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Applications of universality limits to zeros and reproducing kernels of orthogonal polynomials

Eli Levin, Doron S. Lubinsky
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ajfp2t4ucrcbxh6e6fdg6pwf74" style="color: black;">Journal of Approximation Theory</a> </i> &nbsp;
We apply universality limits to asymptotics of spacing of zeros {x kn } of orthogonal polynomials, for weights with compact support and for exponential weights.  ...  A typical result is lim n→∞ x kn − x k+1,n K n (x kn , x kn ) = 1 under minimal hypotheses on the weight, withK n denoting a normalized reproducing kernel.  ...  Acknowledgments The authors thank the referees for helpful comments, in particular pointing out a gap in a proof of one result.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jat.2007.05.003">doi:10.1016/j.jat.2007.05.003</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ambycclttvebjhqrwfuinitore">fatcat:ambycclttvebjhqrwfuinitore</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171005102326/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/f1a/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAyMTkwNDUwNzAwMTAyNQ%3D%3D.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c3/9f/c39f58f25643d5535f2f262cc42fac2437c21b67.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jat.2007.05.003"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles

Doron S. Lubinsky
<span title="2016-08-10">2016</span> <i title="SIGMA (Symmetry, Integrability and Geometry: Methods and Application)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/tkgxczxugbe7dobnlqzuoueefi" style="color: black;">Symmetry, Integrability and Geometry: Methods and Applications</a> </i> &nbsp;
We survey the current status of universality limits for m-point correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels  ...  In particular, we consider underlying measures on compact intervals, and fixed and varying exponential weights, as well as universality limits for a variety of orthogonal systems.  ...  I owe Percy a great deal: it was Percy's 60th birthday conference at the Courant Institute that inspired me to try apply classical methods of orthogonal polynomials to universality limits.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3842/sigma.2016.078">doi:10.3842/sigma.2016.078</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bstonrh7ybb7ppb3phk4ci4g7y">fatcat:bstonrh7ybb7ppb3phk4ci4g7y</a> </span>
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Page 6383 of Mathematical Reviews Vol. , Issue 90K [page]

<span title="">1990</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
One of these results states necessary and sufficient conditions, in terms of the zeros, for a polynomial to be the nth orthogonal polynomial corresponding to an indefinite metric.  ...  Krein on orthogonal polynomials for the non- stationary case (pp. 65-78); Harry Dym, Hermitian block Toeplitz matrices, orthogonal polynomials, reproducing kernel Pontryagin spaces, interpolation and extension  ... 
<span class="external-identifiers"> </span>
<a target="_blank" rel="noopener" href="https://archive.org/details/sim_mathematical-reviews_1990-11_90k/page/6383" title="read fulltext microfilm" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Archive [Microfilm] <div class="menu fulltext-thumbnail"> <img src="https://archive.org/serve/sim_mathematical-reviews_1990-11_90k/__ia_thumb.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a>

Orthogonal polynomials and exact correlation functions for two cut random matrix models

Nivedita Deo
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2y36cnqlfjfbjkv7jqmcepxs5u" style="color: black;">Nuclear Physics B</a> </i> &nbsp;
The oscillating and smooth parts of the two point correlator are extracted and the universality of local fine grained and smoothed global correlators is established.  ...  An asymptotic form for orthogonal polynomials for arbitrary polynomial potentials that support a Z_2 symmetric distribution is obtained.  ...  Acknowledgements: I would like to thank E. Brezin and S. Jain for discussions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0550-3213(97)00561-0">doi:10.1016/s0550-3213(97)00561-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/o2rbkrrqhvghffcectv2wldcem">fatcat:o2rbkrrqhvghffcectv2wldcem</a> </span>
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Locally Supported Kernels for Spherical Spline Interpolation

Michael Schreiner
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ajfp2t4ucrcbxh6e6fdg6pwf74" style="color: black;">Journal of Approximation Theory</a> </i> &nbsp;
By the use of locally supported basis functions for spherical spline interpolation the applicability of this approximation method is spread out since the resulting interpolation matrix is sparse and thus  ...  We show how spherical spline interpolation with polynomial precision can be managed with locally supported kernels, thus giving the possibility to combine approximation techniques based on spherical harmonic  ...  It will be shown in this paper, that for a large class of locally supported kernels the zeros of the Legendre coefficients correspond to the zeros of certain Gegenbauer polynomials.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jath.1997.3037">doi:10.1006/jath.1997.3037</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5zmghaxak5eepa5shbzixtfnm4">fatcat:5zmghaxak5eepa5shbzixtfnm4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180721071413/https://kluedo.ub.uni-kl.de/frontdoor/deliver/index/docId/572/file/gruen_140.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/dd/89/dd895901c5ca4e51db1f45294dd7000babec6903.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jath.1997.3037"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

Anna Maltsev
<span title="2010-06-26">2010</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4t2npfwxdfchbkc6v2dtzlglle" style="color: black;">Communications in Mathematical Physics</a> </i> &nbsp;
In particular, we define a reproducing kernel S L for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic  ...  We deduce that the eigenvalues near ξ 0 in a large box of size L are spaced asymptotically as 1 Lρ . We adapt the methods used to show similar results for orthogonal polynomials. vi  ...  In particular it is used to show interlacing of zeros of orthogonal polynomials, and the link between universality and clock behavior of zeros, as discussed later.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00220-010-1074-z">doi:10.1007/s00220-010-1074-z</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/n5ec72fnz5e5zdgkwoiwmm5xt4">fatcat:n5ec72fnz5e5zdgkwoiwmm5xt4</a> </span>
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Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems [article]

Catherine Bénéteau, Dmitry Khavinson, Constanze Liaw, Daniel Seco,, Brian Simanek
<span title="2016-06-28">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk.  ...  As a consequence, we obtain detailed information regarding zeros of reproducing kernels in weighted spaces of analytic functions.  ...  The authors would like to thank A. Sola for useful discussions, and the National Science Foundation for their support of the SEAM conference in 2016, where some of this work was carried out.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1606.08615v1">arXiv:1606.08615v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/g4my2lnkdbeklmyyklsa2cg5na">fatcat:g4my2lnkdbeklmyyklsa2cg5na</a> </span>
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Universality of random matrices in the microscopic limit and the Dirac operator spectrum

G. Akemann, P.H. Damgaard, U. Magnea, S. Nishigaki
<span title="">1997</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2y36cnqlfjfbjkv7jqmcepxs5u" style="color: black;">Nuclear Physics B</a> </i> &nbsp;
We prove the universality of correlation functions of chiral unitary and unitary ensembles of random matrices in the microscopic limit.  ...  The essence of the proof consists in reducing the three-term recursion relation for the relevant orthogonal polynomials into a Bessel equation governing the local asymptotics around the origin.  ...  Acknowledgements P.H.D. would like to thank J. Verbaarschot for an early discussion, and P. Nevai and W. Van Assche for e-mail correspondences.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0550-3213(96)00713-4">doi:10.1016/s0550-3213(96)00713-4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/enqg7h7r4nh2tcefjiecr4giwa">fatcat:enqg7h7r4nh2tcefjiecr4giwa</a> </span>
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Some recent methods for establishing universality limits

D.S. Lubinsky
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ctxlg7dwwbdybeqr3eeyjuprcm" style="color: black;">Nonlinear Analysis</a> </i> &nbsp;
These include Levin's method using a Markov-Bernstein inequality, a comparison inequality of the author, and a method based on complex analysis and reproducing kernels.  ...  We survey some recent methods for establishing universality limits for random matrices in the unitary case.  ...  to a technical limit involving orthogonal polynomials.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.na.2009.06.023">doi:10.1016/j.na.2009.06.023</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gxrcjhljlnhhhpiw5ymren5liy">fatcat:gxrcjhljlnhhhpiw5ymren5liy</a> </span>
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On the computation of density and two-point correlation functions of a class of random matrix ensembles [article]

Kazi Alam, Swapnil Yadav, K. A. Muttalib
<span title="2020-10-18">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We demonstrate a method to solve a general class of random matrix ensembles numerically.  ...  We reproduce standard results for a variety of well-known ensembles and show some new results for the Muttalib-Borodin ensembles and recently introduced γ-ensemble for which analytic results are not yet  ...  He also calculated respective biorthogonal polynomials for each family. Finally, the limit kernels were also obtained. Limit kernel for Laguerre and Jacobi ensemble turned out to be the same.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2010.10330v1">arXiv:2010.10330v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/f3cpe5hqinfbfmocp5eyigkkdq">fatcat:f3cpe5hqinfbfmocp5eyigkkdq</a> </span>
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Microscopic universality of complex matrix model correlation functions at weak non-Hermiticity

G. Akemann
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/xefhbpc6n5fzpn6t7c45s6623i" style="color: black;">Physics Letters B</a> </i> &nbsp;
The universality of the correlation functions then follows from that of the kernel of orthogonal polynomials and a mapping of massive to massless correlators.  ...  To this aim we establish the universality of the asymptotics of orthogonal polynomials in the complex plane.  ...  Acknowledgments: I am indebted to P. Di Francesco, E. Kanzieper and G. Vernizzi for enlightening discussions and correspondence.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0370-2693(02)02737-5">doi:10.1016/s0370-2693(02)02737-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3gaxzsojerc3pk55jrhy3gxbke">fatcat:3gaxzsojerc3pk55jrhy3gxbke</a> </span>
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Estimating linear response statistics using orthogonal polynomials: An RKHS formulation [article]

He Zhang, John Harlim, Xiantao Li
<span title="2020-12-08">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This error bound provides a means to understand the feasibility as well as limitation of the kernel embedding with Mercer-type kernels.  ...  While the resulting representation is analogous to Polynomial Chaos Expansion (PCE), the connection to the reproducing kernel Hilbert space (RKHS) theory allows one to establish the uniform convergence  ...  Acknowledgment The research of XL was supported under the NSF grant DMS-1819011 and JH was supported under the NSF grant DMS-1854299.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1912.11110v3">arXiv:1912.11110v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dlkeehc7anbl7lmzsr2ons5rii">fatcat:dlkeehc7anbl7lmzsr2ons5rii</a> </span>
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Kernel-Induced Sampling Theorem

Akira Tanaka, Hideyuki Imai, Masaaki Miyakoshi
<span title="">2010</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gkn2pu46ozb4tmkxczacnmtvkq" style="color: black;">IEEE Transactions on Signal Processing</a> </i> &nbsp;
The key idea of our work is adopting the reproducing kernel Hilbert space corresponding to the Gramian matrix of the kernel and the given set of sampling points as the range space of a sampling operator  ...  A necessary and sufficient condition for the corresponding reproducing kernel and the given set of sampling points to perfectly recover the functions is obtained in this paper.  ...  ACKNOWLEDGMENT The authors are most grateful to anonymous reviewers for their fruitful comments that improve the quality of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tsp.2010.2046637">doi:10.1109/tsp.2010.2046637</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nhsyblxjgvdubdghnx5rxalp6u">fatcat:nhsyblxjgvdubdghnx5rxalp6u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180722045238/https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/43185/1/TSP58-7_3569-3577.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ab/87/ab87a7ef6995d92aa6470840781ec3528f187b91.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/tsp.2010.2046637"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Applications of new Geronimus type identities for real orthogonal polynomials

D. S. Lubinsky
<span title="2010-02-03">2010</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
Moreover, we apply the formula to deduce a weak convergence result and a discrepancy estimate, and also to establish a Gauss quadrature associated with µ with nodes at the zeros of p n (a) p n−1 (t) −  ...  Let µ be a positive measure on the real line, with associated orthogonal polynomials {p n }. Let Ima = 0.  ...  Acknowledgement Vili Totik provided useful references and perspectives during the OPSFA conference in Leuven in July 2009.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9939-10-10276-7">doi:10.1090/s0002-9939-10-10276-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/27o5mgjcrfeetpyvvqziwrj65q">fatcat:27o5mgjcrfeetpyvvqziwrj65q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170816204133/http://www.ams.org/journals/proc/2010-138-06/S0002-9939-10-10276-7/S0002-9939-10-10276-7.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c5/47/c547270b6bf6e5f2ec2eb4de96d684d6432a6309.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1090/s0002-9939-10-10276-7"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Orthogonal polynomials of several variables [article]

Yuan Xu
<span title="2021-05-04">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Preliminary version of Chapter 2 in the book "Encyclopedia of Special functions: The Askey-Bateman Project, Vol. 2: Multivariate special functions", T. H. Koornwinder and J. V.  ...  .), Cambridge University Press, 2021.  ...  Let P H n (·, ·) denote the reproducing kernel of H d n (H) and let P n (·, ·) denote the reproducing kernel of V d n with respect to (1 − x 2 ) − 1 2 W H (x).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1701.02709v2">arXiv:1701.02709v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pso2qnkkgzc6xitj3nq4rlc3da">fatcat:pso2qnkkgzc6xitj3nq4rlc3da</a> </span>
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