# Ronald DeVore

**The Walter E. Koss Professor and Distinguished Professor of Mathematics**

## Education

## Professional Experience

## Visiting Positions

## Awards and Prizes

## Ph.D. Students

## Professional Service

### Invited Talks:

Over 200 colloquia and 100 international conferences

American Mathematical Society Invited Plenary Address, 1990

SIAM Invited Plenary Addresses, 1992, 2000, 2009, 2014

Canadian Mathematical Society Invited Plenary Address, 1994

Foundations of Computational Mathematics Invited Plenary Addresses, 1998, 2002

International Congress of Mathematicians Invited Plenary Address, 2006

### Review Panels:

NSF Classical Analysis Program Review, 1994

NSF Panel Reviews, 1997, 2000, 2002

South Carolina EPSCoR Committee for NSF ASEE, 1995

Science Foundation of Ireland Mathematics Program Review, 2005

### Conferences Co-Organized:

### Editorial and Advisory Boards:

### Special Issues:

### Committees:

### Other:

## Publications

### Monographs

*Approximation of Continuous Functions by Positive Linear Operators*, Springer Lecture Notes in Mathematics, 293 (1972).(with R. Sharpley),

*Maximal Functions Measuring Smoothness*, Memoirs of AMS, 293 (1984).(with G.G. Lorentz),

*Constructive Approximation*, Springer Grundlehren, 303 (1993).

### Expository Articles

*Freud's work in contructive function theory*, Journal of Approximation Theory, 46 (1) (1986) 32-37.*Approximation of functions*, Approximation Theory, AMS Short Course, Jan. 1986 (C. de Boor, Ed.), Proc. Symp. Applied Mathematics, AMS 36 (1986) 1-20.(with Brad Lucier)

*Wavelets*, Acta Numerica, 1 (1991), 1-56.*Nonlinear Approximation*, Acta Numerica, 7 (1998), 51-150.(with D. Donoho, M. Vetterli, and I. Daubechies),

*Data Compression and harmonic analysis*, IEEE Transactions on Information Theory Numerica, 44 (6) (1998), 2435-2476.*Adaptive Numerical Methods for PDEs*, 32nd Congress National D'Analyse Numerique, Canum 2000, Port d'Albret, June 2000, A51-A65(with A. Cohen and W. Dahmen),

*Adaptive Wavelet Techniques in Numerical Simulation*, Encyclopedia of Computational Mathematics, John Wiley & Sons (Stein, deBorst, and Hughes, Eds.) Vol. 1, 2004.*Optimal Computation*, Proceedings of the International Congress of Mathematicians, Madrid, (2007), 187-215.*Nonlinear approximation and its application*, in: Multiscale, Nonlinear, and Adaptive Approximation, R. DeVore and A Kunoth, Spring, pages 169-201, 2009.*Foundations of compressed sensing, capturing functions in high dimensions, and capturing functions in infinite dimensions*, Course notes, Paris, 2009.*The Theoretical Foundation of Reduced Basis Methods*, Model Reduction and Approximation: Theory and Algorithms. (P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, Eds.), SIAM, Philadelphia, PA, (2017), 137-168.(with A. Cohen)

*Approximation of high-dimensional parametric PDEs*, Acta Numerica, 24 (2015), 1-159.(with M.A. Davenport and R. Baraniuk)

*Compressive Sensing*, Connexions, Houston, TX, 2011.(with A. Ron)

*Developing a Computation-Friendly Mathematical Foundation for Spline Functions*, SIAM News, 38 (4) (2005).*Some Fundamental Issues in Computational Mathematics*, Foundations of Computational Mathematics, Minneapolis, (2002).*Lorentz spaces*, International Congress of Mathematicians, (2022).

### Book Reviews

*Interpolation Theory, Function Spaces, Differential Operators*, by Hans Triebel, Bull. AMS, 2 (1980), 339-345.*N-widths*, by Allan Pinkus, Jahresbericht, 86 (1985), 43-44.

### Research Articles

(with R. Bojanic),

*On polynomials of best one-sided approximation*, Enseignement Math., 12 (1966), 139-164.R. A. DeVore,

*One-sided approximation of functions*, J. Approximation Theory, 1 (1) (1968), 11-25.R. A. DeVore,

*On L*_{p}*approximation of functions whose m*^{th}*derivative is of bounded variation*, Studia Scientiarum Math. Hung., 3 (1968) 443-450.R. A. DeVore,

*On Jackson's theorem*, J. Approx. Theory, 1 (3) (1968) 314-318.R. A. DeVore,

*Multipliers of uniform convergence*, Enseignement Math., 14 (1968) 175-188.(with R. Bojanic),

*A proof of Jackson's theorem*, Bull. AMS, 75 (1969) 364-367.R. A. DeVore,

*Optimal convergence of positive linear operators*, in "Proc. Symp. on the Constructive Theory of Functions", eds. G Alexits and S.B. Stechkin, Akademia Kiado, Budapest, (1969) 101-119.R. A. DeVore,

*Saturation of positive convolution operators*, J. Approx. Theory, 3 (4) (1970) 410-429.R. A. DeVore,

*On a saturation theorem of Tureckii*, Tohoku Math. J., 23 (3) (1971) 353-362.R. A. DeVore,

*On the direct theorem of saturation*Tohoku Math. J., 23 (3) (1971) 363-370.R. A. DeVore,

*A pointwise "o" saturation theorem for positive convolution operators*, "Linear Operators and Approximation," (P.L. Butzer, J.P. Kahane, and B. Sz-Nagy, Eds.) ISNM 20, Birkhauser, Basel (1972), 364-370.R. A. DeVore,

*Inverse theorems for approximation by positive linear operators*, in "Spline Functions and Approximation Theory," (A. Meir and A. Sharma, Eds.) ISNM 21, Birkhauser, Basel, (1973) 371-376.(with F. Richards),

*Saturation and inverse theorems for spline approximation*, in "Spline Functions and Approximation Theory", (A. Meir and A. Sharma, Eds.) ISNM 21, Birkhauser, Basel, (1973) 73-82.(with A. Sharma and A. Meir),

*Strongly and weakly non-poised HB-interpolation problems*, Canadian J. Math., 25 (1973) 1040-1050.(with F. Richards),

*The degree of approximation by Chebyshevian splines*, Transactions AMS, 181 (1973) 401-418.R. A. DeVore,

*An extension of Bernstein's inequality*, in "Approximation Theory," ed. G.G. Lorentz, Academic Press, New York, 1973, 125-129.R. A. DeVore,

*A property of Chebyshev polynomials*, J. Approx. Theory, 12 (4) (1974) 418-419.R. A. DeVore,

*Minimum moduli of differential operators from the viewpoint of approximation theory*, J. Approx. Theory, 13 (2) (1975) 128-135.R. A. DeVore,

*Degree of monotone approximation*, in "Linear Operators and Approximation II," eds. P.L. Butzer and B. Sz. Nagy, ISNM 25, Birkhauser, Basel, 1974, 337-351.(with J. Szabados),

*Saturation theorems for discretized linear operators*, Analysis Mathematica, 1 (2) (1975) 81-89.R. A. DeVore,

*Monotone approximation by splines*, SIAM J. Math. Anal., 8 (5) (1977) 891-905.R. A. DeVore,

*Monotone approximation by polynomials*, SIAM J. Math. Anal., 8 (5) (1977) 906-921.R. A. DeVore,

*Pointwise approximation by polynomials and splines*,"Proceedings of the Conference on Constructive Function Theory, Kaluga," Academy of Science USSR, Moscow, 1975, 132-144.R. A. DeVore,

*Degree of approximation*, in "Approximation II," eds. G.G. Lorentz, C.K. Chui, and L.L. Schumaker, Academic Press, New York, (1976) 117-162.(with H. Berens),

*Quantitative Korovkin theorems for positive linear operators on L*_{p}*spaces*, Trans. AMS, 245 (1978) 349-361.(with K. Scherer),

*A constructive theory for approximation by splines with an arbitrary sequence of knot sets*, in "Approximation Theory," eds. R. Schaback and K. Scherer, Springer Lecture Notes in Math., Vol. 556, Springer, Berlin, (1976) 167-183.(with S. Riemenschneider and R. Sharpley),

*Weak interpolation in Banach spaces*, J. Functional Analysis, 33 (1) (1979) 58-94.R. A. DeVore,

*L*_{p}*[-1, 1] approximation by algebraic polynomials*, in "Linear Spaces and Approximation," eds. P.L. Butzer and B. Sz. Nagy, ISNM. Vol. 40, Birkhauser, Basel, (1978) 397-406.(with K. Scherer),

*Interpolation of linear operators on Sobolev spaces*, Annals of Math. 109 (1979) 583-589.(with W. Dahmen and K. Scherer),

*Multidimensional spline approximation*, SIAM J. Numer. Anal., 17 (3) (1980) 380-402.(with H. Berens),

*A characterization of Bernstein polynomials*, in "Approximation III", ed. E.W. Cheney, Academic Press, New York, 1980, 213-220.(with C. de Boor and K. Höllig),

*Mixed norm n-widths*, Proceedings AMS, 80 (4) (1980) 577-583.R. A. DeVore,

*Embeddings of Besov spaces into BMO*, in "Approximation and Function Spaces," ed, Z. Ciesielski, Virdirldki Rfiyot, Warsaw, 1979, 259-263.(with K. Scherer),

*Variable knot, variable degree spline approximation to xΒ*, in "Quantitative Approximation", (R. DeVore and K. Scherer, Eds.) Academic Press, New York, 1980, 121-131.(with C. Bennett and R. Sharpley),

*Weak L∞ and BMO*, Annals of Math., 113 (1981) 601-611.(with C. Bennett and R. Sharpley),

*Maximal singular integrals on L∞*, in "Functions, Series, Operators", Colloquium Math. Soc. Janos Bolyai, Budapest, 1980, 233-236.(with C. de Boor),

*Approximation by smooth multivariate splines*, Transactions AMS, 276 (2) (1983) 775-788.(with L. R. Scott),

*Error bounds for Gaussian quadrature and weighted-L1 polynomial approximation*, SIAM J. Num. Anal., 21 (2) (1984) 400-412.R. A. DeVore,

*Maximal functions and their application to rational approximation*, in "Approximation Theory", (Z. Ditzian, A. Meir, S. Riemenschneider and A. Sharma, Eds.) CMS Conf. Proc., Vol. 3, CMS-AMS, Providence, (1983) 143-155.(with C. de Boor and K. Höllig),

*Approximation order from smooth bivariate pp functions*, in "Approximation IV", eds. C. Chui, L. Schumaker, and J. Ward, Academic Press, New York, 1983, 353-357.(with R. Sharpley),

*Shorter Notes: On the differentiability of functions in ℜn*(preprint), Proceedings AMS, 91 (2) (1984) 326-328.(with S. Riemenschneider and R. Sharpley),

*n-widths for C*_{p}^{α}*spaces*, in "Anniversary Volume on Approximation Theory and Functional Analysis," (P.L. Butzer, R. Stens, and B. Sz. Nagy, Eds.) ISNM, Vol.65, Birkhauser, (1984) 213-222.R. A. DeVore,

*The n-width of BV ∩ Lipα*, in "Anniversary Volume on Approximation Theory and Functional Analysis", eds. P.L. Butzer, R. Stens, and B. Sz. Nagy, ISNM, Vol.65, Birkhauser, (1984) 223-230.R. A. DeVore,

*The K-functional for H1 and BMO*, in "Proc. of the Conference on Interpolation Spaces and Allied Topics in Analysis" (M. Cwikel and J. Peetre, Eds.) Springer Lecture Notes, Vol 1070, Springer, Berlin, (1984) 66-79.(with C. de Boor),

*Partitions of unity and approximation*, Proceedings AMS, 93 (4) (1985) 705-709.(with X.M. Yu),

*Pointwise estimates for monotone approximation*, Constructive Approximation, 1 (1) (1985) 323-331.(with C. de Boor),

*A geometric proof of total positivity for spline interpolation*, Mathematics of Computation, 45 (1985) 497-504.(with X.M. Yu),

*Mulitivariate rational approximation*, Transactions AMS, 293 (1) (1986) 161-169.(with Z. Yan),

*Error analysis for piecewise curve fitting algorithms*, Computer Aided Geometric Design, 3 (3) (1986) 205-215.R. A. DeVore,

*Approximation by rational functions*, Proceedings AMS, 98 (4) (1986) 601-604.R. A. DeVore,

*A note on adaptive approximation*, Approx. Theory & its Appl., 3 (4) (1987) 74-78.(with V. Popov),

*Interpolation spaces and non-linear approximation*, in "Function Spaces and Applications, (M. Cwikel, J. Peetre, Y. Sagher, and H. Wallin, Eds.) Springer Lecture Notes in Math., Vol. 1302, Springer, Berlin, (1988) 191-205.(with V. Popov),

*Interpolation of Besov spaces*, Transactions AMS, 305 (1) (1988) 397-414.(with V. Popov),

*Free multivariate splines*, Constructive Approximation, 3 (1) (1987) 239-248.(with H. Kierstead and G.G. Lorentz),

*A proof of Borsuk's theorem*, in "Functional Analysis", (E. Odell and H. Rosenthal, Eds.) Springer Lecture Notes in Math., Vol. 1332, Springer, Berlin, (1988) 195–202.(with V. Popov),

*Interpolation of approximation spaces*, in "Constructive Theory of Functions 87," Publ. House of Bulg. Acad. Sci., Sofia, (1988) 110-119.(with B. Lucier),

*High order regularity for conservation laws*, Indiana Math. J., 39 (2) (1990) 413-430.(with R. Howard and C. Micchelli),

*Optimal non-linear approximation*, Manuscripta Math., 63 (4) (1989) 469-478.(with B. Lucier),

*High order regularity for the solution of the inviscid Burger's equation*(preprint), in Nonlinear Hyperbolic Problems, Proceedings of the Advanced Workshop held in Bordeaux, France, June, 13-17, 1988, Springer Lecture Notes in Mathematics, Vol. 1402, (C. Carrasso, B. Hanouzet, J-L. Joy, Eds.), Springer Verlag, New York, (1989) 147-154.(with B. Jawerth and V. Popov),

*Compression of wavelet decompositions*, American Journal of Mathematics, 114 (4) (1992) 737-785.R. A. DeVore,

*Degree of nonlinear approximation*, in "Approximation VI," eds. C. Chui, J. Ward, and L. Schumaker Academic Press, New York, (1989) 175-201.(with X.M. Yu),

*Nonlinear n-widths of Besov Spaces*, in "Approximation VI, eds. C. Chui, J. Ward, and L. Schumaker, Academic Press, New York, 1989, pp. 203.(with W. Dahmen and C. Micchelli),

*On monotone extensions of boundary data*, Numerische Mathematik, 60 (1) (1992) 477-492.(with X.M. Yu),

*Degree of adaptive approximation*, Math. Comp., 55 (1990) 625-635.(with X.M. Yu),

*The K-functional for Besov spaces*, J. Approximation Theory, 67 (1) (1991) 38-50.(with D. Leviatan and X.M. Yu),

*L*_{p}*Approximation by reciprocals of trigonometric and algebraic polynomials*, Canadian Math. Bull., 33 (4) (1990) 460-469.(with B. Jawerth and B. Lucier),

*Surface compression*, Computer Aided Geometric Design, 9 (3) (1992) 219-239.(with B. Jawerth and B. Lucier),

*Image compression through wavelet transform coding*, IEEE Proceedings on Information Theory, 38 (2) (1992) 719-746.(with B. Jawerth and B. Lucier),

*Data compression using wavelets: error, smoothness, and quantization*, in DCC-91, Data Compression Conference, J.A. Storer and J.H. Reif, ed, IEEE Computer Society Press, Los Alamitos, CA, 1991, 186-195.(with D. Leviatan and X.M. Yu),

*Polynomial approximation in L*_{p}*(0 ≤ p ≤ 1)*, Constructive Approximation, 8 (2) (1992) 187-201.(with R. Sharpley),

*Besov spaces on domains in R*^{d}, TAMS, 335 (2) (1993) 843-864.(with P. Petrushev and X.M. Yu),

*Nonlinear wavelet approximation in the space C*, Progress in Approximation Theory, Springer Verlag, New York, 1992, 261-283.(with D. Leviatan),

*Convex polynomial approximation in L*_{p}*(0 ≤ p ≤ 1)*, J. Approximation Theory, 75 (1) (1993).(with C. de Boor and A. Ron),

*Approximation from shift invariant subspaces of L*_{2}*(R*^{d}*)*, TAMS, 341 (2) (1994) 787-806.(with C. de Boor and A. Ron),

*On the construction of multivariate (pre)wavelets.*, Constructive Approximation, 9 (2-3) (1993) 123-166.(with C. de Boor and A. Ron),

*The structure of finitely generated shift invariant spaces in L*_{2}*(R*^{d}*)*, J. Functional Analysis, 119 (1) (1994) 37-78.(with B. Lucier),

*On the size and smoothness of solutions to nonlinear hyperbolic conservation laws*, SIAM J. Math. Analysis, 27 (3) (1996) 684-707.(with G. Kyriazis, D. Leviatan, and V.M. Tikhomirov),

*Wavelet compression and nonlinear n-widths*, Advances in Computational Math., 1 (2) (1993) 197-214.(with B. Lucier),

*Fast wavelet techniques for near-optimal image processing*. in 1992 IEEE Military Communications Conference, IEEE Communications Society, (1992) 1129-1135.(with A. Chambolle, B. Lucier, and Y. Lee ),

*Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage*, IEEE Image Processing, 7 (3) (1998) 319-335.(with B. Lucier),

*Smoothness spaces and wavelet decomposition*, SPIE 1830 (1992) 2-12.(with B. Lucier, M. Kallergi, Wei Qian, R. Clark, E. Saff, and L.P. Clarke),

*Wavelet compression and segmentation of mammographic images*, Journal of Digital Imaging, 7 (1994) 27-38.(with P. Petrushev and V. Temlyakov),

*Multivariate trigonometric polynomial approximations with frequencies from the hyperpolic cross*, Math. Zametki, 56 (3) (1994) 36-63.(with V. Temlyakov),

*Nonlinear approximation by trigonometric sums*, Journal of Fourier Analysis and Applications, 2 (1) (1995) 29-48.(with S. Konjagin and V. Temlyakov),

*Hyperbolic wavelet approximation*, Constructive Approximation, 14 (1) (1998) 1-26.R. A. DeVore,

*Adaptive wavelet bases for image compression,*in Wavelets, Images, and Surface Fitting (P.J. Laurent, A. Le Mehaute, and L.L. Schumaker, eds.), A K Peters, Boston, 1994, 197-219.(with Z. Yang, M. Kallergi, B. Lucier, W. Qian, R. A. Clark, and L.P. Clarke),

*The effect of wavelet bases on compression of digital mammograms*, IEEE Engineering in Medicine and Biology, 14 (5) (1995) 570-577.(with B. Lucier),

*Classifying the smoothness of images: theory and applications to wavelet image processing*, in Proceedings of ICIP-94, Vol. II, IEEE Computer Society Press, (1994) 6-10.(with Y.K. Hu and D. Leviatan),

*Convex polynomial and spline approximation in L*_{p}*, 0 ≤ p ≤ ∞*, Constructive Approximation, 12 (3) (1996) 409-422.(with A. Cohen and W. Dahmen),

*Multiscale decompositions on bounded domains*, Transactions AMS, 352 (8) (2000) 3651-3685.(with V. Temlyakov),

*Some remarks on greedy algorithms*, Advances in Computational Math., 5 (1) (1996) 173-187.(with K. Oskolkov and P. Petrushev),

*Approximation by feed-forward neural networks*, Annals of Numerical Mathmetics, 4 (1996) 261-288.(with B. Lucier and Z. Yang),

*Feature extraction in digital mammography*, in Wavelets in Biology and Medicine (Eds. Akram Aldroubi and Michael Unser), CRC, Boca Rotan, Florida, 1996, 145-161.(with S. Dahlke),

*Besov Regularity for Elliptic Boundary Value Problems*, Comm. Partial Differential Equations, 22 (1-2) (1997) 1-16.(with C. de Boor and A. Ron),

*Approximation orders of FSI spaces in L*_{2}*(ℜ*^{d}*)*, Constructive Approximation, 14 (4) (1998) 631-652.(with S. Dahlke and E. Dahmen),

*Nonlinear approximation and adaptive techniques for solving elliptic operator equations*, in Multiscale Techniques for PDEs (Eds. W. Dahmen, A. Kurdila, and P. Oswald), Academic Press, San Diego, 1997, 237-283.(with V. Temlyakov),

*Nonlinear approximation in finite dimensional spaces*, Journal of Complexity, 13 (4) (1997) 489-508.(with G. Kyriazis and P. Wang),

*Multiscale characterizations of Besov spaces on bounded domains*, J. Approximation Theory, 93 (2) (1998) 273-292.(with A. Cohen and R. Hochmuth),

*Restricted nonlinear approximation*, Constructive Approximation, 16 (1) (2000) 85-113.(with A. Cohen, P. Petrushev, and H. Xu),

*Nonlinear approximation and the space BV(ℜ*^{2}*)*, American Journal of Math., 121 (3) (1999) 587-628.(with W. Shao, J. Pierce, K. Kaymaz, B. Lerner, W. Campbell),

*Using nonlinear wavelet compression to enhance image registration*, SPIE, 3078 (1997) 539-551.(with H. Wang, J. Liu and H. Xu),

*A CFL-free explicit scheme with compression for linear hyperbolic equations*, Lecture Notes in Physics, 552 (2000) 116-123.(with D. Leviatan and I. A. Shevchuk),

*Approximation of monotone functions: A counter example*, 1996 Proceedings of the Chamonix Conference (Eds. A. Le Mehaute, C. Rabut, and L.L. Schumaker), Vanderbilt University Press, 1997, 95-102.(with A. Cohen, W. Dahmen),

*Adaptive wavelet methods for elliptic operator equations: convergence rates*, Math. Comp., 70 (233) (2000) 27-75.(with A. Cohen, W. Dahmen, and I. Daubechies),

*Tree Approximation and Optimal Encoding*, ACHA, 11 (2) (2001) 192-226.(with A. Cohen, G. Kerkyacharian, and D. Picard),

*Maximal Spaces with given rate of convergence for thresholding algorithms*, ACHA, 11 (2) (2001) 167-191.(with G. Petrova),

*The Averaging Lemma*, J. Am. Math. Soc., 14 (2) (2000) 279-296.(with R. Baraniuk, G. Kyriazis, and X.M. Yu),

*Near Best tree approximation*, Advances in Comp. Math, 16 (4) (2002) 357-373.(with A. Cohen, W. Dahmen),

*Adaptive wavelet methods II - Beyond the elliptic case*, J. FoCM, 2 (3) (2002) 203-245.(with A. Cohen, W. Dahmen, and I. Daubechies),

*Harmonic Analysis of the Space BV*, Revista Matematica Iberoamericana, 19 (2003) 235-263.(with L.S. Johnson, C. Pan, and R. Sharpley),

*Optimal entropy encoders for mining multiply resolved data*, Data Mining II (N. Ebecken and C.A. Brebbia, Eds.), WIT Press, Boston, 2000, 73-82.(with I. Daubechies, C. S. Gunturk, and V. Vaishampayan),

*A/D Conversion with an Imperfect Quantizers*, IEEE Trans. Inf. Theory, 52 (2006) 874-885.(with I. Daubechies, C. S. Gunturk, and V. Vaishampayan),

*Beta Expansions: A New Approach to Digitally Corrected A/D Conversion*, Proceedings of the IEEE Intl. Symposium of Circuits and Systems, Scottsdale, Arizona, May 2002, 26-29.(with I. Daubechies),

*Approximating a bandlimited function from very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order*(preprint), Annals of Mathematics, 158 (2003) 679-710.(with A. Cohen and W. Dahmen),

*Sparse Evaluation of Compositions of Functions Using Multiscale Expansions*, SIAM J. Math. Anal., 35 (2) (2003) 279-303.(with G. Petrova and V. Temlyakov),

*Best Basis Selection for Approximation in L*_{p}, J. of FoCM, 3 (2003) 161-185.(with A. Cohen and W. Dahmen),

*Adaptive Wavelet Schemes for Nonlinear Variational Problems*, SIAM J. Numer. Anal., 41 (5) (2003) 1785-1823.(with Peter Binev),

*Fast Computation in Adaptive Tree Approximation*, Numerische Mathematik, 97 (2004) 193-217.(with Peter Binev and W. Dahmen),

*Adaptive Finite Element Methods with Convergence Rates*, Numerische Mathematik, 97 (2004) 219-268.(with A. Petukhov and R. C. Sharpley),

*Motion Estimation with the Redundant Wavelet Transform*, The Third International Workshop on Digital and Computational Video, M.A. Bayoumi (ed.), IEEE Proceedings, St. Petersburg, Fl., 2002, 53-59.(with Peter Binev, W. Dahmen, and P. Petrushev),

*Approximation Classes for Adaptive Methods*, Serdica Math. J., 28 (4) (2002) 391-416.(with A. Cohen, W. Dahmen, and M. Campos-Pinto),

*On the stability of nonlinear conservation laws in the Hausdorff metric*, J. Hyperbolic Differential Equations, 2 (2005) 25-38.(with P. Bechler, A. Kamont, G. Petrova, and P. Wojtaszczyk),

*Greedy Wavelet Projections are Bounded on BV*, Trans. Amer. Math. Soc., 359 (2007) 637-648.(with P. Binev, W. Dahmen, and N. Dyn),

*Adaptive Approximation of Curves*, in: Approximation Theory: A volume dedicated to Borislav Bojanov (D.K. Dimitrov, G. Nikolov, and R. Uluchev, Eds.), Marin Drinov Academic Publishing House, Sofia (2004), 43-57.(with G. Kerkyacharian, D. Picard and V. Temlyakov),

*Approximation Methods for Supervised Learning*(preprint), J. of FOCM, 6 (2006) 3-58.(with P. Binev, A. Cohen, W. Dahmen, and V. Temlyakov),

*Universal Algorithms for Learning Theory Part I: Piecewise Constant Functions*, J. Machine Learning, 6 (2005) 1297-1321.(with A. Kurdila, M. Nechyba, R. Lind, P. Ifju, P. Binev, W. Dahmen, and R. Sharpley),

*Vision-Based Control of Micro-Air-Vehicles: Progress and Problems in Estimation*, 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, December 2004, 1636-1642.(with H. Narfi Stefansson, Kevin W. Eliceiri, Charles F. Thomas, Amos Ron, Robert Sharpley, and John G. White),

*Wavelet Compression of Three-Dimensional Time-Lapse Biological Image Data*, Microsc. Microanal., 11 (2005) 9-17.(with P. Binev, A. Cohen, and W. Dahmen),

*Universal Algorithms for Learning Theory Part II: Piecewise Polynomials*, Constructive Approximation, 26, No. 2 (2007) 127-152.(with G. Petrova and P. Wojtaszczyk),

*Anisotropic Smoothness Spaces via Level Sets*, Comm. Pure and Appl. Math., b61 (2008), 1264-1297.(with R. Baraniuk, M. Davenport, and M. Waken),

*A simple proof of the restricted isometry property for random matrices*, Constructive Approximation, 28 (2008), 253-263.(with A. Barron A. Cohen, and W. Dahmen),

*Approximation and Learning by Greedy Algorithms*, Annals of Statistics, 36 (2008) 64-94.(with A. Cohen and W. Dahmen),

*Compressed sensing and best k-term approximation*, JAMS, 22 (2009), 211-231(with A. Ron),

*Approximation using Scattered Shifts of a Multivariate Function*, TAMS, 362 (2010), 6205-6229.(with A. Cohen and W. Dahmen),

*Instance Optimal Decoding by Thresholding in Compressed Sensing*, in: Harmonic Analysis and Partial Differential Equations, Contemporary Mathematics, 505 (2010), 1-28.(with P. Binev, A. Cohen, and W. Dahmen),

*Universal Piecewise Polynomial Estimators for Machine Learning*in: Curves and Surface Design, Proceedings of the Avignon Conference (2006), (A. Cohen, J.L. Merrien, L. Shumaker, Eds.), Nashboro Press, 2007, 48-78.R. A. DeVore,

*Deterministic constructions of compressed sensing matrices*, Journal of Complexity, 23 (2007) 918-925.(with A. Cohen and W. Dahmen),

*A taste of compressed sensing*, Proceedings of the SPIE Conference, 6576 (2007).(with A. Cohen, and C. Schwab),

*Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs*, Foundations of Computational Mathematics, 10 (2010), 615-646.(with G. Petrova and Przemek Wojtaszczyk),

*Instance optimality in probability with and**l*_{1}*-minimization decoder*, Appl. Comput. Harmon. Anal., 27 (2009), 275-288.(with I. Daubechies, M. Fornasier and S. Gunturk),

*Iteratively re-weighted least squares minimization for sparse recovery*, Communications on Pure and Applied Mathematics, 63, No. 1 (2010), 1-38.(with G. Petrova and P. Wojtaszczyk),

*Approximating Functions of Few Variables in High Dimensions*, Constructive Approximation, 33 (2011), 125-143.(with A. Cohen and C. Schwab),

*Analytic Regularity and Polynomial Approximation of Parametric Stochastic Elliptic PDEs*, Analysis and Applications, 9 (2011), 11-47.(with A. Cohen, I. Daubechies, G. Kerkyacharian, and D. Picard),

*Capturing Ridge Functions in High Dimensions from Point Queries*, Constructive Approximation, 35 (2012), 225-243.(with P. Binev, A. Cohen, W. Dahmen, G. Petrova, and P. Wojtaszczyk),

*Convergence Rates for Greedy Algorithms in Reduced Basis Methods*(preprint), SIAM J. Math. Anal. 43 (2011), 1457-1472.(with P. Binev, W. Dahmen, P. Lamby, D. Savu, R. Sharpley)

*Compressed Sensing and Electron Microscopy*, Modeling Nanoscale Imaging in Electron Microscopy, Springer, (2011), 73-126.(with A. Belochitski, P. Binev, M. Fox-Rabinovitz, V. Krasnopolski, and P. Lamby)

*Tree Approximation of the Long Wave Radiation Parameterization in the NCAR CAM Global Climate Model*, Journal of Computation and Applied Mathematics, 236 (2011) 447-460.(with A. Chkifa, A. Cohen, R. DeVore, and C. Schwab),

*Sparse Adaptive Taylor Approximation Algorithms for Parametric and Stochastic Elliptic PDEs*, M2AN Math. Model. Numer. Anal., 47 (2013), 253-280.(with G. Petrova, M. Hielsberg, L. Owens, B. Clack, and A. Sood)

*Processing Terrain Point Cloud Data*, SIAM J. Imaging Sci., 6(1)(2013), 1-31.(with G. Petrova and P. Wojtaszczyk)

*Greedy Algorithms for Reduced Bases in Banach Spaces*, Constructive Approximation, 37 (2013), 455-466.(with P. Binev, A. Cohen and W. Dahmen)

*Classification Algorithms using Adaptive Partitioning*, Annals of Statisitics, 42(6)(2014), 2141-2163.(with A. Cohen and R. Nochetto)

*Convergence Rates of AFEM with H*^{-1}*Data*, JofCM 12 (2012), 671-718.(with A. Bonito and R. Nochetto),

*Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients*, SIAM J. Numer. Anal. 51-6 (2013), 3106-3134.(with A. Cohen, G. Petrova, and P. Wojtaszczyk),

*Finding the minimum of a function*, Methods and Applications of Analysis, 20(4)(2013), 365-382.(with M. Bachmayr, W. Dahmen, and L. Grasedyck),

*Approximation of High-Dimensional Rank One Tensors*, Constructive Approximation, 39 (2014), 385-395.(with E. Zuazua),

*Recovery of initial temperature from discrete sampling*, M3AS, 24(12)(2014), 2487-2501.(with V. Temlyakov),

*Convex optimization on Banach Spaces*(preprint), Foundations of Computational Mathematics, 16(2)(2016), 369-394.(with A. Cohen)

*Kolmogorov widths under holomorphic mappings*(preprint), IMA J. Numer. Anal., 36(1)(2016), 1-12.(with W. Dahmen, L. Grasedyck, and E. Süli)

*Tensor-Sparsity of Solution to High-Dimensional Elliptic Partial Differential Equations*(preprint), Foundations of Computational Mathematics, 16(4)(2016), 813-874.(with I. Daubechies, M. Fornasier and S. Gunturk),

*Iteratively Re-weighted Least Squares minimization: Proof of faster than linear rate for sparse recovery*, Information Sciences and Systems, (2008), 26-29.(with A. Cohen and W. Dahmen),

*Orthogonal Matching Pursuit under the Restricted Isometry Property*, Constructive Approximation, 45(1) (2017), 113-127.(with P. Binev, A. Cohen, W. Dahmen, G. Petrova, and P. Wojtaszczyk),

*Data Assimilation in Reduced Modeling*(preprint), SIAM UQ, 5(2017), 1-29.(with A. Cohen, S. Foucart, H. Rauhut),

*Recovery of functions of many variables via compressive sensing*, In Proc. SampTA 2011, Singapore, (2010).(with G. Petrova and P. Wojtaszczyk),

*Data assimilation and sampling in Banach spaces*(preprint, arXiv:1602.06342), Calcolo, 54(2017), 963-1007.(with M. Bachmayr, A. Cohen, and G. Migliorati),

*Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients*(preprint), M2AN, 51(1)(2016), 341-363.(with A. Bonito, A. Cohen, G. Petrova, and G. Welper),

*Diffusion Coefficients Estimation for Elliptic Partial Differential Equations*, (preprint), SIAM J. Math. Anal., 49(2)(2017), 1570-1592.(with S. Foucart, G. Petrova, and P. Wojtaszczyk),

*Computing a quantity of interest from observational data*, (preprint) Constructive Approximation, 49 (3) (2019), 461-508.(with A. Cohen, W. Dahmen and J. Nichols),

*Reduced Basis Greedy Selection Using Random Training Sets*, (preprint), M2AN Math. Model. Numer. Anal., 54 (2020), 1509-1524.(with A. Cohen, W. Dahmen, J. Fadili, O. Mula and J. Nichols),

*Optimal reduced model algorithms for data-based state estimation*, (preprint), SIAM Journal on Numerical Analysis, 58 (6), 3355-3381.(with I. Daubechies, S. Foucart, B. Hanin and G. Petrova),

*Nonlinear Approximation and (Deep) ReLU Networks*, arXiv preprint arXiv:1905.02199.(with A. Bonito, D. Guignard, P. Jantsch and G. Petrova),

*Polynomial Approximation of Anisotropic Analytic Functions of Several Variables*, (preprint), Constructive Approximation, (2020), 1-30.(with A. Bonito, A. Cohen, D. Guignard, P. Jantsch and G. Petrova),

*Nonlinear Methods for Model Reduction*, arXiv preprint, arXiv:2005.02565.(with A. Cohen and W. Dahmen),

*Adaptive Wavelet Techniques in Numerical Simulation*, Encyclopedia of Computational Mechanics Second Edition, (2018), 1-41.(with A. Cohen and W. Dahmen),

*State Estimation--The Role of Reduced Models*, Recent Advances in Industrial and Applied Mathematics, 2022, 57-77.(with A. Cohen, G. Petrova, and P. Wojtaszczyk),

*Optimal Stable Nonlinear Approximation*, Foundations of Computational Mathematics 22(2022), 607-648.(with B. Hanin and G. Petrova),

*Neural Network Approximation*, Acta Numerica 30(2021), 327-444.(with with I. Daubechies, N. Dym, S. Faigerbaum-Golovin, S.Z. Kovalsky, J. Park, and G. Petrova, B. Sobel), Neural network approximation of refinable functions, IEEE Transactions on Information Theory, to appear.

(with P. Binev, A. Bonito, and G. Petrova, Optimal learning, ArXiv: 2203 15994.