Continuous wavelet transform of Schwartz distributions. (English) Zbl 1428.42074
Summary: In this paper we extend the continuous wavelet transform to Schwartz distributions and derive the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak distributional sense. But the uniqueness theorem for our wavelet inversion formula is valid for the space \(\mathcal{D}'_F\) obtained by filtering (deleting) (i) all non-zero constant distributions from the space \(\mathcal{D}'\), (ii) all non-zero constants that appear with a distribution as a union, as for example, for \({x^2}/(1+x^2) = 1-1/(1+x^2), 1\) is deleted and \(-1/(1+x^2)\) is retained. The kernel of our wavelet transform is an element of \(\mathcal{D}\) which when integrated along the real line vanishes, but none of its moments of order \(m\ge 1\) along the real line is zero. The set of such kernels will be denoted by \(D_m\).
MSC:
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 46F12 | Integral transforms in distribution spaces |
| 46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
| 46F10 | Operations with distributions and generalized functions |
Keywords:
wavelet transform; continuous wavelet transform; window functions; integral transform of generalized functions; Schwartz distributionsReferences:
| [1] | N.I. Akhiezer and I.M. Glazman, Linear operators in Hilbert space, Ungar Publishing Company, New York (1966). · Zbl 0098.30702 |
| [2] | A. Bogess and F.J. Narcowich, A first course in wavelets with Fourier analysis, Prentice Hall, New Jersey (2001). · Zbl 1177.42001 |
| [3] | C.K. Chui, Introduction to wavelets, Academic Press (1992). · Zbl 0925.42016 |
| [4] | I. Daubechies, Ten lectures on wavelets, Mathematics Department, CBMS-NSF Regional Conf. Series in Appl. Math., SIAM, Philadelphia (1992). · Zbl 0776.42018 |
| [5] | L. Ehrenpreis, Analytic functions and the Fourier transform of distributions, J. Ann. Math. 63 (1956), 129-159. · Zbl 0139.08004 · doi:10.2307/1969993 |
| [6] | I.M. Gelfand and G.E. Shilov, Generalized functions, vol. 1, Academic Press, New York (1964). Chapter 2. · Zbl 0115.33101 |
| [7] | I.M. Gelfand and G.E. Shilov, vol. 2, Academic Press, New York (1965). Chapter 3. |
| [8] | B. Han, Pairs of frequency-based nonhomogeneous dual wavelet frames in the distribution space, Appl. Comp. Harmonic Anal. 29 (2010), no. 3, 330-353. · Zbl 1197.42021 · doi:10.1016/j.acha.2010.01.004 |
| [9] | B. Han, Nonhomogeneous wavelet systems in high dimensions, Appl. Comp. Harmonic Anal. 32 (2012), no. 2, 169-196. · Zbl 1241.42028 · doi:10.1016/j.acha.2011.04.002 |
| [10] | I.M. Holschneider, Wavelets an analysis tool, Oxford Science Publications, Clarendon Press (1995). · Zbl 0874.42020 |
| [11] | F. Keinert, Wavelets and multiwavelets, Chapman and Hall/CRC (2003). |
| [12] | E. Kreyszig, Advanced engineering mathematics, 2nd ed. (1968). · Zbl 0103.27803 |
| [13] | Y. Meyer, Wavelets and multiwavelets, Cambridge Studies in Advanced Mathematics 37 (1992). · Zbl 0776.42019 |
| [14] | J. Navarro, Singularities of distributions via the wavelet transform, SIAM J. Math. Anal. 30 (1999), no. 2, 454-467. · Zbl 0968.42024 · doi:10.1137/S0036141095292494 |
| [15] | J.N. Pandey, Shannon wavelet series expansion of a class of generalized functions, J. Wavelet Theory Appl. 1 (2007), no. 7, 65-81. |
| [16] | J.N. Pandey, The Hilbert transform of Schwartz distributions and applications, Wiley (1996). · Zbl 0844.46022 |
| [17] | J.N. Pandey and S.K. Upadhyay, The continuous wavelet transform and window functions, Proc. Amer. Math. Soc. 143 (2015), no. 11, 4750-4763. · Zbl 1338.46052 · doi:10.1090/proc/12590 |
| [18] | J.N. Pandey and S.K. Upadhyay, Continuous wavelet transform of Schwartz tempered distribution I, Cogent Math. Stat., 6 (2019), no. 1, Art. ID 1623647. · Zbl 1427.46026 |
| [19] | J.N. Pandey, N.K. Jha and O.P. Singh, The continuous wavelet transform in \(n\)-dimensions, Int. J. Wavelets Multiresolution Inf. Process. 14 (2016), no. 5, art. id. 1650037, 13 pp. · Zbl 1350.42054 |
| [20] | J.N. Pandey, J.S. Maurya, S.K. Upadhyay and H.M. Srivastava, Continuous wavelet transform of Schwartz tempered distributions in \(S'(R^n)\), Symmetry 11 (2019), art. id. 235, 8 pp. · Zbl 1416.46043 |
| [21] | R.S. Pathak, The wavelet transform, Atlantis Press, Paris, and World Scientific, Hackensack, NJ (2009). · Zbl 1192.44004 |
| [22] | E.B. Postnikov, E.A. Lebedeva and A.I. Lavrova, Computational implementation of inverse continuous wavelet transform without a requirement of admissibility conditions, Appl. Math. Comp. 282 (2016), 158-136. · Zbl 1410.42033 · doi:10.1016/j.amc.2016.02.013 |
| [23] | M.H. Protter and C.B. Morrey Jr., Modern mathematical analysis, Addison-Wesley (1964). · Zbl 0135.25603 |
| [24] | H.H. Schaefer, Topological vector spaces, Springer-Verlag (1971). · Zbl 0217.16002 |
| [25] | L. Schwartz, Théorie des distributions, Hermann, Paris (1966). · Zbl 0149.09501 |
| [26] | G.E. Shilov, Generalized functions and partial differential equations, Gordon and Breach, New York (1968). · Zbl 0177.36302 |
| [27] | G. Strang and T. Nyguyen, Wavelets and filter banks, Wellesley-Cambridge Press (1997). |
| [28] | F. Trèves, Topological vector spaces, distributions and kernels, Academic Press (1966). · Zbl 1111.46001 |
| [29] | V.S. Vladimirov, Generalized functions in mathematical physics, Mir, Moscow (1979). · Zbl 0515.46033 |
| [30] | G.G. Walter and X. Shen, Wavelets and other orthogonal systems, 2nd ed., Chapman and Hall/CRC (2009). · Zbl 1005.42018 |
| [31] | F. Weisz, Convergence of the inverse continuous wavelet transform in Wiener amalgam spaces, Analysis 35 (2015), no. 1, 33-46. · Zbl 1316.42045 · doi:10.1515/anly-2014-1267 |
| [32] | F. Weisz, Inversion formulas for the continuous wavelet transform, Acta. Math. Hungar. 138 (2013), no. 3, 237-258. · Zbl 1289.42096 · doi:10.1007/s10474-012-0263-y |
| [33] | F. Weisz, Inverse continuous wavelet transform in Pringsheim’s sense in Wiener amalgam spaces, Acta Math. Hungar. \bf145, no. 2, 392-415. · Zbl 1363.42078 · doi:10.1007/s10474-015-0492-y |
| [34] | A.H. Zemanian, Distribution theory and transform analysis, McGraw-Hill (1965). · Zbl 0127.07201 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
