IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

# Pencho Petrushev

 Professor Department of Mathematics University of South Carolina Office: LC 427 Phone: 803-777-6686 Email: pencho@math.sc.edu Homepage: http://www.math.sc.edu/~pencho/

## Education

 Doc. of Sci. Mathematics Sofia University, Bulgaria 1983 Ph.D. Mathematics Sofia University, Bulgaria 1977 B.S. Mathematics Sofia University, Bulgaria 1972

## Experience

 2009 – Present Director Interdisciplinary Mathematics Institute, Univ. of South Carolina 1996 – Present Professor Department of Mathematics, Univ. of South Carolina 1986 – 1996 Professor Inst. of Mathematics, Bulgarian Acad. of Sciences 1982 – 1986 Senior Research Fellow Inst. of Mathematics, Bulgarian Acad. of Sciences 1977 – 1982 Research Fellow Inst. of Mathematics, Bulgarian Acad. of Sciences

## Research

### Research Interests

General: Approximation Theory, Harmonic Analysis, Numerical Analysis
Specific:

• Spaces of distributions such as Hardy, Besov and Triebel-Lizorkin spaces in harmonic, nonclassical, anisotropic and geometric settings;
• Construction of bases and frames (needlets) for Besov and Triebel-Lizorkin spaces in nonclassical and geometric settings:
• Nonlinear approximation from rational functions, splines, frames, ridge functions and more general dictionaries;
• Nonlinear $n$-term approximation of harmonic functions from linear combinations of shifts of the fundamental solution of the Laplace equation (Newtonian kernel);
• Application of spherical needlets to approximation and fast evaluation of quantities represented in high degree surface or solid spherical harmonics;
• Reproduction of sound from optical sound tracks of digital scans of motion picture films.

### Current Projects

• Approximation and evaluation of quantities represented in spherical harmonics - This is a joint project with Kamen Ivanov (Bulgarian Academy of Sciences) for application of highly localized reproducing kernels (spherical needlets) to approximation and evaluation of quantities represented in surface or solid spherical harmonics. The main targeted application of this development is to Geopotential and Geomagnetic field modeling. This research is supported by NGA/DOD: grant HM01771210004 "Highly Effective Compression and Evaluation of Geodetic Quantities" (August 15, 2012 - August 14, 2017). The purpose of this project is to develop algorithms and software for fast, accurate, stable, and memory efficient evaluation of geodetic quantities represented in high degree (> 2,000) surface or solid spherical harmonics at many (millions) scattered points on the surface of the Earth or in the space above its surface.

## Teaching Activities

### Current Courses

• Math 554 - Analysis I

### Previous Courses

• Undergraduate courses: Calculus I (Math 141), Calculus II (Math 142), Vector Calculus (Math 241), Elementary Differential Equations (Math 242), Ordinary Differential Equations (Math 520), Linear Algebra (Math 544), Analysis I (Math 554), Analysis II (Math 555).
• Graduate courses: Analysis I (Math 703), Analysis II (Math 704), Applied Mathematics I (Math 720), Applied Mathematics II (Math 721), Approximation Theory (Math 725), Nonlinear Approximation (Math 729), Fourier Analysis (Math 750), Maximal Operators, Littlewood-Paley Theory, and Bases (Math 758P).

## Honors and Other Special Scientific Recognition

• The 2010 USC Educational Foundation Award for Research in Science, Mathematics, and Engineering
• The 2010 Manuel Garcia Prize from the International Association of Logopedics and Phoniatrics for outstanding contributions to the official journal of IALP and to the field of communication sciences and disorders, jointly with D. Deliyski, H. Bonilha, T. Gerlach, B. Martin-Harris, and R. Hillman
• The 1989 Bulgarian Mathematical Scienece Award "N. Obreshkov'' awarded by the Bulgarian Academy of Sciences and Sofia University

## 5 Selected Publications

• S. Dekel, G. Kerkyacharian, G. Kyriazis, and P. Petrushev, Hardy spaces associated with non negative self-adjoint operators, Studia Math. 239 (2017), no. 1, 17-54.
• G. Kerkyacharian, P. Petrushev, Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces, Trans. Amer. Math. Soc. 367 (2015), 121–189.
• F. Narcowich, P. Petrushev, and J. Ward, Decomposition of Besov and Triebel-Lizorkin spaces on the sphere, J. Funct. Anal. 238 (2006), 530–564.
• G. Kyriazis, P. Petrushev, New bases for Triebel-Lizorkin and Besov spaces, Trans. Amer. Math. Soc. 354 (2002), 749–776.
• A. Cohen, R. DeVore, P. Petrushev, and H. Xu, Nonlinear approximation and the space $BV(R^2)$, Amer. J. Math. 121 (1999), 587–628.