Given a distribution on the sphere we define, in analogy to the work of Łojasiewicz, the value of at a point of the sphere and we show that if has the value at , then the Fourier-Laplace series of at is Abel-summable to .
We consider the multiplicative algebra P(𝒢₊') of continuous scalar polynomials on the space 𝒢₊' of Roumieu ultradistributions on [0,∞) as well as its strong dual P'(𝒢₊'). The algebra P(𝒢₊') is densely embedded into P'(𝒢₊') and the operation of multiplication possesses a unique extension to P'(𝒢₊'), that is, P'(𝒢₊') is also an algebra. The operation of differentiation on these algebras is investigated. The polynomially extended Laplace transformation and its connections with the differentiation...
In this paper we study the Radon transform on the set of horocycles of a homogeneous tree , and describe its image on various function spaces. We show that the functions of compact support on that satisfy two explicit Radon conditions constitute the image under of functions of finite support on . We extend these results to spaces of functions with suitable decay on , whose image under satisfies corresponding decay conditions and contains distributions on that are not defined pointwise....
We prove that ridgelet transform and adjoint ridgelet transform are continuous, where . We also define the ridgelet transform on the space of tempered distributions on , adjoint ridgelet transform on and establish that they are linear, continuous with respect to the weak-topology, consistent with , respectively, and they satisfy the identity , .
We obtain real-variable and complex-variable formulas for the integral of an integrable distribution in the n-dimensional case. These formulas involve specific versions of the Cauchy kernel and the Poisson kernel, namely, the Euclidean version and the product domain version. We interpret the real-variable formulas as integrals of S’-convolutions. We characterize those tempered distribution that are S’-convolvable with the Poisson kernel in the Euclidean case and the product domain case. As an application...