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:
“Beside all the small reasons we assign, there is a great reason for the existence of every extant fact; a reason which lies grand and immovable, often unsuspected behind it in silence.”
—Ralph Waldo Emerson (1803–1882)
“Truth exists. The sole purpose of this proposition is to assert the existence of truth against imbeciles and sceptics.”
—Edward Herbert Of Cherbury, Lord (1583–1648)
“I have always thought the suicide shd/ bump off at least one swine before taking off for parts unknown.”
—Ezra Pound (1885–1972)
“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)