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List of Publications

Books:

  1. Weighted Polynomial Approximation, World Scientific, Singapore, 1996.
  2. Guest Editor : Mathematical Aspects of Neural Networks, Special Issue of Advances in Computa-tional Mathematics, 5 (1996) (With C. A. Micchelli).
  3. Fundamentals of Approximation Theory, Narosa Publishing Co., Delhi, 2000 (With D. V. Pai).
  4. Local analysis of spectral and scattered data; in preparation, to be publised by Atlantis Publications (Springer Verlag).
  5. Representation of Functions on Big Data, in preparation, Atlantis Press (Springer Verlag) (With C. K. Chui).
  6. Co-Editor: Wavelet Analysis and Applications, Proceedings of the international workshop in Delhi, 1999, Narosa Publishing Co., Delhi, 2001. (With P. K. Jain, M. Krishnan, J. Prestin, and D. Singh).
  7. Co-Editor: Frontiers in interpolation and approximation, Dedicated to the memory of Ambikeshwar Sharma, Chapman and Hall/CRC, Boca Raton, Florida, 2006 (With N. K. Govil, R. N. Mohapatra, Z. Nashed, and J. Szabados).
  8. Co-Editor: Special volumes of Journal of Approximation Theory in memory of George G. Lorentz, Volumes 158, 159 (two issues each), Elsevier, 2009 (With C. K. Chui)

Articles:

  1. Weighted polynomial approximation in rearrangement invariant Banach function spaces on the whole real line; Indian J. Math., 22 (3)(1980), 209-224 (with G. Freud).
  2. K-functionals and moduli of continuity in weighted polynomial approximation; Arkiv for Mat., 21 (1983), 145-161 (with G. Freud).
  3. Weighted analogues of Nikolskii-type inequalities and their applications; in Proc. Conference on Harmonic Analysis in Honor of A. Zygmund (Becker, et al., eds.), Vol. II (1983), Wadsworth International: Belmont, 783-801.
  4. Weighted polynomial approximation of entire functions, I; J. of Approximation Theo., 35 (1982), 203-213.
  5. Weighted polynomial approximation of entire functions, II; J. of Approximation Theo., 33 (1981), 59-68.
  6. Comonotone approximation by splines of piecewise monotone functions: J. of Approximation Theo. 35 (1982) 364-369 (with D. Leviatan).
  7. The rate of monotone spline approximation in the Lp-norm; SIAM J. of Math. Anal., 13(5) 1982, 866-874 (with D. Leviatan).
  8. On a problem of G. Freud; J. of Math, Anal. and Appl., 96 (1983), 395-404.
  9. On the domain of convergence of expansions in polynomials orthogonal with respect to generalweight functions on the whole real line; Acta. Math. Acad. Sci. Hungar., 44(3-4) (1984), 223-227.
  10. Extremal problems associated with polynomials with exponential weights; Trans. Amer. Math.Soc., 285 (1984), 223-234 (with E.B. Sa↵).
  11. Extremal problems for polynomials with Laguerre weights; in Approximation Theory, IV; College Station Texas, 1983, (Chui, et. al. eds.), Academic Press, 1983, 619-624 (with E.B. Sa↵).
  12. Polynomials with Laguerre weights in Lp; in Proc. Conf. Rational approximation and interpolation, Tampa, Florida, 1983, (Graves-Morris, et. al. eds.) 511-523 (with E.B. Sa↵).
  13. On the smoothness of Fourier transform; in the Proc. International Symposium on Interpolation spaces, Lund. Sweden, 1983 (Cwickel, Peetre, eds.), Lecture Notes, 1070, Springer Verlag, Berlin, 1984, 202-207.
  14. A trace theorem for caloric functions; International Journal of Math and Mathematical Sciences, B(1) (1985), 29-35.
  15. Extensions of Dirichlet-Jordan convergence criterion to a general class of orthogonal polynomial expansions; J. of Approx. Theo., 42 (1984), 138-148.
  16. Where does the sup norm of a weighted polynomial live? (A Generalization of incomplete polyno- mials); Constructive Approximation, 1 (1985), 71-91 (with E.B. Sa↵).
  17. Weighted polynomials on finite and infinite intervals; A unified approach; Bulletin American Math- ematical Society 11 (1984), 351-354 (with E.B. Sa↵).
  18. Weighted polynomial approximation; J. of Approx. Theo., 46 (1986), 100-110 (invited survey paper).
  19. A Weierstrass-type theorem for certain weighted polynomials; in Approximation Theory and Ap- plications (S.P. Singh ed.), Pitman, Boston, 1985, 115-123 (with E.B. Sa↵).
  20. A quantitative Dirichlet-Jordan type theorem for orthogonal polynomial expansions; SIAM J. of Math., Anal., 19 (1988), 484-492.
  21. Where does the Lp-norm of weighted polynomial live?, Trans. Amer. Math. Soc., 303 (1987), 109-124, Errata: 308 (1988), 431. (with E.B. Sa↵).
  22. Freud’s conjecture for exponential weights, Bull. Amer. Math. Soc., 15 (1986), 217-221 (with D.S. Lubinsky and E.B. Sa↵).
  23. A proof of Freud’s conjecture for exponential weights, Const. Approx. 4 (1988), 65-83. (With D.S. Lubinsky and E.B. Sa↵).
  24. A weighted transfinite diameter, in Approximation Theory V (Chui et. al. eds.), Academic Press, 1986, 479-481.
  25. A rate of convergence theorem for expansions in Freud polynomials, J. Approx. Theo. 55 (1988), 150-171.
  26. Some Discrepancy theorems; in “Approximation Theory, Tampa” (E.B. Sa↵ ed.) Vol. 1287, Lecture Notes in Mathematics, Springer Verlag, Berlin, 1987, 117-131.
  27. Weighted analogue of capacity, transfinite diameter and Chebyshev constant; Constr. Approx., 8 (1991), 105-124. (with E.B. Sa↵)
  28. The distribution of zeros of asymptotically extremal polynomials; J. Approx. Theo. 65 (1991), 279–300 (with E.B.Sa↵).
  29. The convergence of Fourier series and a K-functional; J. Math. Anal. Appl., 154 (1991), 134-141.
  30. Approximation in certain intermediate spaces; J. Approx. Theory, 62 (1990), 110-132.
  31. On the distribution of zeros of polynomials orthogonal on the unit circle; J. Approx. Theory, 63 (1990), 30-38. (with E.B. Sa↵).
  32. Bounds for certain Freud polynomials; J. Approx. Theory, 63 (1990), 238-254.
  33. On the n-width for weighted approximation of entire functions; in Approximation Theory VI, (Chui et al eds.), Academic Press, 1989, 429-432. (With C.A. Micchelli.)
  34. Hermite interpolation at the zeros of Freud polynomials; Acta Math. Hung., 60 (1992), 225-240 (With Y. Xu).
  35. The rate of convergence of a Hermite interpolation process; in Approximation Theory VI, (Chui et al eds.), Academic Press, 1989, 433-436. (With Y. Xu)
  36. A general study of maximal robust stability regions; Circuits, Systems, and Signal Processing, 10 (1991), 15-30 (With C.K. Chui).
  37. The mean convergence of expansions in Freud-type orthogonal polynomials; SIAM J. Math. Anal., 22 (1991), 847-855 (With Y. Xu).
  38. On multivariate robust stability; SIAM J. Control and Optimization, 30 (1992), 1190-1199 (With C.K. Chui).
  39. General Markov-Bernstein and Nikolskii-type inequalities; Approximation Theory and its Applica- tions, 6:4 (1990), 107–117.
  40. Finite-infinite range inequalities in the complex plane; Int. J. Math. and Math. Sci., 14 (1991), 625-638.
  41. Weighted polynomials, radial basis functions and potentials in locally compact spaces; Num. Funct. Anal. and Optimization, 11 (9& 10), 1990-91, 987–1017.
  42. A general discrepancy theorem; Arkiv f ̈or Mathematik, 31 (1993), 219-246 (With H.–P. Blatt).
  43. Detection of singularities using segment approximation; Mathematics of Computation, 59 (1992),533-540 (With R. Grothmann).
  44. On trigonometric wavelets; Constr. Approx., 9 (1993), 167-190 (With C.K. Chui).
  45. Approximation by superposition of a sigmoidal function and radial basis functions; Advances in Applied Mathematics, 13 (1992), 350-373 (With C.A. Micchelli).
  46. Weighted approximation of entire functions on unbounded subsets of the complex domain; Bull. Canadian Math. Soc., 36 (1993), 303-313.
  47. Approximation properties of a multilayerd feedforward artificial neural network; Advances in Com- putational Mathematics, 1 (1993), 61-80.
  48. Neural networks for localized approximation; Mathematics of Computation, 63 (1994), 607-623 (With C. K. Chui and X. Li).
  49. Neural networks for localized approximation of real functions; In “Neural Networks for Signal Processing III”, (C.A. Kamm et. al. eds.), IEEE, New York, 1993, 190-196.
  50. Dimension-independent bounds on approximation by neural networks; IBM J. of Research and Development,38 (1994), 277-284 (With C. A. Micchelli).
  51. How to choose an activation function; in “Neural Information Processing Systems, 6”, (J. D. Cowan, G. Tesauro, J. Alspector Eds.), Morgan Kaufmann Publishers, San Fransisco, 1993, pp. 319-326 (With C. A. Micchelli).
  52. A discrepancy theorem concerning polynomials of best approximation in Lpw[1,1]; Monatshafte Mathematik, 120 (1995), 91-103. (With H.–P. Blatt).
  53. Approximation of real functions using neural networks; in “Proc. of Int. Conf. on Computational Mathematics, New Delhi, India, 1993”, (C. A. Micchelli ed.), World Scientific, 1994, pp. 267-278.
  54. Degree of approximation by neural and translation networks with a single hidden layer, Advances in Applied Mathematics, 16 (1995), 151-183. (With C. A. Micchelli).
  55. Neural networks for optimal approximation of smooth and analytic functions; Neural Computation, 8 (1996), 164- 177.
  56. Versatile Gaussian networks; Proceedings of IEEE Workshop on Nonlinear Image and Signal Pro- cessing, (I. Pitas Editor), Halkidiki, Greece, June, 1995, IEEE, pp.70-73.
  57. Some discrepancy theorems in the theory of weighted polynomial approximation; Journal of Math- ematical Analysis and Applications, 219 (1998), 312–330 (With H.–P. Blatt).
  58. Limitations of the approximation capabilities of a neural network with a single hidden layer, Ad- vances in Computational Mathematics, 5 (1996), 233-243. (With C. K. Chui and X. Li).
  59. On smooth activation functions; in “Mathematics of Neural Networks, Models, Algorithms, and Applications”, (S. W. Ellacott, J. C. Mason, and I. J. Anderson Eds.), Kluwer Academic Publishers, 1997, pp. 275–279.
  60. Neural networks for function approximation; in “Neural networks for signal processing, V”, (F. Girosi, J. Makhoul, E. Manolakos, E. Wilson Eds.), IEEE, New York, 1995, pp.21-29. (With L. Khachikyan).
  61. Neural networks for functional approximation and system identification; Neural Computation, 9 (1997), 143–159. (With N. Hahm)
  62. System identification using neural networks; in “Neural networks for signal processing, VI”, (S. Usui, Y. Tohkura, S. Katagiri, and E. Wilson Eds.), pp. 82–88, IEEE, New York, 1996, (With N. Hahm).
  63. Bounded quasi-interpolatory polynomial operators; Journal of Approximation Theory, 96 (1999), 67–85. (With J. Prestin).
  64. On Marcinkiewicz-Zygmund-Type Inequalities; in “Approximation theory: in memory of A. K. Varma”, (N. K. Govil, R. N. Mohapatra, Z. Nashed, A. Sharma, and J. Szabados Eds.), Marcel Dekker, 1998, pp.389–404. (With J. Prestin)
  65. On a choice of sampling nodes for optimal approximation of smooth functions by generalized trans- lation networks; in Artificial Neural Networks, Conference Publication No. 440 (IEE), 1997, pp. 210-215. (With J. Prestin)
  66. Neural beam-steering and direction finding; in Neural Networks in Engineering Systems, (A. B. Bulsari and S. Kallio Eds.), Royal Institute of Technology, Stockholm, 1997, pp. 269-272. (With H. Southall)
  67. Approximation of smooth functions by neural networks; Invited survey, in “Dealing with complexity: A neural network approach”, (K. Warwick et. al. eds), “Perspectives in Neural Computing”, Springer Verlag, London, 1998, pp.189–204.
  68. Polynomial frames for the detection of singularities; in “Wavelet Analysis and Multiresolution Methods” (Ed. Tian-Xiao He), Lecture Notes in Pure and Applied Mathematics, Vol. 212, Marcel Decker, 2000, 273–298. (With J. Prestin).
  69. On a sequence of fast decreasing polynomial operators; in: Applications and Computation of Or- thogonal Polynomials (Eds. W. Gautschi, G.H. Golub, G. Opfer) Internat. Ser. Numer. Math., Birkh ̈auser, Basel, 1999, 165-178. (With J. Prestin).
  70. On the detection of singularities of a periodic function; Advances in Computational Mathematics, 12 (2000), 95–131 (With J. Prestin).
  71. Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature. Math. Comp. 70 (2001), no. 235, 1113–1130. (With F. J. Narcowich and J. D. Ward). (Corrigendum: Math. Comp. 71 (2001), 453–454.)
  72. Approximation Properties of Zonal Function Networks Using Scattered Data on the Sphere; Ad- vances in Computational Mathematics, 11 (1999), 121–137 (With F. J. Narcowich and J. D. Ward).
  73. Representing and analyzing scattered data on the sphere; in “Multivariate approximation and applications” (A. Pinkus, D. Leviatan, N. Dyn, and D. Levin Eds.), Cambridge University Press, Cambridge, 2001, pp. 44–72. (With F. J. Narcowich and J. D. Ward).
  74. On a build-up polynomial frame for the detection of singularities; in “Self-Similar Systems” (V. B. Priezzhev and V. P. Spiridonov Eds.), Joint Institute for Nuclear Research, Dubna, Russia, 1999, pp. 98–109. (With J. Prestin).
  75. Quasi-interpolation in shift invariant spaces; J. Math. Anal. and Appl. 251 (2000), 356-363. (With F. J. Narcowich and J. D. Ward).
  76. Polynomial frames on the sphere; Adv. Comput. Math. 13 (2000), no. 4, 387–403. (With F. J. Narcowich, J. Prestin, and J. D. Ward).
  77. Zonal function network frames on the sphere; Neural Networks, 16(2) (2003), 183–203 (With F. J. Narcowich and J. D. Ward).
  78. Approximation theory and neural networks; in ”Wavelet Analysis and Applications, Proceedings of the international workshop in Delhi, 1999” (P. K. Jain, M. Krishnan, H. N. Mhaskar J. Prestin, and D. Singh Eds.), Narosa Publishing, New Delhi, India, 2001, 247–289.
  79. Neural network frames on the sphere; in “Neural networks for signal processing, X, (B. Widrow, L. Guan, K. Paliwa, T. Adall, J. Larsen, E. Wilson, and S. Douglas Eds.), IEEE, New York, 2000, pp. 175–184. (With F. J. Narcowich and J. D. Ward).
  80. Approximation with Interpolatory Constraints; Proc. Amer. Math. Soc. 130 (2002), no. 5, 1355–1364 (With F. J. Narcowich, N. Sivakumar, and J. D. Ward).
  81. A local discrepancy theorem; Indiagationes Math., 12(1), (2001), 23–39. (With V. V. Andrievskii and H.-P. Blatt).
  82. On the representation of band limited functions using finitely many bits; Journal of Complexity, 18 (2002), no. 2, 449–478.
  83. On the degree of approximation in multivariate weighted approximation; in “Advanced Problems in Constructive Approximation” (Proceedings of the IDOMAT 2001 conference) (M.D. Buhmann and D.H. Mache Eds.), ISNM 142, Birkh ̈auser, Basel, 2002, pp.129–141.
  84. On the representation of band-dominant functions on the sphere using finitely many bits; Advances in Computational Mathematics, 21 (2004), 127–146. (With F. J. Narcowich and J. D. Ward).
  85. When is approximation by Gaussian networks necessarily a linear process?; Neural Networks, 17 (2004), 989–1001.
  86. Local quadrature formulas on the sphere; Journal of Complexity, 20 (2004), 753–772 (Among the top 25 most downloaded articles, October–December 2004)).
  87. A tribute to G ́eza Freud; Journal of Approximation Theory, 126 (2004), 1–15. (Invited paper).
  88. On the tractability of multivariate integration and approximation by neural networks; Journal of Complexity, 20 (2004), 561–590 (Among the top 25 most downloaded articles, October– December 2004)).
  89. Local quadrature formulas on the sphere, II; in “Advances in Constructive Approximation” (M. Neamtu and E. B. Sa↵ eds), Nashboro Press, Nashville, 2004, pp. 333–344.
  90. On local smoothness classes of periodic functions; Journal of Fourier Analysis and Applications, 11 (3) (2005), 353 – 373 (with J. Prestin).
  91. Polynomial operators and local smoothness classes on the unit interval; Journal of Approximation Theory, 131(2004), 243-267.
  92. On the representation of smooth functions on the sphere using finitely many bits; Applied and Computational Harmonic Analysis 18, Issue 3 , May 2005, Pages 215-233 (Among the top 25 most downloaded articles, April-September 2005)..
  93. A Markov–Bernstein inequality for Gaussian networks; in “Trends and applications in constructive approximation” (M. G. de Bruin, D. H. Mache, and J. Szabados eds.), ISNM 105, Birkh ̈auser Verlag, Bassel, 2005, pp. 165–180.
  94. Polynomial frames: a fast tour; in “Approximation Theory XI, Gatlinburg, 2004” (C. K. Chui, M. Neamtu, and L. Schumaker Eds.), Nashboro Press, Brentwood, 2005, 287–318. (With J. Prestin) (Invited paper).
  95. Polynomial operators and local approximation of solutions of pseudo-di↵erential equations on the sphere; Numerische Mathematik, 103 (2006), 299–322. (With Q. T. Le Gia).
  96. Matrix–free interpolation on the sphere; SIAM J. Numer. Analysis 44 (3) (2006), pp. 1314–1331. (With M. Ganesh).
  97. Weighted quadrature formulas and approximation by zonal function networks on the sphere; Journal of Complexity Theory, 22 (3), June 2006, 348–370 (Among the top 25 most downloaded articles, July–September 2005).
  98. Quadrature–free quasi–interpolation on the sphere; Electronic Transactions on Numerical Analy- sis,25 (2006), 101–114 (With M. Ganesh).
  99. On bounded interpolatory and quasi-interpolatory polynomial operators; in “Frontiers in interpo- lation and approximation” (N. K. Govil, H. N. Mhaskar, R. N. Mohapatra, Z. Nashed, and J. Szabados eds.), Chapman and Hall/CRC, Boca Raton, 2006, pp. 345–364 (Invited paper).
  100. Quadrature in Besov spaces on the Euclidean sphere; Journal of Complexity, 23 (2007), 528–552 (With K. Hesse and I. H. Sloan) (Among the top 25 most downloaded articles, October– December 2007)).
  101. Polynomial operators for spectral approximation of piecewise analytic functions; Appl. Comput. Harmon. Anal. 26 (2009) 121–142 (With J. Prestin).
  102. Di↵usion polynomial frames on metric measure spaces; Applied and Computational Harmonic Anal- ysis, Volume 24, Issue 3, May 2008, Pages 329-353 (With M. Maggioni) (Among the top 25 most downloaded articles, April-June 2008).
  103. Localized linear polynomial operators and quadrature formulas on the sphere; SIAM J. Numer. Anal. 47 (1) (2008), 440–466. (With Q. T. Le Gia).
  104. Polynomial operators and local smoothness classes on the unit interval, II; Ja ́en J. of Approx., 1 (1) (2009), 1–25. (Invited paper)
  105. On a filter for exponentially localized kernels based on Jacobi polynomials; Journal of Approxima- tion Theory 160 (2009), pp. 256-280 (With F. Filbir and J. Prestin)
  106. A quadrature formula for di↵usion polynomials corresponding to a generalized heat kernel; Journal of Fourier Analysis and Applications 16 (2010), 629–657 (with F. Filbir).
  107. Lp Bernstein estimates and approximation by spherical basis functions; Math. Comp. 79 (2010), no. 271, 1647–1679. (With F. Narcowich, J. Prestin, and J. D. Ward)
  108. Eignets for function approximation; Appl. Comput. Harmon. Anal. 29 (2010) 63–87.
  109. MRA Contextual-Recovery Extension of Smooth Functions on Manifolds; Appl. Comput. Harmon. Anal. 28 (2010) 104–113 (With C. K. Chui).
  110. Minimum Sobolev norm interpolation with trigonometric polynomials on the torus; Journal of Computational Physics, 249 (2013) 96–112 (With S. Chandrasekaran and K. R. Jayaraman).
  111. Minimum Sobolev Norm schemes and applications in image processing. In IS&T/SPIE Electronic Imaging, International Society for Optics and Photonics, (2010, February), pp. 753507-753507 (with Chandrasekaran, S., Jayaraman, K. R., Mott, J., and Pauli, S.).
  112. Higher order numerical discretization methods with Sobolev norm minimization, Accepted for pub- lication in the Proceedings of International Conference on Computational Science, ICCS 2011, Procedia Computer Science, Elsevier (With S. Chandrasekaran, K. R. Jayaraman, M. Gu, and J. Mott).
  113. A generalized di↵usion frame for parsimonious representation of functions on data defined manifolds; Neural Networks 24 (2011) 345–359.
  114. Wiener type theorems for Jacobi series with nonnegative coecients; Proceedings of the American Mathematical Society, 140(3) (2012), 977–986 (With S. Tikhonov).
  115. Marcinkiewicz–Zygmund measures on manifolds; Journal of Complexity, 27 (2011), 568–596 (With F. Filbir).
  116. On the problem of parameter estimation in exponential sums; Constructive Approximation 35 (3)(2012), 323–343 (With F. Filbir and J. Prestin).
  117. Learning Biomedical Data Locally using Di↵usion Geometry Techniques; in Proceeding (771) Imag- ing and Signal Processing in Health Care and Technology / 772: Human–Computer Interaction / 773: Communication, Internet and Information Technology – 2012, (R. Merrell, D.-G. Shin, M.H. Hamza Eds) Baltimore, 2012, pp. 125–131. (With M. Ehler, F. Filbir)
  118. Quadrature formulas for integration of multivariate trigonometric polynomials on spherical tri- angles; International Journal on Geomathematics, 3 (2012), 119–138 (With J. Beckman and J. Prestin).
  119. Locally learning biomedical data using di↵usion frames; Journal of Computational Biology, 19, (11) (2012), 1251–1264 DOI: 10.1089/cmb.2012.0187 (With M. Ehler, F. Filbir).
  120. Smooth function extension based on high dimensional unstructured data; Mathematics of Compu- tation, 290(83)(2014), 2865–2891 (With C. K. Chui).
  121. Applications of classical approximation theory to RBF networks and computational harmonic anal- ysis; Bull. Math. Sci. 3 (2013), 485–549 (With P. Nevai and E. Shvarts).
  122. Legendre filters for numerical di↵erentiation at boundary point; Applied Mathematics and Compu- tation 224 (2013) 835–847 (With Valeriya Naumova and Sergei V. Pereverzyev)
  123. Representation of functions on big data: graphs and trees; Applied and Computational Harmonic Analysis, Volume 38, Issue 3, May 2015, Pages 489-509 (With C. K. Chui and F. Filbir).
  124. Local numerical integration on the sphere; Int. J. Geomath 5 (2014),143–162 (with J. Beckman and J. Prestin).
  125. A minimum Sobolev norm technique for the numerical discretization of PDEs; Submitted for pub- lication (with S. Chandrasekaran).
  126. 126. Signal decomposition and analysis via extraction of frequencies; Applied and Computational Har- monic Analysis,Published online, January 13, 2015 (with C. K. Chui).