Abstract
R. Boas obtained a characterization of the class of all square integrable functions vanishing on some interval of the form [−σ, σ] in terms of the behavior of its Fourier transform. An analogue of Boas’ theorem is obtained. A characterization of the image of the space of square integrable functions vanishing on some interval under various integral transformations is derived. The class of integral transformations considered is related to singular Sturm-Liouville boundary-value problems on a half line.
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References
Bang H.H. A property of infinitely differentiable functions. Proc. Amer. Math. Soc. 108 (1990), pp.73–76.
Boas R. Some theorems on Fourier transforms and conjugate trigonometric integrals. Trans. Amer. Math. Soc. 40(1936), pp. 287–308.
Levitan B.M. Eigenfunction Expansions (Russian). Moscow-Leningrad, 1950.
Levitan B.M. and Sargsjan I.S. Sturm-Liouville and Dirac Operators. Kluwer Acad. Publ., Dordrecht, 1991.
Paley R.E.A.C. and Wiener N. Fourier Transforms in the Complex Domain. Colloq. Publ. Amer. Math. Soc., 1934.
Titchmarsh E.C. Eigenfunction Expansions Associated with Second-Order Differential Equations. Part 1. Clarendon Press, Oxford, 1962.
Vu Kim Tuan. Supports of functions and integral transforms. Proceedings of International Workshop on the Recent Advances in Applied Mathematics (RAAM’ 96), held in Kuwait, May 4–7, 1996, pp. 507–521.
Vu Kim Tuan. On the range of the Y-transform. Bull Austral. Math. Soc. 54(1996), pp. 329–345.
Vu Kim Tuan. On the range of the Hankel and extended Hankel transforms. J. Math. Anal. Appl. 209(1997), pp. 460–478.
Vu Kim Tuan. New type Paley-Wiener theorems for the modified multidimensional Mellin transform. J. Fourier Anal Appl. 4(1998), no. 3., pp. 317–328.
Vu Kim Tuan. Airy integral transform and the Paley-Wiener theorem. Proceedings of the 2nd International Workshop on Transform Methods and Special Functions, Varna, Bulgaria, 23–30 August 1996. Eds.: Rusev D., Dimovski I., and Kiryakova V., IMI-BAS, Sofia, 1998, pp. 523–531.
Vu Kim Tuan and Zayed A.I. Paley-Wiener-type theorems for a class of integral transforms (submitted).
Zayed A.I. Function and Generalized Function Transformations. CRC Press, Boca Raton, Florida, 1996.
Zayed A.I. Advances in Shannon’s Sampling Theory. CRC Press, Boca Raton, Florida, 1993.
Zayed A.I. and Walter G. On the inversion of integral transforms associated with Sturm-Liouville problems. J. Math. Anal. Appl. 164 (1992), pp. 285–306.
Zayed A.I. On integral transforms whose kernels are solutions of singular Sturm-Liouville problems. Proc. Royal Soc. Edinburgh 108A (1988), pp. 201–228.
Zayed A.I. Real singularities of Sturm-Liouville expansions. SIAM J. Math. Anal. 18 (1987), pp. 219–227.
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Tuan, V.K., Zayed, A.I. Generalization of a Theorem of Boas to a Class of Integral Transforms. Results. Math. 38, 362–376 (2000). https://doi.org/10.1007/BF03322017
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DOI: https://doi.org/10.1007/BF03322017