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As explained in Riesz & Sz.-Nagy (1990), every non-decreasing non-negative function F can be decomposed uniquely as a sum of a jump function f and a continuous monotone function g: the jump function f is constructed by using the jump data of the original monotone function F and it is easy to check that g = F − f is continuous and monotone. [10]
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 ...
A function that is not monotonic. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [1] [2] [3] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
In this way any monotone function can be written in a unique way as the sum of a continuous monotone function and a jump function. Since the formula for H ( x ) {\displaystyle H(x)} is a positive combination of characteristic functions, it is a uniformly convergent sum, so the analysis of Riesz & Sz.-Nagy (1990 , pp. 13–15) is particularly ...
In theoretical computer science, monotone dualization is a computational problem of constructing the dual of a monotone Boolean function.Equivalent problems can also be formulated as constructing the transversal hypergraph of a given hypergraph, of listing all minimal hitting sets of a family of sets, or of listing all minimal set covers of a family of sets.
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...
A function : defined on an interval is said to be operator monotone if whenever and are Hermitian matrices (of any size/dimensions) whose eigenvalues all belong to the domain of and whose difference is a positive semi-definite matrix, then necessarily () where () and () are the values of the matrix function induced by (which are matrices of the same size as and ).
SBV functions i.e. Special functions of Bounded Variation were introduced by Luigi Ambrosio and Ennio De Giorgi in the paper (Ambrosio & De Giorgi 1988), dealing with free discontinuity variational problems: given an open subset of , the space is a proper linear subspace of (), since the weak gradient of each function belonging to it ...