|
Absolutely And Completely Monotonic Functions And Sequences
In mathematics, the notions of an absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very strong monotonicity properties. Both types of functions have derivatives of all orders. In the case of an absolutely monotonic function, the function as well as its derivatives of all orders must be non-negative in its domain of definition which would imply that the function as well as its derivatives of all orders are monotonically increasing functions in the domain of definition. In the case of a completely monotonic function, the function and its derivatives must be alternately non-negative and non-positive in its domain of definition which would imply that function and its derivatives are alternately monotonically increasing and monotonically decreasing functions. Such functions were first studied by S. Bernshtein in 1914 and the terminology is also due to him. There are several other related notions like the concepts of alm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernstein's Theorem On Monotone Functions
In real analysis, a branch of mathematics, Bernstein's theorem states that every real number, real-valued function (mathematics), function on the half-line that is totally monotone is a mixture of exponential functions. In one important special case the mixture is a weighted average, or expected value. Total monotonicity (sometimes also ''complete monotonicity'') of a function means that is continuous function, continuous on , infinitely differentiable on , and satisfies (-1)^n \frac f(t) \geq 0 for all nonnegative integers and for all . Another convention puts the opposite inequality (mathematics), inequality in the above definition. The "weighted average" statement can be characterized thus: there is a non-negative finite Borel measure on with cumulative distribution function such that f(t) = \int_0^\infty e^ \, dg(x), the integral being a Riemann–Stieltjes integral. In more abstract language, the theorem characterises Laplace transforms of positive Borel measures on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |