# On Smoothing Operations and their Generating Functions

• I. J. Schoenberg
Part of the Contemporary Mathematicians book series (CM)

## Abstract

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.

## Keywords

Entire Function Laurent Series Frequency Function Real Zero Vertical Strip
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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