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Published: 6 September 2023

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1

Dyukarev, Yu M. "Degenerate Nevanlinna-Pick problem." Ukrainian Mathematical Journal 57, no. 10 (October 2005): 1559–70. http://dx.doi.org/10.1007/s11253-006-0014-8.

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2

El-Sabbagh, A. A. "On the solution of Nevanlinna Pick problem with selfadjoint extensions of symmetric linear relations in Hilbert space." International Journal of Mathematics and Mathematical Sciences 20, no. 3 (1997): 457–64. http://dx.doi.org/10.1155/s0161171297000628.

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The representation of Nevanlinna Pick Problem is well known, see [7], [8] and [11]. The aim of this paper is to find the necessary and sufficient condition for the solution of Nevanlinna Pick Problem and to show that there is a one-to-one correspondence between the solutions of the Nevanlinna Pick Problem and the minimal selfadjoint extensions of symmetric linear relation in Hilbert space. Finally, we define the resolvent matrix which gives the solutions of the Nevanlinna Pick Problem.
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3

Fisher, Stephen D., and Dmitry Khavinson. "Extreme Pick-Nevanlinna Interpolants." Canadian Journal of Mathematics 51, no. 5 (October 1, 1999): 977–95. http://dx.doi.org/10.4153/cjm-1999-043-5.

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AbstractFollowing the investigations of B. Abrahamse [1], F. Forelli [11], M. Heins [14] and others, we continue the study of the Pick-Nevanlinna interpolation problem inmultiply-connected planar domains. One major focus is on the problem of characterizing the extreme points of the convex set of interpolants of a fixed data set. Several other related problems are discussed.
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4

Costara, Constantin. "On the spectral Nevanlinna–Pick problem." Studia Mathematica 170, no. 1 (2005): 23–55. http://dx.doi.org/10.4064/sm170-1-2.

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5

Davidson, Kenneth R., Vern I. Paulsen, Mrinal Raghupathi, and Dinesh Singh. "A constrained Nevanlinna-Pick interpolation problem." Indiana University Mathematics Journal 58, no. 2 (2009): 709–32. http://dx.doi.org/10.1512/iumj.2009.58.3486.

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6

Hartz, Michael. "On the Isomorphism Problem for Multiplier Algebras of Nevanlinna-Pick Spaces." Canadian Journal of Mathematics 69, no. 1 (February 1, 2017): 54–106. http://dx.doi.org/10.4153/cjm-2015-050-6.

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AbstractWe continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with the restrictions of a universal space, namely theDrury-Arveson space. Instead, we work directly with theHilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic.This generalizes results of Davidson, Ramsey,Shalit, and the author.
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7

Yücesoy, Veysel, and Hitay Özbay. "On the real, rational, bounded, unit interpolation problem in ℋ∞ and its applications to strong stabilization." Transactions of the Institute of Measurement and Control 41, no. 2 (April 23, 2018): 476–83. http://dx.doi.org/10.1177/0142331218759598.

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One of the most challenging problems in feedback control is strong stabilization, i.e. stabilization by a stable controller. This problem has been shown to be equivalent to finding a finite dimensional, real, rational and bounded unit in [Formula: see text] satisfying certain interpolation conditions. The problem is transformed into a classical Nevanlinna–Pick interpolation problem by using a predetermined structure for the unit interpolating function and analysed through the associated Pick matrix. Sufficient conditions for the existence of the bounded unit interpolating function are derived. Based on these conditions, an algorithm is proposed to compute the unit interpolating function through an optimal solution to the Nevanlinna–Pick problem. The conservatism caused by the sufficient conditions is illustrated through strong stabilization examples taken from the literature.
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8

Stray, A. "Interpolating sequences and the Nevanlinna Pick problem." Publicacions Matemàtiques 35 (July 1, 1991): 507–16. http://dx.doi.org/10.5565/publmat_35291_14.

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9

Ionescu, A. "On the Operator-Valued Nevanlinna-Pick Problem." Zeitschrift für Analysis und ihre Anwendungen 14, no. 3 (1995): 431–39. http://dx.doi.org/10.4171/zaa/632.

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10

Derkach, V. A. "On Schur–Nevanlinna–Pick Indefinite Interpolation Problem." Ukrainian Mathematical Journal 55, no. 10 (October 2003): 1567–87. http://dx.doi.org/10.1023/b:ukma.0000022069.69507.bc.

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11

Dyukarev, Yu M. "Nevanlinna–Pick Problem for Stieltjes Matrix Functions." Ukrainian Mathematical Journal 56, no. 3 (March 2004): 446–65. http://dx.doi.org/10.1023/b:ukma.0000045689.16180.31.

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12

Kosiński, Łukasz, and Włodzimierz Zwonek. "Nevanlinna-Pick interpolation problem in the ball." Transactions of the American Mathematical Society 370, no. 6 (November 14, 2017): 3931–47. http://dx.doi.org/10.1090/tran/7063.

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13

Li, Kin Y. "Nevanlinna-Pick problem on reproducing kernel spaces." 1 13 (2018): 33–40. http://dx.doi.org/10.12988/ijcms.2018.71240.

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14

Agler, J., and N. J. Young. "The two-point spectral Nevanlinna-Pick problem." Integral Equations and Operator Theory 37, no. 4 (December 2000): 375–85. http://dx.doi.org/10.1007/bf01192826.

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15

Agler, Jim, and John E. MCarthy. "Nevanlinna-Pick interpolation on the bidisk." Journal für die reine und angewandte Mathematik (Crelles Journal) 1999, no. 506 (January 15, 1999): 191–204. http://dx.doi.org/10.1515/crll.1999.506.191.

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Abstract We solve the matrix-valued Nevanlinna-Pick problem in the space of bounded analytic functions on the bidisk, and give a description of all the interpolating functions. We also prove the Toeplitz-corona theorem on the bidisk.
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16

Takahashi, Sechiko. "A sufficient condition for Nevanlinna parametrization and an extension of Heins theorem." Nagoya Mathematical Journal 153 (1999): 87–100. http://dx.doi.org/10.1017/s0027763000006905.

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AbstractAn extended interpolation problem on a Riemann surface is formulated in terms of local rings and ideals. A sufficient condition for Nevanlinna parametrization is obtained. By means of this, Heins theorem on Pick-Nevanlinna interpolation in doubly connected domains is generalized to extended interpolation.
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17

Kosiński, Łukasz. "Three-point Nevanlinna–Pick problem in the polydisc." Proceedings of the London Mathematical Society 111, no. 4 (October 2015): 887–910. http://dx.doi.org/10.1112/plms/pdv045.

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18

COSTARA, CONSTANTIN. "THE $2\times 2$ SPECTRAL NEVANLINNA–PICK PROBLEM." Journal of the London Mathematical Society 71, no. 03 (May 24, 2005): 684–702. http://dx.doi.org/10.1112/s002461070500640x.

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19

Agler, Jim, and N. J. Young. "The two-by-two spectral Nevanlinna-Pick problem." Transactions of the American Mathematical Society 356, no. 2 (September 22, 2003): 573–85. http://dx.doi.org/10.1090/s0002-9947-03-03083-6.

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20

Zheleznyak, Аlexander V. "Multiplicative property of series used in the Nevanlinna-Pick problem." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 9, no. 1 (2022): 37–45. http://dx.doi.org/10.21638/spbu01.2022.104.

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In the paper we obtained substantially new sufficient condition for negativity of coefficients of power series inverse to series with positive ones. It has been proved that element-wise product of power series retains this property. In particular, it gives rise to generalization of the classical Hardy theorem about power series. These results are generalized for cases of series with multiple variables. Such results are useful in Nevanlinna – Pick theory. For example, if function k(x, y) can be represented as power series Pn≥0 an(x¯y)n, an > 0, and reciprocal function 1/k(x, y) can be represented as power series Pn≥0 bn(x¯y)n such that bn < 0, n > 0, then k(x, y) is a reproducing kernel function for some Hilbert space of analytic functions in the unit disc D with Nevanlinna – Pick property. The reproducing kernel 1/(1 − x¯y) of the classical Hardy space H2(D) is a prime example for our theorems.
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21

Saitoh, Saburou. "A counter example in the Pick-Nevanlinna interpolation problem." Archiv der Mathematik 51, no. 2 (August 1988): 164–65. http://dx.doi.org/10.1007/bf01206474.

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22

Luxemburg, Leon A., and Philip R. Brown. "The scalar Nevanlinna–Pick interpolation problem with boundary conditions." Journal of Computational and Applied Mathematics 235, no. 8 (February 2011): 2615–25. http://dx.doi.org/10.1016/j.cam.2010.11.013.

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23

Baribeau, Line, Patrice Rivard, and Elias Wegert. "On Hyperbolic Divided Differences and the Nevanlinna-Pick Problem." Computational Methods and Function Theory 9, no. 2 (December 18, 2008): 391–405. http://dx.doi.org/10.1007/bf03321735.

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24

Oloomi, H. M. "Nevanlinna-Pick interpolation problem for two frequency scale systems." IEEE Transactions on Automatic Control 40, no. 1 (1995): 169–73. http://dx.doi.org/10.1109/9.362876.

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25

Tran Duc, Anh. "A NOTE ON S. TAKAHASHI’S ARTICLE." Journal of Science Natural Science 67, no. 3 (October 2022): 10–16. http://dx.doi.org/10.18173/2354-1059.2022-0036.

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26

Bultheel, Adhemar, and Andreas Lasarow. "SCHUR–NEVANLINNA SEQUENCES OF RATIONAL FUNCTIONS." Proceedings of the Edinburgh Mathematical Society 50, no. 3 (October 2007): 571–96. http://dx.doi.org/10.1017/s0013091504001063.

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AbstractWe study certain sequences of rational functions with poles outside the unit circle. Such kinds of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur–Nevanlinna algorithm for Schur functions on the one hand, and to orthogonal rational functions on the unit circle on the other. We shall see that rational functions belonging to a Schur–Nevanlinna sequence can be used to parametrize the set of all solutions of an interpolation problem of Nevanlinna–Pick type for Schur functions.
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27

Tolokonnikov, V. "Extremal functions of the Nevanlinna-Pick problem and Douglas algebras." Studia Mathematica 105, no. 2 (1993): 151–58. http://dx.doi.org/10.4064/sm-105-2-151-158.

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28

Chandel, Vikramjeet Singh. "The three-point Pick–Nevanlinna interpolation problem on the polydisc." Complex Variables and Elliptic Equations 63, no. 9 (September 8, 2017): 1341–52. http://dx.doi.org/10.1080/17476933.2017.1370461.

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29

Ball, Joseph A., Vladimir Bolotnikov, and Sanne ter Horst. "A constrained Nevanlinna-Pick interpolation problem for matrix-valued functions." Indiana University Mathematics Journal 59, no. 1 (2010): 15–52. http://dx.doi.org/10.1512/iumj.2010.59.3776.

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30

Baribeau, Line, and Adama S. Kamara. "A Refined Schwarz Lemma for the Spectral Nevanlinna-Pick Problem." Complex Analysis and Operator Theory 8, no. 2 (May 16, 2013): 529–36. http://dx.doi.org/10.1007/s11785-013-0306-6.

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31

Alpay, D., V. Bolotnikov, and A. Dijksma. "On the Nevanlinna-Pick interpolation problem for generalized stieltjes functions." Integral Equations and Operator Theory 30, no. 4 (December 1998): 379–408. http://dx.doi.org/10.1007/bf01257874.

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32

Zheleznyak, A. V. "Multiplicative Property of the Series Used in the Nevanlinna–Pick Problem." Vestnik St. Petersburg University, Mathematics 55, no. 1 (March 2022): 27–33. http://dx.doi.org/10.1134/s1063454122010162.

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33

Dyukarev, Yu M., and A. E. Choque Rivero. "Criterion for the complete indeterminacy of the Nevanlinna-Pick matrix problem." Mathematical Notes 96, no. 5-6 (November 2014): 651–65. http://dx.doi.org/10.1134/s0001434614110042.

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34

Langer, Heinz, and Andreas Lasarow. "Solution of a multiple Nevanlinna–Pick problem via orthogonal rational functions." Journal of Mathematical Analysis and Applications 293, no. 2 (May 2004): 605–32. http://dx.doi.org/10.1016/j.jmaa.2004.01.022.

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35

Scheinker, David. "Hilbert function spaces and the Nevanlinna–Pick problem on the polydisc." Journal of Functional Analysis 261, no. 8 (October 2011): 2238–49. http://dx.doi.org/10.1016/j.jfa.2011.06.015.

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36

Scheinker, David. "Hilbert function spaces and the Nevanlinna–Pick problem on the polydisc II." Journal of Functional Analysis 266, no. 1 (January 2014): 355–67. http://dx.doi.org/10.1016/j.jfa.2013.07.027.

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37

Kheifets, A. Ya. "The generalized bitangential Schur-Nevanlinna-Pick problem and the related parseval equality." Journal of Soviet Mathematics 58, no. 4 (February 1992): 358–64. http://dx.doi.org/10.1007/bf01097288.

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38

Piekarski, Marian S., and Marceli Uruski. "Tangential nevanlinna-pick interpolation problem for bounded real matrices—A network approach." International Journal of Circuit Theory and Applications 21, no. 6 (November 1993): 513–38. http://dx.doi.org/10.1002/cta.4490210603.

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39

Sabadini, Irene, Daniel Alpay, Vladimir Bolotnikov, and Fabrizio Colombo. "Self-mappings of the quaternionic unit ball: multiplier properties, Schwarz-Pick inequality, and Nevanlinna--Pick interpolation problem." Indiana University Mathematics Journal 64, no. 1 (2015): 151–80. http://dx.doi.org/10.1512/iumj.2015.64.5456.

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40

Dyukarev, Yu M., and I. Yu Serikova. "Complete indeterminacy of the Nevanlinna-Pick problem in the class S[a,b]." Russian Mathematics 51, no. 11 (November 2007): 17–29. http://dx.doi.org/10.3103/s1066369x07110035.

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41

Hamilton, Ryan, and Mrinal Raghupathi. "The Toeplitz corona problem for algebras of multipliers on a Nevanlinna-Pick space." Indiana University Mathematics Journal 61, no. 4 (2012): 1393–405. http://dx.doi.org/10.1512/iumj.2012.61.4685.

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42

Arov, D. Z. "THE GENERALIZED BITANGENT CARATHÉODORY-NEVANLINNA-PICK PROBLEM, AND $ (j,J_0)$-INNER MATRIX-VALUED FUNCTIONS." Russian Academy of Sciences. Izvestiya Mathematics 42, no. 1 (February 28, 1994): 1–26. http://dx.doi.org/10.1070/im1994v042n01abeh001525.

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43

Gombani, A., and György Michaletzky. "On the Nevanlinna-Pick interpolation problem: Analysis of the McMillan degree of the solutions." Linear Algebra and its Applications 425, no. 2-3 (September 2007): 486–517. http://dx.doi.org/10.1016/j.laa.2006.02.045.

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44

Ball, Joseph A., and Vladimir Bolotnikov. "Interpolation Problems for Schur Multipliers on the Drury-Arveson Space: from Nevanlinna-Pick to Abstract Interpolation Problem." Integral Equations and Operator Theory 62, no. 3 (October 8, 2008): 301–49. http://dx.doi.org/10.1007/s00020-008-1626-1.

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45

Chen, Gong-ning. "The general rational interpolation problem and its connection with the Nevanlinna-Pick interpolation and power moment problem." Linear Algebra and its Applications 273, no. 1-3 (April 1998): 83–117. http://dx.doi.org/10.1016/s0024-3795(97)00346-7.

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46

Baranov, Anton, and Rachid Zarouf. "H ∞ interpolation constrained by Beurling–Sobolev norms." Moroccan Journal of Pure and Applied Analysis 9, no. 2 (May 1, 2023): 157–67. http://dx.doi.org/10.2478/mjpaa-2023-0012.

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Abstract We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space H 2 into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.
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47

Kosiński, Łukasz, and Włodzimierz Zwonek. "Nevanlinna–Pick Problem and Uniqueness of Left Inverses in Convex Domains, Symmetrized Bidisc and Tetrablock." Journal of Geometric Analysis 26, no. 3 (April 10, 2015): 1863–90. http://dx.doi.org/10.1007/s12220-015-9611-9.

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48

Bolotnikov, Vladimir, and Stephen P. Cameron. "The Nevanlinna–Pick problem on the closed unit disk: Minimal norm rational solutions of low degree." Journal of Computational and Applied Mathematics 236, no. 13 (July 2012): 3123–36. http://dx.doi.org/10.1016/j.cam.2012.02.009.

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49

Pascoe, J. E. "An Inductive Julia-Carathéodory Theorem for Pick Functions in Two Variables." Proceedings of the Edinburgh Mathematical Society 61, no. 3 (April 10, 2018): 647–60. http://dx.doi.org/10.1017/s0013091517000396.

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AbstractClassically, Nevanlinna showed that functions from the complex upper half plane into itself which satisfy nice asymptotic conditions are parametrized by finite measures on the real line. Furthermore, the higher order asymptotic behaviour at infinity of a map from the complex upper half plane into itself is governed by the existence of moments of its representing measure, which was the key to his solution of the Hamburger moment problem. Agler and McCarthy showed that an analogue of the above correspondence holds between a Pick function f of two variables, an analytic function which maps the product of two upper half planes into the upper half plane, and moment-like quantities arising from an operator theoretic representation for f. We apply their ‘moment’ theory to show that there is a fine hierarchy of levels of regularity at infinity for Pick functions in two variables, given by the Löwner classes and intermediate Löwner classes of order N, which can be exhibited in terms of certain formulae akin to the Julia quotient.
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50

Chen, Gong-ning, and Yong-jian Hu. "On the multiple Nevanlinna-Pick matrix interpolation in the class Cp and the Carathéodory matrix coefficient problem." Linear Algebra and its Applications 283, no. 1-3 (November 1998): 179–203. http://dx.doi.org/10.1016/s0024-3795(98)10095-2.

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