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In this chapter, we first define the Mellin transform in the Hilbert spaceSpaceHilbert \(X_c^2\) . Then, using the Riesz–Thorin interpolation theoremTheoremRiesz–Thorin (Theorem 1.22 ), we extend the definition to the spaces \(X^p_c\) for \(1 < p < 2.\) The case \(p>2\) can be treated within the theory of distributions. However, we will not cover this topic in the present book but give a brief outline in the notes.

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The Mellin Transform in Spaces $$X_c^p$$ for $$1

  • 章节

  • 发表日期：

    2025年

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  • Carlo Bardaro,
  • Paul L. Butzer,
  • Ilaria Mantellini,
  • Gerhard Schmeisser

摘要

In this chapter, we first define the Mellin transform in the Hilbert spaceSpaceHilbert \(X_c^2\) . Then, using the Riesz–Thorin interpolation theoremTheoremRiesz–Thorin (Theorem 1.22 ), we extend the definition to the spaces \(X^p_c\) for \(1 < p < 2.\) The case \(p>2\) can be treated within the theory of distributions. However, we will not cover this topic in the present book but give a brief outline in the notes.