Fourier Transform

Contents:

• Important Examples
  • General Tips
  • Rectangular Function
  • Triangle Function
  • Gaussian function
• Duality
  • Exploration
  • Formulas
  • Application
• Operations
  • Linear Combination
  • Shift
  • Stretch
  • Differentiation
  • Convolution
• Convolution: Introduction
  • Interpreted by Filtering
  • Differentiability
• Convolution: Cooling of a Rod
  • Problem
  • Solution
• Convolution: Central Limit Theorem
  • Gaussian Curve
  • Sum of Random Variables
  • Central Limit Theorem
• Generalized Fourier Transform: Schwartz Space
  • Lemma
  • Differentiation Formulas
  • Closure under Fourier Transform
  • Parseval’s Theorem of Fourier Transform
• Generalized Fourier Transform: Tempered Distribution
  • Delta Distribution
  • Function Induced Distribution
  • Fourier Transform of Distribution
  • Derivitive of Distribution
  • Multiplication of Distribution
  • Convolution of Distribution
• Generalized Fourier Transform: Delta Distribution
  • Sampling Property of Delta
  • Convolution Property of Delta
  • Scaling Property of Delta
• Application One: Diffraction
  • Problem Introduction
  • Solutioin

Reference

1. The Fourier Transform and its Applications

2. Tempered Distributions

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