Convolution
Smooth or combine functions.
Key idea
Smooth or combine functions.
In depth
(f ∗ g)(x) = ∫ f(y)g(x − y) dy blends two functions, with g flipped and slid. It models smoothing, filtering and sums of independent random variables. The Fourier transform converts it to multiplication.
Worked example
If X and Y are independent with densities f and g, X + Y has density f ∗ g.
Formula
Real-world uses
- Blurring and sharpening photos, applying audio reverb and running filters in signal processing
Common mistake
Forgetting to flip one function.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Convolution?
Smooth or combine functions. It is a topic at the expert level in Analysis & Topology.
What is the formula for Convolution?
The key formula is (f∗g)(x) = ∫ f(t)g(x−t)dt.
Where is Convolution used in real life?
Real-world examples: Blurring and sharpening photos, applying audio reverb and running filters in signal processing.
What should I learn before Convolution?
Make sure you are comfortable with Fourier Transform, the previous topic in Analysis & Topology.
What is a common mistake with Convolution?
Forgetting to flip one function.
How can I practise Convolution?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Convolution?
Move on to Sobolev Spaces Intro next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.