Properties
- S(Rn) is a Fréchet space over the complex numbers.
- From Leibniz' rule, it follows that S(Rn) is also closed under pointwise multiplication: if f, g ∈ S(Rn), then fg ∈ S(Rn).
- If 1 ≤ p ≤ ∞, then S(Rn) ⊂ Lp(Rn).
- The space of all bump functions, C∞
c(Rn), is included in S(Rn).
- The Fourier transform is a linear isomorphism S(Rn) → S(Rn).
- If f ∈ S(R), then f is uniformly continuous on R.
Read more about this topic: Schwartz Space
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)