Konvolvera

Konvolvera is a verb used primarily in mathematics and signal processing to describe performing a convolution operation on two functions or sequences. The term is used in several European languages, including Swedish, where konvolvera is used for this action. It shares roots with the noun 'konvolution' and the Latin 'convolvere', meaning to roll together.

In continuous time, the convolution of two functions f(t) and g(t) is (f * g)(t) = ∫_{-∞}^{∞} f(τ) g(t-τ) dτ, provided the integral exists. In discrete time, (f * g)[n] = ∑_{k=-∞}^{∞} f[k] g[n-k]. Convolution is commutative for many common function classes, i.e., f * g = g * f, and it is associative.

Applications: Convolution is foundational in linear time-invariant systems, acting as a filter with impulse response h(t) or h[n]. In image processing, convolving with a kernel blurs, sharpens, or detects features. In probability, the convolution of density functions yields the distribution of the sum of independent random variables. In deep learning, convolutional layers implement discrete convolutions to extract features.

Computational aspects: Convolution can be computed directly or efficiently via the Fast Fourier Transform (FFT), especially for long sequences. In digital signal processing, zero-padding is used to prevent circular convolution; kernel separability reduces complexity by breaking a 2D kernel into a product of 1D kernels.

See also: convolution (mathematics).