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:
“A leaf that is supposed to grow is full of wrinkles and creases before it develops; if one doesn’t have the patience and wants the leaf to be as smooth as a willow leaf from the start, then there is a problem.”
—Johann Wolfgang Von Goethe (1749–1832)
“As a medium of exchange,... worrying regulates intimacy, and it is often an appropriate response to ordinary demands that begin to feel excessive. But from a modernized Freudian view, worrying—as a reflex response to demand—never puts the self or the objects of its interest into question, and that is precisely its function in psychic life. It domesticates self-doubt.”
—Adam Phillips, British child psychoanalyst. “Worrying and Its Discontents,” in On Kissing, Tickling, and Being Bored, p. 58, Harvard University Press (1993)