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In 1922, Harald Bohr and Johannes Mollerup established a remarkable
characterization of the Euler gamma function using its
log-convexity property. A decade later, Emil Artin investigated
this result and used it to derive the basic properties of the gamma
function using elementary methods of the calculus. Bohr-Mollerup's
theorem was then adopted by Nicolas Bourbaki as the starting point
for his exposition of the gamma function. This open access book
develops a far-reaching generalization of Bohr-Mollerup's theorem
to higher order convex functions, along lines initiated by Wolfgang
Krull, Roger Webster, and some others but going considerably
further than past work. In particular, this generalization shows
using elementary techniques that a very rich spectrum of functions
satisfy analogues of several classical properties of the gamma
function, including Bohr-Mollerup's theorem itself, Euler's
reflection formula, Gauss' multiplication theorem, Stirling's
formula, and Weierstrass' canonical factorization. The scope of the
theory developed in this work is illustrated through various
examples, ranging from the gamma function itself and its variants
and generalizations (q-gamma, polygamma, multiple gamma functions)
to important special functions such as the Hurwitz zeta function
and the generalized Stieltjes constants. This volume is also an
opportunity to honor the 100th anniversary of Bohr-Mollerup's
theorem and to spark the interest of a large number of researchers
in this beautiful theory.
In 1922, Harald Bohr and Johannes Mollerup established a remarkable
characterization of the Euler gamma function using its
log-convexity property. A decade later, Emil Artin investigated
this result and used it to derive the basic properties of the gamma
function using elementary methods of the calculus. Bohr-Mollerup's
theorem was then adopted by Nicolas Bourbaki as the starting point
for his exposition of the gamma function. This open access book
develops a far-reaching generalization of Bohr-Mollerup's theorem
to higher order convex functions, along lines initiated by Wolfgang
Krull, Roger Webster, and some others but going considerably
further than past work. In particular, this generalization shows
using elementary techniques that a very rich spectrum of functions
satisfy analogues of several classical properties of the gamma
function, including Bohr-Mollerup's theorem itself, Euler's
reflection formula, Gauss' multiplication theorem, Stirling's
formula, and Weierstrass' canonical factorization. The scope of the
theory developed in this work is illustrated through various
examples, ranging from the gamma function itself and its variants
and generalizations (q-gamma, polygamma, multiple gamma functions)
to important special functions such as the Hurwitz zeta function
and the generalized Stieltjes constants. This volume is also an
opportunity to honor the 100th anniversary of Bohr-Mollerup's
theorem and to spark the interest of a large number of researchers
in this beautiful theory.
Aggregation is the process of combining several numerical values
into a single representative value, and an aggregation function
performs this operation. These functions arise wherever aggregating
information is important: applied and pure mathematics
(probability, statistics, decision theory, functional equations),
operations research, computer science, and many applied fields
(economics and finance, pattern recognition and image processing,
data fusion, etc.). This is a comprehensive, rigorous and
self-contained exposition of aggregation functions. Classes of
aggregation functions covered include triangular norms and conorms,
copulas, means and averages, and those based on nonadditive
integrals. The properties of each method, as well as their
interpretation and analysis, are studied in depth, together with
construction methods and practical identification methods. Special
attention is given to the nature of scales on which values to be
aggregated are defined (ordinal, interval, ratio, bipolar). It is
an ideal introduction for graduate students and a unique resource
for researchers.
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