01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
22.11.1955
E-mail:
,
Keywords:
best approximation; hyperbolic Fourier sum; Kolmogorov width; linear width; best trigonometric approximation; bilinear approximation; trigonometric width; classes of periodic functions.
Subject:
Exact order estimates are obtained of approximation of Besov classes $B^r_{p, \theta}$ of periodic functions of several variables by trigonometric polynomials with harmonics from hyperbolic crosses. The orders are established of Kolmogorov, linear and trigonometric widthes of classes $B^r_{p, \theta}$ in space $L_p$, $1 \leq p, q \leq \infty$. Best $M$-term trigonometric and bilinear approximations of mentioned classes are investigated; in passing some results by Sobolev and Nikolsky in this direction are supplemented and specified. The algorithm is proposed of construction of subspaces of trigonometric polynomials realizing the orders of Kolmogorov widthes of classes of functions of several variables defined by generalized derivative.
Biography
Graduated from Faculty of Mathematics and Mechanics of I. Franko Lvov State University in 1978 (department of theory of functions). Ph. D. thesis was defended in 1988. D. Sci. thesis was defended in 1996. A list of my works contains more than 50 titles.
Main publications:
A. S. Romanyuk, “Nailuchshie $M$-chlennye trigonometricheskie priblizheniya klassov Besova periodicheskikh funktsii mnogikh peremennykh”, Izv. RAN. Ser. matem., 67:2 (2003), 61–100
A. S. Romanyuk, “Priblizhenie klassov $B_{p,\theta}^r$ periodicheskikh funktsii mnogikh peremennykh lineinymi metodami i nailuchshie priblizheniya”, Matem. sb., 195:2 (2004), 91–116
A. S. Romanyuk, “Best Trigonometric and Bilinear Approximations of Classes of Functions of Several Variables”, Mat. Zametki, 94:3 (2013), 401–415; Math. Notes, 94:3 (2013), 379–391
A. S. Romanyuk, “Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables”, Mat. Zametki, 87:3 (2010), 429–442; Math. Notes, 87:3 (2010), 403–415
A. S. Romanyuk, “Best approximations and widths of classes of periodic functions of several variables”, Mat. Sb., 199:2 (2008), 93–114; Sb. Math., 199:2 (2008), 253–275
A. S. Romanyuk, “Best Trigonometric Approximations for Some Classes of Periodic Functions of Several Variables in the Uniform Metric”, Mat. Zametki, 82:2 (2007), 247–261; Math. Notes, 82:2 (2007), 216–228
A. S. Romanyuk, “Bilinear and trigonometric approximations of periodic functions
of several variables of Besov classes $B_{p, \theta}^r$”, Izv. RAN. Ser. Mat., 70:2 (2006), 69–98; Izv. Math., 70:2 (2006), 277–306
A. S. Romanyuk, “Kolmogorov and trigonometric widths of the Besov classes $B^r_{p,\theta}$ of multivariate periodic functions”, Mat. Sb., 197:1 (2006), 71–96; Sb. Math., 197:1 (2006), 69–93
A. S. Romanyuk, “Approximability of the classes $B_{p,\theta}^r$ of periodic functions
of several variables by linear methods and best approximations”, Mat. Sb., 195:2 (2004), 91–116; Sb. Math., 195:2 (2004), 237–261
A. S. Romanyuk, “Best $M$-term trigonometric approximations of Besov classes of periodic functions of several variables”, Izv. RAN. Ser. Mat., 67:2 (2003), 61–100; Izv. Math., 67:2 (2003), 265–302
A. S. Romanyuk, “Approximation of Classes of Periodic Functions in Several Variables”, Mat. Zametki, 71:1 (2002), 109–121; Math. Notes, 71:1 (2002), 98–109
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