Recent zbMATH articles in MSC 33Chttps://zbmath.org/atom/cc/33C2024-09-13T18:40:28.020319ZWerkzeugExplicit transformations for generalized Lambert series associated with the divisor function \(\sigma_a^{(N)}(n)\) and their applicationshttps://zbmath.org/1540.111112024-09-13T18:40:28.020319Z"Banerjee, Soumyarup"https://zbmath.org/authors/?q=ai:banerjee.soumyarup"Dixit, Atul"https://zbmath.org/authors/?q=ai:dixit.atul"Gupta, Shivajee"https://zbmath.org/authors/?q=ai:gupta.shivajeeThe main goal of this paper is to generalize (1.13) and (1.14) obtained in [\textit{A. Dixit} et al., Res. Math. Sci. 9, No. 2, Paper No. 34, 54 p. (2022; Zbl 1492.33006)] in the setting of \(\sigma_a^{(N)}(n)e^{-ny}\), where \(\sigma_a^{(N)}(n)\) is the generalized sum-of-divisors function with \(N\)th powers of divisors. This has been studied by the authors in many places starting from [\textit{A. Dixit} et al., Nagoya Math. J. 239, 232--293 (2020; Zbl 1462.11064)] based on the functional equation.
In this paper, the authors use the Voronoi summation formula [\textit{A. Dixit} et al., ``Voronoi summation formula for the generalized divisor function $\sigma_{z}^{(k)}(n)$'', Preprint, \url{arXiv:2303.09937}] to prove transformation formulas for Lambert series. It seems that the same results could be obtained by the use of the functional equation and the Hecke gamma transform and that the main interest lies in finding new special functions which express the results in a closed form.
Reviewer: Shigeru Kanemitsu (Kitakyūshū)Identities for Rankin-Cohen brackets, Racah coefficients and associativityhttps://zbmath.org/1540.170082024-09-13T18:40:28.020319Z"Labriet, Q."https://zbmath.org/authors/?q=ai:labriet.quentin"Poulain d'Andecy, L."https://zbmath.org/authors/?q=ai:poulain-dandecy.loicThis paper establishes an infinite family of identities satisfied by the Rankin-Cohen brackets involving the Racah polynomials and provides a natural interpretation of these identities via the representation theory of the Lie algebra \(\mathfrak{sl}_2\). As an application, the paper proposes a short new proof of the associativity of the Eholzer product on the space of all modular forms. Furthermore, it is shown that, in some sense, all algebraic identities satisfied by the Rankin-Cohen brackets are consequences of the identities established in this paper together with the well-known anti-symmetry property.
Reviewer: Volodymyr Mazorchuk (Uppsala)On the asymptotics of matrix coefficients of representations of \(\mathrm{SL}(2, \mathbb{R})\)https://zbmath.org/1540.220182024-09-13T18:40:28.020319Z"Losert, Viktor"https://zbmath.org/authors/?q=ai:losert.viktorSummary: We study matrix coefficients of the unitary (and also the completely bounded) representations of \(\mathrm{SL}(2, \mathbb{R})\) and its universal covering group. We describe the asymptotic distribution of column vectors in terms of Whittaker functions, exhibiting also a relationship to the Fourier transform.The trace and the deltoid of \(\mathrm{SU}(3)\)https://zbmath.org/1540.220252024-09-13T18:40:28.020319Z"Lachaud, Gilles"https://zbmath.org/authors/?q=ai:lachaud.gillesSummary: This article is dedicated to describing random variables on the group \(\mathrm{SU}(3)\) associated to the character \(\tau\) of the standard representation, that is:
\begin{itemize}
\item[1.] The complex-valued character itself \(\tau\), and the description of its image, the Deltoid \(\Delta\subset\mathbb{C}\).
\item[2.] The length \(\lambda=|\tau|\).
\item[3.] The norm \(\nu=|\tau|^2\).
\item[4.] The adjoint representation \(\alpha(g)=\nu(g)-1\).
\item[5.] The real part \(\xi=2\operatorname{Re}\tau\) and the imaginary part \(\eta=2\operatorname{Im}\tau\).
\end{itemize}
For the entire collection see [Zbl 1475.11004].On the log-convexity of a Bernstein-like polynomials sequencehttps://zbmath.org/1540.260052024-09-13T18:40:28.020319Z"Girjoaba, Adrian"https://zbmath.org/authors/?q=ai:girjoaba.adrianSummary: We prove that the sequence of the sum of the squares of the Bernstein polynomials is pointwise log-convex. There are given two proofs of this result: one by relating our sequence to the Legendre polynomials sequence and one by induction. I know of this problem from Professor Ioan Rasa, Cluj-Napoca. This work was presented at the International Conference on Approximation Theory and its Applications, Sibiu, 2022, dedicated to the scientific work of Professor Ioan Rasa on the occasion of his 70th anniversary.On the Dotsenko-Fateev complex twin of the Selberg integral and its extensionshttps://zbmath.org/1540.330032024-09-13T18:40:28.020319Z"Neretin, Yury A."https://zbmath.org/authors/?q=ai:neretin.yuri-aAs the title of the paper suggests, the author studies what can be considered the Dotsenko-Fateev complex twin of the Selberg integral and its extensions, thus establishing various identities and domains of convergence of such integrals.
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)On some new inequalities and fractional kinetic equations associated with extended Gauss hypergeometric and confluent hypergeometric functionhttps://zbmath.org/1540.330042024-09-13T18:40:28.020319Z"Chandola, Ankita"https://zbmath.org/authors/?q=ai:chandola.ankita"Pandey, Rupakshi Mishra"https://zbmath.org/authors/?q=ai:pandey.rupakshi-mishraSummary: Fractional kinetic equations are of immense importance in describing and solving numerous intriguing problems of physics and astrophysics. Inequalities are important topics in special functions. In this paper, we studied the monotonicity of the extended Gauss and confluent hypergeometric function that are derived by using the inequalities on generalized beta function involving Appell series and Lauricella function. We also establish generalized fractional kinetic equation involving extended hypergeometric and confluent hypergeometric functions. The solutions of generalized fractional kinetic equation is derived and studied as an application of extended hypergeometric and confluent hypergeometric function using the General integral transform. The results obtained here are general and can be used to derive many new solutions of fractional kinetic equations involving various types of special functions.Recurrent identities for two special functions of hypergeometric typehttps://zbmath.org/1540.330052024-09-13T18:40:28.020319Z"Podkletnova, Svetlana Vladimirovna"https://zbmath.org/authors/?q=ai:podkletnova.svetlana-vladimirovnaSummary: The article presents conclusions and proofs of Gauss-type identities for two known hypergeometric type functions. For the derivation and justification of formulas, the representation of functions in the form of a series is used, as well as an integral representation of the functions under consideration. The article uses the definition and properties of gamma and beta functions, the hypergeometric Gauss function, as well as known identities for these functions. Hypergeometric functions are widely used in solving various types of differential equations. The presence of identities connecting the functions involved in the resulting formulas of solutions greatly simplifies both the final formulas and intermediate calculations in many problems related to solving hyperbolic, elliptic and mixed types of equations.A new determinant form of Bessel polynomials and applicationshttps://zbmath.org/1540.330062024-09-13T18:40:28.020319Z"Altomare, M."https://zbmath.org/authors/?q=ai:altomare.m"Costabile, F. A."https://zbmath.org/authors/?q=ai:costabile.francesco-aldoSummary: In this paper a new determinant form of Bessel polynomials is determined. New recurrence formula, applications to interpolation and new determinant form of Bessel functions of half-integer order are also derived.Analytical and geometrical approach to the generalized Bessel functionhttps://zbmath.org/1540.330072024-09-13T18:40:28.020319Z"Bulboacă, Teodor"https://zbmath.org/authors/?q=ai:bulboaca.teodor"Zayed, Hanaa M."https://zbmath.org/authors/?q=ai:zayed.hanaa-mousa(no abstract)Proof of the Kresch-Tamvakis conjecturehttps://zbmath.org/1540.330082024-09-13T18:40:28.020319Z"Caughman, John S."https://zbmath.org/authors/?q=ai:caughman.john-s-iv"Terada, Taiyo S."https://zbmath.org/authors/?q=ai:terada.taiyo-sSummary: In this paper we resolve a conjecture of \textit{A. Kresch} and \textit{H. Tamvakis} [Duke Math. J. 110, No. 2, 359--376 (2001; Zbl 1072.14514)]. Our result is the following.
{Theorem}: For any positive integer \(D\) and any integers \(i,j (0\leq i,j\leq D)\), the absolute value of the following hypergeometric series is at most 1:
\[
{_4F_3}\bigg[\begin{matrix} -i, i+1, -j, j+1 \\ 1, D+2, -D \end{matrix};1\bigg].
\]
To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem.Weisner's methodic study of modified Jacobi polynomialhttps://zbmath.org/1540.330092024-09-13T18:40:28.020319Z"Chongdar, Asit Kumar"https://zbmath.org/authors/?q=ai:chongdar.asit-kumar"Chongdar, Amartya"https://zbmath.org/authors/?q=ai:chongdar.amartyaLet \(P_n^{(\alpha, \beta)}(x) \) denote the Jacobi polynomials of degree \(n.\) This article is concerned with the modified Jacobi polynomials \(P_n^{(\alpha-n, \beta)}(x),\) which are given by
\[
P_n^{(\alpha-n, \beta)}(x) =\frac{(1+\alpha -n)_n}{n!} \,_ 2 F_1 \left[ -n, 1+\alpha +\beta; 1+\alpha -n, \frac{1-x}{2} \right],
\]
and satisfy the differential equation
\[
(1-x^2) y''+\left[\beta-\alpha +n -(2+\alpha +\beta -n)x \right]y' +n[n+\alpha +\beta]y=0.
\]
The authors use Weisner's group theoretic method, known to be used to derive generating functions for classical orthogonal polynomials and their various modifications, to derive some generating functions for the modified Jacobi polynomials. As an example, the following generating function is here obtained:
\[
(1-t)^n P_n^{(\alpha-n, \beta)}\left( \frac{x+t}{1-t} \right)=\sum_{k=0}^ \infty \frac{(-t)^k}{k!}(-\beta -n)_k P_{n-k}^{(\alpha-n+k, \beta)}(x).
\]
Other similar formulas that are too long to be stated here are obtained. It is also shown that a number of theorems and results on bilateral, mixed trilateral, and trilateral generating functions obtained previously by other authors can be obtained as special cases of results obtained in this article.
Reviewer: Ahmed I. Zayed (Chicago)Duality and difference operators for matrix valued discrete polynomials on the nonnegative integershttps://zbmath.org/1540.330102024-09-13T18:40:28.020319Z"Eijsvoogel, Bruno"https://zbmath.org/authors/?q=ai:eijsvoogel.bruno"Morey, Lucía"https://zbmath.org/authors/?q=ai:morey.lucia"Román, Pablo"https://zbmath.org/authors/?q=ai:roman.pablo-manuelIn this extensive article, the authors introduce a notion of duality for matrix-valued orthogonal polynomials (MVOP) with respect to a measure supported on the non-negative integers which are eigenfunctions of a second-order differential operator.
More precisely, they consider the space of all \(N\times N\) matrix-valued polynomials, \(M_{N}(\mathbb{C})[x]\), and a weight function \(W:\mathbb{Z}\to M_{N}(\mathbb{C})[x]\) such that \(W(x)\) is a positive definite \(\forall x\in \mathbb{N}_0\), \((\mathbb{N}_0=\mathbb{Z}_{\geq 0})\), \(W(x)=0\) \(\forall x\in -\mathbb{N}\) and \(W\) has finite moments of all orders. Then, \(W\) defines a matrix-valued inner product on \(M_{N}(\mathbb{C})[x]\) by
\[
\langle P,Q\rangle_W=\sum_{x=0}^\infty P(x)W(x)Q^*(x),
\]
where * denotes the conjugate transpose. In this context, two sequences \((P_n)_n\) and \((Q_n)_n\) are dual if they are related by
\[
P_n(x)=P_n(0)Q_x(\rho(n))\Upsilon(x), \qquad n,x\in \mathbb{N}_0,
\]
for a certain matrix-valued function \(\Upsilon(x)\) and a \(M_N(\mathbb{C})[x]\)-valued function \(\rho\) which is called eigenvalue function.
The article consists of nine sections. Sections 2 and 3 are introductory to the MVOP theory. Then, they start by introducing difference operators, discrete Fourier algebras, weak Pearson equations, strong Pearson equations, as well as their connection with difference operators and shift operators. A result worth mentioning is Theorem 2.1 where for the the family of weights, they find squared norms, a Rodriguez formula and three recurrence relations.
In Section 3 they introduce the notion of duality for MVOP, thus obtaining a characterization of dual families, and a version of the Christoffel-Darboux identity. The authors consider also dual Fourier algebras and study the adjoint operators on these algebras and on their subalgebras.
Matrix-valued Charlier polynomials are particular cases considered in Section 4, where the weight becomes \[W(x)=A^x\frac{W(0)}{x!}B^{-x},\] with \(x\in \mathbb{N}_0\) and \(A,B\) are invertible constants matrices. Furthermore, if \(B=(aA^*)^{-1}\), \(a>0\), then \[W(x)=\frac{a^x}{x!}A^xW(0)(A^*)^x.\] In the Section 5 the authors consider a one-parameter family of matrix Charlier polynomials. Then, they find a LDU factorization for all the squared norms. First, they establish the properties of the matrix \(L\) and after, they give a second-order difference operator that has the polynomials \(P^\lambda_n\) as eigenfunctions.
Shift operators and recurrence relations for the matrix-valued polynomials are obtained in Section 6. Here the authors construct a one-parameter family of matrix weights \(W^\lambda\), \(\lambda\in \mathbb{N}_0\), with explicit shift operators whose squared norms and three-term recurrence relations are given explicitly.
Section 7 is focused on finding explicit expressions for the entries of the matrix Charlier polynomials.
Section 8 focuses on the study of dual polynomials through their dual orthogonal relations. The authors construct dual squared norms, dual shift operators and study Lie algebras of difference operators associated to the Charlier weight.
Finally, dual-dual polynomials are considered in Section 9, through the study of the 4-tuples \((\tilde{P}^\lambda_n,\tilde{M}_1,\tilde{M}_2, \tilde{\upsilon}^{(\lambda)})\) which are dual of the dual 4-tuple \((Q_x^{(\lambda)},P_n^{\lambda}(0),\Upsilon_1^{\lambda},\rho^{(\lambda)})\).
Reviewer: Iris Athamaica López Palacios (Caracas)Comparison between computational cost of fractals using line-doubletshttps://zbmath.org/1540.330112024-09-13T18:40:28.020319Z"Hussain, Sardar Muhammad"https://zbmath.org/authors/?q=ai:hussain.sardar-muhammad"Shah, Hasrat Hussain"https://zbmath.org/authors/?q=ai:shah.hasrat-hussain"Ro, Jong-Suk"https://zbmath.org/authors/?q=ai:ro.jong-sukSummary: In this work, we study the two-dimensional groundwater flow of fractured porous media with the help of analytic element method. A numerical solution formed by line elements called line-doublets based on a series expansion has been presented in the literature. In this solution, each fracture has an influence that may expand in a series that obeys Laplace's equation exactly. The unknown coefficients are found from the discharge potentials of all other elements that are related to the expansion coefficients in series expansion. This work aims to make comparison between the computational cost for fractals by the analytic element method with different techniques like: Iterative and Matrix method.Two-dimensional \((p, q)\)-heat polynomials of Gould-Hopper typehttps://zbmath.org/1540.330122024-09-13T18:40:28.020319Z"Ghanmi, Allal"https://zbmath.org/authors/?q=ai:ghanmi.allal"Lamsaf, Khalil"https://zbmath.org/authors/?q=ai:lamsaf.khalilThis paper introduces a novel class of holomorphic polynomials that extend the classical Gould-Hopper polynomials to two complex variables. These polynomials encompass various known polynomials as special cases, such as the one-dimensional and two-dimensional holomorphic and polyanalytic Itô-Hermite polynomials. The study focuses on exploring their operational representation, different generating functions, and recurrence relations. Special identities, including multiplication, Runge addition, and Nielson types formulas, are established.
Furthermore, the paper delves into the analysis of higher-order partial differential equations associated with these polynomials and investigates their connection to Gould-Hopper polynomials and hypergeometric functions.
Reviewer: Norbert Hounkonnou (Cotonou)Expansion formulas for a class of function related to incomplete Fox-Wright functionhttps://zbmath.org/1540.330132024-09-13T18:40:28.020319Z"Mehrez, Sana"https://zbmath.org/authors/?q=ai:mehrez.sana"Miraoui, Mohsen"https://zbmath.org/authors/?q=ai:miraoui.mohsen"Agarwal, Praveen"https://zbmath.org/authors/?q=ai:agarwal.praveenThe main goal of this paper is to introduce some properties involving a new class of functions associated with the incomplete Fox-Wright function. The authors derive several properties of these functions, such as differentiation formulas with respect to parameters and fractional integration formulas in terms of the class of function associated with the Fox-Wright function. As an application, the authors present a new summation formula involving the incomplete gamma function and some other special functions. In particular, new identities (involving error functions) for a certain class of functions related to the Fox-Wright functions are derived.
Reviewer: Fahreddin Abdullayev (Mersin)Towards a change of variable formula for ``hypergeometrization''https://zbmath.org/1540.330142024-09-13T18:40:28.020319Z"Blaschke, Petr"https://zbmath.org/authors/?q=ai:blaschke.petrThis paper deals with the application of a linear operator called ``hypergeometrization''. Consider a convergent power series \(f(x)=\sum_{n=0}^{\infty}f_nx^n\) where the coefficients \(f_n\) are in the complex domain. For \(c\ne 0,-1,-2,\dots\), consider the operator \(H_c^a\) such that
\[
H_c^af(x)=\sum_{n=0}^{\infty}f_n\frac{(a)_n}{(c)_n}x^n,
\]
where \((a)_n\) and \((c)_n\) are Pochhammer symbols, \((a)_n=a(a+1)\dots(a+n-1),(a)_0=1,a\ne 0\). The operator \(H_c^a\) inserts a ratio of two Pochhammer symbols in a convergent power series, as shown above. For example,
\[
H_c^a(1-x)^{-b}=H_c^a\sum_{n=0}^{\infty}\frac{(b)_n}{n!}x^n=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n}\frac{x^n}{n!}={_2F_1}(a,b;c;x),
\]
where \(|x|<1\) and \({_2F_1}(\cdot)\) is Gauss hypergeometric function. As another illustration, consider
\[
H_c^a\mathrm{e}^x=H_c^a\sum_{n=0}^{\infty}\frac{x^n}{n!}=\sum_{n=0}^{\infty}\frac{(a)_n}{(c)_n}\frac{x^n}{n!}={_1F_1}(a;c;x)
\]
where \({_1F_1}(\cdot)\) is the confluent hypergeometric function. The author recovers a large number of known results and establishes new results on elementary hypergeometric functions such as \({_1F_0},{_0F_1}, {_1F_1}, {_2F_1},{_3F_2}\) and generalized special functions such as Appell functions, Lauricella functions, Horn functions. Reduction formulae and interconnections among these functions are also established with the help of the hypergeometrization operator \(H_c^a\).
People working in the area of special functions will find in this paper a treasure of results on univariate, bivariate and multivariate hypergeometric functions, several reduction formulae and various interconnections among them.
Reviewer: Arakaparampil M. Mathai (Montréal)Hankel transform, \(\mathcal{K}\)-Bessel functions and zeta distributions in the Dunkl settinghttps://zbmath.org/1540.330152024-09-13T18:40:28.020319Z"Brennecken, Dominik"https://zbmath.org/authors/?q=ai:brennecken.dominikThe paper presents a set of properties
of different functions such as Dunkl kernels, Bessel
functions and Dunkl-type \(K\)-Bessel functions. The author
also discusses the conditions under which these functions exist and
are continuous, as well as their behavior under certain transformations
(e.g., Hankel transform, Fourier transform, Dunkl transform) and
constraints.
One of the key points discussed in the paper is the existence of the
Dunkl-type \(K\)-Bessel functions. Some properties are discussed as well, as for example their holomorphy in certain domains.
Reviewer: Oğuz Yağcı (Kırıkkale)Wick-Fourier-Hermite series in the theory of linear and nonlinear transformations of Gaussian distributionshttps://zbmath.org/1540.330162024-09-13T18:40:28.020319Z"Chernousova, E."https://zbmath.org/authors/?q=ai:chernousova.elena"Molchanov, S."https://zbmath.org/authors/?q=ai:molchanov.stanislav-alekseevich"Shiryaev, A."https://zbmath.org/authors/?q=ai:shiryaev.a-v|shiryaev.albert-n|shiryaev.a-a|shiryaev.a-s|shiryaev.a-kSummary: This article provides information on Hermite polynomials and its application to some problems in risk theory and site percolation.Linear transform that preserve real roots of polynomialshttps://zbmath.org/1540.330172024-09-13T18:40:28.020319Z"Dhaouadi, Lazhar"https://zbmath.org/authors/?q=ai:dhaouadi.lazhar"Saidani, Islem"https://zbmath.org/authors/?q=ai:saidani.islemSummary: In the present paper we introduce a mechanism for generation a new class of linear transformations that preserve real roots of polynomials by using the theory of variation diminishing kernel.On self-adjoint operators generated by the fourth-order Laguerre type differential expressionhttps://zbmath.org/1540.330202024-09-13T18:40:28.020319Z"Anderson, Drew"https://zbmath.org/authors/?q=ai:anderson.drew"Lang, Alan"https://zbmath.org/authors/?q=ai:lang.alan"Laurel, Marcus"https://zbmath.org/authors/?q=ai:laurel.marcus"Matter, Elizabeth"https://zbmath.org/authors/?q=ai:matter.elizabeth"Quintero-Roba, Alejandro"https://zbmath.org/authors/?q=ai:quintero-roba.alejandro"da Silva, Pedro Takemura Feitosa"https://zbmath.org/authors/?q=ai:da-silva.pedro-takemura-feitosaSummary: In this paper we address the problem of finding all self-adjoint operators generated by a Lagrangian symmetrizable differential expression in a Hilbert space with a mixed Sobolev inner product strictly discrete on the derivatives. We start giving survey on the initial results in the literature: the Glazman-Krein-Naimark (GKN) theory, the later developed GKN-EM theory as an extension of the first, and the works of \textit{L. L. Littlejohn} and \textit{R. Wellman} [Oper. Matrices 13, No. 3, 667--704 (2019; Zbl 1426.47003)]
on the subject.
We apply those to the fourth-order Laguerre type differential expression known for having a sequence of orthogonal polynomials as eigenfunctions and a symmetrizable differential expression. As a result, we give a complete description of the operator space defined from the GKN-EM theory, and also a characterization of all possible self-adjoint operators in the Sobolev space where the operator is defined. We validate these results with known examples previously studied in the literature.Laguerre type twice-iterated Appell polynomialshttps://zbmath.org/1540.330212024-09-13T18:40:28.020319Z"Biricik, Neslihan"https://zbmath.org/authors/?q=ai:biricik.neslihan"Özarslan, Mehmet Ali"https://zbmath.org/authors/?q=ai:ozarslan.mehmet-ali"Çekim, Bayram"https://zbmath.org/authors/?q=ai:cekim.bayramSummary: In this study, we use discrete Appell convolution to define the sequence of Laguerre type twice-iterated Appell polynomials. We obtain explicit representation, recurrence relation, determinantal representation, lowering operator, integro-partial raising operator and integro-partial differential equation. In addition, the special cases of this new family are investigated using Euler and Bernoulli numbers. We also state their corresponding characteristic properties.Exact time-integral inversion via Čebyšëv quintic approximations for nonlinear oscillatorshttps://zbmath.org/1540.340782024-09-13T18:40:28.020319Z"Boschi, Martina"https://zbmath.org/authors/?q=ai:boschi.martina"Ritelli, Daniele"https://zbmath.org/authors/?q=ai:ritelli.daniele"Spaletta, Giulia"https://zbmath.org/authors/?q=ai:spaletta.giuliaSummary: The focus of this work is the solution of a fundamental problem that arises in non-dissipative nonlinear oscillators and related applications, namely the rare possibility of explicitly inverting the associated time-integral. Here, the inversion issue is treated by near-minimax approximation of the restoring force via fifth-order Čebyšëv polynomials on a normalised integration interval: this gives rise to a Duffing-type quintic oscillator, whose solutions effectively represent those of the original problem. Indeed, when an odd function describes the restoring force, the elliptic time-integral associated with the quinticate oscillator can be inverted in closed form. This is obtained here, by observing that the integrand involves a quadratic polynomial, built on the quinticate oscillator coefficients, and by studying its discriminant. Based on these findings, we provide a novel solution procedure, implemented within the \textit{Mathematica} scientific environment, that exploits elliptic integrals of the first kind and whose effectiveness is tested on three well-known conservative nonlinear oscillator models.Hadamard's example and solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equationhttps://zbmath.org/1540.352662024-09-13T18:40:28.020319Z"Kalmenov, Tynysbek S."https://zbmath.org/authors/?q=ai:kalmenov.tynysbek-sharipovich"Rogovoy, Alexander V."https://zbmath.org/authors/?q=ai:rogovoy.alexander-v"Kabanikhin, Sergey I."https://zbmath.org/authors/?q=ai:kabanikhin.sergei-iSummary: In the theory of partial differential equations, an example constructed by J. Hadamard, which shows the instability of the solution of the Cauchy problem for the Laplace equation with respect to small changes in the initial data, is of great importance. Hadamard's example served as the beginning of a systematic study of ill-posed problems in mathematical physics. On the other hand, the study of the Cauchy problem for the Laplace equation arises from problems of geophysics. At the same time, the question arises whether the Cauchy problem is correct for other elliptic equations including degenerate elliptic equations. We have constructed analogs of Hadamard's example and established the incorrectness of the solution of the Cauchy problem for the Gellerstedt equation in two-dimensional and multidimensional cases. The condition of strong solvability of the mixed Cauchy problem for the multidimensional Gellerstedt equation in a cylindrical domain is found. The proof is based on the spectral properties of the Laplace operator and the properties of special functions.Heat coefficients for magnetic Laplacians on the complex projective space \(\mathbb{P}(\mathbb{C})\)https://zbmath.org/1540.352672024-09-13T18:40:28.020319Z"Ahbli, K."https://zbmath.org/authors/?q=ai:ahbli.khalid"Hafoud, A."https://zbmath.org/authors/?q=ai:hafoud.ali"Mouayn, Z."https://zbmath.org/authors/?q=ai:mouayn.zouhairSummary: We denote by \(\Delta_\nu\) the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to \(\nu\). When acting on bounded functions on the complex projective \(n\)-space, this operator has a discrete spectrum consisting on eigenvalues \(\beta_m\), \(m\in \mathbb{Z}_+\). For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of \(\Delta_\nu\). Using a suitable polynomial decomposition of the multiplicity of each \(\beta_m\), we write down a trace formula for the heat operator associated with \(\Delta_\nu\) in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as \(t\searrow 0^+\) by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with \(\Delta_\nu\).A note on fractional powers of the Hermite operatorhttps://zbmath.org/1540.353262024-09-13T18:40:28.020319Z"Thangavelu, Sundaram"https://zbmath.org/authors/?q=ai:thangavelu.sundaramSummary: We give a very short proof of a result proved by \textit{M. Cappiello} et al. [Commun. Partial Differ. Equations 40, No. 6, 1096--1118 (2015; Zbl 1318.35091)] on the Weyl symbol of the inverse of the Harmonic oscillator. We also extend their results to fractional powers.
For the entire collection see [Zbl 1537.35003].Time domain model order reduction of discrete-time bilinear systems with Charlier polynomialshttps://zbmath.org/1540.390062024-09-13T18:40:28.020319Z"Li, Yanpeng"https://zbmath.org/authors/?q=ai:li.yanpeng"Jiang, Yaolin"https://zbmath.org/authors/?q=ai:jiang.yaolin"Yang, Ping"https://zbmath.org/authors/?q=ai:yang.pingSummary: This paper investigates time domain model order reduction of discrete-time bilinear systems with inhomogeneous initial conditions. The state of the system is approximated by the power series associated with the Charlier polynomials and the recurrence relation of the expansion coefficients is derived. The expansion coefficients are orthogonalized to construct the projection matrix by the modified multi-order Arnoldi method. The output of the resulting reduced order system maintains a certain number of expansion coefficients of the original output, and the error estimation of the reduced order system is briefly discussed. Due to the fact that the projection matrix involves the information of initial conditions, the proposed method can well reduce discrete-time bilinear systems with inhomogeneous initial conditions. Two numerical examples are employed to illustrate the effectiveness of the proposed method.Unique special solution for discrete Painlevé IIhttps://zbmath.org/1540.390192024-09-13T18:40:28.020319Z"Van Assche, Walter"https://zbmath.org/authors/?q=ai:van-assche.walterIn this paper, it is shown that the discrete Painlevé II equation of the form
\[
x_{n+1}+x_{n-1}=\frac{\alpha nx_{n}}{1-x_{n}^2},\qquad \alpha\in\mathbb{R},
\]
with starting value \(a_{-1}=-1\) has a unique solution for which \(-1<a_{n}<1\) for every \(n\geq 0\). The proof relies on the use of orthogonal polynomials. An upper bound for this special solution is also given.
Reviewer: Ioannis P. Stavroulakis (Ioannina)On a special Kapteyn serieshttps://zbmath.org/1540.400062024-09-13T18:40:28.020319Z"Janssen, A. J. E. M."https://zbmath.org/authors/?q=ai:janssen.augustus-josephus-elizabeth-mariaThe author studies the Kapteyn series of the first kind:
\[
\sum_{n=0}^{\infty}\,\alpha_nJ_{n+\nu}((n+\nu)z),\qquad z\in\mathbb{C}.
\]
Here, \(J_{\mu}\) is the Bessel function of the first kind and order \(\mu\). These series were systematically investigated by \textit{W.~Kapteyn} [Ann. Sci. Éc. Norm. Supér. (3) 10, 91--122 (1893; JFM 25.0846.01)], who, among other things, proved results on the representation of analytic series in connection with Kepler's problem.
The author uses classical notations as in [\textit{G.~N. Watson}, A treatise on the theory of Bessel functions. Cambridge: University Press (1922; JFM 48.0412.02)]. The main object of interest is the eccentric anomaly \(E\), related to the mean anomaly \(M\) (associated with normalized time) through Kepler's equation
\[
M=E-\varepsilon\sin{E},\qquad M\in\mathbb{R}.
\]
In this context, the special Kapteyn series
\[
T(\varepsilon)=\sum_{n=1}^{\infty}\,\frac{1}{n}\,J_n(n\varepsilon),
\]
is studied extensively.
Reviewer: Marcel G. de Bruin (Heemstede)Determining the first radii of meromorphy via orthogonal polynomials on the unit circlehttps://zbmath.org/1540.410302024-09-13T18:40:28.020319Z"Bosuwan, Nattapong"https://zbmath.org/authors/?q=ai:bosuwan.nattapongSummary: Applying a result concerning a convergence of modified orthogonal Padé approximants constructed from orthogonal polynomials on the unit circle, we prove an analogue of Hadamard's theorem for determining the radius of 1-meromorphy of a function holomorphic on the closed unit disk. Furthermore, we apply our result to study analytic properties of the reciprocal of Szegő functions when their corresponding sequence of Verblunsky coefficients has exponential decay.Approximation with Szász-Chlodowsky operators employing general-Appell polynomialshttps://zbmath.org/1540.410532024-09-13T18:40:28.020319Z"Raza, Nusrat"https://zbmath.org/authors/?q=ai:raza.nusrat"Kumar, Manoj"https://zbmath.org/authors/?q=ai:kumar.manoj.5|yadav.manoj-kumar|kumar.manoj|kumar.manoj.15|kumar.manoj.11|kumar.manoj.16|kumar.manoj.2|kumar.manoj.3|kumar.manoj.13|kumar.manoj.24|kumar.manoj.1|kumar.manoj.10|kumar.manoj.14|kumar.manoj.25"Mursaleen, M."https://zbmath.org/authors/?q=ai:mursaleen.mohammad|mursaleen.mohammad-ayman(no abstract)A new class of solutions to the van Dantzig problem, the Lee-Yang property, and the Riemann hypothesishttps://zbmath.org/1540.420122024-09-13T18:40:28.020319Z"Konstantopoulos, Takis"https://zbmath.org/authors/?q=ai:konstantopoulos.takis"Patie, Pierre"https://zbmath.org/authors/?q=ai:patie.pierre"Sarkar, Rohan"https://zbmath.org/authors/?q=ai:sarkar.rohanSummary: The purpose of this paper to carry out an in-depth analysis of the intriguing van Dantzig problem. We start by observing that the celebrated Lee-Yang property and the Riemann hypothesis can be both rephrased in terms of this problem, and, more specifically, in terms of functions in the Laguerre-Pólya class. Motivated by these facts, we proceed by identifying several non-trivial closure properties enjoyed by the set of solutions to this problem. Not only does this revisit but also, by means of probabilistic techniques, deepens the fascinating and intensive studies of functions in the Laguerre-Pólya class. We continue by providing a new class of entire functions that are solutions to the van Dantzig problem. We also characterize the pair of the corresponding van Dantzig random variables. Finally, we investigate the possibility that the Riemann \(\xi\) function belongs to this class.A note on commutators of singular integrals with BMO and VMO functions in the Dunkl settinghttps://zbmath.org/1540.420282024-09-13T18:40:28.020319Z"Dziubański, Jacek"https://zbmath.org/authors/?q=ai:dziubanski.jacek"Hejna, Agnieszka"https://zbmath.org/authors/?q=ai:hejna.agnieszkaIf \(\mathfrak{ x}=(x_1,\dots,x_N)\), \(\mathfrak{ y}=(y_1,\dots,y_N)\) belong to the Euclidean space \(\mathbb{R}^N\), then we put \(\langle\mathfrak{ x},\mathfrak{ y}\rangle=\sum_{i=1}^N x_iy_i\) and \(\|\mathfrak{ x}\|^2= \langle\mathfrak{ x},\mathfrak{ y}\rangle.\) A finite set \(R\subset \mathbb{R}^N \setminus \{0\}\) is a normalized root system if, for all \(\alpha \in R,\) \(R\cap \alpha R=\{\pm \alpha\}, \|\alpha\|=\sqrt{2},\) and \(\sigma_\alpha(R)=R\), where
\[
\sigma_\alpha({\mathfrak{x}}):={\mathfrak{x}}-2\frac{\langle{\mathfrak{x}},\alpha\rangle}{\|\alpha\|^2}\,\alpha.
\]
The finite group \(G\) generated by the reflections \(\sigma_\alpha\), \(\alpha \in R\), is called the Coxeter group (reflection group) of the root system.
A multiplicity function is a \(G\)-invariant function \(k:R \rightarrow \mathbb{C}\). In the paper the multiplicity function \(k\) is fixed and \(k\ge0.\) The associated measure \(dw\) is given by \(dw=w({\mathfrak{x}})\,d{\mathfrak{x}}\), where
\[
w({\mathfrak{x}})=\prod_{\alpha \in R} |\langle{\mathfrak{x}},\alpha\rangle|^{k(\alpha)}.
\]
If \(f \in L^1_{\mathrm{loc}}(dw)\) and \(E \subset \mathbb{R}^N\) is a measurable bounded set, then
\[
f_E:=\frac{1}{w(E)}\int f({\mathfrak{x}})\,dw({\mathfrak{x}}).
\]
The space BMO is defined by
\[
\mathrm{BMO}:=\left\{b \in L^1_{\mathrm{loc}}(dw): \|b\|_{\mathrm{BMO}} < \infty\right\},
\]
where
\[
\|b\|_{\mathrm{BMO}}:= \sup_B \frac{1}{w(B)} \int_B |b(\mathfrak{x})-b_B| \,dw({\mathfrak{x}})
\]
and the supremum extends over all Euclidean balls \(B=B(\mathfrak{y}, r)=\{z \in \mathbb{R}^N:\ \|\mathfrak{y}-\mathfrak{z}\|<r\}\).
Moreover, the space VMO is the closure in BMO of compactly supported Lipschitz functions.
If \(\xi \in \mathbb{R}^N\), the Dunkl operator \(T_\xi\) is the following \(k\)-modification of the directional derivative \(\partial_\xi\):
\[
T_\xi f(\mathfrak{x})=\partial_\xi f(\mathfrak{x})-\sum_{\alpha \in R} \frac{k(\alpha)}{2}\,\langle\alpha, \xi\rangle\, \frac{f(\mathfrak{x})-f(\sigma_\alpha(\mathfrak{x}))}{\langle\alpha,\mathfrak{x}\rangle}.
\]
Given a fixed \(\mathfrak{y} \in \mathbb{R}^N\), the Dunkl kernel \(\mathfrak{x}\mapsto E(\mathfrak{x},\mathfrak{y)}\) is a unique solution of the system
\[
T_\xi f=\langle\xi, \mathfrak{y)}\rangle f, \qquad f(0)=1.
\]
The Dunkl transform \(\mathcal{F}f\) of \(f \in L^1_{\mathrm{loc}}(dw)\) is defined by
\[
\mathcal{F}f(\xi):=c_k^{-1} \int_{\mathbb{R}^N} f(\mathfrak{x}) \,E(\mathfrak{x},-i\xi) \,dw({\mathfrak{x}}), \ \ {\text{where\ \ }}c_k:= \int_{\mathbb{R}^N} e^{-\|\mathfrak{x}\|^2/2} \,dw({\mathfrak{x}}).
\]
(Note that the Dunkl transform generalizes the Fourier transform.) The inverse \(\mathcal{F}^{-1}\) of \(\mathcal{F}\) satisfies
\[
\mathcal{F}^{-1}f({\mathfrak{x}})= c_k^{-1} \int_{\mathbb{R}^N} f(\xi) \,E(i\xi, \mathfrak{x}) \,dw(\xi) \ \ \, {\text {for}} \ f \in L^1(dw).
\]
The authors assume that the (nonradial) kernel \(K=K(\mathfrak{x})\) has similar properties as that from the classical theory of singular integrals and then they define the Dunkl convolution operator \(\mathcal{T}f:=f\ast K\) associated with the kernel \(K\).
Supposing that \(b \in \mathrm{BMO}\) they consider the commutator
\[
Cf(\mathbf{x})=b(\mathfrak{x})\mathcal{T}f(\mathfrak{x})- \mathcal{T}(bf)(\mathfrak{x}).
\]
The first main result of the paper states that the commutator \(C\) is bounded on the space \(L^p(dw)\) if \(1<p<\infty\), while the second one claims that the assumption \(b \in\mathrm{VMO}\) implies that the commutator \(C\) is compact on the space \(L^p(dw)\) if \(1<p<\infty\).
Reviewer: Bohumír Opic (Praha)On skyburst polynomials and their zeroshttps://zbmath.org/1540.420462024-09-13T18:40:28.020319Z"Cantero, María José"https://zbmath.org/authors/?q=ai:cantero.maria-jose"Iserles, Arieh"https://zbmath.org/authors/?q=ai:iserles.ariehSummary: We consider polynomials orthogonal on the unit circle with respect to the complex-valued measure \(z^{\omega -1} \mathrm{d}z\), where \(\omega \in \mathbb{R}\setminus\{0\}\). We derive their explicit form, a generating function and several recurrence relations. These polynomials possess an intriguing pattern of zeros which, as \(\omega\) varies, are reminiscent of a firework explosion. We prove this pattern in a rigorous manner.Sobolev orthogonal polynomials and spectral methods in boundary value problemshttps://zbmath.org/1540.420472024-09-13T18:40:28.020319Z"Fernández, Lidia"https://zbmath.org/authors/?q=ai:fernandez.lidia"Marcellán, Francisco"https://zbmath.org/authors/?q=ai:marcellan-espanol.francisco"Pérez, Teresa E."https://zbmath.org/authors/?q=ai:perez.teresa-e"Piñar, Miguel A."https://zbmath.org/authors/?q=ai:pinar.miguel-aSummary: In the variational formulation of a boundary value problem for the harmonic oscillator, Sobolev inner products appear in a natural way. First, we study the sequences of Sobolev orthogonal polynomials with respect to such an inner product. Second, their representations in terms of a sequence of Gegenbauer polynomials are deduced as well as an algorithm to generate them in a recursive way is stated. The outer relative asymptotics between the Sobolev orthogonal polynomials and classical Legendre polynomials is obtained. Next we analyze the solution of the boundary value problem in terms of a Fourier-Sobolev projector. Finally, we provide numerical tests concerning the reliability and accuracy of the Sobolev spectral method.Coherent pairs and Sobolev-type orthogonal polynomials on the real line: an extension to the matrix casehttps://zbmath.org/1540.420482024-09-13T18:40:28.020319Z"Fuentes, Edinson"https://zbmath.org/authors/?q=ai:fuentes.edinson"Garza, Luis E."https://zbmath.org/authors/?q=ai:garza.luis-eSummary: In this contribution, we extend the concept of coherent pair for two quasi-definite matrix linear functionals \(\mathfrak{u}_0\) and \(\mathfrak{u}_1\). Necessary and sufficient conditions for these functionals to constitute a coherent pair are determined, when one of them satisfies a matrix Pearson-type equation. Moreover, we deduce algebraic properties of the matrix orthogonal polynomials associated with the Sobolev-type inner product
\[
\langle p,q\rangle_{\mathrm{s}} = \langle p,q\rangle_{\mathfrak{u}_0} + \langle p^{\prime} \mathfrak{M}_1, q^{\prime} \mathfrak{M}_2 \rangle_{\mathfrak{u}_1},
\]
where \(\mathfrak{M}_1\) and \(\mathfrak{M}_2\) are \(m \times m\) non-singular matrices and \(p\), \(q\) are matrix polynomials.A note on orthogonal Dirichlet polynomials with rational weighthttps://zbmath.org/1540.420502024-09-13T18:40:28.020319Z"Lubinsky, Doron S."https://zbmath.org/authors/?q=ai:lubinsky.doron-sSummary: Let \(\{\lambda_j\}^\infty_{j =1}\) be a strictly increasing sequence of positive numbers with \(\lambda_1>0\). We find an explicit formula for the orthogonal Dirichlet polynomials \( \{\phi_ n \}\) formed from linear combinations of \(\{\lambda_j^{-it}\}^n_{j=1}\), associated with rational weights
\[
w(t)=\sum^L_{j=1} =\frac{c_j}{\pi (1+(b_jt)^2)},
\]
where \(0<b_1<b_2<\dots\), and the \(\{c_j\}\) are appropriately chosen. Only \(\{\lambda_j^{-it}\}^n_{j=n-L}\) appear in the formula. In the case \(L=2\), we show that the weight can always be taken positive in \(\mathbb{R}\).The effect of adding endpoint masspoints on bounds for orthogonal polynomialshttps://zbmath.org/1540.420512024-09-13T18:40:28.020319Z"Lubinsky, D. S."https://zbmath.org/authors/?q=ai:lubinsky.doron-sSummary: Let \(\nu\) be a positive measure supported on \([-1,1]\), with infinitely many points in its support. Let \(\{p_n (\nu, x)\}_{n\geq 0}\) be its sequence of orthonormal polynomials. Suppose we add masspoints at \(\pm 1\), giving a new measure \(\mu =\nu +M\delta_1 +N\delta_{-1}\). How much larger can \(|p_n (\mu, 0)|\) be than \(|p_n (\nu, 0)|\)? We study this question for symmetric measures, and give more precise results for ultraspherical weights. Under quite general conditions, such as \(\nu\) lying in the Nevai class, it turns out that the growth is no more than \(1+o(1)\) as \(n\rightarrow\infty\).An analogue of Ingham's theorem on the Heisenberg grouphttps://zbmath.org/1540.430082024-09-13T18:40:28.020319Z"Bagchi, Sayan"https://zbmath.org/authors/?q=ai:bagchi.sayan"Ganguly, Pritam"https://zbmath.org/authors/?q=ai:ganguly.pritam"Sarkar, Jayanta"https://zbmath.org/authors/?q=ai:sarkar.jayanta"Thangavelu, Sundaram"https://zbmath.org/authors/?q=ai:thangavelu.sundaramThe famous uncertainty principle on \(\mathbb{R}^n\) says that a function \(f\) and its Fourier transform \(\hat{f}\) cannot both have rapid decay. There are many generalisations of this principle like Heisenberg-Pauli-Weyl inequality, Paley-Wiener theorem and Hardy's uncertainty principle. The authors prove an exact analogue of the lesser known generalisation of uncertainty principle on \(\mathbb{R}^n\) called Ingham's uncertainty principle for the group Fourier transform on the Heisenberg group. They explicitly construct compactly supported functions on the Heisenberg group whose operator valued Fourier transforms have suitable Ingham type decay. They also prove an analogue of Chernoff's theorem for the family of special Hermite operators.
Reviewer: Sanjiv Gupta (Masqaṭ)Computation of the Bell-Laplace transformshttps://zbmath.org/1540.440052024-09-13T18:40:28.020319Z"Cesarano, Clemente"https://zbmath.org/authors/?q=ai:cesarano.clemente"Caratelli, Diego"https://zbmath.org/authors/?q=ai:caratelli.diego"Ricci, Paolo Emilio"https://zbmath.org/authors/?q=ai:ricci.paolo-emilioSummary: An extension of the Laplace transform by using Bell polynomials was recently introduced. In the present paper computational techniques for approximating the transformed functions are derived. The theoretical approach exploits the generating function method, but for the numerical experiments the matrix pencil method has been used, since it proved to be more effective.Singular value decomposition for longitudinal, transverse and mixed ray transforms of 2D tensor fieldshttps://zbmath.org/1540.440072024-09-13T18:40:28.020319Z"Polyakova, Anna P."https://zbmath.org/authors/?q=ai:polyakova.anna-petrovna"Svetov, Ivan E."https://zbmath.org/authors/?q=ai:svetov.ivan-evgenyevichSummary: The operators of longitudinal, transverse and mixed ray transforms acting on two-dimensional symmetric tensor fields of arbitrary degree \(m\) in an unit disk are considered in the article. The singular value decompositions of the operators for a parallel scheme of data acquisition are constructed. Orthogonal bases in original spaces and image spaces are constructed using harmonic, Jacobi and Gegenbauer polynomials. Based on the obtained decompositions the polynomial expressions for the (pseudo)inverse and adjoint operators are obtained.
{{\copyright} 2023 IOP Publishing Ltd}The twofold Ellis-Gohberg inverse problem for rational matrix functions on the real linehttps://zbmath.org/1540.470252024-09-13T18:40:28.020319Z"ter Horst, Sanne"https://zbmath.org/authors/?q=ai:ter-horst.sanne"Kaashoek, M. A."https://zbmath.org/authors/?q=ai:kaashoek.marinus-a"van Schagen, F."https://zbmath.org/authors/?q=ai:van-schagen.frederikSummary: A twofold Ellis-Gohberg inverse problem for rational matrix functions on the real line is considered in this paper. It is assumed that the data functions of the inverse problem are given by finite dimensional state space realizations. Necessary and sufficient conditions for the existence of a solution are given in terms of the matrices appearing in the state space realizations and solutions to associated Lyapunov equations. In case a solution exists, it is unique. We also provide explicit descriptions of this solution in terms of the matrices and Lyapunov
equation solutions associated with the data functions.
For the entire collection see [Zbl 1446.00025].Reflective prolate-spheroidal operators and the adelic Grassmannianhttps://zbmath.org/1540.470682024-09-13T18:40:28.020319Z"Casper, W. Riley"https://zbmath.org/authors/?q=ai:casper.w-riley"Grünbaum, F. Alberto"https://zbmath.org/authors/?q=ai:grunbaum.francisco-alberto"Yakimov, Milen"https://zbmath.org/authors/?q=ai:yakimov.milen-t"Zurrián, Ignacio"https://zbmath.org/authors/?q=ai:zurrian.ignacio-nahuelSummary: Beginning with the work of Landau, Pollak and Slepian in the 1960s on time-band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point \(W\) of Wilson's infinite dimensional adelic Grassmannian \(\mathrm{Gr}^{\mathrm{ad}}\) gives rise to an integral operator \(T_W\), acting on \(L^2(\Gamma)\) for a contour \(\Gamma \subset \mathbb{C}\), which reflects a differential operator with rational coefficients \(R(z, \partial_z)\) in the sense that \(R(-z,-\partial_z) \circ T_W = T_W \circ R(w, \partial_w)\) on a dense subset of \(L^2(\Gamma)\). By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function \(\psi_W(x,z)\). The exact size of this algebra with respect to a bifiltration is in turn determined using algebro-geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time-band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions \(\psi_W(x,-z)\) always reflect a differential operator. A~\(90^\circ\) rotation argument is used to prove that the time-band limited operators of the generalized Fourier transforms with kernels \(\psi_W(x, iz)\) admit a commuting differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.
{\copyright} 2023 Wiley Periodicals LLC.Integral operators, bispectrality and growth of Fourier algebrashttps://zbmath.org/1540.470692024-09-13T18:40:28.020319Z"Casper, W. Riley"https://zbmath.org/authors/?q=ai:casper.w-riley"Yakimov, Milen T."https://zbmath.org/authors/?q=ai:yakimov.milen-tSummary: In the mid 1980s it was conjectured that every bispectral meromorphic function \(\psi(x,y)\) gives rise to an integral operator \(K_{\psi}(x,y)\) which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions \(\psi(x,y)\) where the commuting differential operator is of order \(\leq 6\). We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second- and first-order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel \(K_{\psi}(x,y)\) leading to a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite-dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.Using parity to accelerate Hermite function computations: zeros of truncated Hermite series, Gaussian quadrature and Clenshaw summationhttps://zbmath.org/1540.650882024-09-13T18:40:28.020319Z"Boyd, John P."https://zbmath.org/authors/?q=ai:boyd.john-paul|boyd.john-philipSummary: Although Hermite functions have been studied for over a century and have been useful for analytical and numerical solutions in a myriad of areas, the theory of Hermite functions has gaps. This article is a unified treatment of all the operations -- quadrature, summation, differentiation, and rootfinding -- that can be accelerated by exploiting parity. Any function \(u(y)\) can be decomposed into its parts that are symmetric and antisymmetric with respect to the origin. Suppose that the quadrature on \(y\in [-\infty,\infty]\) is symmetric in the sense that if \(y_j\) is a quadrature abscissa with weight \(w_j\), then \(-y_j\) is also a quadrature point with weight \(w_j\). The number of multiplications is halved by evaluating the quadrature as \(\int_{-\infty}^\infty u(y) dy \approx w_0 u(0) + \sum_{n=1}^M w_n (u(y_n) + u(-y_n))\). Parity is equally useful in computing the zeros, maxima and minima of a truncated Hermite series of degree \(N\) for the important special cases that the series terms are all either symmetric or antisymmetric. The zeros and critical points are the eigenvalues of a companion matrix whose dimension is \(N/2\) instead of \(N\). In addition, for Hermite functions, we show that parity exploitation halves the dimension of the Jacobi matrix (a special case of the companion matrix) whose eigenvalues are the abcissas of Hermite-Gauss quadrature. The number of floating point operations for the recursion for the weights can likewise be halved. Lastly, the same is true for Chenshaw summation of a Hermite series.