Definition
The Schwartz space or space of rapidly decreasing functions on Rn is the function space
where α, β are multi-indices, C∞(Rn) is the set of smooth functions from Rn to C, and
Here, sup denotes the supremum, and we again use multi-index notation.
To put common language to this definition, we could note that a rapidly decreasing function is essentially a function f(x) such that f(x), f′(x), f′′(x), ... all exist everywhere on R and go to zero as x → ±∞ faster than any inverse power of x. Especially, S(Rn) is a subspace of the function space C∞(Rn) of infinitely smooth functions.
Read more about this topic: Schwartz Space
Famous quotes containing the word definition:
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)