Existence of Bump Functions
It is possible to construct bump functions "to specifications". Stated formally, if is an arbitrary compact set in n dimensions and is an open set containing there exists a bump function which is 1 on and 0 outside of Since can be taken to be a very small neighborhood of this amounts to being able to construct a function that is 1 on and falls off rapidly to 0 outside of while still being smooth.
The construction proceeds as follows. One considers a compact neighborhood of contained in so The characteristic function of will be equal to 1 on and outside of so in particular, it will be 1 on and outside of This function is not smooth however. The key idea is to smooth a bit, by taking the convolution of with a mollifier. The latter is just a bump function with a very small support and whose integral is 1. Such a mollifier can be obtained, for example, by taking the bump function from the previous section and performing appropriate scalings.
Read more about this topic: Bump Function
Famous quotes containing the words existence of, existence, bump and/or functions:
“The Frenchman Jean-Paul ... Sartre I remember now was his last name had a dialectical mind good as a machine for cybernetics, immense in its way, he could peel a nuance like an onion, but he had no sense of evil, the anguish of God, and the possible existence of Satan.”
—Norman Mailer (b. 1923)
“The very existence of society depends on the fact that every member of it tacitly admits he is not the exclusive possessor of himself, and that he admits the claim of the polity of which he forms a part, to act, to some extent, as his master.”
—Thomas Henry Huxley (1825–95)
“Physics investigates the essential nature of the world, and biology describes a local bump. Psychology, human psychology, describes a bump on the bump.”
—Willard Van Orman Quine (b. 1908)
“Mark the babe
Not long accustomed to this breathing world;
One that hath barely learned to shape a smile,
Though yet irrational of soul, to grasp
With tiny finger—to let fall a tear;
And, as the heavy cloud of sleep dissolves,
To stretch his limbs, bemocking, as might seem,
The outward functions of intelligent man.”
—William Wordsworth (1770–1850)