Articles – Allan Pinkus

The alternating algorithm in a uniformly convex and uniformly smooth Banach space, Journal of Mathematical Analysis and Applications 421 (2015), 747-753. [pdf]. This article may be viewed at Science Direct.
(with G. Kutyniok, H. Rauhut and V. Temlyakov), Preface to the Special Issue on Sparse Approximate Solution of Linear Systems, in Linear Algebra and its Applications 441 (2014), 1-3. [pdf]. This preface may be viewed at Science Direct.
Smoothness and Uniqueness in Ridge Function Representation, Indagationes Mathematicae 24 (2013), 725-738.[pdf]. This article may be viewed at Science Direct. See errata.
(with V. E. Ismailov) Interpolation on Lines by Ridge Functions, Journal of Approximation Theory 175 (2013), 91-113. [pdf]. This article may be viewed at Science Direct.
On Ridge Functions, Azerbaijan Journal of Mathematics 3 (2013), 122-130. [pdf].
(with A. Kroó) On Stability of the Metric Projection Operator, SIAM Journal on Mathematical Analysis 45 (2013), 639-661.[pdf]. This article may be viewed at SIAM Journal on Mathematical Analysis.
On Best Rank n Matrix Approximations, Linear Algebra and its Applications 437 (2012), 2179-2199.[pdf]. This article may be viewed at Science Direct.
(with Y. Benyamini and A. Kroó) L¹–Approximation and Finding Solutions with Small Support, Constructive Approximation 36 (2012), 399-431.[pdf]. This article can be viewed at Springer.
(with J.M. Quesada) On Chebyshev-Markov-Krein Inequalities, Journal of Approximation Theory 164 (2012), 1262-1282. [pdf]. This article may be viewed at Science Direct.
Sparse Representations and Approximation Theory, Journal of Approximation Theory 163 (2011), 388-412 [pdf]. This article may be viewed at Science Direct.
(edited with A. Kroó and Y. Xu) In Memoriam Borislav D. Bojanov: 18 November 1944–8 April 2009, Journal of Approximation Theory 162 (2010), 1739-1765. [pdf]. This article may be viewed at Science Direct.
(with R. Pinchasi) Dominating Subsets under Projections, SIAM Journal on Discrete Mathematics 24 (2010), 910–920 [pdf]. This article may be viewed directly from SIDMA.
(with A. Kroó) Strong Uniqueness, Surveys in Approximation Theory 5 (2010), 1-91. [pdf] and Surveys in Approximation Theory.
Zero Minors of Totally Positive Matrices, Electronic Journal in Linear Algebra 17 (2008), 532-542. [pdf]. This article may be viewed directly from ELA.
(with D. Hershkowitz) On Nonnegative Sign Equivalent and Sign Similar Factorizations of Matrices, Electronic Journal in Linear Algebra 16 (2007), 162-170. [pdf]. This article may be viewed directly from ELA.
(with D. Wulbert) Extending n-Convex Functions, Studia Math. 171 (2005), 125-152. [pdf].
(with E. L. Ortiz) Herman Müntz: A Mathematician’s Odyssey, Mathematical Intelligencer 27 (2005), 22-31. [pdf].
Density in Approximation Theory, Surveys in Approximation Theory 1 (2005), 1-45. [pdf], and Surveys in Approximation Theory.
Strictly Hermitian Positive Definite Functions, Journal d’Analyse Math. 94 (2004), 293-318. [pdf].
Interpolation by Matrices, Electronic Journal in Linear Algebra 11 (2004), 281-291. [pdf]. This article may be viewed directly from ELA.
Density Methods and Results in Approximation Theory, Orlicz Centenary Volume, Banach Center Publications, Volume 64, Z. Ciesielski, A. Pelczynski, L. Skrzypczak (eds.), 173-192, Institute of Mathematics, Polish Academy of Sciences, Warszawa, 2004. [pdf].
Strictly Positive Definite Functions on a Real Inner Product Space, Advances in Computational Mathematics 20 (2004), 263-271. [pdf]. This article may be viewed at Springer.
(with C. de Boor) The B-spline Recurrence Relations of Chakalov and Popoviciu, Journal of Approximation Theory 124 (2003), 115-123. [pdf]. This article may be viewed at Science Direct.
Negative Theorems in Approximation Theory, American Mathematical Monthly 110 (2003), 900-911. [pdf]. Some corrections regarding this paper are in this file. This paper is available from JSTOR.
(with A. Atzmon) Rank Restricting Functions, Linear Algebra and its Applications 372 (2003), 305-323. [pdf]. This article may be viewed at Science Direct.
(with Uri Elias) Non-Linear Eigenvalue-Eigenvector Problems for STP Matrices, Proceedings of the Royal Society of Edinburgh: Section A 132 (2002), 1307-1331. [pdf]. See Cambridge journals.
(with Uri Elias) Non-Linear Eigenvalue Problems for a Class of Ordinary Differential Equations, Proceedings of the Royal Society of Edinburgh: Section A 132 (2002), 1333-1359. [pdf]. See Cambridge journals.
Some Remarks on Zero-Increasing Transformations, Approximation Theory X: Abstract and Classical Analysis, C. K. Chui, L. L. Schumaker, J. Stoeckler (eds.), 333-352, Vanderbilt Univ. Press., Nashville, 2002. [pdf].
(with J. M. Carnicer and J. M. Pena) On some Zero-Increasing Operators, Acta Mathematica Hungarica 94 (2002), 173-190. [pdf].
(with J. M. Carnicer and J. M. Pena) On Zero-Preserving Linear Transformations, Journal of Mathematical Analysis and Applications 266 (2002), 237-258. [pdf]. This article may be viewed at Science Direct.
Weierstrass and Approximation Theory, Journal of Approximation Theory 107 (2000), 1-66. [pdf]. This article may be viewed at Science Direct. Some additions/corrections regarding this paper are in this file.
On a Problem of G. G. Lorentz, Journal of Approximation Theory 103 (2000), 29-54. [pdf]. This article may be viewed at Science Direct. Some additional comments on this paper are here.
Approximation Theory of the MLP Model in Neural Networks, Acta Numerica 8 (1999), 143-196. [pdf]. This article may be viewed at Cambridge Online.
(with Vitaly Maiorov) Lower Bounds for Approximation by MLP Neural Networks, Neurocomputing 25 (1999), 81-91. [pdf]. This article may be viewed at Science Direct.
(with Martin D. Buhmann) Identifying Linear Combinations of Ridge Functions, Advances in Applied Mathematics 22 (1999), 103-118. [pdf]. This article may be viewed at Science Direct.
An Interlacing Property of Eigenvalues of Strictly Totally Positive Matrices, Linear Algebra and its Applications 279 (1998), 201-206.[pdf]. This article may be viewed at Science Direct.
Uniqueness of Smoothest Interpolants, East Journal of Approximation 3 (1997), 377-380. [pdf].
Approximating by Ridge Functions, in Surface Fitting and Multiresolution Methods, A. Le Mehaute, C. Rabut, L. L. Schumaker (eds.), pp. 279-292, Vanderbilt University press, Nashville, 1997. [pdf].
(with Oleg Davydov) Best Approximation and Cyclic Variation Diminishing Kernels, Journal of Approximation Theory 89 (1997), 380-423. [pdf]. This article may be viewed at Science Direct.
(with Martin Buhmann) On a Recovery Problem, Annals of Numerical Mathematics 4 (1997), 129-142. The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin. [pdf].
TDI-Subspaces of C(R^d) and Some Density Problems from Neural Networks, Journal of Approximation Theory 85 (1996), 269-287. [pdf]. This article may be viewed at Science Direct.
Spectral Properties of Totally Positive Kernels and Matrices, in Total Positivity and its Applications, M. Gasca, C. A. Micchelli (eds.), pp. 477-511, Kluwer Academic Publishers, 1996. [pdf].
Some Density Problems in Multivariate Approximation, in Approximation Theory: Proceedings of the International Dortmund Meeting IDOMAT 95, M. W. Muller, M. Felten, D. H. Mache (eds.), pp. 277-284, Akademie Verlag, 1995. [pdf].
(with B. Wajnryb) A Problem of Approximation Using Multivariate Polynomials, Uspekhi Mat. Nauk 50 (1995), 89-110, (in Russian) = Russian Math. Surveys 50 (1995), 319–340 (in English). [pdf].
(with B. Wajnryb) Multivariate Polynomials: A Spanning Question, Constructive Approximation 11 (1995), 165-180. [pdf]. This paper may be viewed at Springer.
(with C. H. FitzGerald and C. A. Micchelli) Functions that Preserve Families of Positive Semidefinite Matrices, Linear Algebra and its Applications 221 (1995), 83-102. [pdf]. This article may be viewed at Science Direct.
(with V. Ya. Lin) Approximation of Multivariate Functions, in Advances in Computational Mathematics, H. P. Dikshit, C. A. Micchelli (eds.), pp. 257-266, World Scientific, 1994. [pdf].
(with C. A. Micchelli) Variational Problems Arising from Balancing Several Error Criteria, Rendiconti di Matematica 14 (1994), 37-86. [pdf].
(with V. Ya. Lin) Fundamentality of Ridge Functions, Journal of Approximation Theory 75 (1993), 295-311. [pdf]. This article may be viewed at Science Direct.
(with D. Braess) Interpolation by Ridge Functions, Journal of Approximation Theory 73 (1993), 218-236. [pdf]. This article may be viewed at Science Direct.
(with M. Leshno, V. Ya. Lin and S. Schocken) Multilayer Feedforward Networks with a Non-Polynomial Activation Function can Approximate any Function, Neural Networks 6 (1993), 861-867. [pdf]. This article may be viewed at Science Direct. Despite what is stated on the paper, as published, it was actually submitted in February of 1992. Publication was delayed for some unknown reason.
Uniqueness in Vector-Valued Approximation, Journal of Approximation Theory 73 (1993), 17-92. [pdf]. This article may be viewed at Science Direct.
(with D. Wulbert) The Multi-Dimensional Van Neumann Alternating Direction Search Algorithm in C(B) and L¹, Journal of Functional Analysis 104 (1992), 121-148. [pdf].
(with C. A. Micchelli) Descartes Systems from Corner Cutting, Constructive Approximation 7 (1991), 161-194. [pdf].
(with H. Strauss) L¹–Approximation with Constraints, Transactions of the American Mathematical Society 322 (1990), 239-261. [pdf]. This article may be viewed at JSTOR.
(with L. Elsner and D. Hershkowitz) Functional Inequalities for Spectral Radii of Non-Negative Matrices, Linear Algebra and its Applications 129 (1990), 103-130. [pdf].
(with C. A. Micchelli) Some remarks on nonnegative polynomials on polyhedra. Probability, Statistics, and Mathematics: Papers in honor of Samuel Karlin, 163-186, Academic Press, Boston, MA, 1989. [pdf].
(with H. Strauss) Best Approximation with Coefficient Constraints, IMA Journal of Numerical Analysis 8 (1988), 1-22. [pdf].
(with B. Wajnryb) Necessary Conditions for Uniqueness in L¹–Approximation, Journal of Approximation Theory 53 (1988), 54-66. [pdf].
Continuous Selections for the Metric Projection on C₁, Constructive Approximation 4 (1988), 85-96. [pdf].
On Smoothest Interpolants, SIAM Journal of Mathematical Analysis 19 (1988), 1431-1441. [pdf].
(with H. Strauss) One-Sided L¹–Approximation to Differentiable Functions, Approximation Theory and its Applications 3 (1987), 81-96. [pdf].
(with V. Totik) One-Sided L¹–Approximation, Canadian Mathematical Bulletin 29 (1986), 84-90. [pdf].
Unicity Subspaces in L¹–Approximation, Journal of Approximation Theory 48 (1986), 226-250. [pdf].
n-Widths and Optimal Recovery, Proceedings of Symposia in Applied Mathematics, Vol. 36, Ed., C. de Boor, 51-66, American Mathematics Society, Providence, 1986. [pdf].
(with J. M. Anderson, P. Erdös and O. Shisha) The Closed Linear Span of {x^k-c_k}₁^∞, Journal of Approximation Theory 43 (1985), 75-80. [pdf].
n–Widths of Sobolev Spaces in L^p, Constructive Approximation 1 (1985), 15-62.[pdf].
Some Extremal Problems for Strictly Totally Positive Matrices, Linear Algebra and its Applications 64 (1985), 141-156. [pdf].
n-Widths in Approximation Theory: A Survey, Approximation Theory IV, Eds., C. K. Chui, L. L. Schumaker, J. D. Ward, 153-186, Academic press, N. Y., 1983. [pdf].
(with G. J. O. Jameson) Positive and Minimal Projections in Function Spaces, Journal of Approximation Theory 37 (1983), 182-195. [pdf].
(with O. Shisha) A Variation on the Chebyshev Theory of Best Approximation, Constructive Function Theory, 81, 479-481, Sofia, 1983. [pdf].
(with C. de Boor and S. Friedland) Inverses of Infinite Sign Regular Matrices, Transactions of the American Mathematics Society 274 (1982), 59-68. [pdf].
(with C. de Boor) The Approximation of a Totally Positive Band Matrix by Strictly Totally Positive One, Linear Algebra and its Applications 42 (1982), 81-98. [pdf].
(with C. de Boor and R. Q. Jia) Structure of Invertible (Bi)-infinite Totally Positive Matrices, Linear Algebra and its Applications 47 (1982), 41-55. [pdf].
(with O. Shisha) Variations on the Chebyshev and L^q Theories of Best Approximation, Journal of Approximation Theory 35 (1982), 148-168. [pdf].
Best Approximations by Smooth Functions, Journal of Approximation Theory 33 (1981), 147-178. [pdf].
Bernstein’s Comparison Theorem and a Problem of Braess, Aequationes Mathematicae 23 (1981), 98-107. [pdf].
(with C. A. Micchelli) On n-Widths in L^∞, II: Some related extremal problems, Constructive Function Theory, 77, 403-416, Sofia, 1980. [pdf].
(with L. Brutman) On the Erdos Conjecture concerning Minimal Norm Interpolation on the Unit Circle, SIAM Journal of Numerical Analysis 17 (1980), 373-375. [pdf].
(with Z. Ziegler) Interlacing Properties of the Zeros of the Error Functions in Best L^p Approximation, Journal of Approximation Theory 27 (1979), 1-18. [pdf].
On n-Widths of Periodic Functions, Journal d’Analyse Mathematique 35 (1979), 209-235. [pdf].
Matrices and n-Widths, Linear Algebra and its Applications 27 (1979), 245-278. [pdf].
(with C. A. Micchelli) The n-Widths of Rank n+1 Kernels, Journal of Integral Equations 1 (1979), 111-130. [pdf].
(with J. W. Lee) Spectral Properties and Oscillation Theorems for Mixed Boundary-Value Problems of Sturm-Liouville Type, Journal of Differntial Equations 27 (1978), 190-213. [pdf].
(with C. A. Micchelli) Some Problems in the Approximation of Functions of Two Variables and n-Widths of Integral Operators, Journal of Approximation Theory 24 (1978), 51-77. [pdf].
(with C. de Boor) Proof of the Conjecture of Bernstein and Erdos concerning the Optimal Nodes for Polynomial Interpolation, Journal of Approximation Theory 24 (1978), 289-303. [pdf].
Some Extremal Problems of Perfect Splines and the Pointwise Landau Problem on the Finite Interval, Journal of Approximation Theory 23 (1978), 37-64. [pdf].
(with C. A. Micchelli) Moment Theory for Weak Chebyshev Systems with Applications to Monosplines, Quadrature Formulae and Best One-Sided L¹ Approximation by Spline Functions with Fixed Knots, SIAM Journal of Mathematical Analysis 8 (1977), 206-230. [pdf].
(with C. A. Micchelli) On n-Widths in L^∞, Transactions of the American Mathematics Society 234 (1977), 139-174. [pdf].
(with C. A. Micchelli) Total Positivity and the Exact n-Width of Certain Sets in L¹, Pacific Journal of Mathematics 71 (1977), 499-515. [pdf].
(with C. A. Micchelli) On a Best Estimator for the Class M^r using only Function Values, Indiana University Mathematics Journal 26 (1977), 751-759. [pdf].
(with C. A. Micchelli) Best Mean Approximation to a 2-dimensional Kernel by Tensor Products, Bulletin of the American Mathematical Society 83 (1977), 400-402. [pdf].
(with C. de Boor) Backward Error Analysis for Totally Positive Linear Systems, Numerische Mathematik 27 (1977), 485-490. [pdf].
One-Sided L¹-Approximation by Splines with Fixed Knots, Journal of Approximation Theory 18 (1976), 130-135. [pdf].
A Simple Proof of the Hobby-Rice Theorem, Proceedings of the American Mathematical Society 60 (1976), 82-84. [pdf].
(with C. A. Micchelli) On n-Widths and Optimal Recovery in M^r, Approximation Theory, II, Eds., G. G. Lorentz, C. K. Chui, L. L. Schumaker, 475-478, Academic Press, N. Y., 1976.[pdf].
(with C. A. Micchelli) The Exact Asymptotic Value for the n-Width of Smooth Functions in L^∞, Approximation Theory, II, Eds., G. G. Lorentz, C. K. Chui, L. L. Schumaker, 469-474, Academic Press, N. Y., 1976. [pdf].
(with S. Karlin) Divided Differences and other Non-linear Existence Problems at Extremal Points, in Studies in Spline Functions and Approximation Theory, S. Karlin, C. A. Micchelli, A. Pinkus, I. J. Schoenberg, 327-352, Academic Press, N. Y., 1976. [pdf].
(with S. Karlin) Interpolation by Splines with Mixed Boundary Conditions, in Studies in Spline Functions and Approximation Theory, S. Karlin, C. A. Micchelli, A. Pinkus, I. J. Schoenberg, 305-325, Academic Press, N. Y., 1976. [pdf].
(with S. Karlin) An Extremal Problem of Multiple Gaussian Nodes, in Studies in Spline Functions and Approximation Theory, S. Karlin, C. A. Micchelli, A. Pinkus, I. J. Schoenberg, 143-162, Academic Press, N. Y., 1976.[pdf].
(with S. Karlin) Gaussian Quadrature Formulae with Multiple Nodes, in Studies in Spline Functions and Approximation Theory, S. Karlin, C. A. Micchelli, A. Pinkus, I. J. Schoenberg, 113-141, Academic Press, N. Y., 1976. [pdf].
Applications of Representation Theorems to problems of Chebyshev Approximation with Constraints, in Studies in Spline Functions and Approximation Theory, S. Karlin, C. A. Micchelli, A. Pinkus, I. J. Schoenberg, 83-111, Academic Press, N. Y., 1976. [pdf].
Asymptotic Minimum Norm Quadrature Formulae, Numerische Mathematik 24 (1975), 163-175. [pdf].
Representation Theorems for Tchebycheffian Polynomials with Boundary Conditions and their Applications, Israel Journal of Mathematics 17 (1974), 11-34. [pdf].
(with S. Karlin) Oscillation Properties of Generalized Characteristic Polynomials for Totally Positive and Positive Definite Matrices, Linear Algebra and its Applications 8 (1974), 281-312. [pdf].