Functions Of Polynomial Growth Research Articles

Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons

Harmonic and Schrödinger functions of polynomial growth on gradient shrinking Ricci solitons

Uniform distribution in nilmanifolds along functions from a Hardy field

We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if X = G/Γ is a nilmanifold, a1, …, ak ∈ G are commuting nilrotations, and f1, …, fk are functions of polynomial growth from a Hardy field then we show that

Trace singularities in obstacle scattering and the Poisson relation for the relative trace

We consider the case of scattering by several obstacles in {mathbb {R}}^d, d ge 2 for the Laplace operator Delta with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators Delta _1 and Delta _2 obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator g(Delta ) - g(Delta _1) - g(Delta _2) + g(Delta _0) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function xi _mathrm {rel}(lambda ) = -frac{1}{pi } {text {Im}}(Xi (lambda )), where Xi (lambda ) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of xi _mathrm {rel}. In particular we prove that {hat{xi }}_mathrm {rel} is real-analytic near zero and we relate the decay of Xi (lambda ) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function Xi (lambda ) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.

Holomorphic Functions on a Class of Kähler Manifolds With Nonnegative Curvature, I

Abstract In this paper, we consider holomorphic functions of polynomial growth on complete Kähler manifolds with nonnegative curvature. We explain how their growth orders are related to the asymptotic behavior of Kähler–Ricci flow. The main result is to determine minimal orders of holomorphic functions on gradient Kähler–Ricci expanding solitons with nonnegative Ricci curvature.

Polynomial Solutions of the Dirichlet Problem for the Tricomi Equation in a Strip

In the paper we consider the Tricomi equation of mixed type. This equation is elliptic in the upper half-plane, hyperbolic in the lower half-plane and parabolically degenerate on the boundary of half-planes. Equations of a mixed type are used in transonic gas dynamics. The Dirichlet problem for an equation of mixed type in a mixed domain is, in general, ill- posed. Many papers has been devoted to the search for conditions for the well-posednes of the Dirichlet problem for a mixed-type equation in a mixed domain.This paper is devoted to finding exact polynomial solutions of the inhomogeneous Tricomi equation in a strip with a polynomial right-hand side. The Fourier transform method shows that the Dirichlet boundary value problem with polynomial boundary conditions has a polynomial solution. An algorithm for constructing this polynomial solution is given and examples are considered. If the strip lies in the ellipticity region of the equation, then this solution is unique in the class of functions of polynomial growth. If the strip lies in a mixed domain, then the solution of the Dirichlet problem is not unique in the class of functions of polynomial growth, but it is unique in the class of polynomials.

Polynomial Solutions of the Boundary Value Problems for the Poisson Equation in a Layer

It is well known that the Dirichlet problem for the Laplace equation in a ball has a unique polynomial solution (harmonic polynomial) in the case if the given boundary value is the trace of an arbitrary polynomial on the sphere. S.M.Nikol'skii generalized this result in the case of a boundary value problem of the first kind for a linear differential self-adjoint operator of the order 2l with constant coefficients (in particular polyharmonic) and for a domain that is an ellipsoid in R n . For a polyharmonic equation in a ball (homogeneous and inhomogeneous), V.V. Karachik proposed the Almansi formula-based algorithm to construct a polynomial solution of the Dirichlet problem. The paper considers the Poisson equation with the polynomial right-hand side in a multidimensional infinite layer bounded by two hyper-planes. Shows that the Dirichlet boundary value problem and the mixed Dirichlet-Neumann boundary value problem with polynomial boundary conditions have a unique solution in the class of functions of polynomial growth, and this solution is a polynomial. Gives an algorithm for constructing this polynomial solution and considers examples. In particular, presents formulas to give exact values of certain integrals (including multi-dimensional ones) and sums of trigonometric series.

Effective and Sampling Sets for Hörmander Spaces

Recently we have described the topological structure of sampling sets for the Hormander algebra \(A^{-\infty }({\mathbb B}^N)\) of holomorphic functions of polynomial growth in the unit ball \({\mathbb B}^N\) of \({\mathbb C}^N\). In this paper we show that our methods work as well in Hormander algebras of the general type. Our study is based on a detailed analysis of weakly sufficient sets in Hormander spaces having the corresponding Hormander algebra as the inductive limit space.

Некоторые свойства главных подмодулей в модуле целых функций экспоненциального типа и полиномиального роста на вещественной оси

Некоторые свойства главных подмодулей в модуле целых функций экспоненциального типа и полиномиального роста на вещественной оси

Определяющие множества для пространства голоморфных в шаре функций полиномиального роста

Определяющие множества для пространства голоморфных в шаре функций полиномиального роста

Linear continuous right inverse to convolution operator in spaces of holomorphic functions of polynomial growth

We consider the convolution operator in spaces of holomorphic functions, defined in convex subdomains of the complex plane, with polynomial growth at a boundary. We prove that if this operator is surjective on the class of all bounded convex domains, then it always has a linear continuous right inverse one.

Generalized functional calculus in lattice-ordered algebras

We construct a functional calculus on unital uniformly complete f-algebras for continuous center-valued functions of polynomial growth and study the connection with the Minkowski duality.

Sampling in shift-invariant spaces of functions with polynomial growth

Under suitable conditions, a function f(t) in a principal shift-invariant space can be recovered from its uniform samples { f(n)} n∈ℤ with a simple sampling formula. Provided that the generator ϕ has compact support, we consider the sampling problem in the bigger space V ϕ ≔ {∑a n ϕ(t − n) : a n ∈ ℂℤ}. In this space, there exist infinite functions with the same samples y n = f(n). We show that polynomial growth conditions give the uniqueness: if y n has polynomial growth, there is a unique function of polynomial growth f ∈ V ϕ, satisfying f(n) = y n , n ∈ ℤ. This function is given by the known sampling formula. The same result is proved also when we consider average samples y n = f ∗ h(n).

Pointwise Fourier inversion of distributions

We show that, given a tempered distribution T whose Fourier transform is a function of polynomial growth and a point x in R n at which T has the value τ (in the sense of Łojasiewicz), the Fourier integral of T at x is summable in Bochner-Riesz means to τ.

On the support of tempered distributions

We show that, given a tempered distribution S whose Fourier transform is a function of polynomial growth, a point x in $${\mathbb{R}^{n}}$$ is outside the support of S if and only if the Fourier integral of S is summable in Bochner-Riesz means to zero uniformly on a neighbourhood of x.

Weakly sufficient sets for $A^{-\infty} (\Bbb D)$

In the space $A^{-\infty} (\Bbb D)$ of functions of polynomial growth, weakly sufficient sets are those such that the topology induced by restriction to the set coincides with the topology of the original space. Horowitz, Korenblum and Pinchuk defined sampling sets for $A^{-\infty} (\Bbb D)$ as those such that the restriction of a function to the set determines the type of growth of the function. We show that sampling sets are always weakly sufficient, that weakly sufficient sets are always of uniqueness, and provide examples of discrete sets that show that the converse implications do not hold.

On the interpolation of some generalized function spaces of different anisotropy

In the paper, the interpolation properties of the spacesHps(v; ℝn) of Sobolev-Liouville type and the spacesBp, qs(μ ℝn) of Nikol'skii-Besov type generated by functions of polynomial growth that are infinitely differentiable outside of the origin are studied. Interpolation formulas for the pairs {H(vo),H(v1)} and {B(μ0),B(μ1)} of spaces of the above types for which the anisotropies of the interpolated spaces do not depend on each other are proved. The investigated spaces, for certain specification of the generating functions, coincide with the classical (isotropic and anisotropic) Sobolev-Liouville and Nikol'skii-Besov spaces.

Further results on the convergence behavior of conjugate-gradients and Ritz values

The usual estimates for conjugate gradients (CG) specify a non trivial rate of convergence right from the beginning. We investigate situations where the same can be said for Ritz values (considered as approximations to eigenvalues). We investigate the effect on the convergence behavior of Ritz values of multiplying the weight functions by certain functions of polynomial growth. This is shown not to change the convergence behavior but only to cause a delay of at most a certain number of steps. We give competitive alternatives for the usual Ritz error estimates. We show that, in a sense, CG errors can be considered as Ritz errors and vice versa. Hence, properties for CG errors should be expected to have parallels for Ritz values and vice versa. Finally, some smaller results are given.

Harmonic functions of polynomial growth on certain complete manifolds

Given a complete manifold with a particular structure at infinity, we give the dimension of the space of harmonic functions with prescribed polynomial growth.

Real Singularities of Singular Sturm–Liouville Expansions

Elliptic equations in polar coordinates lead to singular Sturm–Liouville problems on $(0,\infty )$ with equations of the form $y'' + (\lambda - q)y + 0$. Let $\phi (x,\lambda )$ be solutions to the problem satisfying the condition $\phi (0,\lambda ) = \sin \beta $, $\phi '(0,\lambda ) = - \cos \beta $. The associated generalized Fourier transform $F(\lambda ) = \int_0^\infty {f(x)\phi (x,\lambda )dx} $ is extended to cases where $F(\lambda )$ is a function of polynomial growth. This enables us to study the location of singularities of the analytic representation $\hat f$ of the generalized function $f(x) = \int_{ - \infty }^\infty {F(\lambda )\phi (x,\lambda )d\rho (\lambda )} $. We do so by comparing them to the location of the singularities of the analytic representation $\hat g$ of the tempered distribution g which is the Fourier transform of $F(s^2 )$.

Holomorphic functions of polynomial growth on bounded domains

Holomorphic functions of polynomial growth on bounded domains