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Journal articles on the topic 'Hermite approximations'

Author: Grafiati

Published: 3 June 2025

Last updated: 31 July 2025

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1

Ky, Nguyen Xuan. "Signal analysis and weighted polynomial approximation." Studia Scientiarum Mathematicarum Hungarica 43, no. 2 (2006): 159–69. http://dx.doi.org/10.1556/sscmath.43.2006.2.2.

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We present applications of Hermite polynomials in signal analysis. Among other result, we give a characterization of the so-called time-frequency window functions in terms of the Hermite--Fourier coefficients, a Bernstein-type theorem for the best approximations of window functions by Hermite-functions, time-frequency approximations. Some analogues for Hankel-transforms will also be considered.

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2

KERMAN, R., M. L. HUANG, and M. BRANNAN. "ERROR ESTIMATES FOR DOMINICI’S HERMITE FUNCTION ASYMPTOTIC FORMULA AND SOME APPLICATIONS." ANZIAM Journal 50, no. 4 (2009): 550–61. http://dx.doi.org/10.1017/s1446181109000273.

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AbstractThe aim of this paper is to find a concrete bound for the error involved when approximating the nth Hermite function (in the oscillating range) by an asymptotic formula due to D. Dominici. This bound is then used to study the accuracy of certain approximations to Hermite expansions and to Fourier transforms. A way of estimating an unknown probability density is proposed.

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3

Singh, Pravin, Nabendra Parumasur, and Shivani Singh. "A Review of Collocation Approximations to Solutions of Differential Equations." Mathematics 10, no. 23 (2022): 4438. http://dx.doi.org/10.3390/math10234438.

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This review considers piecewise polynomial functions, that have long been known to be a useful and versatile tool in numerical analysis, for solving problems which have solutions with irregular features, such as steep gradients and oscillatory behaviour. Examples of piecewise polynomial functions used include splines, in particular B-splines, and Hermite functions. Spline functions are useful for obtaining global approximations whilst Hermite functions are useful for approximation over finite elements. Our aim in this review is to study quintic Hermite functions and develop a numerical colloca

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4

Hu, Jiaxin, Chenglong Yu, and Kangyun Zhou. "Padé Approximations and Irrationality Measures on Values of Confluent Hypergeometric Functions." Mathematics 12, no. 16 (2024): 2516. http://dx.doi.org/10.3390/math12162516.

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Padé approximations are approximations of holomorphic functions by rational functions. The application of Padé approximations to Diophantine approximations has a long history dating back to Hermite. In this paper, we use the Maier–Chudnovsky construction of Padé-type approximation to study irrationality properties about values of functions with the form f(x)=∑k=0∞xkk!(bk+s)(bk+s+1)⋯(bk+t), where b,t,s are positive integers and obtain upper bounds for irrationality measures of their values at nonzero rational points. Important examples includes exponential integral, Gauss error function and Kum

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5

Burova, I. G. "The Hermite-Birkhoff Problem and Local Spline Approximation." WSEAS TRANSACTIONS ON MATHEMATICS 23 (October 2, 2024): 591–98. http://dx.doi.org/10.37394/23206.2024.23.62.

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This paper discusses the use of local spline approximations to solve the Hermite-Birkhoff problem. The solution to a specific problem using polynomial and non-polynomial local splines of the third order of approximation is considered. Here we discuss the case when the values of the function 𝑢(𝑥) and its derivative 𝑢’(𝑥) are given at the nodes of the grid in an alternative way: … , 𝑢(𝑥𝑗), 𝑢′(𝑥𝑗+1), 𝑢(𝑥𝑗+2), … . Note that when using polynomial and non-polynomial spline approximations, it is possible to obtain acceptable solutions in several interesting cases that are impossible when we use the c

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6

Kadakal, Huriye, and Mahir Kadakal. "Inverse trigonometrically convexity and better approximations." Miskolc Mathematical Notes 26, no. 1 (2025): 305. https://doi.org/10.18514/mmn.2025.4519.

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In this paper, we introduce and study the concept of inverse trigonometrically convex functions and their some algebraic properties. We prove some Hermite-Hadamard type integral inequalities for the newly introduced class of functions. We also obtain some refinements of the Hermite-Hadamard inequality for functions whose first derivative in absolute value is inverse trigonometrically convex. Moreover, we proved that Hölder-İşcan and improved power-mean integral inequalities give a better approach than Hölder and power-mean inequalities.

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7

Rosenkilde, Johan, and Arne Storjohann. "Algorithms for simultaneous Hermite–Padé approximations." Journal of Symbolic Computation 102 (January 2021): 279–303. http://dx.doi.org/10.1016/j.jsc.2019.07.026.

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8

Komargodski, Z., and D. Levin. "Hermite type moving-least-squares approximations." Computers & Mathematics with Applications 51, no. 8 (2006): 1223–32. http://dx.doi.org/10.1016/j.camwa.2006.04.005.

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9

Aguirre, Julián, and Judith Rivas. "Hermite pseudospectral approximations. An error estimate." Journal of Mathematical Analysis and Applications 304, no. 1 (2005): 189–97. http://dx.doi.org/10.1016/j.jmaa.2004.09.013.

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10

Suetin, Sergey Pavlovich. "Convergence of Hermite-Padé rational approximations." Russian Mathematical Surveys 78, no. 5 (2023): 967–69. http://dx.doi.org/10.4213/rm10144e.

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11

Väänänen, Keijo, and Wen Wu. "On linear independence measures of the values of Mahler functions." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 6 (2018): 1297–311. http://dx.doi.org/10.1017/s0308210518000148.

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We estimate the linear independence measures for the values of a class of Mahler functions of degrees 1 and 2. For this purpose, we study the determinants of suitable Hermite–Padé approximation polynomials. Based on the non-vanishing property of these determinants, we apply the functional equations to get an infinite sequence of approximations that is used to produce the linear independence measures.

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12

Staravoitov, A. P., and N. V. Ryabchenko. "Uniqueness of the solutions of the Hermite – Pade problems." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, no. 4 (2020): 445–56. http://dx.doi.org/10.29235/1561-2430-2019-55-4-445-456.

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New concepts are introduced in the present work. They are a quite normal index and a quite perfect system of functions. Using these concepts, the uniqueness criterion for solution of two Hermite – Pade problems is proved, the explicit determinant representations of type I and II Hermite – Padé polynomials for an arbitrary system of power series are obtained. The results obtained complement and generalize the well-known result in the theory of Hermite – Padé approximations.

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13

Sergeyev, A. V. "A recursive algorithm for Padé-Hermite approximations." USSR Computational Mathematics and Mathematical Physics 26, no. 2 (1986): 17–22. http://dx.doi.org/10.1016/0041-5553(86)90003-0.

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14

Kandayan, A. A., and V. N. Sorokin. "Multipoint Hermite—Padé approximations for beta functions." Mathematical Notes 87, no. 1-2 (2010): 204–17. http://dx.doi.org/10.1134/s000143461001027x.

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15

Sheppard, Colin J. R., and Miguel A. Porras. "Comparison between the Propagation Properties of Bessel–Gauss and Generalized Laguerre–Gauss Beams." Photonics 10, no. 9 (2023): 1011. http://dx.doi.org/10.3390/photonics10091011.

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The connections between Laguerre–Gauss and Bessel–Gauss beams, and between Hermite–Gauss and cosine-Gauss beams are investigated. We review different asymptotic expressions for generalized Laguerre and Hermite polynomials of large radial/transverse order. The amplitude variations of generalized Laguerre–Gauss beams, including standard and elegant Laguerre–Gauss beams as special cases, are compared with Bessel–Gauss beams. Bessel–Gauss beams can be well-approximated by elegant Laguerre–Gauss beams. For non-integral values of the Laguerre function radial order, a generalized Laguerre–Gauss beam

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16

Mundewadi, R. A., and S. Kumbinarasaiah. "Numerical Solution of Abel′s Integral Equations using Hermite Wavelet." Applied Mathematics and Nonlinear Sciences 4, no. 1 (2019): 169–80. http://dx.doi.org/10.2478/amns.2019.1.00017.

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AbstractA numerical method is developed for solving the Abel′s integral equations is presented. The method is based upon Hermite wavelet approximations. Hermite wavelet method is then utilized to reduce the Abel′s integral equations into the solution of algebraic equations. Illustrative examples are included to demonstrate the validity, efficiency and applicability of the proposed technique. Algorithm provides high accuracy and compared with other existing methods.

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Mundewadi, R. A., and Kumbinarasaiah S. "Numerical Solution of Abel′s Integral Equations using Hermite Wavelet." Applied Mathematics and Nonlinear Sciences 4, no. 2 (2019): 395–406. http://dx.doi.org/10.2478/amns.2019.2.00037.

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AbstractA numerical method is developed for solving the Abel′s integral equations is presented. The method is based upon Hermite wavelet approximations. Hermite wavelet method is then utilized to reduce the Abel′s integral equations into the solution of algebraic equations. Illustrative examples are included to demonstrate the validity, efficiency and applicability of the proposed technique. Algorithm provides high accuracy and compared with other existing methods.

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18

Mattila, Keijo Kalervo, Luiz Adolfo Hegele Júnior, and Paulo Cesar Philippi. "High-Accuracy Approximation of High-Rank Derivatives: Isotropic Finite Differences Based on Lattice-Boltzmann Stencils." Scientific World Journal 2014 (2014): 1–16. http://dx.doi.org/10.1155/2014/142907.

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We propose isotropic finite differences for high-accuracy approximation of high-rank derivatives. These finite differences are based on direct application of lattice-Boltzmann stencils. The presented finite-difference expressions are valid in any dimension, particularly in two and three dimensions, and any lattice-Boltzmann stencil isotropic enough can be utilized. A theoretical basis for the proposed utilization of lattice-Boltzmann stencils in the approximation of high-rank derivatives is established. In particular, the isotropy and accuracy properties of the proposed approximations are deri

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19

Butt, S. I., T. Rasheed, D. Pecaric, and J. Pecaric. "Upper and lower estimations of Popoviciu’s difference via weighted Hadamard inequality with applications." Filomat 37, no. 22 (2023): 7641–62. http://dx.doi.org/10.2298/fil2322641b.

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We consider differences coming from Popoviciu?s inequality and give upper and lower bounds by employing weighted Hermite-Hadamard inequality along with the approximations of Montgomery two point formula. We also give bounds for Popoviciu?s inequality by employing weighted Hermite-Hadamard inequality along with the approximations of Montgomery one point formula. We testify this scenario by utilizing the theory of n-times differentiable convex functions. Our results hold for all n ? 2 and we provide explicit examples to show the correctness of the bounds obtained for special cases. Last but not

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20

Matala-aho, Tapani, and Louna Seppälä. "Hermite–Thue equation: Padé approximations and Siegel's lemma." Journal of Number Theory 191 (October 2018): 345–83. http://dx.doi.org/10.1016/j.jnt.2018.03.014.

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21

Aptekarev, Alexander I., and Arno Kuijlaars. "Hermite-Padé approximations and multiple orthogonal polynomial ensembles." Russian Mathematical Surveys 66, no. 6 (2011): 1133–99. http://dx.doi.org/10.1070/rm2011v066n06abeh004771.

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22

Dũng, Dinh. "Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 3 (2021): 1163–98. http://dx.doi.org/10.1051/m2an/2021017.

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By combining a certain approximation property in the spatial domain, and weighted 𝓁2-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in Bachmayr et al. [ESAIM: M2AN 51 (2017) 341–363] and Bachmayr et al. [SIAM J. Numer. Anal. 55 (2017) 2151–2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We construct such methods and prove convergence rates of the approxim

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23

Kadakal, Huriye. "Better Approximations for Quasi-Convex Functions." Studia Universitatis Babes-Bolyai Matematica 69, no. 2 (2024): 267–81. http://dx.doi.org/10.24193/subbmath.2024.2.02.

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In this paper, by using Hölder-İşcan, Hölder integral inequality and a general identity for differentiable functions we can get new estimates on generalization of Hadamard, Ostrowski and Simpson type integral inequalities for functions whose derivatives in absolute value at certain power are quasi-convex functions. It is proved that the result obtained Hölder-İşcan integral inequality is better than the result obtained Hölder inequality. Keywords: Hölder-İşcan scan inequality, Hermite-Hadamard inequality, Simpson and Ostrowski type inequality, midpoint and trapezoid type inequality, quasi-conv

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24

Ray, Shashwati, and Vandana Chouhan. "Electrocardiogram reconstruction based on Hermite interpolating polynomial with Chebyshev nodes." Indonesian Journal of Electrical Engineering and Computer Science 36, no. 2 (2024): 837. http://dx.doi.org/10.11591/ijeecs.v36.i2.pp837-845.

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Electrocardiogram (ECG) signals generate massive volume of digital data, so they need to be suitably compressed for efficient transmission and storage. Polynomial approximations and polynomial interpolation have been used for ECG data compression where the data signal is described by polynomial coefficients only. Here, we propose approximation using hermite polynomial interpolation with chebyshev nodes for compressing ECG signals that consequently denoises them too. Recommended algorithm is applied on various ECG signals taken from MIT-BIH arrhythmia database without any additional noise as th

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25

Shashwati, Ray Vandana Chouhan. "Electrocardiogram reconstruction based on Hermite interpolating polynomial with Chebyshev nodes." Indonesian Journal of Electrical Engineering and Computer Science 36, no. 2 (2024): 837–45. https://doi.org/10.11591/ijeecs.v36.i2.pp837-845.

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Electrocardiogram (ECG) signals generate massive volume of digital data, so they need to be suitably compressed for efficient transmission and storage. Polynomial approximations and polynomial interpolation have been used for ECG data compression where the data signal is described by polynomial coefficients only. Here, we propose approximation using hermite polynomial interpolation with chebyshev nodes for compressing ECG signals that consequently denoises them too. Recommended algorithm is applied on various ECG signals taken from MIT-BIH arrhythmia database without any additional noise as th

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26

Zhao, Feng, and Shao-Lun Huang. "On the Universally Optimal Activation Function for a Class of Residual Neural Networks." AppliedMath 2, no. 4 (2022): 574–84. http://dx.doi.org/10.3390/appliedmath2040033.

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While non-linear activation functions play vital roles in artificial neural networks, it is generally unclear how the non-linearity can improve the quality of function approximations. In this paper, we present a theoretical framework to rigorously analyze the performance gain of using non-linear activation functions for a class of residual neural networks (ResNets). In particular, we show that when the input features for the ResNet are uniformly chosen and orthogonal to each other, using non-linear activation functions to generate the ResNet output averagely outperforms using linear activation

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27

Jackson, Darrell R., and Eric I. Thorsos. "Tests of Wiener‐Hermite approximations for rough‐surface scattering." Journal of the Acoustical Society of America 84, S1 (1988): S221. http://dx.doi.org/10.1121/1.2026206.

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28

de la Calle Ysern, B., and J. Mínguez Ceniceros. "Zero distribution of incomplete Padé and Hermite–Padé approximations." Journal of Approximation Theory 201 (January 2016): 13–29. http://dx.doi.org/10.1016/j.jat.2015.08.005.

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29

Matala-aho, Tapani. "Type II Hermite–Padé Approximations of Generalized Hypergeometric Series." Constructive Approximation 33, no. 3 (2010): 289–312. http://dx.doi.org/10.1007/s00365-010-9111-x.

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30

Anshul, Srivastava, Yadav Avinash Kumar, Sinha Brijesh Kumar, Heshamuddine Md., and Rao Nadeem. "Direct approximations of Szász-Beta-Schurer operators using Hermite polynomial." Annals of the University of Craiova Mathematics and Computer Science Series 51, no. 2 (2024): 382–97. https://doi.org/10.52846/ami.v51i2.1837.

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The aim of present article is to introduce the Szász-Beta-Schurer operators in terms of Hermite Polynomial. We calculate some estimates and then discuss convergence theorems and order of approximation in terms of Korovkin theorem and first order modulus of smoothness respectively. Next, we study pointwise approximation results in terms of Peetre’s K-functional, second order modulus of smoothness, Lipschitz type space and rth order Lipschitz type maximal function. Lastly, weighted approximation results and statistical approximation theorems are proved.

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31

Butt, Saad Ihsan, Muhammad Nasim Aftab, Hossam A. Nabwey, and Sina Etemad. "Some Hermite-Hadamard and midpoint type inequalities in symmetric quantum calculus." AIMS Mathematics 9, no. 3 (2024): 5523–49. http://dx.doi.org/10.3934/math.2024268.

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<abstract><p>The Hermite-Hadamard inequalities are common research topics explored in different dimensions. For any interval $ [\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re $, we construct the idea of the Hermite-Hadamard inequality, its different kinds, and its generalization in symmetric quantum calculus at $ \mathrm{b_{0}}\in[\mathrm{b_{0}}, \mathrm{b_{1}}]\subset\Re $. We also construct parallel results for the Hermite-Hadamard inequality, its different types, and its generalization on other end point $ \mathrm{b_{1}} $, and provide some examples as well. Some justification with g

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32

Shahood, Ahmed* Jitendra Jayant. "IMPROVING CLASSICAL LUMPED MODEL FOR TRANSIENT HEAT CONDUCTION IN SLAB USING HERMITE APPROXIMATION." INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY 6, no. 6 (2017): 84——93. https://doi.org/10.5281/zenodo.802834.

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The present work aims at applying the ideas on the analysis of improved lumped-parameter model for transient heat conduction in a slab with temperature-dependent thermal conductivity. The transient temperature is found to depend on various model parameters, namely, Biot number, heat source parameter and time. Polynomial Approximation Method (PAM) has been possible to derive a unified relation for the transient thermal behaviour of solid (slab and tube) with both internal generation and boundary heat flux. In all the cases, a closed form solution is obtained between temperature, Biot number, he

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33

Kadakal, Mahir, İmdat İşcan, Praveen Agarwal, and Mohamed Jleli. "Exponential trigonometric convex functions and Hermite-Hadamard type inequalities." Mathematica Slovaca 71, no. 1 (2021): 43–56. http://dx.doi.org/10.1515/ms-2017-0410.

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Abstract In this manuscript, we introduce and study the concept of exponential trigonometric convex functions and their some algebraic properties. We obtain Hermite-Hadamard type inequalities for the newly introduced class of functions. We also obtain some refinements of the Hermite-Hadamard inequality for functions whose first derivative in absolute value, raised to a certain power which is greater than one, respectively at least one, is exponential trigonometric convex function. It has been shown that the result obtained with Hölder-İşcan and improved power-mean integral inequalities give be

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34

Nasyrov, Semen Rafailovich. "Nuttall decomposition of a three-sheeted torus." Izvestiya: Mathematics 88, no. 5 (2024): 873–929. http://dx.doi.org/10.4213/im9561e.

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With the help of the Weierstrass elliptic functions, we study the problem of describing the Nuttall decomposition of a three-sheeted compact Riemann surface of genus $1$ related to an Abelian integral on the surface. This decomposition plays an important role in investigation of Hermite-Padé diagonal approximations.

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35

Rivoal, Tanguy. "Simultaneous Polynomial Approximations of the Lerch Function." Canadian Journal of Mathematics 61, no. 6 (2009): 1341–56. http://dx.doi.org/10.4153/cjm-2009-063-6.

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Abstract We construct bivariate polynomial approximations of the Lerch function that for certain specialisations of the variables and parameters turn out to be Hermite–Padé approximants either of the polylogarithms or ofHurwitz zeta functions. In the former case, we recover known results, while in the latter the results are new and generalise some recent works of Beukers and Prévost. Finally, we make a detailed comparison of our work with Beukers’. Such constructions are useful in the arithmetical study of the values of the Riemann zeta function at integer points and of the Kubota–Leopold p-ad

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36

Liu, Tao, and Stanford Shateyi. "Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem." Mathematics 12, no. 7 (2024): 1121. http://dx.doi.org/10.3390/math12071121.

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An effective strategy to enhance the convergence order of nodal approximations in interpolation or PDE problems is to increase the size of the stencil, albeit at the cost of increased computational burden. In this study, our goal is to improve the convergence orders for approximating the first and second derivatives of sufficiently differentiable functions using the radial basis function-generated Hermite finite-difference (RBF-HFD) scheme. By utilizing only three equally spaced points in 1D, we are able to boost the convergence rate to four. Extensive tests have been conducted to demonstrate

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37

Kalsoom, Humaira, Muhammad Amer Latif, Muhammad Idrees, Muhammad Arif, and Zabidin Salleh. "Quantum Hermite-Hadamard type inequalities for generalized strongly preinvex functions." AIMS Mathematics 6, no. 12 (2021): 13291–310. http://dx.doi.org/10.3934/math.2021769.

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<abstract><p>In accordance with the quantum calculus, the quantum Hermite-Hadamard type inequalities shown in recent findings provide improvements to quantum Hermite-Hadamard type inequalities. We acquire a new $ q{_{\kappa_1}} $-integral and $ q{^{\kappa_2}} $-integral identities, then employing these identities, we establish new quantum Hermite-Hadamard $ q{_{\kappa_1}} $-integral and $ q{^{\kappa_2}} $-integral type inequalities through generalized higher-order strongly preinvex and quasi-preinvex functions. The claim of our study has been graphically supported, and some special

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38

Anastassiou, George A. "Approximations by multivariate sublinear and Max-product operators under convexity." Demonstratio Mathematica 51, no. 1 (2018): 85–105. http://dx.doi.org/10.1515/dema-2018-0008.

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AbstractHere we search quantitatively under convexity the approximation of multivariate function by general multivariate positive sublinear operators with applications to multivariate Max-product operators. These are of Bernstein type, of Favard-Szász-Mirakjan type, of Baskakov type, of sampling type, of Lagrange interpolation type and of Hermite-Fejér interpolation type. Our results are both: under the presence of smoothness and without any smoothness assumption on the function to be approximated which fulfills a convexity assumption.

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Mitchell, Rory, Eibe Frank, and Geoffrey Holmes. "An Empirical Study of Moment Estimators for Quantile Approximation." ACM Transactions on Database Systems 46, no. 1 (2021): 1–21. http://dx.doi.org/10.1145/3442337.

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We empirically evaluate lightweight moment estimators for the single-pass quantile approximation problem, including maximum entropy methods and orthogonal series with Fourier, Cosine, Legendre, Chebyshev and Hermite basis functions. We show how to apply stable summation formulas to offset numerical precision issues for higher-order moments, leading to reliable single-pass moment estimators up to order 15. Additionally, we provide an algorithm for GPU-accelerated quantile approximation based on parallel tree reduction. Experiments evaluate the accuracy and runtime of moment estimators against t

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40

Funes, Jorge Olivares, and Elvis Valero Kari. "Numerical solution of Hermite differential equation using the spline method of order 1 with GeoGebra." Journal of Physics: Conference Series 2090, no. 1 (2021): 012091. http://dx.doi.org/10.1088/1742-6596/2090/1/012091.

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Abstract In this paper we will show the visualization of the approximations that can be obtained by means of the order 1 spline method for Hermite differential equations with well-interactive examples of GeoGebra applets. Here, we will dedicate ourselves to publisize, the great benefit that can be obtained, in the process of generating new mathematical knowledge for learning and teaching the numerical solutions of differential equations.

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41

Chu, Wenchang. "Harmonic number identities and Hermite–Padé approximations to the logarithm function." Journal of Approximation Theory 137, no. 1 (2005): 42–56. http://dx.doi.org/10.1016/j.jat.2005.07.008.

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42

Black, Kelly, and John B. Geddes. "Complex Valued Spectral Hermite Approximations for the Actively Mode-Locked Laser." Journal of Scientific Computing 32, no. 3 (2007): 427–48. http://dx.doi.org/10.1007/s10915-007-9138-8.

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43

Suetin, S. P. "Hermite–Padé polynomials and Shafer quadratic approximations for multivalued analytic functions." Russian Mathematical Surveys 75, no. 4 (2020): 788–90. http://dx.doi.org/10.1070/rm9954.

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44

Baker, George A., and D. S. Lubinsky. "Convergence theorems for rows of differential and algebraic Hermite-Padé approximations." Journal of Computational and Applied Mathematics 18, no. 1 (1987): 29–52. http://dx.doi.org/10.1016/0377-0427(87)90054-9.

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45

Driver, Kathy A., and Nico M. Temme. "On Polynomials Related with Hermite–Padé Approximations to the Exponential Function." Journal of Approximation Theory 95, no. 1 (1998): 101–22. http://dx.doi.org/10.1006/jath.1998.3195.

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46

Ishkhanyan, Artur M. "Exact solution of the 1D Dirac equation for the inverse-square-root potential 1/x$1/\sqrt{x}$." Zeitschrift für Naturforschung A 75, no. 8 (2020): 771–79. http://dx.doi.org/10.1515/zna-2020-0083.

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AbstractWe present the exact solution of the 1D Dirac equation for the inverse-square-root potential $1/\sqrt{x}$ for several configurations of vector, pseudo-scalar, and scalar fields. Each fundamental solution of the problem can be written as an irreducible linear combination of two Hermite functions of a scaled and shifted argument. We derive the exact equations for bound-state energy eigenvalues and construct accurate approximations for the energy spectrum.

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47

Kawamoto, Tatsuro. "Entropy of microcanonical finite-graph ensembles." Journal of Physics: Complexity 4, no. 3 (2023): 035005. http://dx.doi.org/10.1088/2632-072x/acf01c.

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Abstract The entropy of random graph ensembles has gained widespread attention in the field of graph theory and network science. We consider microcanonical ensembles of simple graphs with prescribed degree sequences. We demonstrate that the mean-field approximations of the generating function using the Chebyshev–Hermite polynomials provide estimates for the entropy of finite-graph ensembles. Our estimate reproduces the Bender–Canfield formula in the limit of large graphs.

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48

Chang, Xiang-Ke. "Hermite–Padé approximations with Pfaffian structures: Novikov peakon equation and integrable lattices." Advances in Mathematics 402 (June 2022): 108338. http://dx.doi.org/10.1016/j.aim.2022.108338.

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49

Luo, Xue, and Stephen S. T. Yau. "Hermite Spectral Method with Hyperbolic Cross Approximations to High-Dimensional Parabolic PDEs." SIAM Journal on Numerical Analysis 51, no. 6 (2013): 3186–212. http://dx.doi.org/10.1137/120896931.

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50

Sorokin, V. N. "Hermite-Padé approximations for Nikishin systems and the irrationality of ζ(3)". Russian Mathematical Surveys 49, № 2 (1994): 176–77. http://dx.doi.org/10.1070/rm1994v049n02abeh002229.

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