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On some applications of Duhamel operators

  • Ramiz Tapdigoglu EMAIL logo and Najla Altwaijry
Published/Copyright: October 16, 2022
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 72 Issue 5

Abstract

Let 𝔻 = {z ∈ C : |z| < 1} be the unit disk and Hol(𝔻 Γ— 𝔻) be the space of all holomorphic functions on the bi-disc 𝔻 Γ— 𝔻. We consider the double convolution operator 𝒦f on the subspace Holzw(𝔻 Γ— 𝔻) := {f ∈ Hol(𝔻 Γ— 𝔻) : f(z,w) = g(zw) for some g ∈ Hol(𝔻)} defined by

K f h ( z w ) = ( f βˆ— h ) ( z w ) := ∫ 0 z ∫ 0 w f ( ( z βˆ’ u ) ( w βˆ’ v ) ) h ( u v ) d v   d u .

We study extended eigenvalues of 𝒦f . We characterize extended eigen vectors of 𝒦f in terms of Duhamel operators. Moreover, we describe cyclic vectors of operator 𝒦f by applying the Duhamel product method.

Keywords: Duhamel product; Duhamel operator; holomorphic function; double convolution operator; double integration operator; extended eigenvalue; extended eigenvector; cyclic vector
MSC 2010: Primary 47A63

This work was supported by Researchers Supporting Project number RSP-2021/187 King Saud University, Riyadh, Saudi Arabia.

  1. (Communicated by Marek Balcerzak)

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Received: 2021-04-23
Accepted: 2021-10-23
Published Online: 2022-10-16
Published in Print: 2022-10-26

Β© 2022 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

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  2. Generalized hyperharmonic number sums with reciprocal binomial coefficients
  3. Gini index on generalized r-partitions
  4. Multiplicative functions of special type on Piatetski-Shapiro sequences
  5. Strengthenings of Young-type inequalities and the arithmetic geometric mean inequality
  6. Generalizations of the steffensen integral inequality for pseudo-integrals
  7. Subordination-implication problems concerning the nephroid starlikeness of analytic functions
  8. Boundedness and almost periodicity of solutions of linear differential systems
  9. On variational approaches for fractional differential equations
  10. Approximity of asymmetric metric spaces
  11. Approximation theorems for the new construction of BalΓ‘zs operators and its applications
  12. On Ξ·-biharmonic hypersurfaces in pseudo-Riemannian space forms
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  15. The Teissier-G family of distributions: Properties and applications
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  18. On two correlated linear models with common and different parameters
  19. On some applications of Duhamel operators
https://doi.org/10.1515/ms-2022-0093
Keywords for this article
Duhamel product; Duhamel operator; holomorphic function; double convolution operator; double integration operator; extended eigenvalue; extended eigenvector; cyclic vector

Articles in the same Issue

  1. Regular Papers
  2. Generalized hyperharmonic number sums with reciprocal binomial coefficients
  3. Gini index on generalized r-partitions
  4. Multiplicative functions of special type on Piatetski-Shapiro sequences
  5. Strengthenings of Young-type inequalities and the arithmetic geometric mean inequality
  6. Generalizations of the steffensen integral inequality for pseudo-integrals
  7. Subordination-implication problems concerning the nephroid starlikeness of analytic functions
  8. Boundedness and almost periodicity of solutions of linear differential systems
  9. On variational approaches for fractional differential equations
  10. Approximity of asymmetric metric spaces
  11. Approximation theorems for the new construction of BalΓ‘zs operators and its applications
  12. On Ξ·-biharmonic hypersurfaces in pseudo-Riemannian space forms
  13. Chen’s first inequality for hemi-slant warped products in nearly trans-Sasakian manifolds
  14. Induced mappings on symmetric products of Hausdorff spaces
  15. The Teissier-G family of distributions: Properties and applications
  16. A new extension of the beta generator of distributions
  17. A new family of compound exponentiated logarithmic distributions with applications to lifetime data
  18. On two correlated linear models with common and different parameters
  19. On some applications of Duhamel operators
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