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Area of Science:

  • Real Analysis
  • Functional Analysis
  • Metric Space Theory

Background:

  • The concept of function smoothness is central to analysis.
  • Lipschitz seminorms provide a quantitative measure of smoothness.
  • Bounded variation (BV) is a classical notion of smoothness for functions on the real line.

Purpose of the Study:

  • To introduce and analyze novel notions of average smoothness for real-valued functions.
  • To compare these new average smoothness notions with the established concept of bounded variation.
  • To investigate the combinatorial properties of the newly defined function classes.

Main Methods:

  • Definition of weak and strong average-Lipschitz seminorms on general metric spaces.
  • Specialization of these seminorms to the standard metric on the real line.
  • Comparison with the class of functions of bounded variation (BV).
  • Utilizing the fat-shattering dimension to analyze combinatorial properties.

Main Results:

  • The weak average-Lipschitz seminorm is strictly weaker than bounded variation.
  • The strong average-Lipschitz seminorm is strictly stronger than bounded variation.
  • The class of functions exhibiting weak average smoothness is combinatorially larger than BV functions, as quantified by the fat-shattering dimension.

Conclusions:

  • The newly introduced average smoothness notions offer a refined perspective on function regularity.
  • These notions provide a richer structure for function spaces compared to bounded variation.
  • The combinatorial analysis reveals significant differences in the size and complexity of function classes.