosh α + \cosh (2 α) + 3 α (\cosh α + 3) csch α - 10) \\ \frac{1}{16 α^{2}} {csch}^{4} (\frac{α}{2}) (α^{2} - 2 \cosh α + 2) \\ - \frac{1}{16 α^{3}} {csch}^{4} (\frac{α}{2}) (3 + (α^{3} (\cosh (α) + 3) - \sinh (3 α)) csch α) \\ \frac{1}{4 α^{2}} {csch}^{2} (\frac{α}{2}) \\ - \frac{1}{α^{3}} \coth^{2} (\frac{α}{2}) \end{array}), \\ M [5, .] & = (\begin{array}{c} \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (\cosh α + 3 α csch α - 4) \\ 0 \\ \frac{1}{16 α^{3}} {csch}^{4} (\frac{α}{2}) (α^{3} csch α - 2 \cosh α + 2) \\ 0 \\ \frac{1}{4 α^{3}} {csch}^{2} (\frac{α}{2}) \end{array}) . \end{align}

Therefore, by (4) and (5), the claim follows.

Remark 1.Formulae (4) and (5) are not appropriate for small values of $α$ . In order to remedy that inconvenience, the following Taylor expansions are used:

\begin{align} {\tilde{f}}_{i} & = (\frac{107 α^{5}}{831600} - \frac{α^{3}}{840} + \frac{α}{105} - \frac{1}{20 α} + 𝒪 (α^{7})) 𝒜_{i - 2} + (- \frac{449 α^{5}}{1663200} + \frac{13 α^{3}}{5040} - \frac{3 α}{140} + \frac{9}{20 α} + 𝒪 (α^{7})) 𝒜_{i - 1} \\ + (\frac{α^{5}}{26400} - \frac{α^{3}}{1680} + \frac{α}{140} + \frac{47}{60 α} + 𝒪 (α^{7})) 𝒜_{i} - (- \frac{73 α^{5}}{332640} + \frac{α^{3}}{560} - \frac{α}{84} + \frac{13}{60 α} + 𝒪 (α^{7})) 𝒜_{i + 1} \\ + (- \frac{193 α^{5}}{1663200} + \frac{α^{3}}{1008} - \frac{α}{140} + \frac{1}{30 α} + 𝒪 (α^{7})) 𝒜_{i + 2} . \end{align}

Expressions similar to (4) would be used for approximating the boundary values f₀, f₁, f_n − 2, f_n − 1 and f_n, but they would involve unavailable values $𝒜_{i}$ . For this reason, the values $𝒜_{0}$ , $𝒜_{1}$ , $𝒜_{2}$ , $𝒜_{3}$ and $𝒜_{4}$ (resp. $𝒜_{n - 5}$ , $𝒜_{n - 4}$ , $𝒜_{n - 3}$ , $𝒜_{n - 2}$ and $𝒜_{n - 1}$ ) will be used to approximate f₀ and f₁ (resp. f_n − 1 and f_n). Also a specific linear combination of $𝒜_{n - 5}$ , $𝒜_{n - 4}$ , $𝒜_{n - 3}$ , $𝒜_{n - 2}$ and $𝒜_{n - 1}$ will allow to closely approximate f_n − 2.

Lemma 2.For a given regular function f, let

\begin{align} {\tilde{f}}_{ℓ} & : = η_{0}^{ℓ} 𝒜_{0} + η_{1}^{ℓ} 𝒜_{1} + η_{2}^{ℓ} 𝒜_{2} + η_{3}^{ℓ} 𝒜_{3} + η_{4}^{ℓ} 𝒜_{4}, ℓ = 0, 1, \\ {\tilde{f}}_{n - ℓ} & : = η_{0}^{ℓ} 𝒜_{n - 1} + η_{1}^{ℓ} 𝒜_{n - 2} + η_{2}^{ℓ} 𝒜_{n - 3} + η_{3}^{ℓ} 𝒜_{n - 4} + η_{4}^{ℓ} 𝒜_{n - 5}, ℓ = 0, 1, \end{align}

()

and

{\tilde{f}}_{n - 2} : = η_{0}^{2} 𝒜_{n - 1} + η_{1}^{2} 𝒜_{n - 2} + η_{2}^{2} 𝒜_{n - 3} + η_{3}^{2} 𝒜_{n - 4} + η_{4}^{4} 𝒜_{n - 5},

()

with

\begin{align} η_{0}^{0} & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 26 \cosh α + 3 α \cosh (3 α) csch α + 23), \\ η_{1}^{0} & : = - \frac{1}{96 α} {csch}^{4} (\frac{α}{2}) csch α (24 \sinh α + 15 \sinh (2 α) - 26 \sinh (3 α) + 6 α (\cosh (2 α) + 3 \cosh (3 α))), \\ η_{2}^{0} & : = \frac{1}{96 α} {csch}^{5} (\frac{α}{2}) (- 13 \sinh (\frac{α}{2}) + 44 \sinh (\frac{3 α}{2}) - 31 \sinh (\frac{5 α}{2}) + 18 α \cosh (\frac{5 α}{2})), \\ η_{3}^{0} & : = \frac{1}{96 α} {csch}^{4} (\frac{α}{2}) csch α (39 \sinh α - 15 \sinh (2 α) - 11 \sinh (3 α) + 6 α (3 \cosh (2 α) + \cosh (3 α))), \\ η_{4}^{0} & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 11 \cosh α + 3 α \cosh (2 α) csch α + 8), \\ η_{0}^{1} & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 11 \cosh α + 3 α \cosh (2 α) csch α + 8), \\ η_{1}^{1} & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 9 \cosh α - 3 α \coth α + \cosh (2 α) (11 - 9 α csch α) + 10), \\ η_{2}^{1} & : = \frac{1}{96 α} {csch}^{5} (\frac{α}{2}) (- 11 \sinh (\frac{α}{2}) + 4 \sinh (\frac{3 α}{2}) + 7 \sinh (\frac{5 α}{2}) - 18 α \cosh (\frac{3 α}{2})), \\ η_{3}^{1} & : = \frac{1}{96 α} {csch}^{4} (\frac{α}{2}) csch α (- 9 \sinh (2 α) - 2 \sinh (3 α) + 6 α (3 \cosh α + \cosh (2 α))), \\ η_{4}^{1} & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 2 \cosh α + 3 α \coth α - 1), \\ η_{0}^{2} & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (3 α \coth α - 2 \cosh α - 1), \\ η_{1}^{2} & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (12 + csch α \sinh (3 α) - 3 \cosh α - 9 α \coth α - 3 α csch α), \\ η_{2}^{2} & : = \frac{1}{48 α} (40 - 6 {csch}^{2} (\frac{α}{2}) + 9 (- 2 + α \coth (\frac{α}{2})) {csch}^{4} (\frac{α}{2})), \\ η_{3}^{2} & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 10 - 3 \cosh α + \cosh (2 α) + 3 α (3 + \cosh α) csch α), \\ η_{4}^{2} & : = \frac{1}{48 α} {csch}^{4} (\frac{α}{2}) (- 4 + \cosh α + 3 α csch α) . \end{align}

()

Then, ${\tilde{f}}_{k} - f_{k} = 𝒪 (α^{6})$ , k = 0,1,n − 2,n − 1,n.

Proof.The proof is similar to that of Lemma 1.

Remark 2.In order to get useful expressions for the knots near the boundary, we apply Taylor expansion to obtain

\begin{align} {\tilde{f}}_{0} & = (\frac{17 α^{5}}{166320} - \frac{α^{3}}{504} + \frac{α}{7} + \frac{137}{60 α} + 𝒪 (α^{7})) 𝒜_{0} + (- \frac{163}{60 α} - \frac{17 α}{42} + \frac{α^{3}}{252} - \frac{α^{5}}{6930} + 𝒪 (α^{7})) 𝒜_{1} \\ + (- \frac{α^{5}}{5544} + \frac{5 α}{14} + \frac{137}{60 α} + 𝒪 (α^{7})) 𝒜_{2} + (- \frac{21}{20 α} - \frac{α}{14} - \frac{α^{3}}{252} + \frac{4 α^{5}}{10395} + 𝒪 (α^{7})) 𝒜_{3} \\ + (- \frac{α^{5}}{6160} + \frac{α^{3}}{504} - \frac{α}{42} + \frac{1}{5 α} + 𝒪 (α^{7})) 𝒜_{4}, \\ {\tilde{f}}_{1} & = (\frac{1}{5 α} - \frac{α}{42} + \frac{α^{3}}{504} - \frac{α^{5}}{6160} + 𝒪 (α^{7})) 𝒜_{0} + (\frac{149 α^{5}}{415800} - \frac{α^{3}}{210} + \frac{13 α}{210} + \frac{77}{60 α} + 𝒪 (α^{7})) 𝒜_{1} \\ + (- \frac{43}{60 α} - \frac{3 α}{70} + \frac{α^{3}}{420} - \frac{α^{5}}{9900} + 𝒪 (α^{7})) 𝒜_{2} + (\frac{17}{60 α} - \frac{α}{210} + \frac{α^{3}}{630} - \frac{31 α^{5}}{138600} + 𝒪 (α^{7})) 𝒜_{3} \\ + (- \frac{1}{20 α} + \frac{α}{105} - \frac{α^{3}}{840} + \frac{107 α^{5}}{831600} + 𝒪 (α^{7})) 𝒜_{4}, \\ {\tilde{f}}_{n - 2} & = (- \frac{1}{20 α} + \frac{α}{105} - \frac{α^{3}}{840} + \frac{107 α^{5}}{831600} + 𝒪 (α^{7})) 𝒜_{n - 1} + (\frac{9}{20 α} - \frac{3 α}{140} + \frac{13 α^{3}}{5040} - \frac{449 α^{5}}{1663200} + 𝒪 (α^{7})) 𝒜_{n - 2} \\ + (\frac{47}{60 α} + \frac{α}{140} - \frac{α^{3}}{1680} + \frac{α^{5}}{26400} + 𝒪 (α^{7})) 𝒜_{n - 3} + (\frac{13}{60 α} - \frac{α}{84} + \frac{α^{3}}{560} - \frac{73 α^{5}}{332640} + 𝒪 (α^{7})) 𝒜_{n - 4} \\ + (\frac{1}{30 α} - \frac{α}{140} + \frac{α^{3}}{1008} - \frac{193 α^{5}}{1663200} + 𝒪 (α^{7})) 𝒜_{n - 5}, \\ {\tilde{f}}_{n - 1} & = (\frac{1}{5 α} - \frac{α}{42} + \frac{α^{3}}{504} - \frac{α^{5}}{6160} + 𝒪 (α^{7})) 𝒜_{n - 1} + (\frac{149 α^{5}}{415800} - \frac{α^{3}}{210} + \frac{13 α}{210} + \frac{77}{60 α} + 𝒪 (α^{7})) 𝒜_{n - 2} \\ + (- \frac{43}{60 α} - \frac{3 α}{70} + \frac{α^{3}}{420} - \frac{α^{5}}{9900} + 𝒪 (α^{7})) 𝒜_{n - 3} + (\frac{17}{60 α} - \frac{α}{210} + \frac{α^{3}}{630} - \frac{31 α^{5}}{138600} + 𝒪 (α^{7})) 𝒜_{n - 4} \\ + (- \frac{1}{20 α} + \frac{α}{105} - \frac{α^{3}}{840} + \frac{107 α^{5}}{831600} + 𝒪 (α^{7})) 𝒜_{n - 5}, \\ {\tilde{f}}_{n} & = (\frac{17 α^{5}}{166320} - \frac{α^{3}}{504} + \frac{α}{7} + \frac{137}{60 α} + 𝒪 (α^{7})) 𝒜_{n - 1} + (- \frac{163}{60 α} - \frac{17 α}{42} + \frac{α^{3}}{252} - \frac{α^{5}}{6930} + 𝒪 (α^{7})) 𝒜_{n - 2} \\ + (- \frac{α^{5}}{5544} + \frac{5 α}{14} + \frac{137}{60 α} + 𝒪 (α^{7})) 𝒜_{n - 3} + (- \frac{21}{20 α} - \frac{α}{14} - \frac{α^{3}}{252} + \frac{4 α^{5}}{10395} + 𝒪 (α^{7})) 𝒜_{n - 4} \\ + (- \frac{α^{5}}{6160} + \frac{α^{3}}{504} - \frac{α}{42} + \frac{1}{5 α} + 𝒪 (α^{7})) 𝒜_{n - 5} . \end{align}

4 Integral quasi-interpolant

The starting point is the discrete AH quasi-interpolant¹⁹ in

Ω_{α, I}

defined as

Q f : = \sum_{i = 1}^{n + 2} μ_{i} (f) N_{i},

()

where the coefficients

μ_{i} (f)

are defined as follows:

\begin{align} μ_{1} (f) & : = f (t_{0}), μ_{2} (f) : = f (t_{1}) + \frac{1}{2 \cosh α + 2} (f (t_{0}) - f (t_{2})), \\ μ_{i} (f) & : = f (t_{i - 2}) + \frac{1}{2 \cosh α + 2} (f (t_{i - 1}) - f (t_{i - 3})), 3 \leq i \leq n, \\ μ_{n + 1} (f) & : = f (t_{n - 1}) + \frac{1}{2 \cosh α + 2} (f (t_{n}) - f (t_{n - 2})), μ_{n + 2} (f) : = f (t_{n}) . \end{align}

()

It is well-known¹⁹ that for any function f in

L_{1}^{3} (I) : = {f : f^{'^{'}} absolutely continuous on I and f^{'^{'}} \in L_{1} (I)}

there exists a constant C independent of

α

such as

{‖ Q f - f ‖}_{\infty, I} \leq C α^{3} .

()

We can replace the point values f_i used in the spline quasi-interpolant Qf defined by (9) and (10) by their approximations

{\tilde{f}}_{i}

given in Lemmas 1 and 2 to obtain a quasi-interpolant

\tilde{Q} f

that uses the values

𝒜_{i}

. More precisely, it is defined as

\tilde{Q} f = \sum_{i = 1}^{n + 2} {\tilde{μ}}_{i} (f) N_{i},

()

where the coefficients

{\tilde{μ}}_{i} (f)

are given as follows:

\begin{align} {\tilde{μ}}_{1} (f) & : = {\tilde{f}}_{0}, {\tilde{μ}}_{2} (f) : = {\tilde{f}}_{1} + \frac{1}{2 \cosh α + 2} ({\tilde{f}}_{0} - {\tilde{f}}_{2}), \\ {\tilde{μ}}_{i} (f) & : = {\tilde{f}}_{i - 2} + \frac{1}{2 \cosh α + 2} ({\tilde{f}}_{i - 1} - {\tilde{f}}_{i - 3}), 3 \leq i \leq n, \\ {\tilde{μ}}_{n + 1} (f) & : = {\tilde{f}}_{n - 1} + \frac{1}{2 \cosh α + 2} ({\tilde{f}}_{n} - {\tilde{f}}_{n - 2}), {\tilde{μ}}_{n + 2} (f) : = {\tilde{f}}_{n} . \end{align}

()

After some computations, we can write

\tilde{Q} f = \sum_{i = 1}^{n + 2} υ_{i} (f) N_{i},

where the first coefficients are defined as

\begin{align} υ_{1} (f) & : = \sum_{j = 0}^{4} 𝒜_{j} η_{j}^{0}, \\ υ_{2} (f) & : = \frac{1}{2 \cosh α + 2} υ_{1} (f) + \sum_{j = 0}^{4} (𝒜_{j} η_{j}^{1} - \frac{1}{2 \cosh α + 2} 𝒜_{4 - j} δ_{j - 2}), \\ υ_{3} (f) & : = - \frac{1}{2 \cosh α + 2} υ_{1} (f) + \sum_{j = 0}^{4} (𝒜_{j} η_{j}^{1} + \frac{1}{2 \cosh α + 2} 𝒜_{4 - j} δ_{j - 2}), \\ υ_{4} (f) & : = - \frac{1}{2 \cosh α + 2} \sum_{j = 0}^{4} 𝒜_{j} η_{j}^{1} + \sum_{j = - 2}^{2} (𝒜_{3 - j} + \frac{1}{2 \cosh α + 2} (𝒜_{4 - j} - 𝒜_{2 - j})) δ_{j}, \end{align}

the last ones as

\begin{align} υ_{n + 2} (f) & : = \sum_{j = 0}^{4} 𝒜_{n - 1 - j} η_{j}^{0}, \\ υ_{n + 1} (f) & : = \frac{1}{2 \cosh α + 2} υ_{n + 2} (f) + \sum_{j = 0}^{4} 𝒜_{n - 1 - j} (η_{j}^{1} - \frac{1}{2 \cosh α + 2} η_{j}^{2}), \\ υ_{n} (f) & : = \sum_{j = 0}^{4} (η_{j}^{2} - \frac{1}{2 \cosh α + 2} (η_{j}^{1} - δ_{j - 2})) 𝒜_{n - 1 - j}, \\ υ_{n - 1} (f) & : = \frac{1}{2 \cosh α + 2} \sum_{j = 0}^{4} 𝒜_{n - 1 - j} η_{j}^{2} + \sum_{j = - 2}^{2} (𝒜_{n - 3 - j} - \frac{1}{2 \cosh α + 2} 𝒜_{n - 4 - j}) δ_{j}, \end{align}

and finally,

υ_{i} (f) : = \sum_{j = - 2}^{2} (𝒜_{i - j - 2} - \frac{1}{2 \cosh α + 2} (𝒜_{i - j - 1} - 𝒜_{i - j - 3})) δ_{j}, 5 \leq i \leq n - 2 .

5 Quasi-interpolation error

The integral quasi-interpolant $\tilde{Q} f$ is a good approximation to an enough regular function f.

Theorem 1.For any function f in $L_{1}^{3} (I)$ , there exists a constant C independent of $α$ such that

{‖ \tilde{Q} f - f ‖}_{\infty, I} \leq C α^{3} .

()

Proof.We have

{‖ \tilde{Q} f - f ‖}_{\infty, I} \leq {‖ \tilde{𝒬} f - 𝒬 f ‖}_{\infty, I} + {‖ 𝒬 f - f ‖}_{\infty, I} .

According to (11), it holds

{‖ 𝒬 f - f ‖}_{\infty,} = 𝒪 (α^{3}) .

Then, to get (14), it suffices to prove that

{‖ \tilde{Q} f - Q f ‖}_{\infty, I} = 𝒪 (α^{6})

. For this purpose, from (9) and (12), and taking into account that

{N_{i}, 1 \leq i \leq n + 2}

is a partition of unity, it holds

{‖ \tilde{Q} f - Q f ‖}_{\infty, I} \leq \max_{1 \leq i \leq n + 2} {‖ {\tilde{μ}}_{i} - μ_{i} ‖}_{\infty} .

For 3 ≤ i ≤ n, (10) and (13) give

μ_{i} (f) = f_{i - 2} + \frac{1}{2 \cosh α + 2} (f_{i - 1} - f_{i - 3})

and

{\tilde{μ}}_{i} (f) = {\tilde{f}}_{i - 2} + \frac{1}{2 \cosh α + 2} ({\tilde{f}}_{i - 1} - {\tilde{f}}_{i - 3}),

respectively. Therefore, by Lemmas 1 and 2, we get

\begin{align} {\tilde{μ}}_{i} (f) - μ_{i} (f) & = {\tilde{f}}_{i - 2} - f_{i - 2} + \frac{1}{2 \cosh α + 2} (({\tilde{f}}_{i - 1} - f_{i - 1}) - ({\tilde{f}}_{i - 3} - f_{i - 3})) \\ = 𝒪 (α^{6}) + \frac{1}{2 \cosh α + 2} 𝒪 (α^{6}) \\ = 𝒪 (α^{6}) . \end{align}

For the remaining values of i, it also holds

{‖ {\tilde{μ}}_{i} - μ_{i} ‖}_{\infty} = 𝒪 (h^{6})

. Then,

{‖ \tilde{Q} f - Q f ‖}_{\infty, I} = 𝒪 (α^{6})

The next result provides the order of the integral errors.

Corollary 1.For any function $f \in L_{1}^{3} (I)$ , and $i \in {0, \dots, n - 1}$ there exists a positive constant C₁ such that

|\int_{t_{i}}^{t_{i + 1}} {\tilde{Q}}_{3} f (x) d x - \int_{t_{i}}^{t_{i + 1}} f (x) d x| \leq C_{1} α^{4} .

Proof.For x ∈ [t_i,t_i + 1], Theorem 1 provides the estimate

|\int_{t_{i}}^{t_{i + 1}} {\tilde{Q}}_{3} f (x) d x - \int_{t_{i}}^{t_{i + 1}} f (x) d x| \leq \int_{t_{i}}^{t_{i + 1}} | {\tilde{Q}}_{3} f (x) - f (x) d x | \leq C_{1} α^{3} \int_{t_{i}}^{t_{i + 1}} d x,

and the claim follows.

6 Numerical tests

In this section, we give numerical tests that illustrate the performance of the above quadratic integro spline quasi-interpolation scheme. Firstly, we compare the numerical method proposed here with the results obtained with different methods in other papers in the literature, although the test functions in those papers are extremely simple, namely $f (t) = e^{t}$ (see References 8, 15, 17), $g (t) = \sin t$ (see Reference 8), $h (t) = \cos π t$ (see Reference 15, 17), and $k (t) = \sin 4 π t$ (see Reference 15).

If the interval $I = [0, 1]$ is decomposed into n = 8 equal parts, then the method proposed here for approximating f gives an error equal to 1.33 × 10⁻¹⁵. It is much better than the error 1.08034 × 10⁻⁴ provided by the numerical scheme in Reference 8. In Table 1, we compare the results in Reference 8 for the test function g with the errors provided by the novel integro spline method. Lemmas 1 and 2 in Reference 8 show that ${\tilde{f}}_{i} = f_{i} + O (h^{4})$ , while the approximations ${\tilde{f}}_{i}$ constructed from the mean values $𝒜_{i}$ fulfill that ${\tilde{f}}_{i} = f_{i} + 𝒪 (h^{6})$ (see Lemmas 1 and 2). This fact, together with the better adaptation of the piecewise Algebraic Hyperbolic functions to treat the test function g, explains the better results shown in Table 1.

TABLE 1. The estimated maximum errors for the test function g

n	UAH Method	Method in Reference 8
8	5.90 × 10⁻⁶	3.47 × 10⁻⁵
16	3.85 × 10⁻⁷	2.38 × 10⁻⁶
32	2.45 × 10⁻⁸	1.55 × 10⁻⁷
64	1.55 × 10⁻⁹	9.86 × 10⁻⁹
128	9.77 × 10⁻¹¹	6.22 × 10⁻¹⁰
256	2.31 × 10⁻¹²	3.90 × 10⁻¹¹

Abbreviation: UAH, Uniform Algebraic Hyperbolic.

In Tables 2 and 3, we list the resulting errors for the approximation of the functions h and k, respectively, by using the integral spline quasi-interpolant given in (12) and those in References 17 and 15. Although the number of parts into which the interval is divided has been chosen to compare the results with those referred, also in these cases the novel numerical scheme improves the results in those papers. A higher value of n would be necessary to clearly observe the good results provided by the proposed approximation method in view of the frequency of the functions. We would like recall that the quasi-interpolation operator $\tilde{𝒬}$ provided by (12) does not need any additional end conditions and can be implemented without solving any system of linear equations.

TABLE 2. The estimated maximum errors for the test function h

n	UAH Method	Method in Reference 17	Method in Reference 15
10	1.83 × 10⁻⁴	6.21 × 10⁻⁴	2.50 × 10⁻⁴
20	1.07 × 10⁻⁵	4.05 × 10⁻⁵	3.11 × 10⁻⁵
40	6.59 × 10⁻⁷	2.55 × 10⁻⁶	3.86 × 10⁻⁶

Abbreviation: UAH, Uniform Algebraic Hyperbolic.

TABLE 3. The estimated maximum errors for the test function k

n	UAH Method	Method in Reference 17	Method in Wu¹⁵
10	6.44 × 10⁻²	3.69 × 10⁻¹	3.37 × 10⁻¹
20	3.14 × 10⁻³	7.51 × 10⁻²	2.53 × 10⁻²
40	1.67 × 10⁻⁴	5.16 × 10⁻³	9.59 × 10⁻⁴

Abbreviation: UAH, Uniform Algebraic Hyperbolic.

The uniform norm on

[0, 1]

of the approximation error has been estimated for a function y and its integral UAH quasi interpolant

{\tilde{Q}}_{n} y

(12) and (13) in the spline space associated with a partition into n equal parts as

E_{1} (y, {\tilde{Q}}_{n}) : = \max_{0 \leq r \leq 200} |y (\frac{r}{200}) - {\tilde{Q}}_{n} y (\frac{r}{200})| .

()

In Tables 4-6 the Numerical Convergence Order (NCO) for the four methods are shown. They are computed by the expression

|\log (\frac{E_{1} (y, {\tilde{Q}}_{n})}{E_{1} (y, {\tilde{Q}}_{2 n})}) / \log 2|

and confirm the theoretical results regarding the orders of convergence of the four methods for the four simple functions we have considered.

TABLE 4. NCO for the test function g using our approach and the approximation in Reference 8

n	UAH Method	Method in Reference 8
16	3.94	3.86
32	3.97	3.94
64	3.98	3.97
128	3.99	3.99

Abbreviations: NCO, Numerical Convergence Order; UAH, Uniform Algebraic Hyperbolic.

TABLE 5. NCO for the test function h using our approach, and the approximations in References 17 and 15

n	UAH Method	Method in Reference 17	Method in Reference 15
20	4.08953	3.9386	3.00694
40	4.02499	3.98935	3.01024

Abbreviations: NCO, Numerical Convergence Order; UAH, Uniform Algebraic Hyperbolic.

TABLE 6. NCO for the test function k using our approach, and the approximations in Reference 15

n	UAH Method	Method in Reference 17	Method in Reference 15
20	4.36	2.30	3.74
40	4.23	3.86	4.72

Abbreviations: NCO, Numerical Convergence Order; UAH, Uniform Algebraic Hyperbolic.

Next, we deal with the following three test functions also defined on

[0, 1]

F_{1} (t) = \cosh t \exp (\sinh t), F_{2} (t) = \frac{\exp (\frac{1}{1 + t^{2}})) \tanh (\frac{t}{10 π})}{1 + 16 t^{3}}, F_{3} (t) = - 2 e^{- t} t \cos (20 t) .

We divide the interval into n = 2^ℓ, 3 ≤ ℓ ≤ 7, equal parts and determine the values

𝒜_{i, s} : = \int_{t_{i}}^{t_{i + 1}} F_{s}

, 0 ≤ i ≤ n − 1 and s = 1,2,3, from which the integral quasi-interpolants

{\tilde{Q}}_{n} F_{s}

are determined to fulfill (2). Their errors are shown in Tables 7-9 along with their NCOs, computed from (15).

TABLE 7. Quasi-interpolation errors and NCOs for the test function F₁

n	UAH Method	NCO	Method in Reference 15	NCO	Method in Reference 24	NCO
8	5.21 × 10⁻⁵	—	3.25 × 10⁻⁴	—	3.83 × 10⁻⁴	—
16	2.85 × 10⁻⁶	4.20	2.69 × 10⁻⁵	3.59	3.16 × 10⁻⁵	3.59
32	1.64 × 10⁻⁷	4.11	1.95 × 10⁻⁶	3.78	2.29 × 10⁻⁶	3.79
64	9.79 × 10⁻⁹	4.06	1.32 × 10⁻⁷	3.88	1.54 × 10⁻⁷	3.90
128	5.97 × 10⁻¹⁰	4.03	8.59 × 10⁻⁹	3.94	1.01 × 10⁻⁸	3.93

Abbreviations: NCO, Numerical Convergence Order; UAH, Uniform Algebraic Hyperbolic.

TABLE 8. Quasi-interpolation errors and NCOs for the test function F₂

n	UAH Method	NCO	Method in Reference 15	NCO	Method in Reference 24	NCO
8	1.24 × 10⁻³	—	1.64 × 10⁻²	—	1.69 × 10⁻²	—
16	7.81 × 10⁻⁵	3.98	1.00 × 10⁻³	4.03	1.01 × 10⁻³	4.27
32	4.88 × 10⁻⁶	4	6.26 × 10⁻⁵	3.99	6.28 × 10⁻⁵	4
64	3.05 × 10⁻⁷	4	3.91 × 10⁻⁶	4	3.92 × 10⁻⁶	4
128	1.90 × 10⁻⁸	4	2.44 × 10⁻⁷	4	2.45 × 10⁻⁷	4

Abbreviations: NCO, Numerical Convergence Order; UAH, Uniform Algebraic Hyperbolic.

TABLE 9. Quasi-interpolation errors and NCOs for the test function F₃

n	UAH Method	NCO	Method in Reference 15	NCO	Method in Reference 24	NCO
16	5.26 × 10⁻²	—	1.61 × 10⁻¹	—	1.61 × 10⁻¹	—
32	2.99 × 10⁻³	4.13	2.08 × 10⁻²	2.95	2.07 × 10⁻²	2.95
64	2.37 × 10⁻⁴	3.65	1.10 × 10⁻³	4.24	1.10 × 10⁻³	4.23
128	2.44 × 10⁻⁵	3.27	5.35 × 10⁻⁵	4.36	5.33 × 10⁻⁵	4.36

Abbreviations: NCO, Numerical Convergence Order; UAH, Uniform Algebraic Hyperbolic.

Table 10 shows estimates to the integral errors between the exact values

𝒜_{i, s}

and the corresponding ones obtained by integrating the associated UAH integral quasi-interpolant, as well as the NCOs. They are computed as follows:

E_{2} (y, {\tilde{Q}}_{n}) : = \max_{0 \leq i \leq n - 1} |\int_{t_{i}}^{t_{i + 1}} {\tilde{Q}}_{n} y (t) d t - \int_{t_{i}}^{t_{i + 1}} y (t) d t| .

Moreover, these results suggest that, having defined a UAH quasi-interpolation operator on a uniform partition, a phenomenon of superconvergence (of order four) arises.

TABLE 10. NCOs for errors

|\int_{t_{i}}^{t_{i + 1}} {\tilde{Q}}_{n} F_{s} (t) d t - \int_{t_{i}}^{t_{i + 1}} F_{s} (t) d t|

n	F₁	NCO F₁	F₂	NCO F₂	F₃	NCO F₃
8	3.97 × 10⁻⁶	−	1.005 × 10⁻⁴	—	4.32 × 10⁻²	−
16	1.13 × 10⁻⁷	5.13	3.14 × 10⁻⁶	5	2.22 × 10⁻³	4.28
32	3.30 × 10⁻⁹	5.09	9.83 × 10⁻⁸	4.99	5.34 × 10⁻⁵	5.37
64	9.89 × 10⁻¹¹	5.06	3.07 × 10⁻⁹	5	1.30 × 10⁻⁶	5.36
128	4.28 × 10⁻¹²	4.53	1.13 × 10⁻¹⁰	4.77	3.67 × 10⁻⁸	5.14

Abbreviation: NCO, Numerical Convergence Order.

7 Conclusion

Approximation from values and integral values is an important subject because of its wide applications in many different fields. In this paper, we have constructed a novel integro quasi-interpolation operator that reproduces polynomials and hyperbolic functions. The numerical results illustrate the good performance of the novel approximation scheme.

Acknowledgements

The authors thank the referees for their helpful remarks and suggestions. The first author is a member of the research group FQM 191 Matemática Aplicada funded by the PAIDI program of the Junta de Andalucía. The second author would like to thank the University of Granada for the financial support for the research stay during which this work was carried out.

Biographies

Domingo Barrera Rosillo received the M.S. degree in mathematics and the Ph.D. degree from the Universidad de Granada, Granada, Spain. He is currently an Associate Professor with the Universidad de Granada. Since 1997, he has been working on multivariate approximation, numerical analysis, and spline interpolation and quasi-interpolation. He is also interested in the application of numerical methods for the solution of several problems arising in Engineering. His current research interest also includes numerical solution of integral equations by spline-based methods.
Salah Eddargani received the M.S. degree in mathematics from the Hassan First University, Settat, Morocco. He is currently a PhD student at both the Hassan First University and the University of Granada, and holder of a scholarship from the CNRST through the Excellence Grants Programme. His research has benefited from grants from the Erasmus+ International Dimension programme. He is mainly interested in Powell-Sabin splines, spline quasi-interpolation, and non-polynomial spline interpolation.
Abdellah Lamnii received the M.S. degree in mathematics and the Ph.D. degree from the Mohammed First University, Oujda, Morocco. He is currently an Associate Professor with the Hassan First University. Since 2008, he has been working on spherical spline approximation, spline wavelet transform, and boundary value problems with spline quasi-interpolation. His current research interests include the subdivision methods for curves and surfaces.

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Uniform algebraic hyperbolic spline quasi-interpolant based on mean integral values

Summary

1 Introduction

2 The approximation problem and the spline space

3 Linking point and integral values

4 Integral quasi-interpolant

5 Quasi-interpolation error

6 Numerical tests

7 Conclusion

Acknowledgements

Biographies

References

Citing Literature

Figures

References

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Uniform algebraic hyperbolic spline quasi-interpolant based on mean integral values

Summary

1 Introduction

2 The approximation problem and the spline space

3 Linking point and integral values

4 Integral quasi-interpolant

5 Quasi-interpolation error

6 Numerical tests

7 Conclusion

Acknowledgements

Biographies

References

Citing Literature

Figures

References

Related

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