Summary
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space . The method of integration by parts holds that for differentiable functions and we have A function u' being the weak derivative of u is essentially defined by the requirement that this equation must hold for all infinitely differentiable functions φ vanishing at the boundary points (). Let be a function in the Lebesgue space . We say that in is a weak derivative of if for all infinitely differentiable functions with . Generalizing to dimensions, if and are in the space of locally integrable functions for some open set , and if is a multi-index, we say that is the -weak derivative of if for all , that is, for all infinitely differentiable functions with compact support in . Here is defined as If has a weak derivative, it is often written since weak derivatives are unique (at least, up to a set of measure zero, see below). The absolute value function , which is not differentiable at has a weak derivative known as the sign function, and given by This is not the only weak derivative for u: any w that is equal to v almost everywhere is also a weak derivative for u. (In particular, the definition of v(0) above is superfluous and can be replaced with any desired real number r.) Usually, this is not a problem, since in the theory of Lp spaces and Sobolev spaces, functions that are equal almost everywhere are identified. The characteristic function of the rational numbers is nowhere differentiable yet has a weak derivative. Since the Lebesgue measure of the rational numbers is zero, Thus is a weak derivative of . Note that this does agree with our intuition since when considered as a member of an Lp space, is identified with the zero function. The Cantor function c does not have a weak derivative, despite being differentiable almost everywhere.
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Related concepts (8)
Weak derivative
In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (strong derivative) for functions not assumed differentiable, but only integrable, i.e., to lie in the Lp space . The method of integration by parts holds that for differentiable functions and we have A function u' being the weak derivative of u is essentially defined by the requirement that this equation must hold for all infinitely differentiable functions φ vanishing at the boundary points ().
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
Locally integrable function
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to Lp spaces, but its members are not required to satisfy any growth restriction on their behavior at the boundary of their domain (at infinity if the domain is unbounded): in other words, locally integrable functions can grow arbitrarily fast at the domain boundary, but are still manageable in a way similar to ordinary integrable functions.
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