Real Paley–Wiener theorems and local spectral radius formulas
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- by Nils Byrial Andersen and Marcel de Jeu PDF
- Trans. Amer. Math. Soc. 362 (2010), 3613-3640 Request permission
Abstract:
We systematically develop real Paley–Wiener theory for the Fourier transform on $\mathbb R^d$ for Schwartz functions, $L^p$-functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley–Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of Fourier type as well. An explanation is also given as to why the easily applied classical Paley–Wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull, whereas real Paley–Wiener theorems can be used to reconstruct the support precisely, albeit at the cost of combinatorial complexity. We indicate a possible application of real Paley–Wiener theory to partial differential equations in this vein, and furthermore we give evidence that a number of real Paley–Wiener results can be expected to have an interpretation as local spectral radius formulas. A comprehensive overview of the literature on real Paley–Wiener theory is included.References
- Luís Daniel Abreu, Real Paley-Wiener theorems for the Koornwinder-Swarttouw $q$-Hankel transform, J. Math. Anal. Appl. 334 (2007), no. 1, 223–231. MR 2332551, DOI 10.1016/j.jmaa.2006.12.050
- Ernst Albrecht and Werner J. Ricker, Local spectral properties of constant coefficient differential operators in $L^p(\textbf {R}^N)$, J. Operator Theory 24 (1990), no. 1, 85–103. MR 1086546
- Ernst Albrecht and Werner J. Ricker, Functional calculi and decomposability of unbounded multiplier operators in $L^p(\textbf {R}^N)$, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 1, 151–166. MR 1317333, DOI 10.1017/S0013091500006271
- E. Albrecht and W. J. Ricker, Local spectral properties of certain matrix differential operators in $L^p(\textbf {R}^N)^m$, J. Operator Theory 35 (1996), no. 1, 3–37. MR 1389641
- E. Albrecht and W. J. Ricker, On $p$-dependent local spectral properties of certain linear differential operators in $L^p(\mathbf R^N)$, Studia Math. 130 (1998), no. 1, 23–52. MR 1623000
- Fadhel A. Al-Musallam, A Whittaker transform over a half-line, Integral Transform. Spec. Funct. 12 (2001), no. 3, 201–212. MR 1872431, DOI 10.1080/10652460108819345
- F. Al-Musallam, The range of finite integral transforms arising from $n$-th order singular self-adjoint differential operators, Fract. Calc. Appl. Anal. 6 (2003), no. 2, 175–186. MR 2035413
- Fadhel Al-Musallam and Vu Kim Tuan, A modified and a finite index Weber transforms, Z. Anal. Anwendungen 21 (2002), no. 2, 315–334. MR 1915264, DOI 10.4171/ZAA/1080
- F. Al-Musallam and V. K. Tuan, A finite and an infinite Whittaker integral transform, Comput. Math. Appl. 46 (2003), no. 12, 1847–1859. MR 2018770, DOI 10.1016/S0898-1221(03)90241-0
- Nils Byrial Andersen, On real Paley-Wiener theorems for certain integral transforms, J. Math. Anal. Appl. 288 (2003), no. 1, 124–135. MR 2019749, DOI 10.1016/S0022-247X(03)00585-7
- Nils Byrial Andersen, Real Paley-Wiener theorems for the inverse Fourier transform on a Riemannian symmetric space, Pacific J. Math. 213 (2004), no. 1, 1–13. MR 2040247, DOI 10.2140/pjm.2004.213.1
- Nils Byrial Andersen, A simple proof of a Paley-Wiener type theorem for the Chébli-Trimèche transform, Publ. Math. Debrecen 64 (2004), no. 3-4, 473–479. MR 2058918
- Nils Byrial Andersen, Real Paley-Wiener theorems, Bull. London Math. Soc. 36 (2004), no. 4, 504–508. MR 2069012, DOI 10.1112/S0024609304003108
- Nils Byrial Andersen, On the range of the Chébli-Trimèche transform, Monatsh. Math. 144 (2005), no. 3, 193–201. MR 2130273, DOI 10.1007/s00605-004-0256-1
- Nils Byrial Andersen, Real Paley-Wiener theorems for the Hankel transform, J. Fourier Anal. Appl. 12 (2006), no. 1, 17–25. MR 2215674, DOI 10.1007/s00041-005-4056-3
- Nils Byrial Andersen, Real Paley-Wiener theorems for the Dunkl transform on $\Bbb R$, Integral Transforms Spec. Funct. 17 (2006), no. 8, 543–547. MR 2246499, DOI 10.1080/10652460500441551
- Nils Byrial Andersen and Marcel de Jeu, Elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line, Int. Math. Res. Not. 30 (2005), 1817–1831. MR 2172939, DOI 10.1155/IMRN.2005.1817
- N.B. Andersen, M.F.E. de Jeu, Local spectral radius formulas on compact Lie groups, J. Lie Theory 19 (2009), 223–230.
- Ha Huy Bang, A property of infinitely differentiable functions, Proc. Amer. Math. Soc. 108 (1990), no. 1, 73–76. MR 1024259, DOI 10.1090/S0002-9939-1990-1024259-9
- Ha Huy Bang, Remarks on a property of infinitely differentiable functions, Bull. Polish Acad. Sci. Math. 41 (1993), no. 3, 197–206 (1994). MR 1414766
- Ha Huy Bang, Functions with bounded spectrum, Trans. Amer. Math. Soc. 347 (1995), no. 3, 1067–1080. MR 1283539, DOI 10.1090/S0002-9947-1995-1283539-1
- Ha Huy Bang, A property of entire functions of exponential type, Analysis 15 (1995), no. 1, 17–23. MR 1322126, DOI 10.1524/anly.1995.15.1.17
- Ha Huy Bang, Asymptotic behaviour of the sequence of norms of derivatives, J. Math. Sci. Univ. Tokyo 2 (1995), no. 3, 611–620. MR 1382522
- Ha Huy Bang, On the Bernstein-Nikolsky inequality. II, Tokyo J. Math. 18 (1995), no. 1, 123–131. MR 1334710, DOI 10.3836/tjm/1270043613
- Kha Zuĭ Bang, On an algebra of pseudodifferential operators, Mat. Sb. 186 (1995), no. 7, 3–14 (Russian, with Russian summary); English transl., Sb. Math. 186 (1995), no. 7, 929–940. MR 1355452, DOI 10.1070/SM1995v186n07ABEH000049
- Kha Zuĭ Bang, The existence of a point spectral radius of pseudodifferential operators, Dokl. Akad. Nauk 348 (1996), no. 6, 740–742 (Russian). MR 1440738
- Ha Huy Bang, The Paley-Wiener-Schwartz theorem for nonconvex domains, Functional analysis and global analysis (Quezon City, 1996) Springer, Singapore, 1997, pp. 14–30. MR 1658037
- Kha Zuĭ Bang, Theorems of Paley-Wiener-Schwartz type, Tr. Mat. Inst. Steklova 214 (1997), no. Issled. po Teor. Differ. Funkts. Mnogikh Perem. i ee Prilozh. 17, 298–319 (Russian); English transl., Proc. Steklov Inst. Math. 3(214) (1996), 291–311. MR 1635074
- Kha Zuĭ Bang, Nonconvex cases of the Paley-Wiener-Schwartz theorem, Dokl. Akad. Nauk 354 (1997), no. 2, 165–168 (Russian). MR 1471836
- Kha Zuĭ Bang, The study of the properties of functions belonging to an Orlicz space depending on the geometry of their spectra, Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 2, 163–198 (Russian, with Russian summary); English transl., Izv. Math. 61 (1997), no. 2, 399–434. MR 1470148, DOI 10.1070/im1997v061n02ABEH000120
- Ha Huy Bang and Mai Thi Thu, A property of entire functions of exponential type for Lorentz spaces, Vietnam J. Math. 32 (2004), no. 2, 219–225. MR 2071019
- Ha Huy Bang and Mitsuo Morimoto, On the Bernstein-Nikolsky inequality, Tokyo J. Math. 14 (1991), no. 1, 231–238. MR 1108170, DOI 10.3836/tjm/1270130503
- Ha Huy Bang and Mitsuo Morimoto, The sequence of Luxemburg norms of derivatives, Tokyo J. Math. 17 (1994), no. 1, 141–147. MR 1279574, DOI 10.3836/tjm/1270128192
- J. J. Betancor, J. D. Betancor, and J. M. R. Méndez, Paley-Wiener type theorems for Chébli-Trimèche transforms, Publ. Math. Debrecen 60 (2002), no. 3-4, 347–358. MR 1898567
- Chirine Chettaoui and Khalifa Trimèche, New type Paley-Wiener theorems for the Dunkl transform on $\Bbb R$, Integral Transforms Spec. Funct. 14 (2003), no. 2, 97–115. MR 1969838, DOI 10.1080/10652460290029635
- C. Chettaoui, Y. Othmani, and K. Trimèchi, On the range of the Dunkl transform on $\Bbb R$, Math. Sci. Res. J. 8 (2004), no. 3, 85–103. MR 2108495
- Charles F. Dunkl, Hankel transforms associated to finite reflection groups, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 123–138. MR 1199124, DOI 10.1090/conm/138/1199124
- Ivan Erdélyi and Sheng Wang Wang, A local spectral theory for closed operators, London Mathematical Society Lecture Note Series, vol. 105, Cambridge University Press, Cambridge, 1985. MR 817715, DOI 10.1017/CBO9780511662249
- I. M. Gelfand and G. E. Schilow, Verallgemeinerte Funktionen (Distributionen). II: Lineare topologische Räume. Räume von Grundfunktionen und verallgemeinerten Funktionen, Hochschulbücher für Mathematik, Band 48, VEB Deutscher Verlag der Wissenschaften, Berlin, 1962 (German). MR 0149275
- L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Second edition. Springer-Verlag, Berlin, 1990.
- M. Jelassi and L. T. Rachdi, On the range of the Fourier transform associated with the spherical mean operator, Fract. Calc. Appl. Anal. 7 (2004), no. 4, 379–402. MR 2251523
- M. F. E. de Jeu, The Dunkl transform, Invent. Math. 113 (1993), no. 1, 147–162. MR 1223227, DOI 10.1007/BF01244305
- Marcel de Jeu, Paley-Wiener theorems for the Dunkl transform, Trans. Amer. Math. Soc. 358 (2006), no. 10, 4225–4250. MR 2231377, DOI 10.1090/S0002-9947-06-03960-2
- Kjeld B. Laursen and Michael M. Neumann, An introduction to local spectral theory, London Mathematical Society Monographs. New Series, vol. 20, The Clarendon Press, Oxford University Press, New York, 2000. MR 1747914
- Hatem Mejjaoli and Khalifa Trimèche, Spectrum of functions for the Dunkl transform on $\Bbb R^d$, Fract. Calc. Appl. Anal. 10 (2007), no. 1, 19–38. MR 2348864
- V. V. Napalkov, Uravneniya svertki v mnogomernykh prostranstvakh, “Nauka”, Moscow, 1982 (Russian). MR 678923
- Youssef Othmani and Khalifa Trimèche, Real Paley-Wiener theorems associated with the Weinstein operator, Mediterr. J. Math. 3 (2006), no. 1, 105–118. MR 2215575, DOI 10.1007/BF03339787
- Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019
- Isaac Pesenson, Sampling of Paley-Wiener functions on stratified groups, J. Fourier Anal. Appl. 4 (1998), no. 3, 271–281. MR 1650917, DOI 10.1007/BF02476027
- Isaac Pesenson, A sampling theorem on homogeneous manifolds, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4257–4269. MR 1707201, DOI 10.1090/S0002-9947-00-02592-7
- Isaac Pesenson, Sampling of band-limited vectors, J. Fourier Anal. Appl. 7 (2001), no. 1, 93–100. MR 1812998, DOI 10.1007/s00041-001-0007-9
- Isaac Pesenson, Deconvolution of band limited functions on non-compact symmetric spaces, Houston J. Math. 32 (2006), no. 1, 183–204. MR 2202361
- Isaac Pesenson, Analysis of band-limited functions on quantum graphs, Appl. Comput. Harmon. Anal. 21 (2006), no. 2, 230–244. MR 2259780, DOI 10.1016/j.acha.2006.02.003
- M. Plancherel, G. Pólya, Fonctions entières et intégrales de Fourier multiples. I, Comment. Math. Helv. 9 (1937), 224–248.
- Bebe Prunaru and Mihai Putinar, The generic local spectrum of any operator is the full spectrum, Bull. London Math. Soc. 31 (1999), no. 3, 332–336. MR 1673412, DOI 10.1112/S0024609398005608
- L. I. Ronkin, Introduction to the theory of entire functions of several variables, Translations of Mathematical Monographs, Vol. 44, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by Israel Program for Scientific Translations. MR 0346175
- L. I. Ronkin, Functions of completely regular growth, Mathematics and its Applications (Soviet Series), vol. 81, Kluwer Academic Publishers Group, Dordrecht, 1992. Translated from the Russian by A. Ronkin and I. Yedvabnik. MR 1196691, DOI 10.1007/978-94-011-2418-8
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- Martin Schechter, Spectra of partial differential operators, North-Holland Series in Applied Mathematics and Mechanics, Vol. 14, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1971. MR 0447834
- Sundaram Thangavelu and Yuan Xu, Convolution operator and maximal function for the Dunkl transform, J. Anal. Math. 97 (2005), 25–55. MR 2274972, DOI 10.1007/BF02807401
- François Trèves, Linear partial differential equations with constant coefficients: Existence, approximation and regularity of solutions, Mathematics and its Applications, Vol. 6, Gordon and Breach Science Publishers, New York-London-Paris, 1966. MR 0224958
- François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
- Truong Van Thuong, Some properties of functions with bounded spectrum, Acta Math. Vietnam. 24 (1999), no. 3, 343–352. MR 1735119
- V.K. Tuan, On the Paley–Wiener theorem, in Theory of Functions and Applications. Collection of Works dedicated to the Memory of M.M. Djrbashian, 193–196, Louys Publishing House, Yerevan, 1995.
- V.K. Tuan, Supports of functions and integral transforms, in Proceedings of International workshop on the recent advances in applied mathematics, RAAM ’96, 507-521, Kuwait Univ., Department of Mathematics and Computer Science, Kuwait, 1996.
- Vu Kim Tuan, On the range of the $Y$-transform, Bull. Austral. Math. Soc. 54 (1996), no. 2, 329–345. MR 1411543, DOI 10.1017/S0004972700017792
- Vu Kim Tuan, On the range of the Hankel and extended Hankel transforms, J. Math. Anal. Appl. 209 (1997), no. 2, 460–478. MR 1474619, DOI 10.1006/jmaa.1997.5351
- Vu Kim Tuan, Airy integral transform and the Paley-Wiener theorem, Transform methods & special functions, Varna ’96, Bulgarian Acad. Sci., Sofia, 1998, pp. 523–531. MR 1667774
- Vu Kim Tuan, New type Paley-Wiener theorems for the modified multidimensional Mellin transform, J. Fourier Anal. Appl. 4 (1998), no. 3, 317–328. MR 1650984, DOI 10.1007/BF02476030
- Vu Kim Tuan, Paley-Wiener-type theorems, Fract. Calc. Appl. Anal. 2 (1999), no. 2, 135–143. MR 1689181
- Vu Kim Tuan, On the supports of functions, Numer. Funct. Anal. Optim. 20 (1999), no. 3-4, 387–394. MR 1691371, DOI 10.1080/01630569908816899
- Vu Kim Tuan, A real-variable Paley-Wiener theorem for the Dunkl transform, Abstract and applied analysis, World Sci. Publ., River Edge, NJ, 2004, pp. 365–371. MR 2095112
- Vu Kim Tuan, Paley-Wiener and Boas theorems for singular Sturm-Liouville integral transforms, Adv. in Appl. Math. 29 (2002), no. 4, 563–580. MR 1943365, DOI 10.1016/S0196-8858(02)00032-5
- Vu Kim Tuan, Ali Ismail, and Megumi Saigo, Plancherel and Paley-Wiener theorems for an index integral transform, J. Korean Math. Soc. 37 (2000), no. 4, 545–563. MR 1770827
- Vu Kim Tuan, Ali Ismail, and Megumi Saigo, On an index integral transform involving the modified Bessel function, Integral Transform. Spec. Funct. 12 (2001), no. 4, 375–388. MR 1872376, DOI 10.1080/10652460108819359
- A. I. Zayed and V. K. Tuan, Paley-Wiener-type theorem for a class of integral transforms arising from a singular Dirac system, Z. Anal. Anwendungen 19 (2000), no. 3, 695–712. MR 1784126, DOI 10.4171/ZAA/975
- V. Kim Tuan and Ahmed I. Zayed, Generalization of a theorem of Boas to a class of integral transforms, Results Math. 38 (2000), no. 3-4, 362–376. MR 1799723, DOI 10.1007/BF03322017
- Vu Kim Tuan and Ahmed I. Zayed, Paley-Wiener-type theorems for a class of integral transforms, J. Math. Anal. Appl. 266 (2002), no. 1, 200–226. MR 1876778, DOI 10.1006/jmaa.2001.7740
- Florian-Horia Vasilescu, Analytic functional calculus and spectral decompositions, Mathematics and its Applications (East European Series), vol. 1, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1982. Translated from the Romanian. MR 690957
- F.-H. Vasilescu, Analytic operators and spectral decompositions, Indiana Univ. Math. J. 34 (1985), no. 4, 705–722. MR 808821, DOI 10.1512/iumj.1985.34.34037
- Pavla Vrbová, On local spectral properties of operators in Banach spaces, Czechoslovak Math. J. 23(98) (1973), 483–492. MR 322536
Additional Information
- Nils Byrial Andersen
- Affiliation: Mads Clausen Institute, University of Southern Denmark, Alsion 2, DK-6400 Sønderborg, Denmark
- Address at time of publication: Alssundgymnasiet Sønderborg, Grundtvigs Allé 86, 6400 Sønderborg, Denmark
- Email: byrial@mci.sdu.dk, nba@ags.dk
- Marcel de Jeu
- Affiliation: Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
- Email: mdejeu@math.leidenuniv.nl
- Received by editor(s): May 12, 2008
- Published electronically: February 15, 2010
- Additional Notes: The first author was supported by a research grant from the European Commission IHP Network: 2002–2006 Harmonic Analysis and Related Problems (Contract Number: HPRN-CT-2001-00273 - HARP)
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 3613-3640
- MSC (2010): Primary 42B10; Secondary 47A11
- DOI: https://doi.org/10.1090/S0002-9947-10-05044-0
- MathSciNet review: 2601602