Distribution Theory

Distributions

• Test Functions and Test Function Spaces
• Convergence in Test Function Spaces
• Distributions, Generalized Functions
  • Dirac Delta Function Precisely Defined as a Distribution
• All Locally Integrable Functions Can Be Extended to Distributions
  • Proof that the Dirac Delta Function is Not a Regular Distribution

Operators on Distributions

• Differentiation of Distributions
  • Weak Derivatives
• Translation of Distributions
• Dilation of Distributions
• Multiplication of Distributions by Smooth Functions
  • Differentiation of the Product of Distributions

Convergence, Convolution, Approximation

• Convergence of Distributions
• Differentiation of Distributions is Continuous with Respect to Weak Convergence
• Convolution of Distributions, Distributions as Functions Defined on the Real Numbers
• Auxiliary Lemma for Convolution of Distributions
• Convergence Theorem for Convolution of Distributions
• Distributions Converging to the Dirac Delta Distribution

Tempered Distributions

• Schwartz Space and Schwartz Functions
• Convergence in Schwartz Space
• Test Function Space is a Proper Subset of Schwartz Space
• Tempered Distributions
• Poisson Summation Formula

References

• Gerald B. Folland, Fourier Analysis and Its Applications (1992)
• Robert A. Adams and John J. F. Foutnier, Sobolev Space (2nd Edition, 2003)
• Daniel Eceizabarrena perez, Distribution Theory and Fundamental Solutions of Differential Operators (2015)