Context. The integrals of highly oscillating functions of many variables are one of the central concepts of digital signal and image processing. The object of research is a digital processing of signals and images using new information operators. 

Objective. The work aims to construct a cubature formula for the approximate calculation of the triple integral of a rapidly oscillating function of a general form.

Method. Modern methods of digital signal processing are characterized by new approaches to obtaining, processing and analyzing information. There is a need to build mathematical models in which information can be given not only by the values of the function at points, but also as a set of traces of the function on the planes and  as a set of traces of the function on the lines. There are algorithms which are optimal by accuracy for calculating the integrals of highly oscillating functions of many variables (regular case), which involve different types of information in their construction. As a solution of a broader problem for the irregular case, the work presents the cubature formula for the approximate calculation of the triple integral of the highly oscillating function in a general case. The presented algorithm for approximate calculation of the integral is based on the application of operators that restore the function of three variables using a set of traces of functions on the mutually perpendicular planes. Operators use piece-wise splines as auxiliary functions. The cubature formula correlates with a formula of the Filon type. An error estimation of the approximation of the integral from the highly oscillating function by the cubature formula on the class of differential functions is obtained.

Results. The cubature formula of the approximate calculation of the triple integral from the highly oscillating function of a general form is researched.

Conclusions. The experiments confirm the obtained theoretical results on the error estimation of the approximation triple integral from the highly oscillating function in a general form by the cubature formula. The prospect of further research is to obtain an estimation of the approximation error on wider classes of functions and to prove that the proposed cubature formula is optimal by the order of accuracy. 

Sergienko I. V., Lytvyn O. M. Novi informacijni operatory v matematychnomu modeljuvanni. Kyiv, Naukova Dumka, 2018, 444 p.

Sergienko I. V., Lytvyn O. M. New Information Operators in Mathematical Modeling (A Review), Cybernetics and Systems

Analysis, 2018, Vol. 54, No. 1, pp. 21–30. DOI: 10.1007/s10559-018-0004-5.

Lytvyn O. M., Nechuiviter O. P. Methods in the Multivariate Digital Signal Processing with Using Spline-interlineation, IASTED International Conferences on Automation, Control and Information Technology (ASIT 2010) : proceedings, 2010, pp. 90–96.

Sergienko I. V., Zadiraka V. K., Lytvyn О. N., Melnikova S. S., Nechuiviter О. P. Optymal’ni algorytmy obchyslennja integraliv vid shvydkooscyljujuchyh funkcij ta i'h zastosuvannja : u 2 t. T. 1. Algorytmy. Kyiv, Naukova Dumka, 2011, 447 p.

Sergienko I. V. Zadiraka V. K., Lytvyn О. N., Melnikova S. S., Nechuiviter О. P. Optymal’ni algorytmy obchyslennja integraliv vid shvydkooscyljujuchyh funkcij ta i’h zastosuvannja : u 2 t. T. 2. Zastosuvannja. Kyiv, Naukova Dumka, 2011, 348 p.

Zadiraka V. K., Melnikova S. S., Luts L. V. Optimal integration of rapidly oscillating functions in the class W2,L,N with the use of different information operators, Cybernetics and Systems Analysis, 2013, Vol. 49, No. 2, pp. 229–238.

Sergienko I. V., Zadiraka V. K., Lytvyn O. M., Nechuiviter O. P. Optymal'ni algorytmy obchyslennja integraliv vid shvydkooscyljujuchyh funkcij iz zastosuvannjam novyh informacijnyh operatoriv. Kyiv, Naukova Dumka, 2017, 336 p.

Filon L. N. G. On a quadrature formula for trigonometric integrals, Proc. RoyalSoc. Edinburgh 49, 1928, pp. 38–47.

Flinn E. A. A modification of Filon’s method of numerical integration, JACM 7, 1960, pp. 181–184.

Iserles A. On the numerical quadrature of highly-oscillating integrals I: Fourier transforms, IMA J. Numer. Anal., 2004, No. 24, pp. 365–391.

Milovanovic G. V., Stanic M. P. Numerical Integration of Highly Oscillating Functions, Analytic Number Theory, Approximation Theory and Special Functions, 2014, pp. 613–649.

Olver S. Numerical Approximation of Highly Oscillatory Integrals. PhD thesis. Cambridge, University of Cambridge, 2008, 172 p.

Iserles A., Norsett S. From high oscillation to rapid approximation III: Multivariate expansions, Tech. Reports Numerical Analysis (NA2007/01). DAMPT, University of Cambridge, 2007, 37 p.

Lytvyn O. M., Nechuiviter O. P. 3D Fourier Coefficients on the Class of Differentiable Functions and Spline Interflatation, Journal of Automation and Information Science. – 2012, Vol. 44, No. 3, pp. 45–56. DOI: 10.1615/JAutomatInfScien.v44.i3.40

Lytvyn O. M., Nechuiviter O. P. Approximate Calculation of Triple Integrals of Rapidly Oscillating Functions with the Use of Lagrange Polynomial Interflatation, Cybernetics and Systems Analysis, 2014, Vol. 50, No. 3, pp. 410–418. DOI: 10.1007/s10559-014-9629-1

Iserles A. : On the numerical quadrature of highly oscillating integrals II: Irregular oscillators, IMA J. Numer. Anal., 2005. No. 25, pp. 25–44.

Xiang S. Efficient Filon-type methods for () () b igx a f x e dx ω∫ , Numerische Mathematik, 2007, Vol. 105, No. 4, pp. 633–658.

Liu Y. Fast Evaluation of Canonical Oscillatory Integrals, Appl. Math. Inf. Sci, 2012, Vol. 6, No. 2, pp. 245–251.

Gao J., Iserles A. Error analysis of the extended Filon-type method for highly oscillatory integrals, Tech. Reports Numerical Analysis (NA2016/03). DAMPT, University of Cambridge, 2016, 25 p.

Gao J., Iserles A. A generalization of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals, Tech. Reports Numerical Analysis (NA2016/04). DAMPT, University of Cambridge, 2016, 12 p.

Gao J., Iserles A. An Adaptive Filon Algorithm for Highly Oscillatory Integrals, Tech. Reports Numerical Analysis (NA2016/05). DAMPT, University of Cambridge, 2016, 16 p.

Khoromskij B., Veit A. Efficient computation of highly oscillatory integrals by using QTT tensor approximation, Computational Methods in Applied Mathematics, 2016, Vol. 16, No. 3, pp. 145–159. DOI: 10.1515/cmam-2015-0033

Gao J., Condon M., Iserles A. Spectral computation of highly oscillatory integral equations in laser theory, Tech. Reports Numerical Analysis (NA2018/04), DAMPT, University of Cambridge, 2018, 30 p.

Mezhuyev V. I. Lytvyn, O. M. Nechuiviter O. P., Pershyna Y. I., Lytvyn O. O., Keita K. V. Cubature formula for ap

proximate calculation of integrals of two-dimensional irregular highly oscillating functions, U.P.B. Sci. Bull., Series A., 2017, Vol. 80, No. 3, pp. 169–182.

Lytvyn O. M. Nechuiviter O. P., Pershyna I. I., Mezhuyev V. I. Input Information in the Approximate Calculation of TwoDimensional Integral from Highly Oscillating Functions (Irregular Case), Recent Developments in Data Science and Intelligent Analysis of Information : XVIII International Conference on Data Science and Intelligent Analysis of Information : proceedings. Kyiv, 2019, pp. 365–373. DOI: 10.1007/978-3-31997885-7_36

1. Сергієнко І. В. Нові інформаційні оператори в математичному моделюванні / І. В. Сергієнко, О. М. Литвин. – Київ : Наукова думка, 2018. – 444 с.

2. Sergienko I. V. New Information Operators in Mathematical Modeling (A Review) / I. V. Sergienko, O. M. Lytvyn // Cybernetics and Systems Analysis. – 2018. – Vol. 54, № 1. – P. 21–30. DOI: 10.1007/s10559-018-0004-5.

3. Lytvyn O. M. Methods in the Multivariate Digital Signal Processing with Using Spline-interlineation / O. M. Lytvyn, O. P. Nechuiviter // IASTED International Conferences on Automation, Control and Information Technology (ASIT 2010) : proceedings. – 2010. – P. 90–96.

4. Сергієнко І. В. Оптимальні алгоритми обчислення інтегралів від швидкоосцилюючих функцій та їх застосування : у 2 т. Т. 1. Алгоритми / [І. В. Сергієнко, В. К. Задірака, О. М. Литвин и др.]. – Київ : Наукова думка, 2011. – 447 с.

5. Оптимальні алгоритми обчислення інтегралів від швидкоосцилюючих функцій та їх застосування : у 2 т. Т. 1. Застосування / [І. В. Сергієнко, В. К. Задірака, О. М. Литвин и др.]. – Київ : Наукова думка, 2011. – 348 с.

6. Zadiraka V. K. Optimal integration of rapidly oscillating functions in the class W2,L,N with the use of different information operators / V. K. Zadiraka, S. S. Melnikova, L. V. Luts // Cybernetics and Systems Analysis. – 2013. – Vol. 49, № 2. – P. 229–238.

7. Оптимальні алгоритми обчислення інтегралів від швидкоосцилюючих функцій із застосуванням нових інформаційних операторів / [І. В. Сергієнко, В. К. Задірака, О. М. Литвин, О. П. Нечуйвітер]. – Київ : Наукова думка, 2017. – 336 с.

8. Filon L. N. G. On a quadrature formula for trigonometric integrals / L. N. G. Filon. – Proc. RoyalSoc. Edinburgh 49, 1928. – P. 38–47.

9. Flinn E. A. A modification of Filon’s method of numerical integration / E. A. Flinn // JACM 7. – 1960. – P. 181–184.

10. Iserles A. On the numerical quadrature of highly-oscillating integrals I: Fourier transforms / A. Iserles // IMA J. Numer. Anal. – 2004. – № 24. – P. 365–391.

11. Milovanovic G. V. Numerical Integration of Highly Oscillating Functions / G. V. Milovanovic, M. P. Stanic // Analytic Number Theory, Approximation Theory and Special Functions. – 2014. – P. 613–649.

12. Olver S. Numerical Approximation of Highly Oscillatory Integrals / S. Olver. PhD thesis. – Cambridge : University of Cambridge, 2008. – 172 p.

13. Iserles A. From high oscillation to rapid approximation III: Multivariate expansions / A. Iserles, S. Norsett // Tech. Reports Numerical Analysis (NA2007/01). – DAMPT: University of Cambridge, 2007. – 37 p.

14. Lytvyn O. M. 3D Fourier Coefficients on the Class of Differentiable Functions and Spline Interflatation / O. M. Lyt
vyn, O. P. Nechuiviter // Journal of Automation and Information Science. – 2012. – Vol. 44, № 3. – P. 45–56. DOI: 10.1615/JAutomatInfScien.v44.i3.40

15. Lytvyn O. M. Approximate Calculation of Triple Integrals of Rapidly Oscillating Functions with the Use of Lagrange Polynomial Interflatation / O. M. Lytvyn, O. P. Nechuiviter // Cybernetics and Systems Analysis. – 2014. – Vol. 50, № 3. – P. 410–418. DOI: 10.1007/s10559-014-9629-1

16. Iserles A.: On the numerical quadrature of highly oscillating integrals II: Irregular oscillators / A. Iserles // IMA J. Numer. Anal. – 2005. – № 25. – P. 25–44.

17. Xiang S. Efficient Filon-type methods for () () b igx a f x e dx ω∫ / S. Xiang // Numerische Mathematik. – 2007. – Vol. 105, № 4. – P. 633–658.

18. Liu Y. Fast Evaluation of Canonical Oscillatory Integrals / Y. Liu // Appl. Math. Inf. Sci. – 2012. – Vol. 6, № 2. – P. 245–251.

19. Gao J. Error analysis of the extended Filon-type method for highly oscillatory integrals / J. Gao, A. Iserles // Tech. Reports Numerical Analysis (NA2016/03). – DAMPT: University of Cambridge, 2016. – 25 p.

20. Gao J. A generalization of Filon-Clenshaw-Curtis quadrature for highly oscillatory integrals / J. Gao, A. Iserles // Tech. Reports Numerical Analysis (NA2016/04). – DAMPT: University of Cambridge, 2016. – 12 p.

21. Gao J. An Adaptive Filon Algorithm for Highly Oscillatory Integrals / J. Gao, A. Iserles // Tech. Reports Numerical Analysis (NA2016/05). – DAMPT : University of Cambridge, 2016. – 16 p.

22. Khoromskij B. Efficient computation of highly oscillatory integrals by using QTT tensor approximation / B. Khoromskij, A. Veit // Computational Methods in Applied Mathematics. – 2016. – Vol. 16, № 3. – P. 145–159. DOI: 10.1515/cmam-2015-0033

23. Gao J. Spectral computation of highly oscillatory integral equations in laser theory / J. Gao, M. Condon, A. Iserles // Tech. Reports Numerical Analysis (NA2018/04). – DAMPT : University of Cambridge, 2018. – 30 p.

24. Cubature formula for approximate calculation of integrals of two-dimensional irregular highly oscillating functions / [V. I. Mezhuyev, O. M. Lytvyn, O. P. Nechuiviter et al.] // U.P.B. Sci. Bull., Series A. – 2017. – Vol. 80, № 3. – P. 169–182.

25. Input Information in the Approximate Calculation of TwoDimensional Integral from Highly Oscillating Functions (Irregular Case) / [O. M. Lytvyn, O. P. Nechuiviter, I. I. Pershyna, V. I. Mezhuyev] // Recent Developments in Data Science and Intelligent Analysis of Information : XVIII International Conference on Data Science and Intelligent Analysis of Information : proceedings. – Kyiv, 2019. – P. 365–373. DOI: 10.1007/978-3-319-97885-7_36