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In mathematics, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, ...
Díaz and Pariguan use these definitions to demonstrate a number of properties of the hypergeometric function. Although Díaz and Pariguan restrict these symbols to k > 0, the Pochhammer k-symbol as they define it is well-defined for all real k, and for negative k gives the falling factorial, while for k = 0 it reduces to the power x n.
Thus, to proceed with the appropriate analysis, it suffices to bound the function of interest with continuous nonincreasing positive definite functions. In other words, when a function belongs to the ( K ∞ {\displaystyle {\mathcal {K}}_{\infty }} ) it means that the function is radially unbounded.
A bump function is a smooth function with compact support.. In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (differentiability class) it has over its domain.
A partial function from X to Y is thus a ordinary function that has as its domain a subset of X called the domain of definition of the function. If the domain of definition equals X, one often says that the partial function is a total function.
then f is called K-bilipschitz (also written K-bi-Lipschitz). We say f is bilipschitz or bi-Lipschitz to mean there exists such a K. A bilipschitz mapping is injective, and is in fact a homeomorphism onto its image. A bilipschitz function is the same thing as an injective Lipschitz function whose inverse function is also Lipschitz.
A typical example of a homogeneous function of degree k is the function defined by a homogeneous polynomial of degree k. The rational function defined by the quotient of two homogeneous polynomials is a homogeneous function; its degree is the difference of the degrees of the numerator and the denominator; its cone of definition is the linear ...
The first requirement ensures that the method of kernel density estimation results in a probability density function. The second requirement ensures that the average of the corresponding distribution is equal to that of the sample used. If K is a kernel, then so is the function K* defined by K*(u) = λK(λu), where λ > 0. This can be used to ...