Interpolation splines minimizing a semi-norm

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

doi:10.1007/s10092-013-0080-x

Cited by 30 publications

(7 citation statements)

References 16 publications

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“…But here we use the method suggested by Sobolev for construction of optimal quadrature formulas in the space L (m) 2 which is based on the discrete analogue of the differential operator d 2m / dx 2m (see, for instance, [35]). This method allows to get explicit formulas for optimal coefficients and reduces the size of the system (11)- (16). Further we demonstrate this method in the case m = 2.…”

Section: Remark 32mentioning

confidence: 88%

“…In the system (11)-(16) the coefficients C β (z), β = 0, 1, ..., N, A(z), B(z) and λ α (z), α = 0, 1, ..., m − 1 are unknowns. The system (11)-(16) has a unique solution and this solution gives the minimum to 2 under the conditions (14)- (16) when N + 3 ≥ m. Here we omitted the proof of the existence and uniqueness of the solution of the system (11)- (16). The existence and uniqueness of the solution of this system can be proved similarly the existence and uniqueness of the solution of the discrete Wiener-Hopf type system for coefficients of optimal quadrature formulas in the space L (m) 2 (0, 1) (see [32,36]).…”

Section: Existence and Uniqueness Of The Optimal Interpolation Formulamentioning

confidence: 99%

“…In the recent work [22], authors studied the reproducing kernel functions in the standard Sobolev space W s 2 and their application to tractability of integration. Using the extremal functions in various Hilbert spaces the optimal integration formulas were studied in the works [15, 17, 27, 29-31, 34, 35] and the interpolation formulas and splines were obtained in [13,14,16,28,33].…”

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

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Optimal interpolation formulas with derivative in the space L(m)2(0,1)

Shadimetov

Hayotov

Nuraliev

2019

Filomat

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The paper studies the problem of construction of optimal interpolation formulas with derivative in the Sobolev space L (m) 2 (0, 1). Here the interpolation formula consists of the linear combination of values of the function at nodes and values of the first derivative of that function at the end points of the interval [0, 1]. For any function of the space L (m) 2 (0, 1) the error of the interpolation formulas is estimated by the norm of the error functional in the conjugate space L (m) * 2 (0, 1). For this, the norm of the error functional is calculated. Further, in order to find the minimum of the norm of the error functional, the Lagrange method is applied and the system of linear equations for coefficients of optimal interpolation formulas is obtained. It is shown that the order of convergence of the obtained optimal interpolation formulas in the space L (m) 2 (0, 1) is O(h m). In order to solve the obtained system it is suggested to use the Sobolev method which is based on the discrete analog of the differential operator d 2m / dx 2m. Using this method in the cases m = 2 and m = 3 the optimal interpolation formulas are constructed. It is proved that the order of convergence of the optimal interpolation formula in the case m = 2 for functions of the space C 4 (0, 1) is O(h 4) while for functions of the space L (2) 2 (0, 1) is O(h 2). Finally, some numerical results are presented.

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Section: Remark 32mentioning

confidence: 88%

Section: Existence and Uniqueness Of The Optimal Interpolation Formulamentioning

confidence: 99%

Section: Introduction and Statement Of The Problemmentioning

confidence: 99%

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Optimal interpolation formulas with derivative in the space L(m)2(0,1)

Shadimetov

Hayotov

Nuraliev

2019

Filomat

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“…where Remark . From Theorem 4.1, when m = 2, we get Theorem 7 of [17] and Theorem 3.1 of [19], and when m = 2, ω = 1 we get Theorem 3.1 of [18].…”

Section: Computation Of Coefficients Of Interpolation Spline (14)mentioning

confidence: 95%

“…(0, 1) and K 2 (P 2 ) Hilbert spaces were constructed in works [8,17,18,19,31,32] by using Sobolev's method. Furthermore, the connection between interpolation spline and optimal quadrature formula in the sense of Sard in L (m) 2 (0, 1) and K 2 (P 2 ) spaces were shown in [8] and [18].…”

Section: Introduction Statement Of the Problemmentioning

confidence: 99%

Construction of Interpolation Splines Minimizing the Semi-norm in the Space K2(Pm)

Hayotov¹

2018

J. Sib. Fed. Univ., Math. phys.

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In the present paper, using S.L. Sobolev's method, interpolation splines that minimize the expression ∫ 1 0 (φ (m) (x)+ω 2 φ (m−2) (x)) 2 dx in the space K2(Pm) are constructed. Explicit formulas for the coefficients of the interpolation splines are obtained. The obtained interpolation splines are exact for monomials 1, x, x 2 ,. .. , x m−3 and for trigonometric functions sin ωx and cos ωx.

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