On smoothing operations and their generating functions

In this paper we are mainly concerned with two kinds of linear transformations, the sequence convolution transformation $$ {y_n} = \sum\limits_{v = - 8}^\infty {{a_{n - v}}{x_v}} $$ (1) and the integral convolution transformation $$ g\left( x \right) = \int_{ - \infty }^\infty {\Lambda \left( {x - 1} \right)f\left( t \right)} dt, $$ (2) where the sequence a n and the function Δ(x) are thought of as given. In §2 we also consider the ordinary linear transformation $$ {y_i} = \sum\limits_{k = 1}^n {{a_{ik}}{x_k}} \quad \quad \left( {i = 1, \cdots ,m} \right). $$ (3) The loosely connected topics to be discussed concerning these transformations are perhaps best brought together under the general subject of smoothing operations.

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