In this paper we characterize those bounded linear transformations $Tf$ carrying $L^{1}( \mathbb {R}^{1}) $ into the space of bounded continuous functions on $\mathbb {R}^{1}$, for which the convolution identity $T(f\ast g) =Tf\cdot Tg $ holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.
Holographic reduced representation is based on a suitable distributive coding of structured information in conceptual vectors, whose elements satisfy normal distribution N(0,1/n). The existing applications of this approach concern various models of associative memory that exploit a simple algebraic operation of the scalar product of distributed representations to measure an overlap between two structured concepts. We have described here a method that uses this representation to model a similarity between different concepts and an inference process based on the rules modus ponens and modus tollens.
In this paper we study Beurling type distributions in the Hankel setting. We consider the space ${\mathcal E}(w)^{\prime }$ of Beurling type distributions on $(0, \infty )$ having upper bounded support. The Hankel transform and the Hankel convolution are studied on the space ${\mathcal E}(w)^{\prime }$. We also establish Paley Wiener type theorems for Hankel transformations of distributions in ${\mathcal E}(w)^{\prime }$.
The incomplete Gamma function γ(α, x) and its associated functions γ(α, x+) and γ(α, x−) are defined as locally summable functions on the real line and some convolutions and neutrix convolutions of these functions and the functions x r and x r − are then found.
The abstract Perron-Stieltjes integral in the Kurzweil-Henstock sense given via integral sums is used for defining convolutions of Banach space valued functions. Basic facts concerning integration are preseted, the properties of Stieltjes convolutions are studied and applied to obtain resolvents for renewal type Stieltjes convolution equations.