Generalized Analytic Functions Research Articles

Wide-angle effects in multi-tracer power spectra with Doppler corrections

We examine the computation of wide-angle corrections to the galaxy power spectrum including redshift-space distortions and relativistic Doppler corrections, and also including multiple tracers with differing clustering, magnification and evolution biases. We show that the inclusion of the relativistic Doppler contribution, as well as radial derivative terms, are crucial for a consistent wide-angle expansion for large-scale surveys, both in the single and multi-tracer cases. We also give for the first time the wide-angle cross-power spectrum associated with the Doppler magnification-galaxy cross correlation, which has been shown to be a new way to test general relativity.In the full-sky power spectrum, the wide-angle expansion allows integrals over products of spherical Bessel functions to be computed analytically as distributional functions, which are then relatively simple to integrate over.We give for the first time a complete discussion and new derivation of the finite part of the divergent integrals of the form ∫0 ∞ drrnjℓ (kr) jℓ' (qr), which are necessary to compute the wide-angle corrections when a general window function is included. This facilitates a novel method for integrating a general analytic function against a pair of spherical Bessel functions.

Laplace Transform of nested analytic functions via Bell’s polynomials

Bell's polynomials have been used in many different fields, ranging from number theory to operators theory. In this article we show a method to compute the Laplace Transform (LT) of nested analytic functions. To this aim, we provide a table of the first few values of the complete Bell's polynomials, which are then used to evaluate the LT of composite exponential functions. Furthermore a code for approximating the Laplace Transform of general analytic composite functions is created and presented. A graphical verification of the proposed technique is illustrated in the last section.

Toeplitz Operators with Radial Symbols on General Analytic Function Spaces

Toeplitz Operators with Radial Symbols on General Analytic Function Spaces

Integral representations for higher-order Fréchet derivatives of matrix functions: Quadrature algorithms and new results on the level-2 condition number

We propose an integral representation for the higher-order Fréchet derivative of analytic matrix functions f(A) which unifies known results for the first-order Fréchet derivative of general analytic matrix functions and for higher-order Fréchet derivatives of A−1. We highlight two applications of this integral representation: On the one hand, it allows to find the exact value of the level-2 condition number (i.e., the condition number of the condition number) of f(A) for a large class of functions f when A is Hermitian. On the other hand, it also allows to use numerical quadrature methods to approximate higher-order Fréchet derivatives. We demonstrate that in certain situations—in particular when the derivative order k is moderate and the direction terms in the derivative have low-rank structure—the resulting algorithm can outperform established methods from the literature by a large margin.

The royal road to automatic noncommutative real analyticity, monotonicity, and convexity

It was shown classically that matrix monotone and matrix convex functions must be real analytic by Löwner and Kraus respectively. Recently, various analogues have been found in several noncommuting variables. We develop a general framework for lifting automatic analyticity theorems in matrix analysis from one variable to several variables, the so-called “royal road theorem.” That is, we establish the principle that the hard part of proving any automatic analyticity theorem lies in proving the one variable theorem. We use our main result to prove the noncommutative Löwner and Kraus theorems over operator systems as examples, including an analogue of the rational “butterfly realization” of Helton-McCullough-Vinnikov for general analytic functions.

Rectifiability of divergence-free fields along invariant 2-tori

We find conditions under which the restriction of a divergence-free vector field B to an invariant toroidal surface S is rectifiable; namely constant in a suitable global coordinate system. The main results are similar in conclusion to Arnold’s Structure Theorems but require weaker assumptions than the commutation [B,nabla times B] = 0. Relaxing the need for a first integral of B (also known as a flux function), we assume the existence of a solution u: S rightarrow {mathbb {R}} to the cohomological equation Bvert _S(u) = partial _n B on a toroidal surface S mutually invariant to B and nabla times B. The right hand side partial _n B is a normal surface derivative available to vector fields tangent to S. In this situation, we show that the field B on S is either identically zero or nowhere zero with Bvert _S/Vert BVert ^2 vert _S being rectifiable. We are calling the latter the semi-rectifiability of B (with proportionality Vert BVert ^2 vert _S). The nowhere zero property relies on Bers’ results in pseudo-analytic function theory about a generalised Laplace-Beltrami equation arising from Witten cohomology deformation. With the use of de Rham cohomology, we also point out a Diophantine integral condition where one can conclude that Bvert _S itself is rectifiable. The rectifiability and semi-rectifiability of Bvert _S is fundamental to the so-called magnetic coordinates, which are central to the theory of magnetically confined plasmas.

Explicit complex-valued solutions of the 2D eikonal equation

We present a method to obtain explicit solutions of the complex eikonal equation in the plane. This equation arises in the approximation of the Helmholtz equation by the WKBJ or Evanescent Wave Tracking methods. We obtain the complex-valued solutions (called eikonals) as parametrizations in a complex variable. We consider both the cases of constant and non-constant index of refraction. In both cases, the relevant parametrizations depend on some holomorphic function. In the case of a non-constant index of refraction, the parametrization also depends on some extra exponential complex-valued function and on a quasi-conformal homeomorphism. This is due to the use of the theory of pseudo-analytic functions and the related similarity principle. The parametrizations give information about the formation of caustics and the light and shadow regions for the relevant eikonals.

Dependence of Solvability of the Neumann Problem for Generalized Analytic Functions in a Disc on the Value of Its Radius

Dependence of Solvability of the Neumann Problem for Generalized Analytic Functions in a Disc on the Value of Its Radius

Задачи типа Гильберта для уравнения Коши–Римана с сингулярной окружностью в младших коэффициентах

The Cauchy-Riemann system of equations occupies a special place in the class of first-order elliptic systems. Such a system with lowest terms and a right-hand side is called the generalized Cauchy-Riemann system (GCRS), which can be conveniently studied by making a transition from a real space into a complex space. There are several different mathematical theories of equations that generalize the methods of the theory of functions of a complex variable. In this regard, the work of L. Bers, in which the integration operations with respect to a complex variable for a generalized system of Cauchy-Riemann type are generalized, should primarily be mentioned. This approach has received a well-known finalization in the theory of pseudo-analytic functions. In the works of G.N. Polozhiy, the theory of p-analytic functions was developed, which is close in its ideas to the works of L. Bers. Another, more progressive research area, called «generalized analytical functions», was developed by the school of I.N. Vekua and his followers (B.V. Boyarsky and others). Here, the idea of correspondence between the functions of a complex variable and the solutions of the generalized Cauchy–Riemann equation using the functional analysis techniques is developed. The Vekua theory is constructed on the assumption that the coefficients of the function’s lowest terms belong to the space of summable functions with degree р > 2. The coefficients of such systems may admit «weak» singularities limited by the requirement of p-integrability. Thus, the Vekua theory does not cover even equations with coefficients having first-order singularities. However, other problems, the lowest coefficients of which admit first- order singularities or «strong singularities», boil down to GCRS. In this study, Hilbert type problems are solved for the GCRS the lowest coefficients of which admit strong singularity in a circle. &nbsp

Modulated electromagnetic fields in inhomogeneous media, hyperbolic pseudoanalytic functions, and transmutations

The time-dependent Maxwell system describing electromagnetic wave propagation in inhomogeneous isotropic media in the one-dimensional case reduces to a Vekua-type equation for bicomplex-valued functions of a hyperbolic variable, see Kravchenko and Ramirez [Adv. Appl. Cliord Algebr. 21(3), 547–559 (2011)]. Using this relation, we solve the problem of the transmission through an inhomogeneous layer of a normally incident electromagnetic time-dependent plane wave. The solution is written in terms of a pair of Darboux-associated transmutation operators [Kravchenko, V. V. and Torba, S. M., J. Phys. A: Math. Theor. 45, 075201 (2012)], and combined with the recent results on their construction [Kravchenko, V. V. and Torba, S. M., Complex Anal. Oper. Theory 9, 379-429 (2015); Kravchenko, V. V. and Torba, S. M., J. Comput. Appl. Math. 275, 1–26 (2015)] can be used for efficient computation of the transmitted modulated signals. We develop the corresponding numerical method and illustrate its performance with examples.

Moutard Transform for Generalized Analytic Functions

We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.

Carleson Measures and Hadamard Products in Some General Analytic Function Spaces

Carleson Measures and Hadamard Products in Some General Analytic Function Spaces

On the uniqueness theorem of Holmgren

We review the classical Cauchy–Kovalevskaya theorem and the related uniqueness theorem of Holmgren, in the simple setting of powers of the Laplacian and a smooth curve segment in the plane. As a local problem, the Cauchy–Kovalevskaya and Holmgren theorems supply a complete answer to the existence and uniqueness issues. Here, we consider a global uniqueness problem of Holmgren’s type. Perhaps surprisingly, we obtain a connection with the theory of quadrature identities, which demonstrates that rather subtle algebraic properties of the curve come into play. For instance, if $$\Omega $$ is the interior domain of an ellipse, and I is a proper arc of the ellipse $$\partial \Omega $$ , then there exists a nontrivial biharmonic function u in $$\Omega $$ which is three-flat on I (i.e., all partial derivatives of u of order $$\le 2$$ vanish on I) if and only if the ellipse is a circle. Another instance of the same phenomenon is that if $$\Omega $$ is bounded and simply connected with $$C^\infty $$ -smooth Jordan curve boundary, and if the arc $$I\subset \partial \Omega $$ is nowhere real-analytic, then we have local uniqueness already with sub-Cauchy data: if a function is biharmonic in $${\mathcal {O}}\cap \Omega $$ for some planar neighborhood $${\mathcal {O}}$$ of I, and is three-flat on I, then it vanishes identically on $${\mathcal {O}}\cap \Omega $$ , provided that $${\mathcal {O}}\cap \Omega $$ is connected. Finally, we consider a three-dimensional setting, and analyze it partially using analogues of the square of the standard $$2\times 2$$ Cauchy–Riemann operator. In a special case when the domain is of periodized cylindrical type, we find a connection with the massive Laplacian [the Helmholz operator with imaginary wave number] and the theory of generalized analytic (or pseudoanalytic) functions of Bers and Vekua.

On a Volterra equation of the second kind with ‘incompressible’ kernel

Solving the boundary value problems of the heat equation in noncylindrical domains degenerating at the initial moment leads to the necessity of research of the singular Volterra integral equations of the second kind, when the norm of the integral operator is equal to 1. The paper deals with the singular Volterra integral equation of the second kind, to which by virtue of ‘the incompressibility’ of the kernel the classical method of successive approximations is not applicable. It is shown that the corresponding homogeneous equation when $|\lambda|>1$ has a continuous spectrum, and the multiplicity of the characteristic numbers increases depending on the growth of the modulus of the spectral parameter $|\lambda|$ . By the Carleman-Vekua regularization method (Vekua in Generalized Analytic Functions, 1988) the initial equation is reduced to the Abel equation. The eigenfunctions of the equation are found explicitly. Similar integral equations also arise in the study of spectral-loaded heat equations (Amangaliyeva et al. in Differ. Equ. 47(2):231-243, 2011).

Analyticity of Semisimple Eigenvalues and Corresponding Eigenvectors of Matrix-Valued Functions

In this paper, we study the analyticity of the semisimple eigenvalue and the corresponding eigenvector functions of general analytic matrix-valued functions which are very important in many engineering applications such as the optimum design of dynamical structures, model updating, damage detecting, quantum mechanics, diffraction grating theory, medical imaging, and social network theory. We establish some new sufficient conditions on existence of analytic eigenvalue and eigenvector functions corresponding to semisimple eigenvalues of general analytic matrix-valued functions, which relaxes the condition in [P. Lancaster, A. S. Markus, and F. Zhou, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 606--626]. We also present a numerical method to compute the derivatives of the eigenvalue and eigenvector functions. Numerical performance of the method is illustrated by some numerical examples.

Initial coefficient bounds for a general class of bi-univalent functions

Recently, Srivastava et al. [22] reviewed the study of coefficient problems for bi-univalent functions. Inspired by the pioneering work of Srivastava et al. [22], there has been triggering interest to study the coefficient problems for the different subclasses of bi-univalent functions (see, for example, [1, 3, 6, 7, 27, 29],). Motivated essentially by the aforementioned works, in this paper we propose to investigate the coefficient estimates for a general class of analytic and bi-univalent functions. Also, we obtain estimates on the coefficients |a2| and |a3| for functions in this new class. Further, we discuss some interesting remarks, corollaries and applications of the results presented here.

Construction of Transmutation Operators and Hyperbolic Pseudoanalytic Functions

A representation for integral kernels of Delsarte transmutation operators is obtained in the form of a functional series with exact formulae for the terms of the series. It is based on the application of hyperbolic pseudoanalytic function theory and recent results on mapping properties of the transmutation operators. The kernel \(K_{1}\) of the transmutation operator relating \(A=-\frac{d^{2} }{dx^{2}}+q_{1}(x)\) and \(B=-\frac{d^{2}}{dx^{2}}\) turns out to be one of the complex components of a bicomplex-valued hyperbolic pseudoanalytic function satisfying a Vekua-type hyperbolic equation of a special form. The other component of the pseudoanalytic function is the kernel of the transmutation operator relating \(C=-\frac{d^{2}}{dx^{2}}+q_{2}(x)\) and \(B\) where \(q_{2}\) is obtained from \(q_{1}\) by a Darboux transformation. We prove an expansion theorem and a Runge-type theorem for this special hyperbolic Vekua equation and using several known results from hyperbolic pseudoanalytic function theory together with the recently discovered mapping properties of the transmutation operators we obtain a new representation for their kernels. Several examples are given. Moreover, approaches for numerical computation of the transmutation kernels and for numerical solution of spectral problems are proposed.

Generalized Growth and Approximation of Pseudoanalytic Functions on the Disk

Generalized Growth and Approximation of Pseudoanalytic Functions on the Disk

A Regularization Process for Electrical Impedance Equation Employing Pseudoanalytic Function Theory

The electrical impedance equation is considered an ill-posed problem where the solution to the forward problem is more easy to achieve than the inverse problem. This work tries to improve convergence in the forward problem method, where the Pseudoanalytic Function Theory by means of the Taylor series in formal powers is used, incorporating a regularization method to make a solution more stable and to obtain better convergence. In addition, we include a comparison between the designed algorithms that perform proposed method with and without a regularization process and the autoadjustment parameter for this regularization process.

On two-dimensional supersymmetric quantum mechanics, pseudoanalytic functions and transmutation operators

Pseudoanalytic function theory is considered to study a two-dimensional supersymmetric quantum mechanics system. Hamiltonian components of the superHamiltonian are factorized in terms of one Vekua and one Bers derivative operators. We show that imaginary and real solutions of a Vekua equation and its Bers derivative are ground state solutions for the superHamiltonian. The two-dimensional Darboux and pseudo-Darboux transformations correspond to Bers derivatives in the complex plane. Results on the completeness of the ground states are obtained. Finally, the superpotential is studied in the separable case in terms of transmutation operators. We show how Hamiltonian components of the superHamiltonian are related to the Laplacian operator using these transmutation operators.