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Distributional convolutors for Fourier transform. (English) Zbl 1126.46028

A distributional Fourier analysis was developed in a series of six papers by K.B.Howell [J. Math.Anal.Appl.168, No.2, 342–350 (1992; Zbl 0776.42012); ibid.173, No.2, 419–429 (1993; Zbl 0776.42013); ibid.175, No.1, 257–267 (1993; Zbl 0776.42014); ibid.180, No.1, 79–92 (1993; Zbl 0811.42004); ibid.187, No.2, 567–582 (1994; Zbl 0819.46028); ibid.193, No.3, 832–838 (1995; Zbl 0869.46017)]. The work done by the authors in the present paper complements this work, i.e., it fills up some gaps in the work done by Howell. This is exemplified by their Theorem 3.1, amongst many other theorems stated and proved; it states the following. Let \(m\in \mathbb Z\), \(m<0\), and \(T\in {\mathcal O}'_{c,G,m}\). Then \[ T(x) = \lim_{r\to\infty} \int^{+r}_{-r} F(T)(z)e^{-ixz}\,dz \] in the sense of convergence in \(G'\). Here, \(G\) and \({\mathcal O}_{c,G,m}\), are Fréchet spaces and \({\mathcal O}'_{c,G,m}\) and \(G'\) are their corresponding dual spaces, respectively.

MSC:

46F12 Integral transforms in distribution spaces
46F10 Operations with distributions and generalized functions
42A85 Convolution, factorization for one variable harmonic analysis
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References:

[1] González, B. J.; Negrín, E., Convolution over the spaces \(S_k^\prime \), J. Math. Anal. Appl., 190, 829-843 (1995) · Zbl 0827.46035
[2] González, B. J.; Negrín, E., Fourier transform over the spaces \(S_k^\prime \), J. Math. Anal. Appl., 194, 780-798 (1995) · Zbl 0867.46032
[3] Howell, K. B., A new theory for Fourier analysis. Part I. The space of test functions, J. Math. Anal. Appl., 168, 342-350 (1992) · Zbl 0776.42012
[4] Howell, K. B., A new theory for Fourier analysis. II. Further analysis on the space of test functions, J. Math. Anal. Appl., 173, 419-429 (1993) · Zbl 0776.42013
[5] Howell, K. B., A new theory for Fourier analysis. Part III. Basic analysis on the dual, J. Math. Anal. Appl., 175, 257-267 (1993) · Zbl 0776.42014
[6] Howell, K. B., A new theory for Fourier analysis. Part IV. Basic multiplication and convolution on the dual space, J. Math. Anal. Appl., 180, 79-92 (1993) · Zbl 0811.42004
[7] Howell, K. B., A new theory for Fourier analysis. V. Generalized multiplication and convolution on the dual space, J. Math. Anal. Appl., 187, 567-582 (1994) · Zbl 0819.46028
[8] Howell, K. B., Exponential bounds on elementary multipliers of generalized functions, J. Math. Anal. Appl., 193, 832-838 (1995) · Zbl 0869.46017
[9] Schwartz, L., Théorie des distributions (1978), Hermann: Hermann Paris
[10] Trèves, F., Topological Vector Spaces, Distributions and Kernels (1967), Academic Press: Academic Press New York · Zbl 0171.10402
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