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schwartz space wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 definition 2 examples of functions in the schwartz space 3 properties toggle properties subsection 3 1 analytic properties 3 2 relation of schwartz spaces with other topological vector spaces 4 see also 5 references toggle references subsection 5 1 sources toggle the table of contents schwartz space 12 languages català čeština deutsch español français italiano 日本語 한국어 polski русский українська 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia function space of all functions whose derivatives are rapidly decreasing for the schwartz space of a semisimple lie group see harish chandra s schwartz space for the schwartz space of a locally compact abelian group see schwartz bruhat function in mathematics schwartz space s displaystyle mathcal s is the function space of all functions whose derivatives of all orders are rapidly decreasing this space has the important property that the fourier transform is an automorphism on this space this property enables one by duality to define the fourier transform for elements in the dual space s displaystyle mathcal s of s displaystyle mathcal s that is for tempered distributions a function in the schwartz space is sometimes called a schwartz function a two dimensional gaussian function is an example of a rapidly decreasing function schwartz space is named after french mathematician laurent schwartz definition edit let n displaystyle mathbb n be the set of non negative integers and for any n n displaystyle n in mathbb n let n n n n n times displaystyle mathbb n n underbrace mathbb n times dots times mathbb n _ n text times be the n displaystyle n fold cartesian product the schwartz space or space of rapidly decreasing functions on r n displaystyle mathbb r n is the function space s r n c f c r n c α β n n f α β displaystyle mathcal s left mathbb r n mathbb c right left f in c infty mathbb r n mathbb c mid forall boldsymbol alpha boldsymbol beta in mathbb n n f _ boldsymbol alpha boldsymbol beta infty right where c r n c displaystyle c infty mathbb r n mathbb c is the function space of smooth functions from r n displaystyle mathbb r n into c displaystyle mathbb c and f α β sup x r n x α d β f x displaystyle f _ boldsymbol alpha boldsymbol beta sup _ boldsymbol x in mathbb r n left boldsymbol x boldsymbol alpha boldsymbol d boldsymbol beta f boldsymbol x right here sup displaystyle sup denotes the supremum and we used multi index notation i e x α x 1 α 1 x 2 α 2 x n α n displaystyle boldsymbol x boldsymbol alpha x_ 1 alpha _ 1 x_ 2 alpha _ 2 ldots x_ n alpha _ n and d β 1 β 1 2 β 2 n β n displaystyle d boldsymbol beta partial _ 1 beta _ 1 partial _ 2 beta _ 2 ldots partial _ n beta _ n to put common language to this definition one could consider a rapidly decreasing function as essentially a function f displaystyle f such that f x displaystyle f x f x displaystyle f x f x displaystyle f prime prime x all exist everywhere on r displaystyle mathbb r and go to zero as x displaystyle x rightarrow pm infty faster than any reciprocal power of x displaystyle x in particular s r n c displaystyle mathcal s left mathbb r n mathbb c right is a subspace of c r n c displaystyle c infty mathbb r n mathbb c examples of functions in the schwartz space edit if α displaystyle boldsymbol alpha is a multi index and a is a positive real number then x α e a x 2 s r n displaystyle boldsymbol x boldsymbol alpha e a vert boldsymbol x vert 2 in mathcal s mathbb r n any smooth function f displaystyle f with compact support is in s r n displaystyle mathcal s left mathbb r n right this is clear since any derivative of f displaystyle f is continuous and supported in the support of f displaystyle f so x α d α f displaystyle boldsymbol x boldsymbol alpha boldsymbol d boldsymbol alpha f has a maximum in r n displaystyle mathbb r n by the extreme value theorem because the schwartz space is a vector space any polynomial ϕ x displaystyle phi boldsymbol x can be multiplied by a factor e a x 2 displaystyle e a vert boldsymbol x vert 2 for a 0 displaystyle a 0 a real constant to give an element of the schwartz space in particular there is an embedding of polynomials into a schwartz space properties edit analytic properties edit from leibniz s rule it follows that s r n displaystyle mathcal s left mathbb r n right is also closed under pointwise multiplication f g s r n displaystyle f g in mathcal s left mathbb r n right implies f g s r n displaystyle fg in mathcal s left mathbb r n right in particular this implies that s r n displaystyle mathcal s left mathbb r n right is an r displaystyle mathbb r algebra more generally if f s r displaystyle f in mathcal s left mathbb r right and h displaystyle h is a bounded smooth function with bounded derivatives of all orders then f h s r displaystyle fh in mathcal s left mathbb r right the fourier transform is a linear isomorphism f s r n s r n displaystyle mathcal f mathcal s left mathbb r n right rightarrow mathcal s left mathbb r n right if f s r n displaystyle f in mathcal s left mathbb r n right then f displaystyle f is lipschitz continuous and hence uniformly continuous on r n displaystyle mathbb r n s r n displaystyle mathcal s left mathbb r n right is a distinguished locally convex fréchet schwartz tvs over the complex numbers both s r n displaystyle mathcal s left mathbb r n right and its strong dual space are also complete hausdorff locally convex spaces nuclear montel spaces ultrabornological spaces reflexive barrelled mackey spaces relation of schwartz spaces with other topological vector spaces edit if 1 p displaystyle 1 leqslant p infty then s r n displaystyle mathcal s left mathbb r n right is a dense subset of l p r n displaystyle l p mathbb r n the space of all bump functions c c r n displaystyle c_ text c infty left mathbb r n right is included in s r n displaystyle mathcal s left mathbb r n right see also edit bump function schwartz bruhat function nuclear space references edit sources edit hörmander l 1990 the analysis of linear partial differential operators i distribution theory and fourier analysis 2nd ed berlin springer verlag isbn 3 540 52343 x reed m simon b 1980 methods of modern mathematical physics functional analysis i revised and enlarged ed san diego academic press isbn 0 12 585050 6 stein elias m shakarchi rami 2003 fourier analysis an introduction princeton lectures in analysis i princeton princeton university press isbn 0 691 11384 x trèves françois 2006 1967 topological vector spaces distributions and kernels mineola n y dover publications isbn 978 0 486 45352 1 oclc 853623322 this article incorporates material from space of rapidly decreasing functions on planetmath which is licensed under the creative commons attribution share alike license v t e functional analysis topics glossary spaces banach besov fréchet hilbert inner product normed nuclear schwartz sobolev topological vector properties barrelled quasi barrelled complete dual algebraic topological locally convex reflexive separable theorems hahn banach riesz representation closed graph uniform boundedness principle kakutani fixed point krein milman min max gelfand naimark banach alaoglu operators adjoint bounded compact hilbert schmidt normal nuclear trace class transpose unbounded unitary algebras banach algebra c algebra spectrum of a c algebra operator algebra group algebra of a locally compact group von neumann algebra open problems invariant subspace problem mahler s conjecture applications hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variations functional calculus integral linear operator jones polynomial topological quantum field theory noncommutative geometry riemann hypothesis distribution or generalized functions advanced topics approximation property balanced set choquet theory weak topology banach mazur distance tomita takesaki theory category retrieved from https en wikipedia org w index php title schwartz_space oldid 1360762580 categories topological vector spaces smooth functions fourier analysis function spaces schwartz distributions hidden categories articles with short description short description matches wikidata wikipedia articles incorporating text from planetmath 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