Furthermore, the connection between dunkl operators and the cartan motion group is established. Section 3 contains the paley wiener theorem for smooth functions and canperhapsserve as a model for a proof of such a theorem in other contexts. To illustrate this we specialize in theorem 2 to the case of a noncompact semisimple lie group. The function f 1g is smooth, and equals 1 on e and 0 on 0ec.
Laplace transform representations and paleywiener theorems. 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. For them our theorem 2 is a simple corollary of their paley wiener theorem, which is rather difficult to prove. 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 paleywienerschwartz theorem characterizes the fourier transforms of distributions with bounded compact support as being exactly the entire functions of exponential type which are slowly increasing. Real paleywiener theorems for the multivariable bessel. The paleywiener theorem for certain nilpotent lie groups 2 i there exists a. Chapter 12 the paleywienerschwartz theorem in infinite. Thus, it is natural to ask for the associated paley wiener theorem. The paley wiener schwartz theorem characterizes the fourier transforms of distributions with bounded compact support as being exactly the entire functions of exponential type which are slowly increasing. A paleywiener theorem for the inverse fourier transform on some homogeneous spaces thangavelu, s. Thus, it is natural to ask for the associated paleywiener theorem.
Schlichtkrull established a paleywiener theorem for reductive symmetric spaces which implied arthurs theorem in the special case of the group. If 0 1 and f 2pwe, then ft 1 d x k2n x n2n bknf t n g t tk. Recently, one of us has found a sampling theorem for the askeywilson function transform 6. The rest of the paper is independent 2000 mathematics subject classi. Therefore, generalizations of the paley wiener theorem to rnis of the rst importance. Autocorrelation function and the wienerkhinchin theorem consider a time series xt signal. A paley wiener theorem for locally compact abelian groups by gunar e. We will discuss an illustrative case of this technique. It is the purpose of this paper to address this question, providing a paleywiener.
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. Frames of exponential functions and interpolation in paley. When x a reductive group, our theorem for cpxqspecializes to the wellknown theorem of harishchandra, and our. In section 4 the inversion formula is derived by using the paleywiener theorem. A paleywiener theorem for the inverse fourier transform on. July 1, 2009 communicated by thomas peternell abstract. Theorem 8 there is an 0 such that whenever j n 2nj theorem is valid. Stability theorems for fourier frames and wavelet riesz bases. Finally, we prove a conjecture about dixmierdouady invariants of continuoustrace subquotients of c g when gis twostep nilpotent. We study fourier transforms of compactly supported k nite distributions on x and characterize the image of the space of such. Therefore, generalizations of the paleywiener theorem to rnis of the rst importance. A new proof of a paleywiener type theorem for the jacobi. Delormes recent proof of a paleywiener theorem for reductive groups in 6 is completely independent of the proofs just mentioned.
The classical shannon sampling theorem is based on the paley wiener theorem. From there he entered trinity college, cambridge where he showed himself as a brilliant student. Real paleywiener theorems for the multivariable bessel transform 91 space lpr, for all n2n, the limit d f lim n. Hence magnitude and phase response are interdependent. 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. By the peter weyl theorem we can choose a complete orthonormal basis. Inverse problems course notes the paleywiener theorem. Theorem 2 was first proved by campoli 5 for the rank one case and in general by arthur 1. The theorem of kroetz et al 9 and our paley wiener theorem both involve a certain peseudodi erential shift operator d.
Paley wiener theorem for the euclidean motion group which was announced in 3, restricting our attention to biinvariant functions with respect to the rotation group we obtain an analogue of the paley wiener theorem for the fourierbessel transform. The paleywiener theorem for the jacobi transform and the. We also show that every paleywiener function on a manifold is uniquely determined by its values on some discrete sets of points. A comparison of paleywiener theorems for real reductive lie. A paleywiener theorem for locally compact abelian groups by gunar e. Fourier transforms of arbitrary distributions of course, the original version paley wiener 1934 referred to l2 functions, not distributions. The following result follows directly from the above theorem of paley and wiener. Introduction the classical paleywiener theorems characterize the fourier transforms of var. One trick which has proved useful is to complete the sum. 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.
Autocorrelation function and the wienerkhinchin theorem. Assuming that this signal is known over an in nitely long interval t. A paley wiener theorem for distributions on reductive symmetric spaces by e. Analogue of the paley wiener theorem 79 finite dimensional irreducible unitary representations of g, respectively.
The paley wiener theorem for the jacobi transform and the local huygens principle for root systems with even multiplicities by thomas branson a, gestur olafsson b and angela pasquale c a department of mathematics, university oflowa. Towards a paleywiener theorem for semisimple symmetric. The theorem describes the multipliers in a form which is remarkably similar to the cutoff functions b. A paleywiener theorem for locally compact abelian groups. This chapter presents a paleywienerschwartz theorem in infinite dimensions. The discrete paleywiener theorem 173 in its new form will be made to derive an analog of the cardinal series for d. Completion in number theory, one often deals with incomplete sums, i. This result, called real paleywiener theorem, has been establish for many other integral transforms, see 3, 7, 8. The theorem of kroetz et al 9 and our paleywiener theorem both involve a certain peseudodi erential shift operator d.
It is the purpose of this paper to address this question, providing a paley wiener theorem for the askeywilson function transform. On some problems suggested by the trace formula, in lie group representations ii, springer lecture notes, vol. Section 3 contains the proof of the paleywiener theorem for all complex and ft. Paleywiener theorem for arbitrary nilpotent lie groups, which in turn establishes the truth of a conjecture of moss. 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. Reconstruction of paleywiener functions on the heisenberg. We consider the problem of representing an analytic function on a vertical strip by a bilateral laplace transform. In our paper, we shall focus our attention only on the class of paley wiener functions pw. Real paleywiener theorems for the multivariable bessel transform. Towards a paleywiener theorem for semisimple symmetric spaces. The paleywiener theorem then follows by combining the classical paleywiener theorem on rp with the results of koornwinder in 12 about the jacobi polynomial transform. Paley wiener theorem for the inverse fourier transform on some homogeneous spaces thangavelu, s.
Extending the paleywiener theorem to locally compact abelian groups involves both finding a suitable laplace transform and a suitable analogue for analytic functions. Raymond edward alan christopher paley 7 january 1907 7 april 1933 was an english mathematician. We conjecture a geometrical form of the paleywiener theorem for the dunkl transform and prove three instances thereof, one of which involves a limit transition from opdams results for the graded hecke algebra. Extending the paley wiener theorem to locally compact abelian groups involves both finding a suitable laplace transform and a suitable analogue for analytic functions. Theorem paley wiener for smooth functions if and then extends analytically to and for all nonnegative integers there exists a constant such that. On the paleywiener theorem in the mellin transform setting. A paleywiener theorem for reductive symmetric spaces. Request pdf on the paleywiener theorem in the mellin transform setting in this paper we establish a version of the paleywiener theorem of fourier analysis in the frame of the mellin transform. A paleywiener theorem for the inverse fourier transform. It is shown how the algebra of radial parts of invariant. S j 1 2 2 1 2 2 x dx x f f d f therefore l r x f x c 2. The paleywiener theorem for the jacobi transform and the local huygens principle for root systems with even multiplicities by thomas branson a, gestur olafsson b and angela pasquale c a department of mathematics, university oflowa. The spherical paleywiener theorem 2651 we regard the lie algebra u of u as the subspace of gc given by u k.
For them our theorem 2 is a simple corollary of their paleywiener theorem, which is. The discrete paley wiener theorem 173 in its new form will be made to derive an analog of the cardinal series for d. Paleywiener theorems for the unspherical transform on the heisenberg group francesca astengo, bianca di blasio, fulvio ricci abstract. When the group gis complex the operator dis simple multiplication by a jacobian factor but otherwise it is quite. 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. Multipliers and a paleywiener theorem for real reductive groups, in representation theory of reductive groups, birkhauser, 1983, pp. As a corollary, we get relative analogs of the smooth and tempered bernstein centers rings of multipliers for spxq and cpxq. For the helgason fourier transform on s we prove the following analogue of hardys theorem. The following result is the stability theorem for hilbert frames. A comparison of paleywiener theorems for real reductive. A paleywiener theorem for distributions on reductive. Let n be a htype group and let sna be an one dimensional solvable extension of n. 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. We give a paleywiener theorem for weighted bergman spaces on the existence of such representa.
A paleywiener theorem for distributions on reductive symmetric spaces by e. The classical shannon sampling theorem is based on the paleywiener theorem. We prove several paleywienertype theorems related to the spherical transform on the gelfand pair h n. A paley wiener theorem for the inverse fourier transform on some homogeneous spaces thangavelu, s. We conjecture a geometrical form of the paley wiener theorem for the dunkl transform and prove three instances thereof, one of which involves a limit transition from opdams results for the graded hecke algebra. This chapter presents a paley wiener schwartz theorem in infinite dimensions. As was shown in 9 the operator is inevitable in characterising the image of the heat kernel transform. Theorem paleywiener for smooth functions if and then extends analytically to and for all nonnegative integers there exists a constant such that. Stability theorem for hilbert frames paleywienerkatochristensensuppose h aseparable complex hilbert space, i a countable index set and f f i i. Paleywiener theorems for the two spaces, characterizing them in terms of their spectral transforms. The main result of the paper is a generalization of the whittakershannon formula for reconstruction of a paleywiener function from its values on a discrete set.
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