Functions| Functions |
In General > s.a. analysis
[continuity classes, examples]; Approximation
Methods; functional analysis [function spaces].
* Idea: A map f
: M → \(\mathbb K\) from a manifold M to a field
\(\mathbb K\) (usually, \(\mathbb K\) = \(\mathbb R\) or \(\mathbb C\),
and M may be \(\mathbb R\) or \(\mathbb C\) too).
@ General references:
Oldham et al 08 [An Atlas of Functions];
Olver et al 10 [NIST handbook].
@ Random functions: Shale JFA(79) [of Poisson type];
Wang & Battefeld JCAP(16)-a1607 [generation, Dyson Brownian Motion algorithm].
Polynomials > s.a. Algebraic
Geometry [decomposition of polynomials]; graph
and knot invariants.
* Lee-Yang circle
theorem: A somewhat mysterious result on the location of
zeros of certain polynomials in statistical mechanics.
* Applications:
Knot theory; Graph counting; Statistical mechanics.
* Operations:
Notice that polynomial multiplication is a form of convolution.
* Monic polynomial:
A univariate polynomial in which the leading coefficient is equal to 1.
* Grace-like polynomial:
A polynomial P(z1,
..., zm,
w1,...,
wn), separately
of degree 1 in each of its m + n arguments, such
that P(z1,
..., wn) ≠ 0 whenever
there is a circle in \(\mathbb C\) separating z1,
..., zm
from w1, ...,
wn.
@ General references: Landau NAMS(87) [factoring];
Milovanović et al 94 [extremal problems, inequalities, zeros];
Ruelle mp/00 [grace-like];
Wang & Yeh JCTA(05) [with real zeroes].
@ Lee-Yang circle theorem: Ruelle PRL(71) [extension].
@ Random: Forrester & Honner JPA(99) [statistics of zeros];
Zelditch mp/00-proc.
@ Other types: Edwards BAMS(09) [solvable];
> s.a. Chebyshev, Hermite,
Jack, Laguerre,
legendre, Macdonald Polynomials.
> Polynomial approximations: see bessel functions.
Other Types > s.a. analytic functions;
Almost Periodic Functions; Meromorphic
Functions; Quasiperiodic Functions; series;
summations.
* Bounded variation:
A function f on an interval [a, b] is of
bounded variation if there is a number M such that, for every
partition a < x1
< x2 < ... <
xn−1
< b of the interval, the sum of the variations
|f(xi) −
f(xi−1)|
over all subintervals does not exceed M.
* Concave: A function
f : [a, b] → \(\mathbb R\) is concave
iff the segment joining any two points in its graph lies below the curve,
or for all x, y ∈ [a, b] and all
λ ∈ [0,1], the values of f inside the interval
satisfy f(λ x + (1−λ) y)
≤ λ f(x) + (1−λ)
f(y) .
* Convex: A function f
: [a, b] → \(\mathbb R\) is convex iff the segment joining
any two points in its graph lies above the curve, or for all x, y
∈ [a, b] and all λ ∈ [0,1], the values
of f inside the interval satisfy f(λ x
+ (1−λ) y) ≥ λ
f(x) + (1−λ) f(y) .
* Positive pure
frequency function: One of the form F(x, t)
= f(x) exp{−iωt}, or satisfying
\(\cal L\)t f
= −iω f, for ω > 0.
* Positive frequency
function: One that can be extended to an analytic function
in the lower complex t-plane.
> Other special types:
see harmonic functions;
L-functions;
Rational Functions; Superoscillating Functions.
> Special functions:
see Airy; bessel;
Elliptic; Gamma;
Hypergeometric; Jost;
Mathieu; Struve;
Theta; Whittaker;
Zeta Function; spherical harmonics.
Examples and Properties > s.a. Germ
of a Function; Hyperbolic Functions;
trigonometry.
* Other examples:
C∞ function of compact support,
f(x):= exp{−(x − x0)2 / [(x − x0)2 − h2]} , ψ(x):= exp{−1/x2(a − x)2} for x ∈ (0, a), 0 otherwise;
C∞ function vanishing for x ≤ 0 and equal to 1 for x ≥ a,
χ(x):= C−1 ∫−∞x ψ(x') dx' , C:= ∫\(\,_{\mathbb R}^~\)ψ(x) dx ;
C∞ function on \(\mathbb R\)n, of compact support, equal to 1 in a square box xi ∈ (αi, βi),
g(x1, ..., xn):= ∏i=1n χ(xi − αi + a) χ(βi − xi + a) .
@ Examples: Sturzu qp/02 [ψ(s) = ∑k=−∞∞ exp(−k2/s2)]; Cvijović PRS(07) [polylogarithm]; Tsionskiy & Tsionskiy a1207-wd [comments on infinitely differentiable function of bounded support]; > s.a. Gaussians; Sigmoid.
Expansions and Operations on Functions
> s.a. fourier analysis; Integral Transforms.
* Convolution: For
two functions f and g on \(\mathbb R\), the
convolution is defined by
(f *g)(x):= ∫\(\,_{\mathbb R}^~\) dy f(x−y) g(y) ;
The Fourier transform of the product of two functions is the convolution of the two Fourier transforms.
Generalizations > see distributions; Extrafunctions.
Online resources > see EquPlus [science and math equations in TeX, MathML, png-image and MathType format].
main page
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send feedback and suggestions to bombelli at olemiss.edu – modified 5 apr 2018