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After an overview of key elements of functional analysis, this chapter is devoted to a specific topic in the theory of Hilbert spaces, namely reproducing kernel Hilbert spaces (RKHSs), also known as native spaces. This chapter provides the reader with all the tools needed to comprehend the key topics of this book, namely the design of control systems and machine learning methods based on native space theory. Initially, scalar-valued RKHSs are examined and their properties are discussed in detail and a tutorial manner. Successively, scalar-valued RKHSs are examined and special emphasis is given to highlighting connections with results for scalar-valued RKHSs. Two sets of results in RKHS theory are analyzed, namely how to correlate different RKHSs and how to bound projection errors in RKHSs.

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Elements of Native Space Theory

  • 章节

  • 发表日期：

    2025年

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  • Andrew J. Kurdila,
  • Andrea L’Afflitto,
  • John A. Burns

摘要

After an overview of key elements of functional analysis, this chapter is devoted to a specific topic in the theory of Hilbert spaces, namely reproducing kernel Hilbert spaces (RKHSs), also known as native spaces. This chapter provides the reader with all the tools needed to comprehend the key topics of this book, namely the design of control systems and machine learning methods based on native space theory. Initially, scalar-valued RKHSs are examined and their properties are discussed in detail and a tutorial manner. Successively, scalar-valued RKHSs are examined and special emphasis is given to highlighting connections with results for scalar-valued RKHSs. Two sets of results in RKHS theory are analyzed, namely how to correlate different RKHSs and how to bound projection errors in RKHSs.