Non-analytic Smooth Function
In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions. One can easily prove that any analytic function of a real argument is smooth. The converse is not true, with this article constructing a counterexample.
One of the most important applications of smooth functions with compact support is the construction of so-called mollifiers, which are important in theories of generalized functions, like e.g. Laurent Schwartz's theory of distributions.
The existence of smooth but non-analytic functions represents one of the main differences between differential geometry and analytic geometry. In terms of sheaf theory, this difference can be stated as follows: the sheaf of differentiable functions on a differentiable manifold is fine, in contrast with the analytic case.
The functions below are generally used to build up partitions of unity on differentiable manifolds.
Read more about Non-analytic Smooth Function: Definition of The Function, The Function Is Smooth, The Function Is Not Analytic, A Smooth Function Which Is Nowhere Real Analytic, Smooth Transition Functions, Application To Taylor Series, Application To Higher Dimensions
Famous quotes containing the words smooth and/or function:
“Nations are possessed with an insane ambition to perpetuate the memory of themselves by the amount of hammered stone they leave. What if equal pains were taken to smooth and polish their manners?”
—Henry David Thoreau (1817–1862)
“The intension of a proposition comprises whatever the proposition entails: and it includes nothing else.... The connotation or intension of a function comprises all that attribution of this predicate to anything entails as also predicable to that thing.”
—Clarence Lewis (1883–1964)