CURRICULUM VITA

Pencho P. Petrushev

Department of Mathematics
University of South Carolina
Columbia, SC 29208


VITA

RESEARCH PUBLICATIONS

GRADUATE EDUCATION:
 
Doc. of Sci.  University of Sofia, Bulgaria, 1983
Ph. D. University of Sofia, Bulgaria, 1977

UNDERGRADUATE EDUCATION:
 
B. S.  University of Sofia, Bulgaria, 1972

PROFESSIONAL EXPERIENCE:
 
Professor Department of Mathematics, University of South Carolina,
1996 -- present
Professor Institute of Mathematics, Bulgarian Academy of Sciences,
1986 -- 1996
Senior Scientist Institute of Mathematics, Bulgarian Academy of Sciences,
1982 -- 1986
Scientist Institute of Mathematics, Bulgarian Academy of Sciences, 
1977 -- 1982

VISITING POSITIONS:
 
University of South Carolina Spring 1996, Fall 1995, Spring 1995, Fall 1994,
Spring 1994, Fall 1993, Spring 1993, Fall 1992
University of South Florida Fall 1991, Fall 1990

GRANT SUPPORT:

NIH/NIDCD (R01DC007640): 2007--2012; ILVR (Research grant): 2006--2007; DARPA/DSA (HM1582-05-2-0001): 2005--2006; NSF DMS-0200665: 2002--2005; ARO-DAAD190210028: 2002--2004; ONR-DEPSCoR N00014-00-1-0470: 2000--2003; ONR/DURIP Equipment Grant: 2001--2002; ONR-ARO: 1997--2000
 

PROFESSIONAL SERVICE:

Conferences Organized:

International Conferences on Constructive function theory in Blagoevgrad 1979, Varna 1981, 1984, 1987, 1991 (in Bulgaria)

Co-organizer (with Fernando Cobos) of Special Session # 21: "Interpolation Theory, Function Spaces and Applications" at the First Joint International Meeting of the American Mathematical Society and the Real Sociedad Matemática Española held in Seville (Spain) on June 18-21, 2003 [http://www.us.es/rsme-ams].

Committee Duties:

Vasil A. Popov Prize Selection Committee, member since 1994 and the chairman of the committee since May 2004

Editorial Boards:

Constructive Approximation

East Journal on Approximations

Approximation Theory and its Applications

Serdica Mathematical Journal
 
 

AREAS OF RESEARCH INTEREST: Approximation theory, Harmonic analysis, and their applications. These include

  1. Nonlinear approximation from rational functions, splines, and wavelets and related function spaces.
  2. Nonlinear approximation from hierarchical anisotropic spline bases and associated generalized Besov spaces (B-spaces). Algorithms for nonlinear spline approximation.
  3. Application of nonlinear approximation to numerical solution of PDEs.
  4. Meshless methods for solving elliptic boundary value problems.
  5. Approximation from ridge functions and in particular from neural networks.
  6. Construction of bases and frames, consisting of e.g. rational functions, for Besov and Triebel-Lizorkin spaces.
  7. Construction of frames and development of the Littlewood-Paley theory on the d-dimensional unit sphere and ball, and on the cube with Jacobi weights. Nonlinear n-term approximation from such frames.
  8. Image and signal processing.
  9. Application of localized frames on the sphere to multiscale representation and approximation of the gravitational potential.
  10. Development of methods for voice assessment of high-speed videoendoscopy.
RESEARCH PUBLICATIONS:

Monographs:

  1. (with V. Popov), "Rational approximation of Real Functions", Cambridge University press, 1987, 371 pages.
Refereed Research Articles:
  1. (with V. Hristov), On the Muntz polynomial approximation in Hausdorff's matric, C. R. Acad. Bulgare Sci. 29 (1976), 955-958, (in Russian).
  2. On the rational approximation of functions with a convex derivative, C. R. Acad. Bulgare Sci. 29 (1976), 1249-1252, (in Russian).
  3. (with S. Tashev), Some inverse theorems in Hausdorff's metric, C. R. Acad. Bulgare Sci. 29 (1976), 1721-1724 (in Russian).
  4. On the rational approximation of functions with convex rth derivative, Acta Math. Hungar. 28 (1976), 315-320.
  5. On the rational approximation of functions, C. R. Acad. Bulgare Sci. 29 (1976), 1405-1408, (in Russian).
  6. (with V. Hristov), An improvement of Dini-Lipschitz condition for uniform convergence of Fourier series, C. R. Acad. Bulgare Sci. 29 (1976), 1579-1582, (in Russian).
  7. On the rational approximation of functions of unbounded variation, Serdica Math. J. (Bulgarian Math. Journal), 2 (1976), 149-153.
  8. (with V. Hristov), On the convergence of Fourier series in Banach spaces, C. R. Acad. Bulgare Sci. 29 (1976), 1099-1102.
  9. On the rational approximation of functions, Proceedings of the International Conference on Approximation Theory, Kaluga 1975, Nauka, 1977, pp. 277-279, (in Russian).
  10. (with V. Hristov), Fourier series convergence in Hausdorff's matric, Pliska (Bulgarian Math. Studies), 1 (1977), 21-36, (in Russian).
  11. (with V. Hristov), Convergence of Fourier series in Banach spaces, Pliska (Bulgarian Math. Studies), 1 (1977), 37-47, (in Russian).
  12. (with V. Popov), The exact order of best uniform rational approximation of convex functions, Mat. Sbornik, 103 (1977), 285-292 (English translation in Math. USSR Sbornik, 32 (1977), 245-251).
  13. Uniform rational approximation of functions of bounded variation, Pliska (Bulgarian Math. Studies), 1 (1977), 145-155, (in Russian).
  14. (with Vl. Hristov), Sufficient conditions for convergence of Fourier series, Proceedings of the International Conference on Approximation Theory, Kaluga 1975, Moscow, Nauka, (1977), 392-396, (in Russian).
  15. The exact order of the best uniform rational approximation of some function classes, Colloq. Math. Soc. Janos Bolyai, 19. Fourier analysis and approximation theory (Proc. Colloq. Budapest, Hungary, 1976, eds. G. Alexits, P. Turan) North-Holland, Amsterdam, 1978, pp. 603-624.
  16. (with V. Hristov), An improvement of Dini-Lipschitz condition, Colloq. Math. Soc. Janos Bolyai, 19. Fourier analysis and approximation theory (Proc. Colloq. Budapest, Hungary, 1976, eds. G. Alexits, P. Turan) North-Holland, Amsterdam, 1978, pp. 255-264.
  17. (with S. Tashev), Converse theorems in Hausdorff's metric, Colloq. Math. Soc. Janos Bolyai, 19. Fourier analysis and approximation theory (Proc. Colloq. Budapest, Hungary, 1976, eds. G. Alexits, P. Turan) North-Holland, Amsterdam, 1978, pp. 624-631.
  18. (with Bl. Sendov and S. Tashev), Characterization of the s-derivatives of Lipschitz functions, Serdica Math. J. 4 (1978), 260-266, (in Russian).
  19. Rational approximation in Hausdorff's metric, C. R. Acad. Bulgare Sci. 31 (1978), 155-158, (in Russian).
  20. Uniform rational approximation of functions from the class V_r, C. R. Acad. Bulgare Sci. 31 (1978), 1535-1538, (in Russian).
  21. Rational approximation of real functions in uniform, Hausdorff's, and integral metrics, Proceedings of the Seventh Spring Conference of the Bulg. Math. Soc., 1978, 95-106, (in Bulgarian).
  22. (with V. Hristov), A generalization of the Dini-Lipschitz condition for uniform convergence of Fourier series, Mat. Zametki, 25 (1979), 557-568 (English translation in Math. Notes, 25 (1979), 289-295).
  23. Uniform rational approximation of the functions from the class V_r, Mat. Sbornik, 108 (1979), 418-432 (English translation in Math. USSR Sbornik, 36 (1980), 389-403).
  24. Rational approximation of functions in uniform, Hausdorff's, and integral metrics, Proceedings of the International Conference on Constructive Function Theory, Blagoevgrad, 1977, Sofia 1980, 111-120, (in Russian).
  25. Best rational approximation in Hausdorff's metric, Serdica Math. J. 6 (1980), 29-41, (in Russian).
  26. Lower bound estimates for the best rational approximation in Hausdorff's metric, Serdica Math. J. 6 (1980), 120-127, (in Russian).
  27. Rational approximation of functions of bounded variation in Hausdorff's and integral metric, Serdica Math. J. 6 (1980), 202-210, (in Russian).
  28. Rational approximation of functions in the class V_r, C. R. Acad. Bulgare Sci. 33 (1980), 1607-1610, (in Russian).
  29. (with V. Hristov), On Tauberian theorems for power series and their application in Fourier series, Approximation and Function spaces, Proc. Inter. Conf. Gdansk, 1979, ed. Z. Cieselski, North-Holland, Amsterdam, 1981, pp. 317-329.
  30. Rational and piece-wise polynomial approximation, C. R. Acad. Bulgare Sci. 34 (1981), 7-10, (in Russian).
  31. Some new characteristics in the theory of rational approximation, Proceedings of the International Conference on Constructive Function Theory, Varna 1981, Sofia 1983, pp. 121-124, (in Russian).
  32. Relations between best rational and spline approximation in L_p metric, Pliska (Bulgarian Math. Studies), 5 (1983), 68-83, (in Russian).
  33. On the relation between rational and spline approximations, Constructive Theory of Functions 1984, Proceedings of the International Conference on Constructive Function Theory, Varna 1984, Sofia 1984, 672-674.
  34. (with S. Troyanski), On the Banach-Mazur theorem for universality of C[0, 1], C. R. Acad. Bulgare Sci. 37 (1984), 283-285, (in Russian).
  35. Relations between rational and spline approximations, Acta Math. Hungar. 44 (1984), 61-83.
  36. Direct and converse theorems for spline approximation and Besov spaces, C. R. Acad. Bulgare Sci. 39 (1986), 25-28, (in Russian).
  37. Relations between rational and spline approximations in L_p metric, J. Approx. Theory, 50 (1987), 141-159.
  38. Direct and converse theorems for rational and spline approximation and Besov spaces, Lecture Notes in Mathematics 1302, Proceedings of the Conference "Function Spaces and Applications", Lund 1986, 1988, pp. 363-377.
  39. (with E. Moskona), Characterization of the rational approximation in uniform metric, C. R. Acad. Bulgare Sci. 42 (1989), 37-40.
  40. (with E. Moskona), Uniform rational approximation of functions with first derivative in the real Hardy space ReH^1, Constr. Approx. 7 (1991), 69-103.
  41. (with R. DeVore and X. Yu), Nonlinear wavelet approximation in the space C(R^d), Progress in Approximation Theory (A. A. Gonchar, E. B. Saff, eds.), New York, Springer-Verlag, 1992, pp. 261-283.
  42. (with P. Binev, E. Saff and O. Trifonov), Distribution of interpolation points of best L_2-approximants (nth partial sums of Fourier series), Constr. Approx. 9 (1993), 445-472.
  43. (with R. DeVore and V. Temlyakov), Multivariate trigonometric polynomial approximation with frequencies from the hyperbolic cross, Mat. Zametki, 56 (1994), 36-63 (English translation in Math. Notes, 56 (1994), 900-918).
  44. (with E. Moskona and E. Saff), Gibbs phenomenon for best L_1-trigonometric polynomial approximation, Constr. Approx. 11 (1995), 391-416.
  45. (with R. DeVore and K. Oskolkov), Approximation by feed-forward neural networks, Ann. Numer. Math. 4 (1997), 261-287.
  46. Approximation by ridge functions and neural networks, SIAM J. Math. Anal. 30 (1998), 155-189.
  47. (with A. Cohen, R. DeVore, and H. Xu), Nonlinear, Approximation and the space BV(R^2), Amer. J. Math. 121 (1999), 587-628.
  48. Bases consisting of rational functions of uniformly bounded degrees or more general functions, J. Funct. Anal. 174 (2000), 18-75.
  49. (with G. Kyriazis), New bases for Triebel-Lizorkin and Besov spaces, Trans. Amer. Math. Soc. 354 (2002), no. 2, 749-776.
  50. (with Peter Binev, Wolfgang Dahmen, and Ronald DeVore), Approximation Classes for Adaptive Methods, Serdica Math. J. 28 (2002), 1001-1026.
  51. Multivariate n-term rational and piecewise polynomial approximation, J. Approx. Theory, 121 (2003), 158-197.
  52. (with B. Karaivanov and R. Sharpley), Nonlinear piecewise polynomial approximation: Algorithms, Trans. Amer. Math. Soc. 355 (2003), 2585-2631.
  53. Nonlinear n-term Approximation from Hierarchical Spline Bases, Constructive Theory of Functions, Varna 2002 (B. Bojanov, Ed.), DARBA, Sofia, 2003, pp. 33-85.
  54. (with D. Delyiski), Methods for Objective Assessment of High-Speed Videoendoscopy, 2003, Proceedings: 6th International Conference: Advances in Quantitative Laryngology, Voice and Speech Research AQL-2003, Hamburg, Germany.
  55. (with B. Karaivanov), Multivariate piecewise polynomial approximation beyond Besov spaces, Appl. Comput. Harmon. Anal. 15 (2003), 177-223.
  56. (with O. Davydov), Nonlinear approximation from differentiable piecewise polynomials, SIAM J. Math. Anal. 35 (2004), 708-758.
  57. (with G. Kyriazis and K. Park), B-spaces and their Characterization via Anisotropic Franklin Bases, Approximation Theory: A volume dedicated to Borislav Bojanov, 145-162, (D. K. Dimitrov, G. Nikolov, and R. Uluchev, Eds.), Prof. M. Drinov Acad. Publ. House, Sofia, 2004.
  58. (with M. Campos-Pinto and A. Cohen), High order geometric smoothness for conservation laws, J. Hyperbolic Diff. Eq., J. of Hyperbolic Differ. Equ. 2 (2005), 39-59.
  59. Anisotropic Spaces and Nonlinear n-term Spline Approximation, Approximation Theory XI: Gatlinburg 2004, 363-394, C.K. Chui, M. Neamtu, and L.L. Shumaker, (eds.), Mod. Methods Math., Nashboro Press, Brentwood, TN, 2005.
  60. (with Yuan Xu), Localized polynomial frames on the interval with Jacobi weights, J. Four. Anal. Appl. 11 (2005), 557-575.
  61. (with G. Kyriazis), On the construction of frames for Triebel-Lizorkin and Besov spaces, Proc. Amer. Math. Soc. 134 (2006), 1759-1770.
  62. (with W. Dahmen), "Push-the-Error'' algorithm for nonlinear n-term approximation, Constr. Approx. (2006) 23, 261-304.
  63. (with G. Kyriazis and K. Park), Anisotropic Franklin bases on polygonal domains, Math. Nachr. 279 (2006), 1099-1127.
  64. (with F. J. Narcowich and J. D. Ward), Localized Tight Frames on Spheres, SIAM J. Math. Anal. 38 (2006), 347-692.
  65. (with F. Narcowich and J. Ward), Decomposition of Besov and Triebel-Lizorkin spaces on the sphere, J. Funct. Anal. 238 (2006), 530-564.
  66. (with G. Kerkyacharian, D. Picard, and T. Willer), Needlet algorithms for estimation in inverse problems, Electronic Journal of Statistics, 1 (2007), 30-76.
  67. (with Yuan Xu), Localized polynomial frames on the ball, preprint.
  68. (with W. Dahmen and S. Dekel), Multilevel Preconditioning for Partition of Unity Methods - Some Analytic Concepts, preprint.
  69. (with G. Kyriazis and Yuan Xu), Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces, preprint.
  70. (with W. Dahmen, S. Dekel), Two-level-split decomposition of anisotropic Besov spaces, preprint.
  71. (with G. Kyriazis and Yuan Xu), Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball, preprint.
  72. (with Yuan Xu), Decomposition of spaces of distributions induced by Hermite expansions, preprint.