pistekohtainen

Pistekohtainen is a Finnish term used in mathematics and related fields to describe properties or operations that are defined or applied at each point in a domain individually, rather than globally or uniformly.

In mathematical analysis, pistekohtainen describes concepts defined by evaluating at individual inputs. A classic example is pointwise convergence: a sequence of functions {f_n} defined on a domain D converges pointwise to a function f if, for every x in D, the limit lim_{n→∞} f_n(x) = f(x). Pointwise operations follow the same principle, applied coordinate by coordinate: (f+g)(x) = f(x) + g(x) and (f·g)(x) = f(x) g(x) for each x in D.

Pointwise notions are contrasted with uniform notions. For instance, pointwise convergence is weaker than uniform convergence and can fail to preserve properties like continuity, whereas uniform convergence of continuous functions on a compact domain guarantees a continuous limit.

Beyond pure math, pistekohtainen is used to describe per-point or per-pixel processing in applied contexts such as image processing or sensor data analysis, where transformations are applied independently to each sample or pixel, for example per-point normalization or thresholding.

Etymology and usage: piste means point and kohta denotes a relation or aspect, yielding “pointwise” in Finnish mathematical language. Related terms include pistekohtainen konvergenssi (pointwise convergence) and contrasts with uniform convergence or almost everywhere convergence in broader analysis.