The main motivation behind distribution theory is to extend the common linear operators on functions, such as the derivative, convolution, and the Fourier transform, so that they also apply to the singular, non-smooth, or non-integrable functions that regularly appear in both theoretical and applied analysis.
Distribution theory also seeks to define suitable structures on the spaces of functions involved to ensure the convergence of suitable approximating functions, and the continuity of certain operators. For example, the limit of derivatives should be equal to the derivative of the limit, with some definition of the limiting operation.
When this program is carried out, inevitably we find that we have to enlarge the space of objects that we would consider as “functions”. For example, the derivative of a step function is the Dirac delta function with a spike at the discontinuous step; the Fourier transform of a constant function is also a Dirac delta function, with the spike representing infinite spectral magnitude at one single frequency. (These facts, of course, had long been used in engineering mathematics.)
Remark: Dirac’s notion of delta distributions was introduced to facilitate computations in Quantum Mechanics, however without having at the beginning a proper mathematical definition. In part as a (negative) reaction to such a state of affairs, von Neumann produced a mathematically well-defined foundation of Quantum Mechanics (http://planetmath.org/QuantumGroupsAndVonNeumannAlgebras) based on actions of self-adjoint operators on Hilbert spaces which is still currently in use, with several significant additions such as Frechét nuclear spaces and quantum groups.
There are several theories of such ‘generalized functions’. In this entry, we describe Schwartz’ theory of distributions, which is probably the most widely used.
while any smooth function on induces a distribution
A note on notation. In distribution theory, the topological vector space of smooth functions with compact support on an open set is traditionally denoted by . Let us also denote by the subset of of functions with support in a compact set .
Definition 1 (Distribution).
A distribution is a linear continuous functional on , i.e., a linear continuous mapping . The set of all distributions on is denoted by .
Suppose is a linear functional on . Then is continuous if and only if is continuous in the origin (see this page (http://planetmath.org/ContinuousLinearMapping)). This condition can be rewritten in various ways, and the below theorem gives two convenient conditions that can be used to prove that a linear mapping is a distribution.
Let be an open set in , and let be a linear functional on . Then the following are equivalent:
is a distribution.
For any compact set in , there are constants and such that for all , we have
where is a multi-index, and is the supremum norm.
Proof The equivalence of (2) and (3) can be found on this page (http://planetmath.org/EquivalenceOfConditions2And3), and the equivalence of (1) and (3) is shown in .
Distributions of order
If is a distribution on an open set , and the same can be used for any in the above inequality, then is a distribution of order . The set of all such distributions is denoted by .
Both usual functions and the delta distribution are of order . One can also show that by differentiating a distribution its order increases by at most one. Thus, in some sense, the order is a measure of how ”smooth” a distribution is.
A common notation for the action of a distribution onto a test function (i.e., with above notation) is . The motivation for this comes from this example (http://planetmath.org/EveryLocallyIntegrableFunctionIsADistribution).
|Date of creation||2013-03-22 13:44:08|
|Last modified on||2013-03-22 13:44:08|
|Last modified by||matte (1858)|
|Defines||distribution of finite order|