A paleywiener theorem for some eigenfunction expansions a paleywiener theorem for some eigenfunction expansions thangavelu, s. Theorem 1 for c which decrease faster than any exponential. The main result is a topological paley wiener theorem for the full space c. A new proof of a paleywiener type theorem for the jacobi.
Paley wiener theorems september 7, 20 note that b x bx goes to 1 as tempered distribution by the more di cult half of paley wiener for test functions, f b. An elementary proof of the paleywiener theorem in ct citeseerx. Finally, in section 3, the paleywiener theorem is proven. A paleywiener theorem for locally compact abelian groups by gunar e. However, paley wiener theorems for the full space c. The theorem of kroetz et al 9 and our paleywiener theorem both involve a certain peseudodi erential shift operator d. Pdf generalizations of paleywieners theorem for entire. Real paleywiener theorems for the multivariable bessel transform 91 space lpr, for all n2n, the limit d f lim n. Koornwindera new proof of a paleywiener type theorem for the jacobi. A comparison of paleywiener theorems for real reductive. Associated with the dirac operator and partial derivatives, this paper establishes some real paleywiener type theorems to characterize the cliffordvalued functions whose clifford fourier transform cft has compact support.
Paleywienertype theorem for analytic functions in tubular domains. Finally, chapter 4 is devoted to the study of the fourier transform. A paleywiener theorem for real reductive groups an open dense subset of g. Real paleywiener theorems for the clifford fourier transform. Pdf in this paper our aim is to establish the paley wiener theorem for the weinstein transform.
Paleywiener theorems for the dunkl transform and dunkl. We give an elementary proof of the paleywiener theorem for smooth functions for the dunkl transforms on the real line, establish a similar theorem for l 2functions, and prove identities in the spirit of bang for l pfunctions. Unlike the classical paleywiener theorem, our theorems use real variable. A paley wiener theorem for the inverse fourier transform on some homogeneous spaces thangavelu, s.
When the group gis complex the operator dis simple multiplication by a jacobian factor but otherwise it is quite. Genchev department of mathematics and informatics, sofia university, sofia 1126, bulgaria and h. Towards a paleywiener theorem for semisimple symmetric. Theorem paley wiener for smooth functions if and then extends analytically to and for all nonnegative integers there exists a constant such that.
Extending the paleywiener theorem to locally compact abelian groups involves both finding a suitable laplace transform and a suitable analogue for analytic functions. It is important to note that the paleywiener integral for a particular element h. Pdf paleywiener theorem for the weinstein transform and. Paleywiener theorems for the two spaces, characterizing them in terms of their spectral transforms. Pdf an interpretation and generalizations of the paleywiener theorem for entire functions of exponential type are given in. 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 l 2functions, and prove identities in the spirit of bang for l pfunctions. The paper also shows that if n is any connected, simply connected nilpotent lie group, then almost all. Schlichtkrull established a paleywiener theorem for reductive symmetric spaces which implied arthurs theorem in the special case of the group.
A paleywiener theorem for real reductive groups project euclid. Paleywiener and boas theorems for the quaternion fourier. The proofs seem to be new also in the special case of the fourier transform. For each of these spaces, we prove a paley wiener theorem, some structure theorems, and provide some applications. A number of authors have proved paleywiener theorems for particular classes of groups. Paleywiener type theorems by transmutations springerlink. The classical paleywiener theorem for functions in l dx 2 relates the growth of the fourier transform.
A paleywiener theorem for real reductive groups by james arthur i. 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. In section 4 the inversion formula is derived by using the paleywiener theorem. Among m m m m the literature, by imbedding r into c r. Our proof is guided by the one for the classical paleywiener theorem cited in 19. This paper is concerned with the classical paleywiener theorems in one and several complex variables, the. Elementary proofs of paleywiener theorems for the dunkl.
A number of generalizations of the classical paleywiener theorem pw the orem to higher dimensional spaces were studied 1, 3, 6, 7 and 9. A paleywiener theorem for the askeywilson function. On the schwartzbruhat space and the paleywiener theorem for. Introduction one of the central theorems of harmonic analysis on r is the paleywiener theorem which characterizes the class of functions on c which are fourier transforms of c. As a corollary, we get relative analogs of the smooth and tempered bernstein centers rings of multipliers for spxq and cpxq. Nov, 2009 the main result of this post, the paley wiener theorem, states that these necessary conditions for a function to be in the range of the fourier transform are in fact sufficient. A test function fsupported on a closed ball b r of radius rat the origin in r has fourier. A paleywiener theorem for some eigenfunction expansions. A local paleywiener theorem for compact symmetric spaces. A paley wiener theorem for distributions on reductive symmetric spaces by e. Then, the classical ndimensional paleywiener theorem is generalized to a case wherein 0 pdf file 127 kb abstract. Let us now return to the main track of the proof of the paleywiener theorem. This paper establishes a real paleywiener theorem to characterize the quaternionvalued functions whose quaternion fourier transform has compact support by the partial derivative and also a boas theorem to describe the quaternion fourier transform of these functions that vanish on a neighborhood of the origin by an integral operator. In this work we obtain paleywiener type theorems where the fourier transform is replaced by transforms associated with selfadjoint operators on l d.
The main result of this post, the paleywiener theorem, states that these necessary conditions for a function to be in the range of the fourier transform are in fact sufficient. On a simple application of paleywiener theorem and related doubts. In section 4 we prove the theorem, using these results. The main theorem establishes sufficient conditions on the dual. In the last section we use theorem 1 to derive a paleywiener type theorem for. We introduce results of this type for the socalled gabor transform and give a full characterization in terms of fourier and wigner transforms for several variables of a paleywiener theorem in this general setting, which is new in the literature.
A paleywiener theorem for reductive symmetric spaces. Based on the riemannlebesgue theorem for the cft, the boas theorem is provided to describe the cft of cliffordvalued functions that vanish on a neighborhood of. Extending the paley wiener theorem to locally compact abelian groups involves both finding a suitable laplace transform and a suitable analogue for analytic functions. Paleywienertype theorems for a class of integral transforms. The discrete exponentials the hilbert space that is important in the following discussion is the l2 space of the unit measure d8 on the real line, where this measure is. Laplace transform representations and paleywiener theorems. A paleywiener theorem for selected nilpotent lie groups. The result covers fourier series on compact symmetric spaces, hermite and special hermite expansions as particular. Paley wiener theorem for the inverse fourier transform on some homogeneous spaces thangavelu, s. Inverse problems course notes the paleywiener theorem. A paleywiener theorem for the inverse fourier transform on. We give a paleywiener theorem for weighted bergman spaces on the existence of such representa. We will in this paper consider paley wiener theorems for the real hyperbolic spaces x p, q. A paley wiener theorem for locally compact abelian groups by gunar e.
The discrete paley wiener theorem 173 in its new form will be made to derive an analog of the cardinal series for d. The paper studies weak paley wiener properties for group extensions by use of mackeys theory. A paleywiener theorem for spherical padic spaces and. Paleywiener theorems and shannon sampling in the clifford. On a simple application of paleywiener theorem and. We give an elemenlary proof of the paleywiener theorem in several complex variables using. We consider the problem of representing an analytic function on a vertical strip by a bilateral laplace transform. The paley wiener theorem in the group case let g be a connected reductive group over a nonarchimedean local eld f. As was shown in 9 the operator is inevitable in characterising the image of the heat kernel transform. A paleywiener theorem for distributions on reductive. Download fulltext pdf a paley wiener theorem for distributions on reductive symmetric spaces article pdf available in journal of the institute of mathematics of jussieu 64 december 2005. A paleywiener theorem for the inverse fourier transform. 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.
A comparison of paleywiener theorems for real reductive lie. The theorem is named for raymond paley 19071933 and norbert wiener 18941964. Let us rst remind the paley wiener theorem of bernstein and heiermann. We study fourier transforms of compactly supported k nite distributions on x and characterize the image of the space of such. Theorem paleywiener for smooth functions if and then extends analytically to and for all nonnegative integers there exists a constant such that. July 1, 2009 communicated by thomas peternell abstract. 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. 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.
Section 3 contains the proof of the paleywiener theorem for all complex and ft. This result, called real paleywiener theorem, has been establish for many other integral transforms, see 3, 7, 8. In chapters 2 and 3 of the book, the authors transfer many results presented in volume 1 to generalized functions corresponding to these more general spaces. When x a reductive group, our theorem for cpxqspecializes to the wellknown theorem of harishchandra, and our. These operators permit also to define and study dunkl translation operators and. We formulate and prove a version of paleywiener theorem for the inverse fourier transforms on noncompact riemannian symmetric spaces and. Real paleywiener theorems for the multivariable bessel. In section 3 we show that the concept of monogenic func tions is a natural. A paleywiener theorem for distributions on reductive symmetric spaces by e. Herein, a weighted version of the paleywienertype theorem for analytic functions in a tubular domain over a regular cone is obtained by using h p space methods.
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point a. The paleywiener theorem for certain nilpotent lie groups 2 i there exists a. This paper concerns itself with the problem of generalizing to nilpotent lie groups a weak form of the classical paleywiener theorem for r n the generalization is accomplished for a large subclass of nilpotent lie groups, as well as for an interesting example not in this subclass. A paley wiener theorem for real reductive groups an interesting consequence of our main result is the construction of an algebra of multipliers for cg, k. The paleywiener theorem describes the properties of the fourier spectrum of a function, which is the. The original theorems did not use the language of distributions, and instead applied to squareintegrable functions.
By a multiplier we mean a linear operator c on cg, k such that cf. A paleywiener theorem for locally compact abelian groups. These operators permit also to define and study dunkl translation operators and dunkl convolution product. The paper studies weak paleywiener properties for group extensions by use of mackeys theory. We define an analogue of the paleywiener space in the context of the askeywilson function transform, compute explicitly its reproducing kernel and prove that the growth of functions in this space of entire functions is of order two and type ln q. This space of functions 5e0 was fhst introduced to the authors in lectures by casselman. The theorem describes the multipliers in a form which is remarkably similar to the cutoff functions b. The classical paleywiener theorem for functions in l dx 2 relates the growth of the fourier transform over the complex plane to the support of the function. We also analyze this type of results when the support of the function is not compact using. Real paleywiener theorems for the multivariable bessel transform. 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. The theorem of kroetz et al 9 and our paley wiener theorem both involve a certain peseudodi erential shift operator d. Paleywiener theorem and the factorization article pdf available in journal of integral equations and applications 62 june 1994 with 40 reads how we measure reads.
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