astmefunktsioonid

Astmefunktsioonid, also known as step functions, are mathematical functions that exhibit a constant value over specific intervals, and then abruptly jump to a new constant value at certain points. The most basic example is the Heaviside step function, often denoted as H(x) or θ(x). This function is typically defined as 0 for x < 0 and 1 for x ≥ 0. Other variations exist, with different values assigned at the jump point.

These functions are fundamental in various fields of mathematics and engineering. In signal processing, they are used to model sudden changes or the onset of signals. For instance, turning on a switch can be represented by a step function. In probability theory, the cumulative distribution function of a discrete random variable is a form of step function.

Mathematically, step functions can be expressed using set notation or piecewise definitions. They are also important in calculus, particularly in the context of integration. The integral of a step function over an interval corresponds to the area under the function within that interval. The Fourier transform of a step function is also a well-studied concept, revealing its frequency domain characteristics. Their discontinuous nature at specific points makes them a subject of interest in analysis.