Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-01T03:20:05.665Z Has data issue: false hasContentIssue false
This chapter is part of a book that is no longer available to purchase from Cambridge Core

References

Apoorva Khare
Affiliation:
Indian Institute of Science, Bangalore
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agler, Jim and McCarthy, John Edward. What can Hilbert spaces tell us about bounded functions in the bidisk? In A Glimpse at Hilbert Space Operators (Axler, S., Rosenthal, P., Sarason, D., Eds.), pp. 8197. Operator Theory: Advances and Applications, Vol. 207, Basel: Birkhäuser, 2010.Google Scholar
Agler, Jim, McCarthy, John Edward, and Young, Nicholas John. Operator Analysis: Hilbert space methods in complex analysis. Cambridge Tracts in Mathematics, Vol. 219, Cambridge: Cambridge University Press, 2020.Google Scholar
Aissen, Michael, Edrei, Albert, Schoenberg, Isaac Jacob, and Whitney, Anne M.. On the generating functions of totally positive sequences. Proc. Natl. Acad. Sci. USA, 37(5): 303307, 1951.Google Scholar
Aissen, Michael, Schoenberg, Isaac Jacob, and Whitney, Anne M.. On the generating functions of totally positive sequences I. J. d’Analyse Math., 2: 93103, 1952.CrossRefGoogle Scholar
Ait-Haddou, Rachid and Mazure, Marie-Laurence. The fundamental blossoming inequality in Chebyshev spaces – I: Applications to Schur functions. Found. Comput. Math., 18(1): 135158, 2018.CrossRefGoogle Scholar
Ait-Haddou, Rachid, Sakane, Yusuke, and Nomura, Taishin. Chebyshev blossominginMüntz spaces: Toward shaping with Young diagrams. J. Comput. Appl. Math., 247: 172208, 2013.Google Scholar
Akhiezer, Naum Ilyich. The classical moment problem and some related questions in analysis. Translated by Kemmer, N.. New York: Hafner Publishing, 1965.Google Scholar
Albert, Arthur. Conditions for positive and nonnegative definiteness in terms of pseudoinverses. SIAM J. Appl. Math., 17(2): 434440, 1969.Google Scholar
Ando, Tsuyoshi. Totally positive matrices. Linear Algebra Appl., 90: 165219, 1987.Google Scholar
Bai, Zhi Dong and Zhang, Li-Xin. Semicircle law for Hadamard products. SIAM J. Matrix Anal. Appl., 29(2): 473495, 2007.CrossRefGoogle Scholar
Bakonyi, Mihály and Woerdeman, Hugo J.. Matrix completions, moments, and sums of Hermitian squares. Princeton Series in Applied Mathematics, Princeton, NJ: Princeton University Press, 2011.Google Scholar
Bapat, Ravindra B. and Raghavan, T.E.S.. Nonnegative matrices and applications. Encyclopedia of Mathematics and Its Applications, Vol. 64, Cambridge: Cambridge University Press, 1997.Google Scholar
Barbosa, Victor Simões and Menegatto, Valdir Antonio. Strict positive definiteness on products of compact two-point homogeneous spaces. Integral Trans. Spec. Funct., 28(1): 5673, 2017.Google Scholar
Beatson, Richard Keith and Castell, Wolfgang zu. Dimension hopping and families of strictly positive definite zonal basis functions on spheres. J. Approx. Theory, 221: 2237, 2017.Google Scholar
Beatson, Richard Keith, Castell, Wolfgang zu, and Xu, Yuan. A Pólya criterion for (strict) positive-definiteness on the sphere. IMAJ.Numer.Anal., 34(2): 550568, 2014.Google Scholar
Belton, Alexander, Guillot, Dominique, Khare, Apoorva, and Putinar, Mihai. Matrix positivity preservers in fixed dimension. I. Adv. Math., 298: 325368, 2016.Google Scholar
Belton, Alexander, Guillot, Dominique, Khare, Apoorva, and Putinar, Mihai. Matrix positivity preservers in fixed dimension. Comptes Rendus Math., 354(2): 143148, 2016.CrossRefGoogle Scholar
Belton, Alexander, Guillot, Dominique, Khare, Apoorva, and Putinar, Mihai. Schur polynomials and positivity preservers (Extended abstract). In FPSAC 2016 Proceedings (Vol. BC), pp. 155166. Nancy, France: Discrete Mathematics and Theoretical Computer Science (DMTCS), 2016.Google Scholar
Belton, Alexander, Guillot, Dominique, Khare, Apoorva, and Putinar, Mihai. Simultaneous kernels of matrix Hadamard powers. Linear Algebra Appl., 576: 142157, 2019.CrossRefGoogle Scholar
Belton, Alexander, Guillot, Dominique, Khare, Apoorva, and Putinar, Mihai. A panorama of positivity. Part I: Dimension free. In Analysis of Operators on Function Spaces (The Serguei Shimorin Memorial Volume; Aleman, A., Hedenmalm, H., Khavinson, D., Putinar, M., Eds.), Parts 1 and 2 (unified), pp. 117165. Trends in Mathematics, Basel: Birkhäuser, 2019. arXiv:math.CA/ 1812.05482Google Scholar
Belton, Alexander, Guillot, Dominique, Khare, Apoorva, and Putinar, Mihai. A panorama of positivity. Part II: Fixed dimension. In Complex Analysis and Spectral Theory, proceedings of the CRM Workshop held at Laval University, QC, May 21–25, 2018 (Dales, G., Khavinson, D., Mashreghi, J., Eds.), Parts 1 and 2 (unified), pp. 109–150. CRM Proceedings – AMS Contemporary Mathematics, Vol. 743, Providence, RI: American Mathematical Society, 2020. arXiv: math.CA/1812.05482Google Scholar
Belton, Alexander, Guillot, Dominique, Khare, Apoorva, and Putinar, Mihai. Totally positive kernels, Pólya frequency functions, and their transforms. J. d’Analyse Math., in press; arXiv:math.FA/2006.16213Google Scholar
Belton, Alexander, Guillot, Dominique, Apoorva Khare, and Mihai Putinar. Matrix compression along isogenic blocks. arXiv, 2020. arXiv:math.RA/ 2010.14429Google Scholar
Belton, Alexander, Guillot, Dominique, Khare, Apoorva, and Putinar, Mihai. Hirschman–Widder densities. arXiv, 2020. arXiv:math.CA/2101.02129Google Scholar
Belton, Alexander, Guillot, Dominique, Khare, Apoorva, and Putinar, Mihai. Moment-sequence transforms. J. Eur. Math. Soc., published online, DOI: 10.4171/jems/1145Google Scholar
Berenstein, Arkady, Fomin, Sergey, and Zelevinsky, Andrei. Parametrizations of canonical bases and totally positive matrices. Adv. Math., 122: 49149, 1996.Google Scholar
Berenstein, Arkady and Zelevinsky, Andrei. Total positivity in Schubert varieties. Comment. Math. Helv., 72(1): 128166, 1997.Google Scholar
Berenstein, Arkady and Zelevinsky, Andrei. Tensor product multiplicities, canonical bases and totally positive varieties. Invent. Math., 143: 77128, 2001.Google Scholar
Berg, Christian, Christensen, Jens Peter Reus, and Ressel, Paul. Positive definite functions on abelian semigroups. Math. Ann., 223(3): 253274, 1976.Google Scholar
Berg, Christian, Christensen, Jens Peter Reus, and Ressel, Paul. Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions. Graduate Texts in Mathematics, Vol. 100, New York: Springer, 1984.Google Scholar
Berg, Christian, Peron, Ana Paula, and Porcu, Emilio. Schoenberg’s theorem for real and complex Hilbert spheres revisited. J. Approx. Theory, 228: 5878, 2018.Google Scholar
Berg, Christian and Porcu, Emilio. From Schoenberg coefficients to Schoenberg functions. Constr. Approx., 45: 217241, 2017.CrossRefGoogle Scholar
Bernstein, Serge. Reçons sur les Propriétés Extrémales et la Meilleure Approximation des Fonctions Analytiques d’une Variable Réelle. Borel series of monographs, Paris: Gauthier-Villars, 1926.Google Scholar
Bernstein, Serge. Sur les fonctions absolument monotones. Acta Math., 52(1): 166, 1929.Google Scholar
Bhatia, Rajendra. Matrix analysis. Graduate Texts in Mathematics, Vol. 169, New York: Springer, 1997.Google Scholar
Bhatia, Rajendra. Positive definite matrices. Princeton Series in Applied Mathematics, Princeton, NJ: Princeton University Press, 2015.Google Scholar
Bhatia, Rajendra and Elsner, Ludwig. Positivity preserving Hadamard matrix functions. Positivity, 11(4): 583588, 2007.Google Scholar
Bickel, Peter John and Levina, Elizaveta. Covariance regularization by thresholding. Ann. Statist., 36(6): 25772604, 2008.Google Scholar
Birkhoff, Garrett. Metric foundations of geometry. I. Trans. Amer. Math. Soc., 55: 465492, 1944.Google Scholar
Blumberg, Henry. On convex functions. Trans. Amer. Math. Soc., 20: 4044, 1919.Google Scholar
Blumenthal, Leonard Mascot. New theorems and methods in determinant theory. Duke Math. J., 2(2): 396404, 1936.CrossRefGoogle Scholar
Boas Jr, Ralph Philip. The Stieltjes moment problem for functions of bounded variation. Bull. Amer. Math. Soc., 45: 399404, 1939.Google Scholar
Boas Jr, Ralph Philip. Functions with positive derivatives. Duke Math. J., 8(1): 163172, 1941.Google Scholar
Boas Jr, Ralph Philip. and Widder, David Vernon. Functions with positive differences. Duke Math. J., 7: 496503, 1940.Google Scholar
Bochner, Salomon. Monotone funktionen, Stieltjessche integrale und harmonische analyse. Math. Ann., 108(1): 378410, 1933.Google Scholar
Bochner, Salomon. Hilbert distances and positive definite functions. Ann. Math. ( 2 ), 42: 647656, 1941.Google Scholar
Bonfim, Rafaela Neves, Guella, Jean Carlo, and Menegatto, Valdir Antonio. Strictly positive definite functions on compact two-point homogeneous spaces: The product alternative. Symm., Integr. Geom.: Meth. Appl. (SIGMA), 14(112): 14 pp., 2018.Google Scholar
de Boor, Carl. On calculating with B-splines. J. Approx. Theory, 6(1): 5062, 1972.CrossRefGoogle Scholar
Borger, James. Witt vectors, semirings, and total positivity. In Absolute Arithmetic and F1-Geometry (Thas, K., Ed.), pp. 273331, Eur. Math. Soc., 2016.Google Scholar
Borger, James and Grinberg, Darij. Boolean Witt vectors and an integral Edrei– Thoma theorem. Selecta Math. (NS), 22: 595629, 2016.Google Scholar
Brenti, Francesco. Unimodal, log-concave, and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., Vol. 413, Providence, RI: American Mathematical Society, 1989.Google Scholar
Brenti, Francesco. Combinatorics and total positivity. J. Combin. Theory Ser. A, 71(2): 175218, 1995.Google Scholar
Bressoud, David Marius and Wei, Shi-Yuan. Combinatorial equivalence of definitions of the Schur function. AMS Contemp. Math., 143: 5964, (A Tribute to Emil Grosswald: Number Theory and Related Analysis), 1993.Google Scholar
Buchstaber, Victor Matveevich and Glutsyuk, Aleksei Antonovich. Total positivity, Grassmannian and modified Bessel functions. AMS Contemp. Math., 733: 97108 (Functional Analysis and Geometry: Selim Grigorievich Krein Centennial), 2019. http://dx.doi.org/10.1090/conm/733Google Scholar
Cayley, Arthur. On a theorem in the geometry of position. Cambridge Math. J., II: 267271, 1841.Google Scholar
Chen, Debao, Menegatto, Valdir Antonio, and Sun, Xingping. A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 131(9): 27332740, 2003.Google Scholar
Christensen, Jens Peter Reus and Ressel, Paul. Functions operating on positive definite matrices and a theorem of Schoenberg. Trans. Amer. Math. Soc., 243: 8995, 1978.Google Scholar
Christensen, Jens Peter Reus and Ressel, Paul. Positive definite kernels on the complex Hilbert sphere. Math. Z., 180(2): 193201, 1982.Google Scholar
Cohn, Henry and de Courcy-Ireland, Matthew. The Gaussian core model in high dimensions. Duke Math. J., 167(13): 24172455, 2018.Google Scholar
Cohn, Henry and Kumar, Abhinav. Universally optimal distribution of points on spheres. J. Amer. Math. Soc., 20(1): 99148, 2007.Google Scholar
Cohn, Henry and Woo, Jeechul. Three-point bounds for energy minimization. J. Amer. Math. Soc., 25(4): 929958, 2012.Google Scholar
Cohn, Henry and Zhao, Yufei. Sphere packing bounds via spherical codes. Duke Math. J., 163(10): 19652002, 2014.Google Scholar
Conway, John Bligh. Functions of one complex variable. Graduate Texts in Mathematics, Vol. 11, New York: Springer, 1978.Google Scholar
Cox, Trevor F. and Cox, Michael A.A.. Multidimensional scaling. In Handbook of Data Visualization (Chen, C.-h, Härdle, W., and Unwin, A., Eds.), pp. 315347, Berlin-Heidelberg: Springer, 2008. http://dx.doi.org/10.1007/978-3-540-33037-0Google Scholar
Cryer, Colin Walker. The LU-factorization of totally positive matrices. Linear Algebra Appl., 7(1): 8392, 1973.Google Scholar
Curry, Haskell Brooks and Schoenberg, Isaac Jacob. On spline distributions and their limits: The Pólya distribution functions. Bull. Amer. Math. Soc., 53(11): 1114, 1947.Google Scholar
Curry, Haskell Brooks and Schoenberg, Isaac Jacob. On Pólya frequency functions IV: The fundamental spline functions and their limits. J. d’Analyse Math., 17: 71107, 1966.CrossRefGoogle Scholar
Cryer, Colin Walker. Some properties of totally positive matrices. Linear Algebra Appl., 15(1): 125, 1976.Google Scholar
Dalestkii, Yuri L’vovich and Krein, Selim Grigorievich. Integration and differentiation of functions of Hermitian operators and applications to the theory of perturbations (Russian). Voronoež. Gos. Univ. Trudy Sem. Funkcional. Anal., Vol. 1, pp. 81105, 1956; AMS Translations: Ser. 2, Vol. 47, pp. 130, 1965.Google Scholar
Davydov, Aleksey Aleksandrovich. Totally positive sequences and R-matrix quadratic algebras. J. Math. Sci., 100(1): 18711876, 2000.Google Scholar
Descartes, Réne. Le Géométrie. Appendix to Discours de la méthode, 1637.Google Scholar
Donoghue Jr, William Francis. Monotone matrix functions and analytic continuation. Grundlehren der mathematischen Wissenschaften, Vol. 207, Berlin: Springer-Verlag, 1974.Google Scholar
Dym, Harry and Katsnelson, Victor Emmanuilovich. Contributions of Issai Schur to analysis. In Studies in Memory of Issai Schur (Joseph, A., Melnikov, A., and Rentschler, R., Eds.), pp. xci–clxxxiii, Progress in Mathematics, Vol. 210, Basel: Birkhäuser, 2003.Google Scholar
Edrei, Albert. On the generating functions of totally positive sequences II. J. d’Analyse Math., 2: 104109, 1952.Google Scholar
Edrei, Albert. Proof of a conjecture of Schoenberg on the generating function of a totally positive sequence. Canad. J. Math., 5: 8694, 1953.Google Scholar
Edrei, Albert. On the generation function of a doubly infinite, totally positive sequence. Trans. Amer. Math. Soc., 74: 367383, 1953.Google Scholar
Efron, Bradley. Increasing properties of Pólya frequency functions. Ann. Math. Statist., 36(1): 272279, 1965.Google Scholar
El Karoui, Noureddine. Operator norm consistent estimation of large-dimensional sparse covariance matrices. Ann. Statist., 36(6): 27172756, 2008.Google Scholar
Emonds, Jan and Führ, Hartmut. Strictly positive definite functions on compact abelian groups. Proc. Amer. Math. Soc., 139(3): 11051113, 2011.Google Scholar
Michael, Shaun Fallat and Charles Royal Johnson. Totally nonnegative matrices. Princeton Series in Applied Mathematics, Princeton, NJ: Princeton University Press, 2011.Google Scholar
Fallat, Shaun Michael, Johnson, Charles Royal, and Sokal, Alan David. Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples. Linear Algebra Appl., 520: 242259, 2017; Corrigendum, Linear Algebra Appl., 613: 393396, 2021.Google Scholar
Fekete, Mihály and Pólya, Georg. Über ein Problem von Laguerre. Rend. Circ. Mat. Palermo, 34: 89120, 1912.Google Scholar
Fiedler, Miroslav and Markham, Thomas. An observation on the Hadamard product of Hermitian matrices. Linear Algebra Appl., 215: 179182, 1995.Google Scholar
FitzGerald, Carl Hanson and Horn, Roger Alan. On fractional Hadamard powers of positive definite matrices. J. Math. Anal. Appl., 61(3): 633642, 1977.Google Scholar
FitzGerald, Carl Hanson, Micchelli, Charles Anthony, and Pinkus, Allan. Functions that preserve families of positive semidefinite matrices. Linear Algebra Appl., 221: 83102, 1995.Google Scholar
Fomin, Sergey and Zelevinsky, Andrei. Double Bruhat cells and total positivity. J. Amer. Math. Soc., 12: 335380, 1999.Google Scholar
Fomin, Sergey and Zelevinsky, Andrei. Total positivity: Tests and parametrizations. Math. Intelligencer, 22(1): 2333, 2000.Google Scholar
Fomin, Sergey and Zelevinsky, Andrei. Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15(2): 497529, 2002.Google Scholar
Fréchet, Maurice René. Les dimensions d’un ensemble abstrait. Math. Ann., 68: 145168, 1910.Google Scholar
Fréchet, Maurice René. Sur la définition axiomatique d’une classe d’espaces vectoriels distanciés applicables vectoriellement sur l’espace de Hilbert. Ann. of Math. (2), 36(3): 705718, 1935.Google Scholar
Friedrichs, Kurt Otto. The identity of weak and strong extensions of differential operators. Trans. Amer. Math. Soc., 55: 132151, 1944.CrossRefGoogle Scholar
Gantmacher, Feliks Ruvimovich. The theory of matrices. Vols. 1, 2. Translated by Hirsch, K.A. New York: Chelsea Publishing Co., 1959.Google Scholar
Gantmacher, Feliks Ruvimovich and Krein, Mark Grigor’evich. Sur les matrices complètement non négatives et oscillatoires. Compositio Math., 4: 445476, 1937.Google Scholar
Gantmacher, Feliks Ruvimovich and Krein, Mark Grigor’evich. Oscillation matrices and kernels and small vibrations of mechanical systems. Translated by Eremenko, A.. New York: Chelsea Publishing Co., 2002.Google Scholar
Garloff, Jürgen. Intervals of almost totally positive matrices. Linear Algebra Appl., 363: 103108, 2003.Google Scholar
Garloff, Jürgen and Wagner, David G.. Hadamard products of stable polynomials are stable. J. Math. Anal. Appl., 202(3): 797809, 1996.CrossRefGoogle Scholar
Gasca, Mariano and Micchelli, Charles A. (Eds.) Total positivity and its applications. Proceedings of Jaca Meeting (1994), Mathematics and its Applications, Vol. 359, Dordrecht: Kluwer, 1996.Google Scholar
Gessel, Ira and Viennot, Xavier. Binomial determinants, paths, and hook length formulae. Adv. Math., 58(3): 300321, 1985.Google Scholar
Gneiting, Tillmann. Strictly and non-strictly positive definite functions on spheres. Bernoulli, 19(4): 13271349, 2013.Google Scholar
Goodearl, Kenneth Ralph, Launois, Stephane, and Lenagan, Tom H.. Totally nonnegative cells and matrix Poisson varieties. Adv. in Math., 226(1): 779826, 2011.Google Scholar
Gröchenig, Karlheinz, Romero, José Luis, and Stöckler, Joachim. Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions. Invent. Math., 211: 11191148, 2018.Google Scholar
Gröchenig, Karlheinz and Stöckler, Joachim. Gabor frames and totally positive functions. Duke Math. J., 162(6): 10031031, 2013.Google Scholar
Guella, Jean Carlo, Menegatto, Valdir Antonio, and Peron, Ana Paula. Strictly positive definite kernels on a product of circles. Positivity, 21: 329342, 2017.Google Scholar
Guillot, Dominique, Khare, Apoorva, and Rajaratnam, Bala. Complete characterization of Hadamard powers preserving Loewner positivity, monotonicity, and convexity. J. Math. Anal. Appl., 425(1): 489507, 2015.Google Scholar
Guillot, Dominique, Khare, Apoorva, and Rajaratnam, Bala. Preserving positivity for matrices with sparsity constraints. Trans. Amer. Math. Soc., 368(12): 89298953, 2016.Google Scholar
Guillot, Dominique, Khare, Apoorva, and Rajaratnam, Bala. Critical exponents of graphs. J. Combin. Theory Ser. A, 139: 3058, 2016.Google Scholar
Guillot, Dominique, Khare, Apoorva, and Rajaratnam, Bala. The critical exponent: a novel graph invariant. Sem. Lothar. Combin., 78B, Article 62, 12 pp., 2017.Google Scholar
Guillot, Dominique, Khare, Apoorva, and Rajaratnam, Bala. Preserving positivity for rank-constrained matrices. Trans. Amer. Math. Soc., 369(9): 61056145, 2017.Google Scholar
Guillot, Dominique and Rajaratnam, Bala. Retaining positive definiteness in thresholded matrices. Linear Algebra Appl., 436(11): 41434160, 2012.CrossRefGoogle Scholar
Guillot, Dominique and Rajaratnam, Bala. Functions preserving positive definiteness for sparse matrices. Trans. Amer. Math. Soc., 367(1): 627649, 2015.Google Scholar
Hadamard, Jacques Salomon. Théorème sur les séries entières. Acta Math., 22: 5563, 1899.Google Scholar
Hamburger, Hans Ludwig. Über eine Erweiterung des Stieltjesschen Momenten-problems. Math. Ann., Part I, Vol. 81, pp. 235319, 1920; Part II, Vol. 82, pp. 120164, 1921; Part III, Vol. 82, pp. 168187, 1921.Google Scholar
Heiligers, Berthold. Totally nonnegative moment matrices. Linear Algebra Appl., 199 (suppl. 1): 213227, 1994.Google Scholar
Helson, Henry, Kahane, Jean-Pierre, Katznelson, Yitzhak, and Rudin, Walter. The functions which operate on Fourier transforms. Acta Math., 102(1): 135157, 1959.Google Scholar
Hero, Alfred and Rajaratnam, Bala. Large-scale correlation screening. J. Amer. Statist. Assoc., 106(496): 15401552, 2011.Google Scholar
Hero, Alfred and Rajaratnam, Bala. Hub discovery in partial correlation graphs. IEEE Trans. Inform. Theory, 58(9): 60646078, 2012.Google Scholar
Hershkowitz, Daniel, Neumann, Michael, and Schneider, Hans. Hermitian positive semidefinite matrices whose entries are 0 or 1 in modulus. Linear Multilinear Algebra, 46(4): 259264, 1999.Google Scholar
Herz, Carl S.. Fonctions opérant sur les fonctions définies-positives. Ann. Inst. Fourier (Grenoble), 13: 161180, 1963.Google Scholar
Hiai, Fumio. Monotonicity for entrywise functions of matrices. Linear Algebra Appl., 431(8): 11251146, 2009.Google Scholar
Hiai, Fumio and Petz, Denes. Introduction to matrix analysis and applications. Universitext, Cham: Springer, viii+332 pp., 2014.Google Scholar
Hilbert, David. Über die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann., 32: 342350, 1888.Google Scholar
Hilbert, David. Über ternäre definite Formen. Acta Math., 17: 169197, 1893.Google Scholar
Hirschman Jr, Isidore Isaac. and Widder, David Vernon. The inversion of a general class of convolution transforms. Trans. Amer. Math. Soc., 66(1): 135201, 1949.Google Scholar
Hirschman Jr, Isidore Isaac. and Widder, David Vernon. The convolution transform. Princeton Legacy Library, Princeton, NJ: Princeton University Press, 1955.Google Scholar
Hô Hai, Phùng. Poincaré series of quantum spaces associated to Hecke operators. Acta Math. Vietnam., 24(2): 235246, 1999.Google Scholar
Horn, Roger Alan. The theory of infinitely divisible matrices and kernels. Trans. Amer. Math. Soc., 136: 269286, 1969.Google Scholar
Horn, Roger Alan and Johnson, Charles Royal. Matrix analysis. Cambridge: Cambridge University Press, 1985.Google Scholar
Horn, Roger Alan and Johnson, Charles Royal. Topics in matrix analysis. Cambridge: Cambridge University Press, 1991.Google Scholar
Hunter, David Boss. The positive-definiteness of the complete symmetric functions of even order. Math. Proc. Camb. Phil. Soc., 82(2): 255258, 1977.CrossRefGoogle Scholar
Hurwitz, Adolf. Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. Math. Ann., 46: 273284, 1895.Google Scholar
Jain, Tanvi. Hadamard powers of some positive matrices. Linear Algebra Appl., 528: 147158, 2017.Google Scholar
Jain, Tanvi. Hadamard powers of rank two, doubly nonnegative matrices. Adv. in Oper. Theory, 5: 839849, 2020 (Rajendra Bhatia volume).Google Scholar
Jameson, Graham James Oscar. Counting zeros of generalised polynomials: Descartes’ rule of signs and Laguerre’s extensions. The Mathematical Gazette, 90(518): 223234, 2006.Google Scholar
Johnson, Charles Royal and Walch, Olivia. Critical exponents: Old and new. Electr. J. Linear Alg., 25: 7283, 2012.Google Scholar
Karlin, Samuel. Total positivity, absorption probabilities and applications. Trans. Amer. Math. Soc., 111: 33107, 1964.Google Scholar
Karlin, Samuel. Total positivity. Vol. 1. Stanford, CA: Stanford University Press, 1968.Google Scholar
Karlin, Samuel and McGregor, James. Coincidence probabilities of birth and death processes. Pacific J. Math., 9(4): 11091140, 1959.Google Scholar
Karlin, Samuel and McGregor, James. Coincidence probabilities. Pacific J. Math., 9(4): 11411164, 1959.Google Scholar
Karlin, Samuel and Rinott, Yosef. Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal., 10(4): 467498, 1980.Google Scholar
Karlin, Samuel and Rinott, Yosef. Total positivity properties of absolute value multinormal variables with applications to confidence interval estimates and related probabilistic inequalities. Ann. Statist., 9(5): 10351049, 1981.Google Scholar
Karlin, Samuel and Ziegler, Zvi. Chebyshevian spline functions. SIAM J. Numer. Anal., 3(3): 514543, 1966.Google Scholar
Khare, Apoorva. Sharp nonzero lower bounds for the Schur product theorem. Proc. Amer. Math. Soc. 149(12): 50495063, 2021.Google Scholar
Khare, Apoorva. Smooth entrywise positivity preservers, a Horn–Loewner master theorem, and symmetric function identities. Trans. Amer. Math. Soc.,inpress, DOI: http://dx.doi.org/10.1090/tran/8563Google Scholar
Khare, Apoorva. Critical exponents for total positivity, individual kernel encoders, and the Jain–Karlin–Schoenberg kernel. arXiv, 2020. arXiv:math .FA/2008.05121Google Scholar
Khare, Apoorva and Tao, Terence. Schur polynomials, entrywise positivity preservers, and weak majorization. Sem. Lothar. Combin., 80B, Article 14, 12 pp., 2018.Google Scholar
Khare, Apoorva and Tao, Terence. On the sign patterns of entrywise positivity preservers in fixed dimension. Amer. J. Math., 143(6): 18631929, 2021.Google Scholar
Kim, Jee Soo and Proschan, Frank. Total positivity.InEncyclopedia of Statistical Sciences (Kotz, S. et al, Eds.), Vol. 14, New York: Wiley, pp. 86658672, 2006.Google Scholar
Kjeldsen, Tinne Hoff. The early history of the moment problem. Historia Math., 20(1): 1944, 1993.Google Scholar
Kodama, Yuji and Williams, Lauren. KP solitons, total positivity and cluster algebras. Proc. Natl. Acad. Sci. USA, 108: 89848989, 2011.Google Scholar
Kodama, Yuji and Williams, Lauren. KP solitons and total positivity for the Grassmannian. Invent. Math., 198(3): 637699, 2014.Google Scholar
Kolmogorov, Andrey Nikolaevich. Kurven in Hilbertschen Raum, die gegenüber einer einparametrigen Gruppe von Bewegungen invariant sind. C.R. Doklady Acad. Sci. U.R.S.S. 26: 69, 1940.Google Scholar
Krantz, Steven George and Parks, Harold Raymond. A primer of real analytic functions. Birkhäuser Advanced Texts, Boston, MA: Birkhäuser, xiii+209 pp., 2002.Google Scholar
Laguerre, Edmond. Mémoire sur la théorie des équations numériques. J. Math. Pures Appl., 9: 9146, 1883.Google Scholar
Lam, Thomas, Postnikov, Alexander E., and Pylyavskyy, Pavlo. Schur positivity and Schur log-concavity. Amer. J. Math., 129(6): 16111622, 2007.Google Scholar
Li, Ai and Horvath, Steve. Network neighborhood analysis with the multi-node topological overlap measure. Bioinformatics, 23(2): 222231, 2007.CrossRefGoogle ScholarPubMed
Liberti, Leo and Lavor, Carlile. Six mathematical gems from the history of distance geometry. Int. Trans. Oper. Res., 23(5): 897920, 2016.Google Scholar
Lindström, Bernt. On the vector representations of induced matroids. Bull. London Math. Soc., 5(1): 8590, 1973.Google Scholar
Loewner, Charles. Über monotone Matrixfunktionen. Math. Z., 38(1): 177216, 1934.Google Scholar
Loewner, Charles. On totally positive matrices. Math. Z., 63: 338340, 1955 (Issai Schur memorial volume).Google Scholar
Loewner, Charles. Determination of the critical exponent of the Green’s function. 1965 Symposium on Function Theory, Erevan, USSR, 1966.Google Scholar
Loewner, Charles. On schlicht-monotonic functions of higher order. J. Math. Anal. Appl., 14(2): 320325, 1966. http://dx.doi.org/10.1016/0022-247X(66)90033-3Google Scholar
Lusztig, George. Total positivity in reductive groups, Lie theory and geometry. Progr. Math., Vol. 123, Boston, MA: Birkhäuser Boston, 1994, pp. 531568.Google Scholar
Lusztig, George. Total positivity and canonical bases. In Algebraic Groups and Lie Groups (Lehrer, G.I, Ed.), Austral. Math. Soc. Lect. Ser., Vol. 9, Cambridge: Cambridge University Press, 1997, pp. 281295.Google Scholar
Macdonald, Ian Grant. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. New York: The Clarendon Press, Oxford University Press, (2nd ed.), 1995. With contributions by Zelevinsky, A., Oxford Science Publications.Google Scholar
Maló, Ernest. Note sur les équations algébriques dont toutes les racines sont réelles. J. Math. Spéc. (Ser. 4), 4: 710, 1895.Google Scholar
Marcus, Marvin David and Khan, Nisar A.. A note on the Hadamard product. Canad. Math. Bull., 2(2): 8183, 1959.Google Scholar
Menegatto, Valdir Antonio and Oliveira, Claudemir Pinheiro. Positive definiteness on products of compact two-point homogeneous spaces and locally compact abelian groups. Canad. Math. Bull., 63(4): 705715, 2020.Google Scholar
Menegatto, Valdir Antonio and Peron, Ana Paula. Positive definite kernels on complex spheres. J. Math. Anal. Appl., 254(1): 219232, 2001.Google Scholar
Menger, Karl. Die Metrik des Hilbertschen Raumes. Anz. Akad. Wissen. Wien, Math.Nat.Kl., 65: 159160, 1928.Google Scholar
Menger, Karl. Untersuchungen über allgemeine Metrik. Math. Ann.: Part I: Vol. 100: 75163, 1928; Part II: Vol. 103: 466501, 1930.Google Scholar
Menger, Karl. New foundation of Euclidean geometry. Amer. J. Math., 53(4): 721745, 1931.Google Scholar
Mercer, James. Functions of positive and negative type and their connection with the theory of integral equations. Phil. Trans. Royal Soc. A, 209: 415446, 1909.Google Scholar
Micchelli, Charles Anthony. Cardinal L-splines. In Studies in spline functions and approximation theory (Karlin, S., Micchelli, C.A., Pinkus, A., and Schoenberg, I.J., Eds.), pp. 203250, New York: Academic Press, 1976.Google Scholar
Motzkin, Theodore Samuel. Beiträge zur Theorie der linearen Ungleichungen. PhD dissertation, Basel, 1933 and Jerusalem, 1936.Google Scholar
Motzkin, Theodore Samuel. Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products. Bull. Amer. Math. Soc., 54(4): 352360, 1948.Google Scholar
Motzkin, Theodore Samuel. The Euclidean algorithm. Bull. Amer. Math. Soc., 55(12): 11421146, 1949.Google Scholar
Motzkin, Theodore Samuel. A proof of Hilbert’s Nullstellensatz. Math. Z., 63: 341344, 1955 (Issai Schur memorial volume).Google Scholar
Motzkin, Theodore Samuel. The arithmetic-geometric inequality. In Inequalities (Shisha, O., Ed., Proc. Sympos. Wright–Patterson Air Force Base, Ohio, 1965), pp. 205224, New York: Academic Press, 1967.Google Scholar
Musin, Oleg R. The kissing number in four dimensions. Ann. of Math. (2), 168(1): 132, 2008.Google Scholar
Ostrowski, Alexander Markowich. Über die Funktionalgleichung der Exponentialfunktion und verwandte Funktionalgleichung. Jber. Deut. Math. Ver., 38: 5462, 1929.Google Scholar
Pascoe, James Eldred. Noncommutative Schur-type products and their Schoenberg theorem. arXiv, 2019. arXiv:math.FA/1907.04480Google Scholar
Pinkus, Allan. Strictly positive definite functions on a real inner product space. Adv. Comput. Math., 20: 263271, 2004.Google Scholar
Pinkus, Allan. Totally positive matrices. Cambridge Tracts in Mathematics, Vol. 181, Cambridge: Cambridge University Press, 2010.Google Scholar
Pitman, Jim. Probabilistic bounds on the coefficients of polynomials with only real zeros. J. Combin. Th. Ser. A, 77(2): 279303, 1997.Google Scholar
Pólya, Georg and Szegő, Gabor. Aufgaben und Lehrsätze aus der Analysis. Band II: Funktionentheorie, Nullstellen, Polynome Determinanten, Zahlentheorie. Berlin: Springer-Verlag, 1925.Google Scholar
Popoviciu, Tiberiu. Sur l’approximation des fonctions convexes d’ordre supérieur. Mathematica (Cluj), 8: 185, 1934.Google Scholar
Porcu, Emilio, Bevilacqua, Moreno, and Genton, Marc G.. Spatio-temporal covariance and cross-covariance functions of the great circle distance on a sphere. J. Amer. Statist. Assoc., 111(514): 888898, 2016.Google Scholar
Postnikov, Alexander E.. Total positivity, Grassmannians, and networks. arXiv, 2006. arXiv:math.CO/0609764Google Scholar
Poulain, Augustine Francois. Théorèmes généraux sur les équations algébriques. Nouv. Ann. Math., 6: 2133, 1867.Google Scholar
Reams, Robert. Hadamard inverses, square roots and products of almost semidefinite matrices. Linear Algebra Appl., 288: 3543, 1999.Google Scholar
Ressel, Paul. Laplace-transformation nichtnegativer und vektorwertiger maße. Manuscripta Math., 13: 143152, 1974.Google Scholar
Rietsch, Konstanze Christina. Quantum cohomology rings of Grassmannians and total positivity. Duke Math. J., 110(3): 523553, 2001.Google Scholar
Rietsch, Konstanze Christina. Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties. J. Amer. Math. Soc., 16(2): 363392, 2003.Google Scholar
Wayne Roberts, A. and Varberg, Dale E.. Convex functions. New York, London: Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], Vol. 57, Pure and Applied Mathematics, 1973.Google Scholar
Tyrrell Rockafellar, R.. Convex analysis. Princeton Mathematical Series, Vol. 28, Princeton, NJ: Princeton University Press, 1970.Google Scholar
Rothman, Adam J., Levina, Elizaveta, and Zhu, Ji. Generalized thresholding of large covariance matrices. J. Amer. Statist. Assoc., 104(485): 177186, 2009.Google Scholar
Rudin, Walter. Positive definite sequences and absolutely monotonic functions. Duke Math. J, 26(4): 617622, 1959.Google Scholar
Schmüdgen, Konrad. Around Hilbert’s 17th problem. Documenta Math., Extra Volume, “Optimization Stories,” pp. 433438, 2012.Google Scholar
Schmüdgen, Konrad. The moment problem. Graduate Texts in Mathematics, Vol. 277, Cham: Springer, 2017.Google Scholar
Schoenberg, Isaac Jacob. Über variationsvermindernde lineare Transformationen. Math. Z., 32: 321328, 1930.Google Scholar
Schoenberg, Isaac Jacob. Zur Abzählung der reellen Wurzeln algebraischer Gleichungen. Math. Z., 38: 546564, 1934.Google Scholar
Schoenberg, Isaac Jacob. Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert.” Ann. of Math. (2), 36(3): 724732, 1935.Google Scholar
Schoenberg, Isaac Jacob. On certain metric spaces arising from Euclidean spaces by a change of metric and their imbedding in Hilbert space. Ann. of Math. (2), 38(4): 787793, 1937.Google Scholar
Schoenberg, Isaac Jacob. Metric spaces and completely monotone functions. Ann. of Math. (2), 39(4): 811841, 1938.Google Scholar
Schoenberg, Isaac Jacob. Metric spaces and positive definite functions. Trans. Amer. Math. Soc., 44(3): 522536, 1938.Google Scholar
Schoenberg, Isaac Jacob. Positive definite functions on spheres. Duke Math. J., 9(1): 96108, 1942.Google Scholar
Schoenberg, Isaac Jacob. Contributions to the problem of approximation of equidistant data by analytic functions. Part A. On the problem of smoothing or graduation. A first class of analytic approximation formulae. Quart. Appl. Math., 4(1): 4599, 1946.Google Scholar
Schoenberg, Isaac Jacob. On totally positive functions, Laplace integrals and entire functions of the Laguerre–Pólya–Schur type. Proc. Natl. Acad. Sci. USA, 33(1): 1117, 1947.Google Scholar
Schoenberg, Isaac Jacob. On variation-diminishing integral operators of the convolution type. Proc. Natl. Acad. Sci. USA, 34(4): 164169, 1948.Google Scholar
Schoenberg, Isaac Jacob. Some analytical aspects of the problem of smoothing. In Studies and Essays presented to R. Courant on his 60th birthday (Friedrichs, K.O., Neugebauer, O.E., and Stoker, J.J., Eds.), pp. 351370, New York: Interscience Publ., 1948.Google Scholar
Schoenberg, Isaac Jacob. On Pólya frequency functions. II. Variation-diminishing integral operators of the convolution type. Acta Sci. Math. (Szeged), 12: 97106, 1950.Google Scholar
Schoenberg, Isaac Jacob. On Pólya frequency functions. I. The totally positive functions and their Laplace transforms. J. d’Analyse Math., 1: 331374, 1951.Google Scholar
Schoenberg, Isaac Jacob. On the zeros of the generating functions of multiply positive sequences and functions. Ann. of Math. (2), 62(3): 447471, 1955.Google Scholar
Schoenberg, Isaac Jacob. A note on multiply positive sequences and the Descartes rule of signs. Rend. Circ. Mat. Palermo, 4: 123131 1955.Google Scholar
Schoenberg, Isaac Jacob and Whitney, Anne M.. On Pólya frequency functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves. Trans. Amer. Math. Soc., 74: 246259, 1953.Google Scholar
Schur, Issai. Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. J. reine angew. Math., 140: 128, 1911.Google Scholar
Schur, Issai. Über Potenzreihen, die im Innern des Einheitskreises beschränkt sind. J. reine angew. Math., 147: 205232, 1917.Google Scholar
Schwarz, Binyamin. Totally positive differential systems. Pacific J. Math., 32(1): 203229, 1970.Google Scholar
Shohat, James Alexander and Tamarkin, Jacob Davidovich. The problem of moments. AMS Mathematical Surveys, New York: American Mathematical Society, 1943.Google Scholar
Sierpińsky, Wacław. Sur l’équation fonctionelle f(x+y) = f(x)+f(y). Fund. Math., 1(1): 116122, 1920.Google Scholar
Simon, Barry Martin. Loewner’s theorem on monotone matrix functions. Grundlehren der mathematischen Wissenschaften, Vol. 354, Berlin: Springer-Verlag, 2019.Google Scholar
Skandera, Mark. Inequalities in products of minors of totally nonnegative matrices. J. Algebraic Combin., 20(2): 195211, 2004.Google Scholar
Skryabin, Serge. On the graded algebras associated with Hecke symmetries, II. The Hilbert series. arXiv, 2019. arXiv:math.RA/1903.09128Google Scholar
Sobolev, Sergei Lvovich. Sur un théorème d’analyse fonctionnelle. Rec. Math. (Mat. Sbornik) N.S. 4(46): 471497, 1938.Google Scholar
Steinwart, Ingo. On the influence of the kernel on the consistency of support vector machines. J. Mach. Learn. Res., 2(1): 6793, 2002.Google Scholar
Stewart, James Drewry. Positive definite functions and generalizations, an historical survey. Rocky Mountain J. Math., 6(3): 409434, 1976.Google Scholar
Stieltjes, Thomas Joannes. Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse, 8(4): 1122, 1894–95.Google Scholar
Sturmfels, Bernd. Totally positive matrices and cyclic polytopes. Linear Algebra Appl., 107: 275281, 1988.Google Scholar
Vapnik, Vladimir Naumovich. The nature of statistical learning theory. Statistics for Engineering and Information Science, New York: Springer, 2000.Google Scholar
Vasudeva, Harkrishan Lal. Positive definite matrices and absolutely monotonic functions. Indian J. Pure Appl. Math., 10(7): 854858, 1979.Google Scholar
Vishwakarma, Prateek Kumar. Positivity preservers forbidden to operate on diagonal blocks. Trans. Amer. Math. Soc., in press, DOI: 10.1090/tran/8256Google Scholar
Neumann, John von and Schoenberg, Isaac Jacob. Fourier integrals and metric geometry. Trans. Amer. Math. Soc., 50: 226251, 1941.Google Scholar
Vybíral, Jan. A variant of Schur’s product theorem and its applications. Adv. Math., 368, Article 107140, 9 pp., 2020.Google Scholar
Wagner, David G.. Total positivity of Hadamard products. J. Math. Anal. Appl., 163(2): 459483, 1992.Google Scholar
Hsien-Chung, Wang. Two-point homogeneous spaces. Ann. of Math. (2), 55(1): 177191, 1952.Google Scholar
Wang, Yi and Yeh, Yeong-Nan. Polynomials with real zeros and Pólya frequency sequences. J. Combin. Theory Ser. A, 109(1): 6374, 2005.Google Scholar
White, Philip and Porcu, Emilio. Towards a complete picture of stationary covariance functions on spheres cross time. Electron. J. Statist., 13(2): 25662594, 2019.Google Scholar
Widder, David Vernon. The Laplace transform. Princeton Legacy Library, Princeton, NJ: Princeton University Press, 1941.Google Scholar
Witsenhausen, Hans Sylvain. Minimum dimension embedding of finite metric spaces. J. Combin. Th. Ser. A, 42(2): 184199, 1986.Google Scholar
Xu., Yuan Positive definite functions on the unit sphere and integrals of Jacobi polynomials. Proc. Amer. Math. Soc., 146(5): 20392048, 2018.Google Scholar
Xu, Yuan and Cheney, Elliott Ward. Strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 116(4): 977981, 1992.Google Scholar
Zhan, Xingzhi. Matrix theory. Graduate Studies in Mathematics, Vol. 147, Providence, RI: American Mathematical Society, 264 pp., 2013.Google Scholar
Zhang, Bin and Horvath, Steve. A general framework for weighted gene co-expression network analysis. Stat. Appl. Genet. Mol. Biol., 4: Article 17, 45 pp (electronic), 2005.Google Scholar
Zhang, Fuzhen. Matrix theory: Basic results and techniques. Universitext, New York: Springer-Verlag, xvii+399 pp., 2011.Google Scholar
Ziegel, Johanna. Convolution roots and differentiability of isotropic positive definite functions on spheres. Proc. Amer. Math. Soc., 142(6): 20632077, 2014.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Apoorva Khare, Indian Institute of Science, Bangalore
  • Book: Matrix Analysis and Entrywise Positivity Preservers
  • Online publication: 10 March 2022
  • Chapter DOI: https://doi.org/10.1017/9781108867122.039
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Apoorva Khare, Indian Institute of Science, Bangalore
  • Book: Matrix Analysis and Entrywise Positivity Preservers
  • Online publication: 10 March 2022
  • Chapter DOI: https://doi.org/10.1017/9781108867122.039
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Apoorva Khare, Indian Institute of Science, Bangalore
  • Book: Matrix Analysis and Entrywise Positivity Preservers
  • Online publication: 10 March 2022
  • Chapter DOI: https://doi.org/10.1017/9781108867122.039
Available formats
×