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A function in the Schwartz space is sometimes called a Schwartz function. A two-dimensional Gaussian function is an example of a rapidly decreasing function. Schwartz space is named after French mathematician Laurent Schwartz.
A function that is absolutely monotonic on [,) can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line. The big Bernshtein theorem : A function f ( x ) {\displaystyle f(x)} that is absolutely monotonic on ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} can be ...
Similarly, a regularly varying function is a function of a real variable whose behaviour at infinity is similar to the behaviour of a power law function (like a polynomial) near infinity. These classes of functions were both introduced by Jovan Karamata , [ 1 ] [ 2 ] and have found several important applications, for example in probability theory .
A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
It can be proved that a real function is of bounded variation in [,] if and only if it can be written as the difference = of two non-decreasing functions and on [,]: this result is known as the Jordan decomposition of a function and it is related to the Jordan decomposition of a measure.
Here are 10 common benefits of decreasing term insurance to consider: Cost-effective: Premiums for decreasing term insurance are generally lower compared to other types of life insurance, making ...
Then f is a non-decreasing function on [a, b], which is continuous except for jump discontinuities at x n for n ≥ 1. In the case of finitely many jump discontinuities, f is a step function. The examples above are generalised step functions; they are very special cases of what are called jump functions or saltus-functions. [8] [9]