Duals of Gelfand-Shilov spaces of type $ K\{M_p\} $ for the Hankel transformation

Samuel García-Baquerín; Isabel Marrero; Samuel García-Baquerín; Isabel Marrero

doi:10.3934/math.2024891

AIMS Mathematics

2024, Volume 9, Issue 7: 18247-18277. doi: 10.3934/math.2024891

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Duals of Gelfand-Shilov spaces of type $ K\{M_p\} $ for the Hankel transformation

Samuel García-Baquerín ¹,
Isabel Marrero ^{2
,
,}

1.
Escuela de Máster y Doctorado, Universidad de La Rioja, 2 Luis de Ulloa St, 26004 Logroño, La Rioja, Spain
2.
Departamento de Análisis Matemático and Instituto de Matemáticas y Aplicaciones (IMAULL), Universidad de La Laguna, P.O. Box 456, 38200 La Laguna (Tenerife), Canary Islands, Spain

Received: 14 January 2024 Revised: 20 May 2024 Accepted: 21 May 2024 Published: 31 May 2024
MSC : 46F05, 46F12

For $ \mu \geq-\frac{1}{2} $, and under appropriate conditions on the sequence $ \{M_p\}_{p = 0}^{\infty} $ of weights, the elements, the (weakly, weakly^*, strongly) bounded subsets, and the (weakly, weakly^*, strongly) convergent sequences in the dual of a space $ \mathcal{K}_\mu $ of type Hankel-$ K\{M_p\} $ can be represented by distributional derivatives of functions and measures in terms of iterated adjoints of the differential operator $ x^{-1} D_x $ and the Bessel operator $ S_\mu = x^{-\mu-\frac{1}{2}} D_x x^{2 \mu+1} D_x x^{-\mu-\frac{1}{2}} $. In this paper, such representations are compiled, and new ones involving adjoints of suitable iterations of the Zemanian differential operator $ N_\mu = $ $ x^{\mu+\frac{1}{2}} D_x x^{-\mu-\frac{1}{2}} $ are proved. Prior to this, new descriptions of the topology of the space $ \mathcal{K}_\mu $ are given in terms of the latter iterations.
- Bessel operator,
- boundedness,
- convergence,
- distribution,
- generalized function,
- $ K\{M_p\} $ space,
- representation,
- test function,
- Zemanian operator,
- Zemanian space
Citation: Samuel García-Baquerín, Isabel Marrero. Duals of Gelfand-Shilov spaces of type $ K\{M_p\} $ for the Hankel transformation[J]. AIMS Mathematics, 2024, 9(7): 18247-18277. doi: 10.3934/math.2024891

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Abstract

For $ \mu \geq-\frac{1}{2} $, and under appropriate conditions on the sequence $ \{M_p\}_{p = 0}^{\infty} $ of weights, the elements, the (weakly, weakly^*, strongly) bounded subsets, and the (weakly, weakly^*, strongly) convergent sequences in the dual of a space $ \mathcal{K}_\mu $ of type Hankel-$ K\{M_p\} $ can be represented by distributional derivatives of functions and measures in terms of iterated adjoints of the differential operator $ x^{-1} D_x $ and the Bessel operator $ S_\mu = x^{-\mu-\frac{1}{2}} D_x x^{2 \mu+1} D_x x^{-\mu-\frac{1}{2}} $. In this paper, such representations are compiled, and new ones involving adjoints of suitable iterations of the Zemanian differential operator $ N_\mu = $ $ x^{\mu+\frac{1}{2}} D_x x^{-\mu-\frac{1}{2}} $ are proved. Prior to this, new descriptions of the topology of the space $ \mathcal{K}_\mu $ are given in terms of the latter iterations.

References

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