Convolution

Convolution combines two functions by integrating or summing products of their shifted values. It describes how one function spreads, filters, or weights the other.

Convolution integral: An integral that combines two continuous functions by multiplying one with shifted copies of the other. This is the continuous formula behind the operation.
Linear time-invariant system: A system whose response is linear and unchanged by shifts in time. Its output is the convolution of its input with its impulse response.
Integral: A mathematical operation that accumulates quantities continuously over an interval or region. Continuous convolution accumulates products using an integral.
Cross-correlation: A measure of similarity between functions or sequences as one is shifted relative to the other. It resembles convolution but differs in how one operand is reversed or conjugated.
Discrete convolution: A sum of products formed from shifted sequences. It is the corresponding operation for sampled or indexed data.
Convolutional neural network: A neural network that applies learned local filters across structured data, especially images. Its core layers use discrete convolution-like operations to detect spatial patterns.
Summation: The addition of a sequence of terms, often indexed by integers. Discrete convolution accumulates products through a sum.
Pointwise multiplication: An operation that multiplies two functions' values at each matching input. Convolution combines values across shifts rather than only matching inputs.
Convolution theorem: A Fourier transform identity that turns convolution into multiplication and multiplication into convolution. It provides a faster route to many convolutions through the frequency domain.
Image convolution: An image-processing operation that combines pixel neighborhoods with a matrix of weights. It uses two-dimensional convolution for smoothing, sharpening, and edge detection.

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