## [Convolution](/convolution-0) Definition A binary operation that combines two functions (or sequences) into a third by integrating (or summing) the product of one function with a shifted copy of the other: (f * g)(t) = ∫ f(τ) g(t−τ) dτ (or discrete analogue). ## Principle Principle Translation-invariant linear aggregation: convolution synthesizes the response of a linear, shift-invariant system by superposing shifted, weighted contributions of an input. ## Demonstration Demonstration Smoothing a noisy time series by convolving it with a narrow, positive kernel reduces high-frequency fluctuations and produces a locally averaged signal. ## Misapplication Misapplication Treating convolution as pointwise multiplication in the original domain; confusing convolution with simple multiplication or with correlation without accounting for the shift and sign convention. ## Consequence Consequence Convolution with suitable kernels yields filtering, smoothing, and linear system responses; repeated convolutions affect spread and regularity of signals or probability densities. ## Reversal Reversal Deconvolution seeks to recover an input given an output and a kernel, and is typically ill-posed and sensitive to noise, contrasting with the stabilizing averaging effect of convolution. ## Boundary Boundary Defined when integrals or sums converge (e.g., functions in L^1 or sequences summable); not all function pairs admit a well-defined convolution and care is needed with distributions or unbounded domains. ## Semantic Tension Semantic Tension Often contrasted with correlation: correlation measures similarity under shift, while convolution inserts a reversal or sign convention in the shift; both are related but serve different inferential roles. ## Synthesis Synthesis Convolution is the translation-invariant linear combination of one function with shifted copies of another that implements filtering, accumulation, or system response when integrability conditions permit.