The scale space representation of a signal obtained by Gaussian smoothing satisfies a number of special properties, scale-space axioms, which make it into a special form of multi-scale representation.
The purpose of this article is to describe a few of these approaches: For one-dimensional signals, there exists quite a well-developed theory for continuous and discrete kernels that guarantee that new local extrema or zero-crossings cannot be created by a convolution operation.
If we on the other hand sacrifice the continuous semi-group structure, there are more options: For discrete signals, the use of generalized binomial kernels provides a formal basis for defining the smoothing operation in a pyramid.
For temporal data, the one-sided truncated exponential kernels and the first-order recursive filters provide a way to define time-causal scale-spaces [2][3] that allow for efficient numerical implementation and respect causality over time without access to the future.
The first-order recursive filters also provide a framework for defining recursive approximations to the Gaussian kernel that in a weaker sense preserve some of the scale-space properties.