Composition of Pseudo-Differential Operators Associated with Jacobi Differential Operator

Research Article
  • 14 Downloads

Abstract

Using inverse Fourier–Jacobi transform two symbols are defined and two pseudo-differential operators (p.d.o.’s) \(\mathcal {P}_{\alpha ,\beta }(x,D) \) and \(\mathsf {Q}_{\alpha ,\beta } (x,D) \) are introduced. Composition of \(\mathcal {P}_{\alpha ,\beta }(x,D) \) and \(\mathsf {Q}_{\alpha ,\beta } (x,D) \) is defined. It is shown that the p.d.o.’s and composition of p.d.o.’s are bounded in a Sobolev type space. Some special cases are discussed.

Keywords

Pseudo-differential operator Fourier–Jacobi differential operators Jacobi function Fourier–Jacobi Convolution 

Mathematics Subject Classification

46F12 

Notes

Acknowledgements

The first author is supported by NBHM, Govt. of India, under Grant No. 2/48(14)/2011/R&D II/3501.

References

  1. 1.
    Flensted-Jensen M (1972) Paley–Wiener type theorems for a differential operator connected with symmetric spaces. Ark Math 10:143–162MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Flensted-Jensen M, Koornwinder TH (1973) The convolution structure for Jacobi function expansions. Ark Math 11:245–262MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Flensted-Jensen M, Koornwinder TH (1979) Jacobi functions: the addition formula and the positivity of the dual convolution structure. Ark Math 17:139–151MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Salem NB (1994) Convolution semigroups and central limit theorem associated with a dual convolution structure. J Theor Probab 7:417–436MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Salem NB, Dachraoui A (1998) Pseudo-differential operators associated with the Jacobi differential operator. J Math Anal Appl 220:365–381MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Salem NB, Dachraoui A (2000) Sobolev type spaces associated with Jacobi differential operators. Integral Transforms Spec Funct 9:163–184MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Betancor JJ, Betancor JD, Méndez JMR (2004) Hypoelliptic Jacobi convolution operators on Schwartz distributions. Positivity 8:407–422MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Pathak RS, Pathak S (2000) Certain pseudo-differential operator associated with the Bessel operator. Indian J Pure Appl Math 31:309–317MathSciNetMATHGoogle Scholar
  9. 9.
    Pathak RS, Pathak S (2002) Product of pseudo-differential operators involving Hankel convolution. Indian J Pure Appl Math 33:367–378MathSciNetMATHGoogle Scholar
  10. 10.
    Zaidman S (1991) Distributions and pseudo-differential operators. Longman, EssexMATHGoogle Scholar
  11. 11.
    Prasad A, Singh MK (2015) Pseudo-differential operators associated with the Jacobi differential operator and Fourier-cosine wavelet transform. Asian Eur J Math 8(1): Article ID:1550010, 16 ppGoogle Scholar
  12. 12.
    Wong MW (2014) An introduction to pseudo-differential operators. World Scientific Publishing, SingaporeCrossRefMATHGoogle Scholar
  13. 13.
    Prasad A, Kumar M (2011) Product of two generalized pseudo-differential operators involving fractional Fourier transform. J Pseudo Differ Oper Appl 2(3):355–365MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Prasad A, Kumar P (2016) Composition of pseudo-differential operators associated with fractional Hankel–Clifford integral transformations. Appl Anal 95(8):1792–1807MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia
  2. 2.Department of Mathematics, St. Xavier’s CollegeRanchi UniversityRanchiIndia

Personalised recommendations