Approximation properties of generalized Fup-functions
Keywords:
function with a compact support, approximation of periodic functions, up-function, Kolmogorov width, best approximation, generalized Fup-function
Abstract
Generalized Fup-functions are considered. Almost-trigonometric basis theorem is proved. Spaces of linear combinations of shifts of the generalized Fup-functions are constructed and an upper estimate of the best approximation of classes of periodic differentiable functions by these spaces in the norm of $L_2[-\pi,\pi]$ is obtained.Downloads
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References
Kolmogoroff A.N. ¨Uber die beste Ann¨aherung von Funktionen einer gegebener Funktionenklasse. // Ann. of Math., 1936. – 37. – P. 107–110.
Rvachev V.A. Compactly supported solutions of functional–differential equations and their applications. // Russian Math. Surveys, 1990. – 45. – P. 87–120.
Rvachev V.L., Rvachev V.A. Non-classical methods of approximation theory in boundary value problem (in Russian). – K.: Naukova Dumka, 1979. – 196 p.
Makarichev V.A. Approximation of periodic functions by mups(x). // Math. Notes, 2013. – 93. – P. 858–880.
Rvachev V.L., Rvachev V.A. A certain finite function (in Ukrainian). // Proc. Ukr. SSR Acad. Sci., Ser. A., 1971. – 8. – P. 705–707.
Rvachev V.A., Starets G.A. Some atomic functions and their apllications (in Ukrainian). // Proc. Ukr. SSR Acad. Sci., Ser. A., 1983. – 11. – P. 22–24.
Makarichev V.A. Asymptotics of the basis functions of generalized Taylor series for the class H;2. // Math. Notes, 2011. – 89. – P. 689–705.
Dyn N., Ron A. Multiresolution analysis by infinitely differentiable compactly supported functions. // Appl. Comput. Harmon. Anal., 1995. – 2. – P. 15–20.
Cooklev T., Berbecel G.I., Venetsanopoulos A.N. Wavelets and differentialdilatation equations. // IEEE Transactions on signal processing, 2000. – 48. – P. 670–681.
Charina M., Stockler J. Tight wavelet frames for irregular multiresolution analysis. // Appl. Comput. Harmon. Anal., 2008. – 25. – P. 98–113.
Makarichev V.A. Applications of the function mups(x). // Progress in analysis. Proceedings of the 8th congress of the International Society for Analysis, its Applications, and Computation (ISAAC), 2012. – 2. – P. 297–304.
Makarichev V.A. The function mups(x) and its applications to the of generalized Taylor series, approximation theory and wavelet theory. // In book: Contemporary problems of mathematics, mechanics and computing sciences. – Kharkiv: Apostrophe, 2011. – P. 279–287.
Brysina I.V., Makarichev V.A. Atomic wavelets. // Radioelectronic and Computer Systems, 2012. – 53. – P. 37–45.
Rvachova T.V. On a nonstationary system of infinitely differentiable wavelets with compact support (in Russian). // Visn. Hark. nac. univ. im. V.N. Karazina, Ser.: Mat. prikl. mat. meh., 2011. – 967. – P. 63–80.
Lazorenko O.V. The use of atomic functions in the Choi–Williams analysis of ultrawideband signals. // Radioelectronics and Communications Systems, 2009. 52. – P. 397–404.
Ulises Moya-Sanchez E., Bayro-Corrochano E. Quaternionic analytic signal using atomic functions. // Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, Lecture Notes in Computer Science, 2012. – 7441. – P. 699–706.
Gotovac H., Andricevic R., Gotovac B. Multi-resolution adaptive modeling of groundwater flow and transport problems. // Adv. Water Resour., 2007. – 30. – P. 1105–1126.
Gotovac H., Cvetkovic V., Andricevic R. Adaptive Fup multi-resolution approach to flow and advective transport in heterogeneous porous media. // Adv. Water Resour., 2009. – 32. – P. 885–905.
Gotovac H., Gotovac B. Maximum entropy algorithm with inexact upper entropy bound based on Fup basis functions with compact support. // J. Comput. Phys., 2009. – 228. – P. 9079–9091.
Basarab M.A. Periodic atomic quasiinterpolation. // Ukrainian Math. J., – 2001. – 53. – P. 1728 – 1734.
Stoyan Y.G., Protsenko V.S., Man’ko G.P., Goncharyuk I.V., Kurpa L.V., Rvachev V.A., Sinekop N.S., Sirodzha I.B., Shevchenko A.N., Sheiko T.I. The theory of R-functions and current problems of applied mathematics (in Russian). – Kiev: Naukova Dumka, 1986. – 264 p.
RvachovaT.V. On a relation between the coefficients and sum of the generalized Taylor series. // Mathematical physics, analysis and geometry, 2003. – 10. – P. 262–268.
Rvachova T.V. On the asymptotics of the basic functions of a generalized Taylor series (in Russian). // Visn. Hark. nac. univ. im. V.N. Karazina, Ser.: Mat. prikl. mat. meh., 2003. – 602. – P. 94–104.
Rvachova T.V. On the rate of approximation of the infinitely differentiable functions by the partial sums of the generalized Taylor series (in Russian). // Visn. Hark. nac. univ. im. V.N. Karazina, Ser.: Mat. prikl. mat. meh., 2010. – 931. – P. 93–98.
Makarichev V.A. On the asymptotics of the basic functions of a generalized Taylor series for some classes of infinitely differentiable functions (in Russian). // Del’nevostochniy mathematicheskiy zhurnal, 2011. – 11. – P. 56–75.
Brysina I.V., Makarichev V.A. On the asymptotics of the generalized Fupfunctions. // Adv. Pure Appl. Math., 2014. – 5. – P. 131–138.
Rvachev V.A. Compactly supported solutions of functional–differential equations and their applications. // Russian Math. Surveys, 1990. – 45. – P. 87–120.
Rvachev V.L., Rvachev V.A. Non-classical methods of approximation theory in boundary value problem (in Russian). – K.: Naukova Dumka, 1979. – 196 p.
Makarichev V.A. Approximation of periodic functions by mups(x). // Math. Notes, 2013. – 93. – P. 858–880.
Rvachev V.L., Rvachev V.A. A certain finite function (in Ukrainian). // Proc. Ukr. SSR Acad. Sci., Ser. A., 1971. – 8. – P. 705–707.
Rvachev V.A., Starets G.A. Some atomic functions and their apllications (in Ukrainian). // Proc. Ukr. SSR Acad. Sci., Ser. A., 1983. – 11. – P. 22–24.
Makarichev V.A. Asymptotics of the basis functions of generalized Taylor series for the class H;2. // Math. Notes, 2011. – 89. – P. 689–705.
Dyn N., Ron A. Multiresolution analysis by infinitely differentiable compactly supported functions. // Appl. Comput. Harmon. Anal., 1995. – 2. – P. 15–20.
Cooklev T., Berbecel G.I., Venetsanopoulos A.N. Wavelets and differentialdilatation equations. // IEEE Transactions on signal processing, 2000. – 48. – P. 670–681.
Charina M., Stockler J. Tight wavelet frames for irregular multiresolution analysis. // Appl. Comput. Harmon. Anal., 2008. – 25. – P. 98–113.
Makarichev V.A. Applications of the function mups(x). // Progress in analysis. Proceedings of the 8th congress of the International Society for Analysis, its Applications, and Computation (ISAAC), 2012. – 2. – P. 297–304.
Makarichev V.A. The function mups(x) and its applications to the of generalized Taylor series, approximation theory and wavelet theory. // In book: Contemporary problems of mathematics, mechanics and computing sciences. – Kharkiv: Apostrophe, 2011. – P. 279–287.
Brysina I.V., Makarichev V.A. Atomic wavelets. // Radioelectronic and Computer Systems, 2012. – 53. – P. 37–45.
Rvachova T.V. On a nonstationary system of infinitely differentiable wavelets with compact support (in Russian). // Visn. Hark. nac. univ. im. V.N. Karazina, Ser.: Mat. prikl. mat. meh., 2011. – 967. – P. 63–80.
Lazorenko O.V. The use of atomic functions in the Choi–Williams analysis of ultrawideband signals. // Radioelectronics and Communications Systems, 2009. 52. – P. 397–404.
Ulises Moya-Sanchez E., Bayro-Corrochano E. Quaternionic analytic signal using atomic functions. // Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, Lecture Notes in Computer Science, 2012. – 7441. – P. 699–706.
Gotovac H., Andricevic R., Gotovac B. Multi-resolution adaptive modeling of groundwater flow and transport problems. // Adv. Water Resour., 2007. – 30. – P. 1105–1126.
Gotovac H., Cvetkovic V., Andricevic R. Adaptive Fup multi-resolution approach to flow and advective transport in heterogeneous porous media. // Adv. Water Resour., 2009. – 32. – P. 885–905.
Gotovac H., Gotovac B. Maximum entropy algorithm with inexact upper entropy bound based on Fup basis functions with compact support. // J. Comput. Phys., 2009. – 228. – P. 9079–9091.
Basarab M.A. Periodic atomic quasiinterpolation. // Ukrainian Math. J., – 2001. – 53. – P. 1728 – 1734.
Stoyan Y.G., Protsenko V.S., Man’ko G.P., Goncharyuk I.V., Kurpa L.V., Rvachev V.A., Sinekop N.S., Sirodzha I.B., Shevchenko A.N., Sheiko T.I. The theory of R-functions and current problems of applied mathematics (in Russian). – Kiev: Naukova Dumka, 1986. – 264 p.
RvachovaT.V. On a relation between the coefficients and sum of the generalized Taylor series. // Mathematical physics, analysis and geometry, 2003. – 10. – P. 262–268.
Rvachova T.V. On the asymptotics of the basic functions of a generalized Taylor series (in Russian). // Visn. Hark. nac. univ. im. V.N. Karazina, Ser.: Mat. prikl. mat. meh., 2003. – 602. – P. 94–104.
Rvachova T.V. On the rate of approximation of the infinitely differentiable functions by the partial sums of the generalized Taylor series (in Russian). // Visn. Hark. nac. univ. im. V.N. Karazina, Ser.: Mat. prikl. mat. meh., 2010. – 931. – P. 93–98.
Makarichev V.A. On the asymptotics of the basic functions of a generalized Taylor series for some classes of infinitely differentiable functions (in Russian). // Del’nevostochniy mathematicheskiy zhurnal, 2011. – 11. – P. 56–75.
Brysina I.V., Makarichev V.A. On the asymptotics of the generalized Fupfunctions. // Adv. Pure Appl. Math., 2014. – 5. – P. 131–138.
Published
2016-12-12
Cited
How to Cite
Брисіна, І. В., & Макарічев, В. О. (2016). Approximation properties of generalized Fup-functions. Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 84, 61-92. https://doi.org/10.26565/2221-5646-2016-84-06
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