Preservers of totally positive kernels and Pólya frequency functions

Belton, Alexander; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai

Mathematics Research Reports, Tome 3 (2022), pp. 35-56.

Fractional powers and polynomial maps preserving structured totally positive matrices, one-sided Pólya frequency functions, or totally positive kernels are treated from a unifying perspective. Besides the stark rigidity of the polynomial transforms, we unveil an ubiquitous separation between discrete and continuous spectra of such inner fractional powers. Classical works of Schoenberg, Karlin, Hirschman, and Widder are completed by our classification. Concepts of probability theory, multivariate statistics, and group representation theory naturally enter into the picture.

Classification : 15B48, 15A15, 39B62, 42A82, 44A10, 47B34
Mots clés : Totally non-negative function, totally positive function, totally non-negative kernel, totally positive kernel, totally non-negative matrix, totally positive matrix, entrywise transformation, Pólya frequency function, Pólya frequency sequence, Hirschman–Widder density, exponential random variable, spherical function, orbital integral, multivariate statistics.

[1] Aissen, Michael; Schoenberg, Isaac Jacob; Whitney, Anne Marie On the generating functions of totally positive sequences. I, J. Analyse Math., Volume 2 (1952), pp. 93-103 | DOI | MR | Zbl

[2] Ando, Tsuyoshi Totally positive matrices, Linear Algebra Appl., Volume 90 (1987), pp. 165-219 | DOI | MR | Zbl

[3] Bakan, Andrew; Craven, Thomas; Csordas, George Interpolation and the Laguerre–Pólya class, Southwest J. Pure Appl. Math., Volume 1 (2001), pp. 38-53 | MR | Zbl

[4] Belton, Alexander; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai Totally positive kernels, Pólya frequency functions, and their transforms, J. d’Analyse Math. (in press)

[5] Belton, Alexander; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai Matrix positivity preservers in fixed dimension. I, Adv. Math., Volume 298 (2016), pp. 325-368 | DOI | MR | Zbl

[6] Belton, Alexander; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai Hirschman–Widder densities, Appl. Comput. Harmon. Anal., Volume 60 (2022), pp. 396-425 | DOI | MR | Zbl

[7] Belton, Alexander; Guillot, Dominique; Khare, Apoorva; Putinar, Mihai Moment-sequence transforms, J. Eur. Math. Soc. (JEMS), Volume 24 (2022) no. 9, pp. 3109-3160 | DOI | MR | Zbl

[8] Berenstein, Arkady; Fomin, Sergey; Zelevinsky, Andrei Parametrizations of canonical bases and totally positive matrices, Adv. Math., Volume 122 (1996) no. 1, pp. 49-149 | DOI | MR | Zbl

[9] Berenstein, Arkady; Zelevinsky, Andrei Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math., Volume 143 (2001) no. 1, pp. 77-128 | DOI | MR | Zbl

[10] Berezin, F. A. Quantization in complex symmetric spaces, Izv. Akad. Nauk SSSR Ser. Mat., Volume 39 (1975) no. 2, p. 363-402, 472 | MR | Zbl

[11] Brenti, Francesco Unimodal, log-concave and Pólya frequency sequences in combinatorics, Mem. Amer. Math. Soc., Volume 81 (1989) no. 413, p. viii+106 | DOI | MR | Zbl

[12] Brenti, Francesco Combinatorics and total positivity, J. Combin. Theory Ser. A, Volume 71 (1995) no. 2, pp. 175-218 | DOI | MR | Zbl

[13] Brown, Lawrence D.; Johnstone, Iain M.; MacGibbon, K. Brenda Variation diminishing transformations: a direct approach to total positivity and its statistical applications, J. Amer. Statist. Assoc., Volume 76 (1981) no. 376, pp. 824-832 | DOI | MR | Zbl

[14] Choudhury, Projesh Nath Characterizing total positivity: single vector tests via Linear Complementarity, sign non-reversal, and variation diminution, Bull. Lond. Math. Soc., Volume 54 (2022) no. 2, pp. 791-811 | DOI | MR

[15] Curry, H. B.; Schoenberg, I. J. On Pólya frequency functions. IV. The fundamental spline functions and their limits, J. Analyse Math., Volume 17 (1966), pp. 71-107 | DOI | MR | Zbl

[16] Descartes, Rene La Géomeétrie: Appendix to Discours de la méthode, 1637

[17] Edrei, Albert On the generating functions of totally positive sequences. II, J. Analyse Math., Volume 2 (1952), pp. 104-109 | DOI | MR | Zbl

[18] Efron, Bradley Increasing properties of Pólya frequency functions, Ann. Math. Statist., Volume 36 (1965), pp. 272-279 | DOI | MR | Zbl

[19] Fallat, Shaun M.; Johnson, Charles R. Totally nonnegative matrices, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2011, xvi+248 pages | DOI | MR

[20] Faraut, Jacques Rayleigh theorem, projection of orbital measures and spline functions, Adv. Pure Appl. Math., Volume 6 (2015) no. 4, pp. 261-283 | DOI | MR | Zbl

[21] Faraut, Jacques; Korányi, Adam Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994, xii+382 pages (Oxford Science Publications) | MR

[22] Farrell, Roger H. Multivariate calculation: Use of the continuous groups, Springer Series in Statistics, Springer-Verlag, New York, 1985, xvi+376 pages | DOI | MR

[23] Fekete, Mihály; Pólya, Georg Über ein Problem von Laguerre, Rend. Circ. Mat. Palermo, Volume 34 (1912), pp. 89-120 | DOI

[24] FitzGerald, Carl H.; Horn, Roger A. On fractional Hadamard powers of positive definite matrices, J. Math. Anal. Appl., Volume 61 (1977) no. 3, pp. 633-642 | DOI | MR | Zbl

[25] Fomin, Sergey; Zelevinsky, Andrei Double Bruhat cells and total positivity, J. Amer. Math. Soc., Volume 12 (1999) no. 2, pp. 335-380 | DOI | MR | Zbl

[26] Fomin, Sergey; Zelevinsky, Andrei Total positivity: tests and parametrizations, Math. Intelligencer, Volume 22 (2000) no. 1, pp. 23-33 | DOI | MR | Zbl

[27] Gantmacher, F. R.; Krein, M. G. Sur les matrices completement non négatives et oscillatoires, Compos. Math., Volume 4 (1937), pp. 445-476 | Zbl

[28] Gantmacher, F. R.; Krein, M. G. Oscillation matrices and small oscillations of mechanical systems, Moscow-Leningrad, 1941, paging unknown pages | MR

[29] Total positivity and its applications, Mathematics and its Applications, 359 (1996), p. x+518 | DOI | MR

[30] Gelfand, I. M.; Naimark, M. A. Unitarnye predstavleniya klassičeskih grupp, Izdat. Nauk SSSR, Moscow-Leningrad, 1950, 288 pages (Trudy Mat. Inst. Steklov. no. 36,) | MR

[31] Gindikin, S. G. Invariant generalized functions in homogeneous domains, Funkcional. Anal. i Priložen., Volume 9 (1975) no. 1, pp. 56-58 | MR

[32] Gröchenig, Karlheinz Schoenberg’s theory of totally positive functions and the Riemann zeta function (2020) (Preprint, available at http://arxiv.org/abs/2007.12889)

[33] Gröchenig, Karlheinz; Romero, José Luis; Stöckler, Joachim Sampling theorems for shift-invariant spaces, Gabor frames, and totally positive functions, Invent. Math., Volume 211 (2018) no. 3, pp. 1119-1148 | DOI | MR | Zbl

[34] Gröchenig, Karlheinz; Stöckler, Joachim Gabor frames and totally positive functions, Duke Math. J., Volume 162 (2013) no. 6, pp. 1003-1031 | DOI | MR | Zbl

[35] Grommer, J. Ganze transzendente Funktionen mit lauter reellen Nullstellen, J. Reine Angew. Math., Volume 144 (1914), pp. 114-166 | DOI | MR | Zbl

[36] Harish-Chandra Differential operators on a semisimple Lie algebra, Amer. J. Math., Volume 79 (1957), pp. 87-120 | DOI | MR | Zbl

[37] Hirschman, I. I.; Widder, D. V. The convolution transform, Princeton University Press, Princeton, N. J., 1955, x+268 pages | MR

[38] Hirschman, I. I. Jr.; Widder, D. V. The inversion of a general class of convolution transforms, Trans. Amer. Math. Soc., Volume 66 (1949), pp. 135-201 | DOI | MR | Zbl

[39] Itzykson, C.; Zuber, J. B. The planar approximation. II, J. Math. Phys., Volume 21 (1980) no. 3, pp. 411-421 | DOI | MR | Zbl

[40] Jain, Tanvi Hadamard powers of rank two, doubly nonnegative matrices, Adv. Oper. Theory, Volume 5 (2020) no. 3, pp. 839-849 | DOI | MR | Zbl

[41] James, Alan T. Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist., Volume 35 (1964), pp. 475-501 | DOI | MR | Zbl

[42] Karlin, Samuel Total positivity, absorption probabilities and applications, Trans. Amer. Math. Soc., Volume 111 (1964), pp. 33-107 | DOI | MR | Zbl

[43] Karlin, Samuel Total positivity. Vol. I, Stanford University Press, Stanford, Calif., 1968, xii+576 pages | MR

[44] Katkova, Olga M. Multiple positivity and the Riemann zeta-function, Comput. Methods Funct. Theory, Volume 7 (2007) no. 1, pp. 13-31 | DOI | MR | Zbl

[45] Khare, Apoorva Critical exponents for total positivity, individual kernel encoders, and the Jain–Karlin–Schoenberg kernel (2020) (Preprint, available at http://arxiv.org/abs/2008.05121)

[46] Khare, Apoorva Smooth entrywise positivity preservers, a Horn-Loewner master theorem, and symmetric function identities, Trans. Amer. Math. Soc., Volume 375 (2022) no. 3, pp. 2217-2236 | DOI | MR | Zbl

[47] Khare, Apoorva; Tao, Terence On the sign patterns of entrywise positivity preservers in fixed dimension, Amer. J. Math., Volume 143 (2021) no. 6, pp. 1863-1929 | DOI | MR | Zbl

[48] Kim, Jee Soo; Proschan, Frank Total positivity, Encyclopedia of Statistical Sciences (Kotz, S. et al, ed.), Volume 14, John Wiley & Sons, 2006, pp. 8665-8672 | DOI

[49] Kodama, Yuji; Williams, Lauren KP solitons and total positivity for the Grassmannian, Invent. Math., Volume 198 (2014) no. 3, pp. 637-699 | DOI | MR | Zbl

[50] Kodama, Yuji; Williams, Lauren K. KP solitons, total positivity, and cluster algebras, Proc. Natl. Acad. Sci. USA, Volume 108 (2011) no. 22, pp. 8984-8989 | DOI | MR | Zbl

[51] Laguerre, Edmond N. Sur les fonctions du genre zéro et du genre un, C. R. Acad. Sci., Volume 95 (1882), pp. 828-831 | Zbl

[52] Laguerre, Edmond N. Mémoire sur la théorie des équations numériques, J. Math. Pures et Appl., Volume 9 (1883), pp. 9-146 | Zbl

[53] Letac, Gérard; Massam, Hélène The Laplace transform ${(dets)}^{-p}exp\mathrm{tr}\left(s^{-1}w\right)$ and the existence of non-central Wishart distributions, J. Multivariate Anal., Volume 163 (2018), pp. 96-110 | DOI | MR | Zbl

[54] Levin, B. Ja. Distribution of zeros of entire functions, Translations of Mathematical Monographs, 5, American Mathematical Society, Providence, R.I., 1980, xii+523 pages (Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman) | MR

[55] Loewner, Charles On totally positive matrices, Math. Z., Volume 63 (1955), pp. 338-340 | DOI | MR | Zbl

[56] Lusztig, G. Total positivity in reductive groups, Lie theory and geometry (Progr. Math.), Volume 123, Birkhäuser Boston, Boston, MA, 1994, pp. 531-568 | DOI | MR | Zbl

[57] Lusztig, George Total positivity and canonical bases, Algebraic groups and Lie groups (Austral. Math. Soc. Lect. Ser.), Volume 9, Cambridge Univ. Press, Cambridge, 1997, pp. 281-295 | MR | Zbl

[58] Mayerhofer, Eberhard On Wishart and noncentral Wishart distributions on symmetric cones, Trans. Amer. Math. Soc., Volume 371 (2019) no. 10, pp. 7093-7109 | DOI | MR | Zbl

[59] Olshanski, Grigori; Vershik, Anatoli Ergodic unitarily invariant measures on the space of infinite Hermitian matrices, Contemporary mathematical physics (Amer. Math. Soc. Transl. Ser. 2), Volume 175, Amer. Math. Soc., Providence, RI, 1996, pp. 137-175 | DOI | MR

[60] Peddada, Shyamal Das; Richards, Donald St. P. Proof of a conjecture of M. L. Eaton on the characteristic function of the Wishart distribution, Ann. Probab., Volume 19 (1991) no. 2, pp. 868-874 | MR | Zbl

[61] Pinkus, Allan Totally positive matrices, Cambridge Tracts in Mathematics, 181, Cambridge University Press, Cambridge, 2010, xii+182 pages | MR

[62] Pólya, Georg Über Annäherung durch Polynome mit lauter reellen Wurzeln, Rend. Circ. Mat. Palermo, Volume 36 (1913), pp. 279-295 | DOI | Zbl

[63] Postnikov, Alexander Total positivity, Grassmannians, and networks (2006) (Preprint, available at http://arxiv.org/abs/0609764v1)

[64] Rietsch, Konstanze Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties, J. Amer. Math. Soc., Volume 16 (2003) no. 2, pp. 363-392 | DOI | MR | Zbl

[65] Rudin, Walter Positive definite sequences and absolutely monotonic functions, Duke Math. J., Volume 26 (1959), pp. 617-622 http://projecteuclid.org/euclid.dmj/1077468771 | MR | Zbl

[66] Schoenberg, I. J. On Pólya frequency functions. I. The totally positive functions and their Laplace transforms, J. Analyse Math., Volume 1 (1951), pp. 331-374 | DOI | MR | Zbl

[67] Schoenberg, I. J. Cardinal spline interpolation, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973, vi+125 pages | MR

[68] Schoenberg, Isaac Jacob Über variationsvermindernde lineare Transformationen, Math. Z., Volume 32 (1930) no. 1, pp. 321-328 | DOI | MR | Zbl

[69] Schoenberg, Isaac Jacob Positive definite functions on spheres, Duke Math. J., Volume 9 (1942), pp. 96-108 http://projecteuclid.org/euclid.dmj/1077493072 | MR

[70] Schoenberg, Isaac Jacob On Pólya frequency functions. II. Variation-diminishing integral operators of the convolution type, Acta Sci. Math. (Szeged), Volume 12 (1950), pp. 97-106 | MR | Zbl

[71] Schoenberg, Isaac Jacob On the zeros of the generating functions of multiply positive sequences and functions, Ann. of Math. (2), Volume 62 (1955), pp. 447-471 | DOI | MR

[72] Schoenberg, Isaac Jacob; Whitney, Anne Marie On Pólya frequence functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves, Trans. Amer. Math. Soc., Volume 74 (1953), pp. 246-259 | DOI | MR | Zbl

[73] Schur, J. Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math., Volume 140 (1911), pp. 1-28 | DOI | MR | Zbl

[74] Schur, J.; Pólya, G. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math., Volume 144 (1914), pp. 89-113 | DOI | MR | Zbl

[75] Sra, Suvrit Positive definite functions of noncommuting contractions, Hua-Bellman matrices, and a new distance metric (2021) (Preprint, available at http://arxiv.org/abs/2112.00056)

[76] Takemura, Akimichi Zonal polynomials, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 4, Institute of Mathematical Statistics, Hayward, CA, 1984, ii+104 pages | MR

[77] Thoma, Elmar Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z., Volume 85 (1964), pp. 40-61 | DOI | MR | Zbl

[78] Tschebotareff, Nikolaj Über die Realität von Nullstellen ganzer transzendenter Funktionen, Math. Ann., Volume 99 (1928) no. 1, pp. 660-686 | DOI | MR | Zbl

[79] Vergne, M.; Rossi, H. E. Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math., Volume 136 (1976) no. 1-2, pp. 1-59 | DOI | MR | Zbl

[80] Vershik, A. M.; Kerov, S. V. Characters and factor-representations of the infinite unitary group, Dokl. Akad. Nauk SSSR, Volume 267 (1982) no. 2, pp. 272-276 | MR

[81] Voiculescu, Dan Représentations factorielles de type II1 de $U\left(\infty \right)$, J. Math. Pures Appl. (9), Volume 55 (1976) no. 1, pp. 1-20 | MR | Zbl

[82] Wallach, Nolan R. The analytic continuation of the discrete series. I, II, Trans. Amer. Math. Soc., Volume 251 (1979), p. 1-17, 19–37 | DOI | MR | Zbl

[83] Whitney, Anne Marie A reduction theorem for totally positive matrices, J. Analyse Math., Volume 2 (1952), pp. 88-92 | DOI | MR