# Curriculum Vitae

## Pencho Petrushev

December 2020

### Graduate Education:

Sofia University, Bulgaria

Doc. of Sci. in Mathematics (Second Doctoral Degree) 1983

Ph.D. 1977 in Mathematics; Thesis Advisor: Vasil A. Popov

### Undergraduate Education:

Sofia University, Bulgaria

B.S. June 1972 in Mathematics

### Professional Employment (Permanent Positions):

1996 – Present | Professor | Department of Mathematics, USC, Columbia |

2009 – 2018 | Director | Interdisciplinary Math. Institute, USC, Columbia |

1986 – 1996 | Professor | Inst. of Mathematics, Bulg. Acad. of Sciences |

1982 – 1986 | Senior Research Fellow | Inst. of Mathematics, Bulg. Acad. of Sciences |

1977 – 1982 | Research Fellow | Inst. of Mathematics, Bulg. Acad. of Sciences |

### Awards and Honors:

The 2010 USC Educational Foundation Award for Research in Science, Mathematics, and Engineering

The 1989 Bulgarian Mathematical Scienece Award "N. Obreshkov" awarded by the Bulgarian Academy of Sciences and Sofia University

### Editorial Boards:

Constructive Approximation - 2004 – present

Journal of Approximation Theory - 2020 – present

SIAM Journal on Mathematical Analysis - 2010 – present

Journal of Fourier Analysis and Applications - 2012 – present

Foundations of Computational Mathematics - 2014 – present

Transactions of Mathematics and Its Applications - 2016 – present

Analysis in Theory and Applications - 1990 – present

Serdica Mathematical Journal - 2004 – 2019

Mathematics - Open Access Journal - 2012 – 2018

### Doctoral Students: at USC

Borislav Karaivanov, Thesis Title: "Nonlinear piecewise polynomial approximation: Theory and Algorithms", completed in December 2001.

Kyungwon Park, Thesis Title: "Bivariate Rational Approximation and Anisotropic Franklin Bases", completed in May 2004.

### Master Students: at USC.

Xiang Yu, Department of Computer Science, USC, Thesis Title: "Clientserver system for progressive image transmission based on Adaptive wavelet bases", completed in May 2001.

Wendy Newcomb, Department of Mathematics, USC, Thesis Title: "Fast Evaluation of Band Limited Functions at Scattered Points on the Sphere", completed in April 2015.

Jared Szi, Department of Mathematics, USC, Thesis Title: "Chebyshev Inversion of the Radon Transform", completed in April 2016

David Hughey, Department of Mathematics, USC, Thesis Title: "Nonequispaced Fast Fourier Transform", completed in August 2017

### Grant Support (last 10 years):

NGA/DOD: HM04761812002

Title: Highly Effective Computational Methods in Geomagnetic Field Modeling

PI: P. PetrushevV
February 22, 2018 - February 21, 2023; $750,000

NSF: DMS-1714369

Title: Nonlinear approximation in geometric, harmonic and anisotropic settings with applications

PI: P. Petrushev

August 16, 2017 - June 30, 2021; $162,941

NGA/DOD: HM01771210004

Title: Highly Effective Compression and Evaluation of Geodetic Quantities

PI: P. Petrushev

August 15, 2012 - August 14, 2017; $750,000

NEH: PR-50207-15

Title: AEO-Light 2.0:An Open Source Application for Image-Based Digital

Reproduction of Optical Film Sound

PI: G. Wilsbacher, Co-PI: P. Petrushev

January 1, 2015 - December 31, 2016; $222,146

NSF: DMS-1211528

Title: Representation and Approximation of Functions in Nonclassical and
Anisotropic Settings with Applications

PI: P. Petrushev

June 1, 2012 - May 31, 2015; $170,011

NEH: PR50122

Title: An Open Source Application for Image-Based Digital Reproduction of
Optical Film Sound

PI: G. Wilsbacher, Co-PIs: P. Petrushev, M. Cooper, B. Karaivanov

May 2011 - May 2014; $362,500

NIH/NIDCD: R01DC007640

Title: Efficacy of Laryngeal High-Speed Videoendoscopy

PI: D. Deliyski, Co-PI: P. Petrushev

February 1, 2007 - February 1, 2014 (extended); $3,078,774

NIH/NIDCD: 5R01DC7640

Title: Throat Clearing, Coughing, and Alternative Behaviors

PI: Bonilha Heather (Medical University of South Carolina)

Subcontract PI: D. Deliyski, Co-PI: P. Petrushev

October 1, 2010 - June 30, 2011, $14,097

NSF: DMS 0709046

Title: Highly efficient methods for surface and solid sphere studies

PI: Pencho Petrushev

July 15, 2007 - September 30, 2010; $143,907

### Conference Organizing or Program Committees:

Nine international Approximation Theory Conferences in Bulgaria;

### Invited Conference Talks (last 10 years):

- Conference on Time-Frequency, STROBL09, Strobl, Austria, June 15 - 20, 2009
- International workshop "Nonlinear and Adaptive Approximation" at Castle Reisensburg (Günzburg), Germany, September 30 - October 3, 2009
- Workshop "Topics in Several Complex Variables and Geometry" University of Cyprus, Nicosia, November 22, 2009
- Oberwolfach Workshop: Wavelet and Multiscale Methods, August 1-7, 2010
- Conference: "From Abstract to Computational Harmonic Analysis" Strobl, Austria, June 13 - 19, 2011
- Conference: "New Trends in Approximation Theory" Ein Gedi, Israel, January 4 - 7, 2012
- Workshop: "New perspectives in Approximation theory and Applications" Laboratoire Jacques-Louis Lions, Universit Pierre et Marie Curie, Paris, June 15, 2012
- Workshop: "Applied Coorbit Theory" Erwin Schroedinger Institute (Univ. Vienna), Austria, September 17 - 21, 2012
- Numerical Methods for PDEs: On Occasion of Raytcho Lazarov's 70th Birthday Texas A&M University, Collage Station, January 25-26, 2013
- Oberwolfach Workshop "Multiscale and High-Dimensional Problems", July 28th - August 3rd, 2013
- Int. Conference CONSTRUCTIVE THEORY OF FUNCTIONS - 2013, Sozopol, Bulgaria, June 9 - 15, 2013
- 8th Internat. Conference "Curves and Surfaces", Paris, June 12-18, 2014
- International Conference Mathematics Days in Sofia, Sofia, Bulgaria, July 7-10, 2014
- Foundations of Computational Mathematics conference, FoCM 2014 - Montevideo, Uruguay, December 11 - 20, 2014
- International Conference on Multivariate Approximation, Giessen, Schloss Rauischholzhausen, March 31 - April 5, 2016
- International Conference Constructive Theory of Functions - 2016, Sozopol, Bulgaria, June 11 - 17, 2016
- Conference: Multiscale and High-Dimensional Problems, Oberwolfach, March 26 - April 1, 2017
- Conference in Cyprus: Contemporary Aspects of Analysis, Protaras, Cyprus, May 1 - 5, 2017
- FoCM 2017 - Foundations of Computational Mathematics Conference, Barcelona, July 10 - 19, 2017
- Fourth Aachen Conference on Computational Engineering Science (AC.CES), Aachen, July 27 - 28, 2017
- International Conference: New perspectives in the theory of function spaces and their applications (NPFSA-2017), Bedlewo, Poland, September 17 - 23, 2017
- Intelligence Community (IC) Academic Research Symposium - 2018, The National Academy of Sciences, Washington, DC, September 25-27, 2018.
- Conference in Cyprus: Contemporary Aspects of Analysis II, Protaras, Cyprus, May 6-10, 2019
- Intelligence Community (IC) Academic Research Symposium - 2019, The National Academy of Sciences, Washington, DC, September 10-12, 2019.

### Service on Professional Panels:

Member of the Vasil A. Popov Prize Selection Committee, since 1994

### 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

### Publications:

110 (1 book, 106 articles in print, 1 in press, and 2 submitted for publication)

### Monograph:

- P. Petrushev, V. Popov, "Rational approximation of Real Functions", Cambridge University Press, 1987, 371 pages.

### Selected Articles:

- P. Petrushev, Relations between rational and spline approximations in Lp metric, J. Approx. Theory, 50 (1987), 141–159.
- P. Petrushev, 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.
- E. Moskona, P. Petrushev, Uniform rational approximation of functions with first derivative in the real Hardy space ReH1, Constr. Approx. 7 (1991), 69–103.
- R. DeVore, P. Petrushev, and X. Yu, Nonlinear wavelet approximation in the space C(Rd), Progress in Approximation Theory (A. A. Gonchar, E. B. Saff, eds.), New York, Springer-Verlag, 1992, pp. 261–283.
- P. Binev, E. Saff, P. Petrushev, and O. Trifonov, Distribution of interpolation points of best L2–approximants (nth partial sums of Fourier series), Constr. Approx. 9 (1993), 445–472.
- R. DeVore, P. Petrushev, 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).
- E. Moskona, P. Petrushev, and E. Saff, Gibbs phenomenon for best L1–trigonometric polynomial approximation, Constr. Approx. 11 (1995), 391–416.
- R. DeVore, K. Oskolkov, and P. Petrushev, Approximation by feedforward neural networks, Ann. Numer. Math. 4 (1997), 261–287.
- P. Petrushev, Approximation by ridge functions and neural networks, SIAM J. Math. Anal. 30 (1998), 155–189.
- A. Cohen, R. DeVore, P. Petrushev, and H. Xu, Nonlinear, Approximation and the space BV (R2), Amer. J. Math. 121 (1999), 587–628.
- P. Petrushev, Bases consisting of rational functions of uniformly bounded degrees or more general functions, J. Funct. Anal. 174 (2000), 18–75.
- G. Kyriazis, P. Petrushev, New bases for Triebel-Lizorkin and Besov spaces, Trans. Amer. Math. Soc. 354 (2002), no. 2, 749–776.
- P. Binev, W. Dahmen, R. DeVore, P Petrushev, Approximation classes for adaptive methods, Serdica Math. J. 28 (2002), 1001–1026.
- P. Petrushev, Multivariate n-term rational and piecewise polynomial approximation, J. Approx. Theory, 121 (2003), 158–197.
- B. Karaivanov, P. Petrushev, Nonlinear piecewise polynomial approximation beyond Besov spaces, Appl. Comput. Harmon. Anal. 15 (2003), 177–223.
- B. Karaivanov, P. Petrushev, and R. Sharpley, Nonlinear piecewise polynomial approximation: Algorithms, Trans. Amer. Math. Soc. 355 (2003), 2585–2631.
- O. Davydov, P. Petrushev, Nonlinear approximation from differentiable piecewise polynomials, SIAM J. Math. Anal. 35 (2003), 708–758.
- P. Petrushev, Nonlinear n-term approximation from hierarchical spline bases, Constructive Theory of Functions , Varna 2002 (B. Bojanov, Ed.), DARBA, Sofia, 2003, pp. 33–85.
- G. Kyriazis, K. Park, and P. Petrushev, B-spaces and their characterization via anisotropic Franklin bases, Approximation Theory: a vol- ume dedicated to Borislav Bojanov (D. K. Dimitrov, G. Nikolov, and R. Uluchev, Eds.), pp. 145–162, Prof. M. Drinov Acad. Publ. House, Sofia, 2004.
- M. Campos-Pinto, A. Cohen, and P. Petrushev, High order geometric smoothness for conservation laws, J. of Hyperbolic Differ. Equ. 2 (2005), 39–59.
- P. Petrushev, Anisotropic spaces and nonlinear n-term approximation, Approximation Theory XI: Gatlinburg 2004, 363–394, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2005.
- P. Petrushev, Y. Xu, Localized polynomial frames on the interval with Jacobi weights, J. Fourier Anal. and Appl. 11 (2005), 557–575.
- W. Dahmen, P. Petrushev, "Push-the-Error" algorithm for nonlinear n-term approximation, Constr. Approx. 23 (2006), 261–304.
- G. Kyriazis, P. Petrushev, On the construction of frames on Triebel- Lizorkin and Besov spaces, Proc. Amer. Math. Soc. 134 (2006), 1759–1770.
- F. J. Narcowich, P. Petrushev, and J. D. Ward, Localized Tight Frames on Spheres, SIAM J. Math. Anal. 38 (2006), 574–594.
- 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, K. Park, and P. Petrushev, Anisotropic Franklin bases on polygonal domains, Math. Nachr. 279 (2006), 1099–1127.
- G. Kerkyacharian, P. Petrushev, D. Picard and T. Willer, Needlet algorithm for estimations in inverse problems, Electronic Journal of Statistics, 1 (2007), 30–76.
- W. Dahmen, S. Dekel, and P. Petrushev, Multilevel preconditioning for partition of unity methods - some analytic concepts, Numer. Math. 107 (2007), 503–532.
- P. Petrushev, Yuan Xu, Localized polynomial frames on the ball, Constr. Approx. 27 (2008), 121–148.
- G. Kyriazis, P. Petrushev, and Yuan Xu, Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces, Studia Math. 186 (2008), 161–202.
- G. Kyriazis, P. Petrushev, and Yuan Xu, Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball, Proc. London Math. Soc. 97 (2008), 477–513.
- P. Petrushev, Yuan Xu, Decomposition of spaces of distributions induced by Hermite expansions, J. Fourier Anal. and Appl. 14 (2008), 372–414.
- G. Kerkyacharian, P. Petrushev, D. Picard, and Yuan Xu, Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions, J. Funct. Anal. 256 (2009), 1137–1188.
- G. Kyriazis, P. Petrushev, On the construction of bases and frames for spaces of distributions, J. Funct. Anal. 257 (2009), 2159–2187.
- S. Dekel, Y. Han, and P. Petrushev, Anisotropic meshless frames on Rn, J. Fourier Anal. and Appl. 15 (2009), 634–662.
- S. Dekel, P. Petrushev, Anisotropic function spaces with applications, in "Multiscale, Nonlinear and Adaptive Approximation", Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday, R. DeVore, A. Kunoth (Eds.) Springer, 2009, ISBN 978-3-642-03412-1.
- W. Dahmen, S. Dekel, and P. Petrushev, Two-level-split Decomposition of Anisotropic Besov Spaces, Constr. Approx. 31 (2010), no. 2, 149–194
- K. Ivanov, P. Petrushev, and Yuan Xu, Sub-exponentially localized kernels and frames induced by orthogonal expansions, Math. Z. 264 (2010), 361–397.
- G. Kerkyacharian, G. Kyriazis, E. Le Pennec, P. Petrushev and D. Picard, Inversion of noisy Radon transform by SVD based needlets, Appl. Comput. Harmon. Anal. 28 (2010), 24–45.
- S. Dekel, P. Petrushev and T. Weissblat, Hardy spaces on Rn with pointwise variable anisotropy, J. Fourier Anal. and Appl. 17 (2011), 1066–1107.
- T. Coulhon, G. Kerkyacharian, P. Petrushev, Heat Kernel Generated Frames in the Setting of Dirichlet Spaces, J. Fourier Anal. Appl. 18 (2012), no. 5, 995–1066.
- K. Ivanov, P. Petrushev, Yuan Xu, Decomposition of spaces of distributions induced by tensor product bases, J. Funct. Anal. 263 (2012), no. 5, 1147–1197.
- G. Kyriazis, P. Petrushev, "Compactly" supported frames for spaces of distributions on the ball, Monatsh. Math. 165 (2012), no. 3-4, 365–391.
- K. Ivanov, P. Petrushev, Highly effective evaluation and reconstruction of geodetic quantities on the sphere, Proceedings of the ASPRS 2013 Annual Conference, Baltimore, March, 2013.
- G. Kyriazis, P. Petrushev, Rational bases for spaces of holomorphic functions in the disc, J. Lond. Math. Soc. (2) 89 (2014), no. 2, 434–460.
- K. Ivanov, P. Petrushev, Irregular sampling of band-limited functions on the sphere, Appl. Comput. Harmon. Anal. 37 (2014), no. 3, 545–562.
- S. Dekel, G. Kerkyacharian, G. Kyriazis, P. Petrushev, Compactly supported frames for spaces of distributions associated with nonnegative self-adjoint operators, Studia Math. 225 (2014), no. 2, 115–163.
- K. Ivanov, P. Petrushev, Fast Evaluation and Irregular Sampling of Band-limited Functions on the Sphere, Proceedings of the International Conference "Constructive theory of functions - 2013" Sozopol, Prof. Marin Drinov, Academic Publishing House, Sofia 2014, pp. 115–140. ISBN 978-954-322-811-9
- 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.
- K. Ivanov, P. Petrushev, Fast memory efficient evaluation of spherical polynomials at scattered points, Adv. Comput. Math. 41 (2015), no. 1, 191–230.
- K. Ivanov, P. Petrushev, Highly effective stable evaluation of bandlimited functions on the sphere, Numer. Algorithms, 71 (2016), no. 3, 585–611.
- K. Ivanov, P. Petrushev, Harmonic Besov and Triebel-Lizorkin spaces on the ball, J. Fourier Anal. Appl. 23 (2017), no. 5, 1062–1096.
- A. G. Georgiadis, G. Kerkyacharian, G. Kyriazis, P. Petrushev, Homogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators, J. Math. Anal. Appl. 449 (2017), no. 2, 1382–1412.
- M. Lind, P. Petrushev, Nonlinear nonnested spline approximation, Constr. Approx. 45 (2017), no. 2, 143–191.
- 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, S. Ogawa, P. Petrushev, and D. Picard, Regularity of Gaussian processes on Dirichlet spaces, Constr. Approx. 47 (2018), 277–320.
- K. G. Ivanov, N. Pavlis, and P. Petrushev, Precise and efficient evaluation of gravimetric quantities at arbitrarily scattered points in space, Journal of Geodesy, 92 (2018), no. 7, 779–796.
- S. Dekel, G. Kerkyacharian, G. Kyriazis, and P. Petrushev, A New proof of the atomic decomposition of Hardy spaces, CONSTRUCTIVE THEORY OF FUNCTIONS, Sozopol 2016 (K. Ivanov, G. Nikolov and R. Uluchev, Eds.), pp. 59–73, Prof. Marin Drinov Academic Publishing House, Sofia, 2018.
- A. G. Georgiadis, G. Kerkyacharian, G. Kyriazis, and P. Petrushev, Atomic and Molecular Decomposition of Homogeneous Spaces of Distributions Associated to Non-negative Self-Adjoint Operators, J. Fourier Anal. Appl. 25 (2019), 3259–3309.
- G. Kerkyacharian, P. Petrushev, and Yuan Xu, Gaussian bounds for the heat kernels on the ball and simplex: Classical approach, Studia Math. 250 (2020), no. 3, 235–252.
- G. Kerkyacharian, P. Petrushev, and Yuan Xu, Gaussian bounds for the weighted heat kernels on the interval, ball, and simplex, Constr. Approx. 51 (2020), no. 1, 73–122.
- K.G. Ivanov, P. Petrushev, Nonlinear n-term approximation of harmonic functions from shifts of the Newtonian kernel, Trans. Amer. Math. Soc. 373 (2020), no. 5, 3117–3176.
- G. Cleanthous, A. Georgiadis, G. Kerkyacharian, P. Petrushev, and D. Picard, Kernel and wavelet density estimators on manifolds and more general metric spaces, Bernoulli 26 (2020), no. 3, 1832–1862.
- K.G. Ivanov, P. Petrushev, Fast and accurate evaluation of geomagnetic field elements at arbitrary points in space, Geophys. J. Int. (2020) 224, 181–190.
- K. G. Ivanov, P. Petrushev, Nonlinear spline approximation in BMO(R), CONSTRUCTIVE THEORY OF FUNCTIONS, Sozopol 2019 (B. Draganov, K. Ivanov, G. Nikolov and R. Uluchev, Eds.), pp. 73–112, Prof. Marin Drinov Publishing House, Sofia, 2020.
- A. Georgiadis, G. Kerkyacharian, P. Petrushev, Product Besov and Triebel-Lizorkin spaces with application to nonlinear approximation, Constr. Approx. https://doi.org/10.1007/s00365-019-09490-1
- K.G. Ivanov, P. Petrushev, Highly localized kernels on the sphere induced by Newtonian kernels, arXiv:1808.08637