Generalized functions, space of
The space dual to the space of test (sufficiently good) functions. The Fréchet–Schwartz spaces (cf. Fréchet space) (type FS) and the spaces strongly dual to them (type DFS) play an important role here. A space of type FS is the projective limit of a directed set of Banach spaces and its dual is a space of type DFS. A space of type DFS is the inductive limit of a directed set of Banach spaces and its dual is a space of type FS. Spaces of types FS and DFS are complete, separable, reflexive, and Montel. In spaces of types FS and DFS weak and strong convergence coincide.
Examples of spaces of test and generalized functions.
1) The spaces and . The space of (rapidly-decreasing) test functions consists of the -functions that together with all their derivatives decrease at infinity faster than any power of . This space is the projective limit of the sequence of Banach spaces , consisting of the -functions with norm
and the inclusion is compact; is of type FS. The dual space (the space of generalized functions of slow growth) is the inductive limit of the sequence of Banach spaces , where the imbedding is compact, so that is of type DFS. If a sequence of generalized functions is (weakly) convergent in , then it converges with respect to the norm of functionals in some . The Fourier transformation is an isomorphism on the spaces and .
2) The spaces and ( an open set in ). The space of test functions consists of the -functions that have compact support in (see Support of a generalized function). It is endowed with the topology of the strong inductive limit of the (increasing) sequence of spaces of type FS, where is a strictly-increasing sequence of open sets that exhausts , , compact, . The space is the projective limit of the (decreasing) sequence of Banach spaces , consisting of the functions with support in and with norm
where the imbedding is compact. Let be the space (strongly) dual to ; and . A sequence of test functions in converges in if it converges in some space . A sequence of generalized functions in converges in if it converges on every element of (weak convergence). For a linear functional on to be a generalized function in it is necessary and sufficient that for any open set there exist numbers and such that
The space is (weakly) complete: If a sequence of generalized functions , is such that for any in the sequence of numbers converges, then the functional
belongs to . A generalized function in has unrestricted "growth" in a neighbourhood of the boundary ; in particular, any function determines a generalized function in by the formula
3) The spaces and . Let be the Banach space of all functions , , that are holomorphic in the tubular neighbourhood , , with norm
the imbedding , , is compact. Let be the inductive limit of the (increasing) sequence of spaces , . The space is of type DFS, and its dual is of type FS. The elements of are Fourier hyperfunctions; is also isomorphic to the space .
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For functional-analytic properties of distribution spaces see [a1].
Test spaces and spaces of generalized functions which satisfy such invariance requirements can be constructed starting from a separable Hilbert space and an unbounded self-adjoint operator on . The test space ( analyticity space) is defined by . The distribution space ( trajectory space) consists of mappings with the property: for all : . The duality pairing is , where is sufficiently small and depends on .
Both spaces are inductive and projective limits of Hilbert spaces. Many Gel'fand–Shilov spaces are of this type.
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|[a2]||A.H. Zemanian, "Generalized integral transformations" , Interscience (1968) MR0423007 Zbl 0181.12701|
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|[a4]||S.J.L. van Eijndhoven, J. de Graaf, "Trajectory spaces, generalized functions and unbounded operators" , Lect. notes in math. , 1162 , Springer (1985) Zbl 0622.46032|
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|[a8]||W. Rudin, "Functional analysis" , McGraw-Hill (1974) MR1157815 MR0458106 MR0365062 Zbl 0867.46001 Zbl 0253.46001|
Generalized functions, space of. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Generalized_functions,_space_of&oldid=28203