Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 4 (2009), 239 -- 252

THE n-DIMENSIONAL CONTINUOUS WAVELET TRANSFORMATION ON GELFAND AND SHILOV TYPE SPACES

S. K. Upadhyay, R. N. Yadav and Lokenath Debnath

Abstract. In this paper the wavelet transformation on Gelfand and Shilov spaces of type WM(⬛n), WΩ(Δn) and WMΩ(Δn) is studied. It is shown that Wψφ : WM(⬛n) → WM(⬛n×⬛+n), Wψφ : WΩ(Δn) → WΩ(Δn×⬛+n) and Wψφ : WMΩ(Δn) → WMΩ(Δn×⬛+n) is linear and continuous where ⬛n and Δn are n-dimensional real numbers and complex numbers. A boundedness result in a generalized Sobolev space is derived.

2000 Mathematics Subject Classification: 42C40; 46F12.
Keywords: Continuous wavelet transformation; Sobolev space; Fourier transformation; W-spaces.

Full text

References

  1. L. Debnath, Wavelet Transforms and Their Applications, Birkhauser Verlag, Boston (2002). MR1877823(2002m:42001). Zbl 1019.94003.

  2. A. Friedman, Generalized Functions and Partial Differential Equations, Prentice Hall, Englewood Cliffs, N. J., 1963. MR0165388(29 #2672). Zbl 0116.07002.

  3. I. M. Gelfand and G. E. Shilov, Generalized Functions, Vol. 3., Academic Press, New York, 1967. MR0435833(55 #8786c).

  4. R. S. Pathak, The wavelet transform of distributions, Tohoku Math. J., 56 (2004) 411-421. MR1185272(94i:46051). Zbl 1078.42029.

  5. R. S. Pathak and S. K. Upadhyay, Wp-spaces and Fourier transformation, Proc. Amer. Math. Soc., 121:3 (1994) 733-738. MR1124154(92h:62104). Zbl 0816.46033.

  6. R. S. Pathak and Girish Pandey, Wavelet transform on spaces of type W, Rocky Mountain Journal of Mathematics, 39 (2009), no. 2, 619--631. MR2491156(2009k:42080). Zbl 1167.42012.




S. K. Upadhyay R. N. Yadav
Department of Applied Mathematics, Department of Mathematics and Statistics,
I. T. and C I M S, D S T, B. H. U., D. D. U. Gorakhpur University,
Varanasi - 221005, Gorakhpur,
India. India.
email: sk_upadhyay2001@yahoo.com

Lokenath Debnath
Department of Mathematics,
The University of Texas-Pan American,
1201 West University Drive,
Edinburg, 78539, USA.
e-mail: debnathl@utpa.edu

http://www.utgjiu.ro/math/sma