Annals of mathematics, a distinguished journal of research papers in pure mathematics, was founded in 1884. The paley wiener schwartz theorem characterizes compactly supported smooth functions bump functions and more generally compactly supported distributions in terms of the decay property of their fourierlaplace transform of distributions. The main ingredient in the proof is the gutzmers formula. The paleywiener theorem we follow the presentation in 1, p. The paleywienerschwartz theorem characterizes compactly supported smooth functions bump functions and more generally compactly supported distributions in terms of the decay property of their fourierlaplace transform of distributions conversely this means that for a general distribution those covectors along which its fourier transform does not suitably decay detect the singular. A paleywiener theorem for the inverse fourier transform on. There are two versions of the construction depending on whether q is congruent to 1 or 3. The classical shannon sampling theorem is based on the paley wiener theorem. As was shown in 9 the operator is inevitable in characterising the image of the heat kernel transform. Fourier transforms of arbitrary distributions of course, the original version paley wiener 1934 referred to l2 functions, not distributions. Ams proceedings of the american mathematical society. A paleywiener theorem for real reductive groups, acta math.
A proof of the paley wiener theorem for hyperfunctions with a convex compact support by the heat kernel method suwa, masanori and yoshino, kunio, tokyo journal of mathematics, 2004. The paleywiener theorem for certain nilpotent lie groups 2 i there exists a. In section 4 the inversion formula is derived by using the paley wiener theorem. Taken together they will tell us that if then the expression is a polynomial in t, whenever the minimum distance from t to the walls. We formulate and prove a version of paleywiener theorem for the inverse fourier transforms on noncompact riemannian symmetric spaces and heisenberg groups. We rst successfully proved the cli ord paley wiener theorem and subsequently accomplished the cli ord analogue of the shannon sampling. A comparison of paleywiener theorems for real reductive. A description of the image of a certain space of functions or generalized functions on a locally compact group under the fourier transform or under some other injective integral transform is called an analogue of the paleywiener theorem. In mathematics, a paleywiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its fourier transform. Patrick delorme iml submitted on 15 may 2009 v1, last revised 31 may 2010 this version, v2. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. A proof of the paleywiener theorem for hyperfunctions with a convex compact support by the heat kernel method suwa, masanori and yoshino, kunio, tokyo journal of mathematics, 2004. Annals of mathematics is published bimonthly with the cooperation of princeton university and the institute for advanced study. The paley wiener theorem we follow the presentation in 1, p.
The original theorems did not use the language of distributions, and instead applied to squareintegrable functions. An example of smooth compactly supported function with nonvanishing fourier transform. A paley wiener theorem for distributions on reductive symmetric spaces by e. A paleywiener theorem for the inverse fourier transform. The paley construction uses quadratic residues in a finite field gf q where q is a power of an odd prime number. In section 3 we will recall this result and show how it applies to the k finite functions in c. Section 3 contains the proof of the paley wiener theorem for all complex and ft. The analogues of the uncertainty principle for wlct are studied in section 4.
A new proof of a paleywiener type theorem for the jacobi transform 147 by substituting zsinht 2 in 2. A paleywiener like theorem in realanalysis mathematics. In mathematics, the paley construction is a method for constructing hadamard matrices using finite fields. A paley wiener theorem for the inverse fourier transform on some homogeneous spaces thangavelu, s. A paleywiener theorem for distributions on reductive symmetric spaces by e. Our proof is guided by the one for the classical paley wiener theorem cited in 24. Paley wiener theorem for compactlysupported distributions e 3. We rst successfully proved the cli ord paleywiener theorem and subsequently accomplished the cli ord analogue of the shannon sampling. For schwartz function gwith the support of bgnot meeting b r, bg for su ciently small. Now to prove f is an entire function, for that it is enough to prove that f. However, i have looked up the proof in paley and wiener and i find it far too technical and nonselfcontained for me to follow with any ease.
Pdf document information annals of mathematics fine hall washington road princeton university princeton, nj 08544, usa phone. Journal of the institute of mathematics of jussieu. It is widely regarded as one of the main mathematics journals in the world. We generalize opdams estimate for the hypergeometric functions in a bigger domain with the multiplicity parameters being not necessarily positive, which is crucial. The construction was described in 1933 by the english mathematician raymond paley.
The construction was described in 1933 by the english mathematician raymond paley the paley construction uses quadratic residues in a finite field gfq where q is a power of an odd prime number. Paleywiener theorem for line bundles over compact symmetric. We study fourier transforms of compactly supported k nite distributions on x and characterize the image of the space of such. The theorem of kroetz et al 9 and our paley wiener theorem both involve a certain peseudodi erential shift operator d. In mathematics, a paley wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its fourier transform. A new proof of a paleywiener type theorem for the jacobi. The classical shannon sampling theorem is based on the paleywiener theorem. A paleywiener theorem for the inverse fourier transform on some homogeneous spaces thangavelu, s. Codimensionp paleywiener theorems yang, yan, qian, tao, and. A paleywiener theorem for distributions on reductive. Pages 9871029 from volume 162 2005, issue 2 by patrick delorme.
The theorem is named for raymond paley 19071933 and norbert wiener 18941964. But avoid asking for help, clarification, or responding to other answers. A twisted invariant paleywiener theorem for real reductive groups delorme, patrick and mezo, paul, duke mathematical journal, 2008. In the present paper short proofs will be given of a paleywiener type theorem. When the group gis complex the operator dis simple multiplication by a jacobian factor but otherwise it is quite. Now to prove f is an entire function, for that it is enough to prove that f is analytic, for proving f. Therefore, generalizations of the paleywiener theorem to rnis of the rst importance. Mathematical research article applied sciences mos subject. We give an elementary proof of the paley wiener theorem for smooth functions for the dunkl transforms on the real line, establish a similar theorem for l2functions and prove identities in the spirit of bang for lpfunctions.
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