Certain Classes Of Functions Research Articles

Some properties of a class of analytic functions

Making use of convolution, we introduce and investigate a certain class of functions which is analytic in the open unit disk. We obtain interesting properties of starlikeness and convexity for this function class. Special cases and some useful consequences of our main results are also mentioned.

Tie-Breaking the Highest Median: Alternatives to the Majority Judgment

The paper deals with voting rules that require voters to rate the candidates on a finite evaluation scale and then elect a candidate whose median grade is maxi- mum. These rules differ by the way they choose among candidates with the same median grade. Call proponents (resp. opponents) of a candidate the voters who rate this candidate strictly above (resp. strictly below) her median grade. A simple rule, called the typical judgment, orders tied candidates by the difference between their share of proponents and opponents. An appealing rule, called the usual judgment, divides this difference by the share of median votes. An alternative rule, called the central judgment, compares the relative shares of proponents and opponents. The usual judgment is continuous with respect to these shares. The majority judgment of Balinski & Laraki (2007) considers the largest of these shares and loses continuity. A result in Balinski & Laraki (2014) aims to characterize the majority judgment and states that only a certain class of functions share some valuable characteristics, like monotonicity. We relativize this result, by emphasizing that it only holds true for continuous scales of grades. Properties remaining specific to the majority judgment in the discrete case are idiosyncratic features rather than universally sought criteria, and other median-based rules exist that are both monotonic and continuous.

Upper Bounds for the Approximation of Certain Classes of Functions of a Complex Variable by Fourier Series in the Space $L_2$ and $n$-Widths

В работе рассматривается задача среднеквадратичного приближения функций комплексного переменного, регулярных в области $\mathscr D\subset\mathbb C$, рядами Фурье по ортогональной в $\mathscr D$ системе функций $\{\varphi_k(z)\}$, $k=0,1,2,…$ . В случае, когда $\mathscr D=z\in\mathbb C:|z|<1\}$, получены точные оценки скорости сходимости рядов Фурье по ортогональной системе $z^k\}$, $k=0,1,2,…$, на классах функций, задаваемых специальным модулем непрерывности $m$-го порядка. Вычислены точные значения ряда $n$-поперечников на указанных классах функций. Библиография: 17 названий.

On some estimates for best approximations of bivariate functions by Fourier–Jacobi sums in the mean

Some problems in computational mathematics and mathematical physics lead to Fourier series expansions of functions (solutions) in terms of special functions, i.e., to approximate representations of functions (solutions) by partial sums of corresponding expansions. However, the errors of these approximations are rarely estimated or minimized in certain classes of functions. In this paper, the convergence rate (of best approximations) of a Fourier series in terms of Jacobi polynomials is estimated in classes of bivariate functions characterized by a generalized modulus of continuity. An approximation method based on “spherical” partial sums of series is substantiated, and the introduction of a corresponding class of functions is justified. A two-sided estimate of the Kolmogorov N-width for bivariate functions is given.

Weak rate of convergence of the Euler–Maruyama scheme for stochastic differential equations with non-regular drift

We consider an Euler–Maruyama type approximation method for a stochastic differential equation (SDE) with a non-regular drift and regular diffusion coefficient. The method regularizes the drift coefficient within a certain class of functions and then the Euler–Maruyama scheme for the regularized scheme is used as an approximation. This methodology gives two errors. The first one is the error of regularization of the drift coefficient within a given class of parametrized functions. The second one is the error of the regularized Euler–Maruyama scheme. After an optimization procedure with respect to the parameters we obtain various rates, which improve other known results.

On average $n$-widths of certain classes of functions in $L_2(\mathbb{R})$ space

На классах функций $\mathcal{W}^r (\omega_m;\Phi)$, где $r, m\in \mathbb{N}$, $\Phi$ — мажоранта, $\omega_m$ — модуль непрерывности m-го порядка, получены оценки сверху и снизу колмогоровского, линейного и бернштейновского средних $\nu$-поперечников. Также приведено условие, при выполнении которого из указанного результата следуют точные значения перечисленных экстремальных характеристик.

Hyperbolic equation with singular coefficient

Dedicated to research existence conditions of quasi-linear differential equations with measurable coefficients, ie the study limitations imposed by the non linearity in which the system will have to research a solution and uniqueness of the solution of a certain class of functions. We consider weak solvability of quasi-linear differential partial differential equations of hyperbolic type with smooth and measurable singular coefficients, research methods based on the theory of semigroups using the method of differential forms. A new class of operators $A_\lambda^p:W_ 1 ^p \rightarrow W_ -1 ^p$ associated with a given differential equation and investigate the properties of these operators.

Wave equations on graded groups and hypoelliptic Gevrey spaces

The overall goal of this dissertation is to investigate certain classical results from harmonic analysis, replacing the Euclidean setting, the abelian structure and the elliptic Laplace operator with a non-commutative environment and hypoelliptic operators. More specifically, we consider wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent non-negative propagation speeds that are H\older-regular or even more so. The corresponding Euclidean problem has been extensively studied in the `80s and some additional results have been recently obtained by Garetto and Ruzhansky in the case of a compact Lie group. We establish sharp well-posedness results in the spirit of the classical result by Colombini, De Giorgi and Spagnolo. In this investigation, a structure reminiscent of Gevrey regularity appears, inspiring deeper investigation of certain classes of functions and a comparison with Gevrey classes. In the latter part of this thesis we discuss such Gevrey spaces associated to the sums of squares of vector fields satisfying the H\ormander condition on manifolds. This provides a deeper understanding of the Gevrey hypoellipticity of sub-Laplacians. It is known that if $\mathcal L$ is a Laplacian on a closed manifold $M$ then the standard Gevrey space $\gamma^s$ on $M$ defined in local coordinates can be characterised by the condition that $\|e^{D\mathcal L^{\frac{1}{2s}}}\phi\|_{L^2(M)} 0$. The aim in this part is to discuss the conjecture that a similar characterisation holds true if $\mathcal L$ is H\ormander's sum of squares of vector fields, with a sub-Laplacian version of the Gevrey spaces involving these vector fields only. We prove this conjecture in one direction, while in the other we show it holds for sub-Laplacians on $SU(2)$ and on the Heisenberg group $\mathbb{H}_n$.

DIVERGENCE OF THE FOURIER SERIES OF CONTINUOUS FUNCTIONS WITH A RESTRICTION ON THE FRACTALITY OF THEIR GRAPHS

We consider certain classes of functions with a restriction on the fractality of their graphs. Modifying Lebesgue’s example, we construct continuous functions from these classes whose Fourier series diverge at one point, i.e. the Fourier series of continuous functions from this classes do not converge everywhere.

Optimal spline spaces of higher degree for [formula omitted] [formula omitted]-widths

In this paper we derive optimal subspaces for Kolmogorov n-widths in the L2 norm with respect to sets of functions defined by kernels. This enables us to prove the existence of optimal spline subspaces of arbitrarily high degree for certain classes of functions in Sobolev spaces of importance in finite element methods. We construct these spline spaces explicitly in special cases.

Asymptotic estimates of p-variational and L p -modules of continuity of functions of certain classes

We obtain sharp estimates for upper bound of p-variational or Lp-modules of continuity on classes of functions with given majorants of trigonometric best approximations or modules of continuity. We also obtain refined direct and inverse theorems of approximation in p-variational metric for conjugate function.

Some combinatorial aspects of discrete non-linear population dynamics

Motivated by issues arising in population dynamics, we consider the problem of iterating a given analytic function a number of times. We use the celebrated technique known as Carleman linearization that turns (for a certain class of functions) this problem into simply taking the power of a real number. We expand this method, showing in particular that it can be used for population models with immigration, and we also apply it to the famous logistic map. We also are able to give a number of results for the invariant density of this map, some being related to the Carleman linearization.

OPERATOR-VALUED FUNCTION SPACE INTEGRALS VIA CONDITIONAL INTEGRALS ON AN ANALOGUE WIENER SPACE II

In the present paper, using a simple formula for the conditional expectations given a generalized conditioning function over an analogue of vector-valued Wiener space, we prove that the analytic operator-valued Feynman integrals of certain classes of functions over the space can be expressed by the conditional analytic Feynman integrals of the functions. We then provide the conditional analytic Feynman integrals of several functions which are the kernels of the analytic operator-valued Feynman integrals.

On exact estimates of the convergence rate of fourier series for functions of one variable in the space L 2[–π, π

The work is devoted to exact estimates of the convergence rate of Fourier series in the trigonometric system in the space of square summable 2π-periodic functions with the Euclidean norm on certain classes of functions characterized by the generalized modulus of continuity. Some N-widths of these classes are calculated, and the residual term of one quadrature formula over equally spaced nodes for a definite integral connected with the issues under consideration is found.

Some characterizations of the natural exponential families in [formula omitted] and related Laplace transforms

We characterize some natural exponential families in R2 via diagonal elements of quadratic variance function. Our proofs rely on Laplace transforms, and along the way, we obtain some new characterizations for Laplace transforms of probability measures in certain classes of functions.

Feynman formulae and phase space Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton functions

Evolution semigroups generated by pseudo-differential operators are considered. These operators are obtained by different (parameterized by a number τ) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains Hamilton functions of particles with variable mass in magnetic and potential fields and more general symbols given by the Lévy-Khintchine formula. The considered semigroups are represented as limits of n-fold iterated integrals when n tends to infinity. Such representations are called Feynman formulae. Some of these representations are constructed with the help of another pseudo-differential operator, obtained by the same procedure of quantization; such representations are called Hamiltonian Feynman formulae. Some representations are based on integral operators with elementary kernels; these are called Lagrangian Feynman formulae. Langrangian Feynman formulae provide approximations of evolution semigroups, suitable for direct computations and numerical modeling of the corresponding dynamics. Hamiltonian Feynman formulae allow to represent the considered semigroups by means of Feynman path integrals. In the article, a family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented as phase space Feynman path integrals with respect to these Feynman pseudomeasures, i.e., different quantizations correspond to Feynman path integrals with the same integrand but with respect to different pseudomeasures. This answers Berezin’s problem of distinguishing a procedure of quantization on the language of Feynman path integrals. Moreover, the obtained Lagrangian Feynman formulae allow also to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures.

Error bounds and estimates for Krylov subspace approximations of Stieltjes matrix functions

When using the Lanczos method to approximate $$f(A){\varvec{b}}$$ , the action of a matrix function on a vector, there is, in contrast to the solution of linear systems, no straightforward way to measure or estimate the error of the current iterate. Therefore, to be able to decide whether the desired accuracy has been reached, several different estimates and bounds for the error have been suggested, all of them specific to certain classes of functions. In this paper, we add to these results by developing a technique to compute error bounds for Stieltjes functions, using a recently suggested integral representation of the error, and we show how these bounds can be computed essentially for free, i.e., with cost independent of the iteration number and the dimension of the matrix A.

Disconjugacy of fourth-order equations on graphs

This paper develops the theory of disconjugacy of fourth-order equations on geometric graphs which arises in modelling rod structures. The disconjugacy of an equation is defined in terms of a special fundamental system of solutions of the homogeneous equation. The disconjugacy property is shown to be related to the positivity property of the Green's functions for certain classes of boundary value problems for a fourth-order equation on a graph. A maximum principle for a fourth-order equation on a graph is formulated, and some properties of differential inequalities are proved. Bibliography: 25 titles.

α-Molecules

Within the area of applied harmonic analysis, various multiscale systems such as wavelets, ridgelets, curvelets, and shearlets have been introduced and successfully applied. The key property of each of those systems are their (optimal) approximation properties in terms of the decay of the L2-error of the best N-term approximation for a certain class of functions. In this paper, we introduce the general framework of α-molecules, which encompasses most multiscale systems from applied harmonic analysis, in particular, wavelets, ridgelets, curvelets, and shearlets as well as extensions of such with α being a parameter measuring the degree of anisotropy, as a means to allow a unified treatment of approximation results within this area. Based on an α-scaled index distance, we first prove that two systems of α-molecules are almost orthogonal. This leads to a general methodology to transfer approximation results within this framework, provided that certain consistency and time–frequency localization conditions of the involved systems of α-molecules are satisfied. We finally utilize these results to enable the derivation of optimal sparse approximation results for a specific class of cartoon-like functions by sufficient conditions on the ‘control’ parameters of a system of α-molecules.

Sliding mode control of continuous time systems with reaching law based on exponential function

In this paper a pseudo-sliding mode control is proposed by introducing a continuous and smooth input signal in order to guarantee both chattering elimination and boundedness of sliding variable derivative in the presence of non-zero external disturbance. For this purpose, having fixed a suitable sliding manifold, a homogeneous differential equation describing the sliding variable evolution is considered. It is discussed later in this paper that the input signal formed on the basis of this equation provides asymptotic convergence of the sliding variable and its derivative to zero as well as the asymptotic stability of the non-linear system in the absence of external disturbance. The dynamics of the system affected by non-zero external disturbance make the state vector converge to domains in a vicinity of the origin at the exponential rate, as the norm of arbitrary trajectory is limited to decreasing exponential function. In order to expand the variety of controllers based on a reaching law and providing the above-mentioned properties, a certain class of functions is presented.