Hardy Spaces on the Plane and Double Fourier Transforms

We provide a direct computational proof of the known inclusion where is the product Hardy space defined for example by R. Fefferman and is the classical Hardy space used, for example, by E.M. Stein. We introduce a third space of Hardy type and analyze the interrelations among these spaces. We give simple sufficient conditions for a given function of two variables to be the double Fourier transform of a function in and respectively. In particular, we obtain a broad class of multipliers on and respectively. We also present analogous sufficient conditions in the case of double trigonometric series and, as a by-product, obtain new multipliers on and respectively.     

Geometric Breathers of the mKdV Equation

In this paper we present some numerical results about the existence of a new family of solutions of the geometric mKdV equation. This equation is characterized by the fact that the curvature evolves according to the focusing mKdV equation. The new type of solutions are similar to the breathers found by M.?Wadati in 1973. We exhibit curves with and without self-intersections, and also some examples of initially simple closed curves that have self-crossings at later times.     

On Fourier Series on the Torus and Fourier Transforms

Mathematical Notes - The question of the representability of a continuous function on $$\mathbb R^d$$ in the form of the Fourier integral of a finite Borel complex-valued measure on $$\mathbb R^d$$...     

与右半平面解析的Laplace-Stiealtjes变换相关联的变换

该文研究了右半平面上解析的Laplace-Stieltjes变换及与之相关联的变换.     

Some Comments on the Solution of Poisson's Equation using Bickley's Formula and Fast Fourier Transforms

The fast Fourier transform method for poisson's equation, usingthe 9-point approximation to the Laplace operator, is outlined.Numerical results for simple test problems indicate that theincreased accuracy of the method, compared with the standard5-point method, is obtained with only a small increase in computerrun-time.     

Fourier Transforms and the Method of Stationary Phase

Some theorems are derived for the asymptotic behaviour of theFourier transform of a generalized function under conditionswhich are likely to occur in applied mathematics and which shouldbe capable of relatively straightforward verification. Thesetheorems are then applied to integrals of the type when the derivative of f is non-vanishing. Furthertheorems are developed to cover the case when the derivativeof f vanishes at a point. In this way the validity of the methodof stationary phase, together with a useful estimate of theerror, is established under circumstances of practical importance.     

mKdV方程N-孤子解的验证

mKdV方程的多孤子解很难直接验证,本文通过证明GLM反散射变换方程导出的Jost解满足两个相容性方程的方法,解决了这个问题.     

Fourier Transforms of Distributions and Hausdorff Measures

Consider a distribution whose support has Hausdorff \(h\) -measure zero. How fast can its Fourier coefficients decay to zero? If \(h\) is close to linear, we can improve on known results by a factor of about \((\log n)^{-1/2}\) .     

Generalized Fourier Transforms Associated with the Oscillator Representation

Journal of Fourier Analysis and Applications - By identifying the Fourier transform $$\mathcal {F}$$ with an operator in the oscillator representation of the metaplectic group...     

On Bounded Solutions of a Class of Nonlinear Integral Equations in the Plane and the Urysohn Equation in a Quadrant of the Plane

Ukrainian Mathematical Journal - We study a class of two-dimensional integral equations in the plane with monotonic nonlinearity. These equations have numerous applications in many various fields...     

Sobolev Stability of Plane Wave Solutions to the Nonlinear Schrödinger Equation

This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case.     

富里算子及共轭变换的弱有界性

石油资源已频临日益枯竭的境地,因而对内燃机代用燃料的研究广泛廾展,植物油做为柴油机代用燃料的研究已遍布世界各国,本文主要介绍国内外植物油燃料的试验研究情况,植物油与其他代用燃料的比较,以及对发展我国植物油燃料的研究谈一些看法,提出一些具体建议。     

On Fourier Transforms of Radial Functions and Distributions

We find a formula that relates the Fourier transform of a radial function on R ⁿ with the Fourier transform of the same function defined on R ⁿ⁺². This formula enables one to explicitly calculate the Fourier transform of any radial function f(r) in any dimension, provided one knows the Fourier transform of the one-dimensional function t?f(|t|) and the two-dimensional function (x ₁,x ₂)?f(|(x ₁,x ₂)|). We prove analogous results for radial tempered distributions.     

关于全平面上解析的Laplace-Stieltjes变换的零级

应用两个零级的型函数,研究了全平面上解析的Laplace-Stieltjes变换的零级,并得到其系数与增长级的关系,推广了Dirichlet级数的相关结论.     

On Radial Functions and Distributions and Their Fourier Transforms

We give formulas relating the Fourier transform of a radial function in $\mathbb{R}^{n}$ and the Fourier transform of the same function in $\mathbb{R}^{n+1}$ , completing the analysis of Grafakos and Teschl (J. Fourier Anal. Appl. 19:167–179, 2013) where the case of $\mathbb{R}^{n}$ and $\mathbb{R}^{n+2}$ was considered.     

Estimates of Best Approximation and Fourier Transforms in Integral Metrics

Under some assumptions on a function F and its Fourier transform we prove new estimates of best approximation of F by entire functions of exponential type σ in L_p( ), 1 ≤ p < 2. The proof is based on some inequalities for in L₁( ) which may be treated as generalizations of results of Bausov and Telyakovskii. As an application we obtain exact estimates of best approximation of some infinitely differentiable functions.     

On the Euler Equation in the Plane

In this work we will establish an existence and uniqueness result for weak Euler's equation for an incompressible fluids in the plane. We assume that the initial vorticity is bounded and furthermore it exists at least a point in which initially the integral related to velocity is absolutely convergent.     

Stability of Nonlinear Subdivision and Multiscale Transforms

Extending upon the work of Cohen, Dyn, and Matei (Appl. Comput. Harmon. Anal. 15:89–116, 2003) and of Amat and Liandrat (Appl. Comput. Harmon. Anal. 18:198–206, 2005), we present a new general sufficient condition for the Lipschitz stability of nonlinear subdivision schemes and multiscale transforms in the univariate case. It covers the special cases (weighted essentially nonoscillatory scheme, piecewise polynomial harmonic transform) considered so far but also implies the stability in some new cases (median interpolating transform, power-p schemes, etc.). Although the investigation concentrates on multiscale transforms $\bigl\{v^0,d^1,\ldots,d^J\bigr\}\longmapsto v^J,\quad J\ge1,$ in ? _∞(?) given by a stationary recursion of the form $v^{j}=Sv^{j-1}+d^{j},\quad j\ge1,$ involving a nonlinear subdivision operator S acting on ? _∞(?), the approach is extendable to other nonlinear multiscale transforms and norms, as well.