Completely monotonic functions

By K. Miller, S. Samko. 2001

357 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/e15e7570eab957f69ae49635954ac9d3f8b89430

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

A class of Matérn-like covariance functions for smooth processes on a sphere (2015, unknown): “According to Miller and Samko (2001) and Gneiting (2013), the Matérn class using great circle distance is valid if and only if the smoothness parameter satisfies 0 ν ≤ 0.5 for 1, 2, and 3 dimensional spheres, and thus it is not mean square differentiable at the origin.”
Strictly and non-strictly positive definite functions on spheres (2011, background): “By Theorem 1 of Miller and Samko (2001) the function φ0(t) = exp(−t)φ1(t) = tνKν(t) is completely monotone, too.”
Choice Probability Generating Functions (2012, unknown): “Examples of completely monotonic functions (see Miller and Samko, 2001) are: • ln(α + βt), with α + βt > 0 and β > 0; • tα, with t ≥ 0 and 0 ≤ α ≤ 1; • −e−αt, with α ≥ 0; • −1/t, with t < 0.”
Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems (2011, methodology): “As for the proof, see Miller and Samko [32], Schneider [47], and also see Pollard [38].”
On Turán type inequalities for modified Bessel functions (2010, background): “functions is given in Widder’s book [36] (see also [29]).”
Spatio-temporal variograms and covariance models (2005, background): “We refer the reader to Miller and Samko (2001) for a recent expository survey of properties of completely monotone functions, as well as various examples.”
AN EXTENSION OF A THEOREM OF SCHOENBERG TO PRODUCTS OF SPHERES (2015, unknown): “Additional examples can be found in [15].”
Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density (2013, background): “For the proofs, more properties, and applications of these functions we refer to [12], [29] and [33].”
Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere (2016, methodology): “Table S1 of the OS) is restricted to the interval (0, 1/2], but it is carefully explained in Gneting (2013), on the basis of the arguments inMiller and Samko (2001), that such a family is not completely monotone for ν > 1/2. Table S2 of the OS reports some possible choices for the function ψ (to be ”
Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics (2011, methodology): “For the Mittag-Leffler functions in one and two-order parameters the conditions to be CM were proved respectively by Pollard [25] in 1948 and by Schneider [27] in 1996, see also Miller and Samko [21, 22] for further details.”
Representation of solutions and large-time behavior for fully nonlocal diffusion equations (2015, unknown): “But since the functions x 7→ Eα,1(−x), x 7→ Eα,α(−x) and x 7→ cx β 2 −1 with c ≥ 0 and β ≤ 2 are known to be completely monotone on R+ [35], we obtain the result.”