What is a compactly supported function?
A function has compact support if it is zero outside of a compact set. Alternatively, one can say that a function has compact support if its support is a compact set. For example, the function in its entire domain (i.e., ) does not have compact support, while any bump function does have compact support.
What defines a smooth function?
Non-smooth Functions. Smooth functions have a unique defined first derivative (slope or gradient) at every point. Graphically, a smooth function of a single variable can be plotted as a single continuous line with no abrupt bends or breaks.
How do you show that a function is compactly supported?
2. A function is said to be compactly supported if its support is a compact set. For convenience, we denote the subspace of Lp that contains all compactly supported functions in Lp by L 0 p and denote the subspace of C0 that contains all compactly supported functions in C0 by C00.
How do you know if a function is smooth?
The function f is said to be infinitely differentiable, smooth, or of class C∞, if it has derivatives of all orders. The function f is said to be of class Cω, or analytic, if f is smooth and if its Taylor series expansion around any point in its domain converges to the function in some neighborhood of the point.
What is compactly supported smooth function?
A bump function (sometimes also called a test function) is a compactly supported smooth function, which is usually supposed to be non-negative, no more than 1, and equals to 1 on a given compact set.
What is a distribution support?
In probability and measure theory In probability theory, the support of a probability distribution can be loosely thought of as the closure of the set of possible values of a random variable having that distribution.
What is a smooth function example?
The function is defined on the open interval (a, b). All derivatives exist up to order n, on the stated interval. For example, a smooth function of class C2 has both a first derivative and a second derivative. If all derivatives exist, the function is called infinitely smooth or infinitely differentiable.
What does it mean when a graph is smooth?
A smooth curve is a curve which is a smooth function, where the word “curve” is interpreted in the analytic geometry context. In particular, a smooth curve is a continuous map from a one-dimensional space to an. -dimensional space which on its domain has continuous derivatives up to a desired order.
What is support of a distribution?
How do you prove a function is beta smooth?
Definition A continuously differentiable function f is β-smooth if the gradient ∇f is β-Lipschitz, that is if for all x, y ∈ X, ∇f (y) − ∇f (x) ≤ βy − x .
When is a function said to be compactly supported?
A function is said to be compactly supported if its support is a compact set. For convenience, we denote the subspace of Lp that contains all compactly supported functions in Lp by L p0 and denote the subspace of C0 that contains all compactly supported functions in C0 by C00. (Note that L p0 (or C00) is not a closed subspace of Lp (or C0 ).)
Is the space of functions with compact support dense?
In good cases, functions with compact support are dense in the space of functions that vanish at infinity, but this property requires some technical work to justify in a given example. As an intuition for more complex examples, and in the language of limits, for any
Which is stronger compact support or vanishing at infinity?
The condition of compact support is stronger than the condition of vanishing at infinity. For example, the function is not compact. Real-valued compactly supported smooth functions on a Euclidean space are called bump functions.
How is compact support used in Poincare duality?
In extending Poincaré duality to manifolds that are not compact, the ‘compact support’ idea enters naturally on one side of the duality; see for example Alexander–Spanier cohomology . Bredon, Sheaf Theory (2nd edition, 1997) gives these definitions.