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Kernels of integral transforms of the form k(xy) are called Fourier kernels. Hardy and Titchmarsh (6) and Watson (15) studied self- reciprocal transforms with Fourier kernels on the positive half-real line R+ and obtained nice characterizing conditions and other properties. Neverthe- less, the Fourier transform has an inherent complex, unitary structure and by treating it on the whole real line more interesting properties can be ex- plored. In this paper, we characterize all unitary integral transforms on L 2 (R) with Fourier kernels. Characterizing conditions are obtained through a natu- ral splitting of the kernel on R+ and R−, and the conversion to convolution integrals. A collection of concrete examples are obtained, which also include "discrete" transforms (that are not integral transforms). Furthermore, we ex- plore algebraic structures for operator-composing integral kernels and identify a certain Abelian group associated with one of such operations. Fractional powers can also be assigned a new interpretation and obtained.

Let H n be the metric space of all bounded domains in C n with the metric equal to the Hausdorff distance between boundaries of domains. We prove that the dimension of the group of automorphisms of domains is an upper semicontinuous func- tion on H n . We also provide theorems and examples regarding the change in topological structure of these groups under small perturbation of a domain in H n .

It is well-known that a real analytic manifold X admits a complexification XC , a complex manifold that contains X as the fixed point set of an antiholomorphic involution. This can be seen as follows.The transition functions defining the manifold X are real-analytic local diffeomorphisms of Rn. The Taylor expansions of these transition functions can be considered as local biholomorphisms of Cn, hence they serve as transition functions of a complex manifold. The germ of the complexification XC is unique. Every sufficiently small tubular neighborhood Ω of X in the tangent bundle TX admits a real analytic diffeomorphism into XC that fixes X . Therefore a sufficiently small tubular neighborhood ofX in TX has a complex structure and can be considered as a complexification of X . In general the complex structure on the tubular neighborhood is not unique since there are manyways to embed it intoXC . There have been a lot of interest in finding canonical complex structures on tubular neighborhoods ofX in TX . With additional datum of a real analytic Riemannian metric g on X a canonical complex structure can be specified for sufficiently small tubular neighborhoods Ω of X in TX (see [GS, LS, S1]). There is a unique complex structure on Ω such that the map f(σ + iτ) = (τγ′(σ))γ(σ), σ + iτ ∈ C, is holomorphic, wherever it is defined

Results are reported of a numerical implementation of the hypcrbolic Fourier transform and the geodesic and horocyclic Radon transforms on the hyperbolic plane, and of their inverses. The study is motivated by the hyperbolic geometry approach to the linearized inverse conductivity problem, suggested by C. A. Berenstein and E. Casadio Tarabusi.

The equations which model the elastostatic shallow circular cylindrical shell (see, e.g., [6,15,23,29]) constitute an important elliptic partial differential equation (PDE) system in the study of shell structures. When the system is subjected to a concentrated point load, the response is described by a fundamental solution of the PDE system. We have found some mathematical inconsistencies in the existing literature. Therefore, in this paper, we discuss these errors, then we use partial fractions and Fourier transform techniques to determine the fundamental solution. Explicit expressions in terms of special functions and convolution integrals are derived and simplified so that the formulas are suitable for algorithmic evaluation and for application elsewhere.

In its proof, a bijective holomorphic mapping σ :Ω → B with σ(q)= 0 has been constructed. However, no explicit arguments showing the complex analyticity of σ−1 had been presented. Indeed, the main purpose of this article is to give it a concise proof. On the other hand, we present it as a separate article since we believe that the arguments here are of a separate interest, in regard to the following unsolved problem

Lempert proved that the Kobayashi metric and the Caratheodory metric of smooth strongly convex domains are smooth away from the zero section of the holomorphic tangent bundle. We will show that if the strong convexity is replaced by the weaker condition of strict convexity the above smoothness result is no longer valid even if the boundary is real analytic. We study the smoothness of the Kobayashi metric of ellipsoids. We prove that the Kobayashi metric of domains of the form , where m≥3/2, is piecewise C 3 off the zero section, but not C 3.

In this paper we prove the best possible Lp estimates for the ∂-b -equation on the boundaries of real ellipsoids in Cn, and provide examples to show why they cannot be improved.

For a domain $D\subset {\Bbb{C}}^n$ we construct a continuous foliation of $D$ into one real dimensional curves such that any function $f\in {C^1(D)}$ which can be extended holomorphically into some neighborhood of each curve in the foliation will be holomorphic on $D$.

We give examples of non-homogeneous bounded domains in C which can exhaust no other domains. And we prove that every bounded domain in C with C boundary can exhaust the ball.

Many problems in structural dynamics involve stabilizing the elastic energy of partial differential equations such as the Euler-Bernoulli beam equation by boundary conditions. Exponential stability is a very desirable property such elastic systems. The energy multiplier method has been successfully applied by several people to establish exponential stability for various PDEs and boundary conditions. However, it has also been found that for certain boundary conditions the energy multiplier method is not effective in proving the exponential stability property. A recent theorem of F. L. Huang introduces a frequency domain method to study such exponential decay problems. In this paper, we derive estimates of the resolvent operator on the imaginary axis and apply Huang's theorem to establish an exponential decay result for an Euler-Bernoulli beam with rate control of the bending moment only. We also derive asymptotic limits of eigenfrequencies, which was also done earlier by P. Rideau. Finally, we indicate the realizability of these boundary feedback stabilization schemes by illustrating some mechanical designs of passive damping devices.