| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
re. 1992.16:
I'm starting a new topic, as the almost periodic functions are not so close
to 1992.0.
The original definition of almost periodic (a.p.) functions was given by
Harald Bohr:
*Definition D1*. A continuous function f:R-->R is said to be almost
periodic if for every epsilon > 0 it is possible to find a quantity L > 0
such that every interval of R of length L contains at least one number
tau such that
|f(x+tau) - f(x)| < epsilon for every real x
Later Bochner found the following equivalent definition:
*Theorem T1*. The continuous function f:R-->R is almost periodic if and
only if from every sequence (f(x+csi(n)))n of functions obtained from
f(x) by means of translations of R one can select a uniformly convergent
subsequence.
You'll remark that almost periodicity is classically defined only for
continuous functions.
Properties of the almost periodic (a.p.) functions:
*Proposition P1*. Every continuous a.p. function is bounded and uniformly
continuous on R .
*Proposition P2*. Continuous periodic functions are also almost periodic.
*Proposition P3*. Sums and products of a.p. functions are also a.p.
*Proposition P4*. For every continuous a.p. function f:R-->R, the
integral means
T
1/(2T) INT f(x+t)dt
-T
tend, as T-->oo , to a limit independent of x , uniformly over R . This
limit, denoted by
M{f(t}
t
is a linear functional defined for continuous a.p. functions.
*Proposition P5*. If f and g are a.p., then the composite function
f * g = M{f(x-t)g(t)}
t
is also a.p., and we have
f * g = g * f
f * (g * h) = (f * g) * h
*Proposition P6*. Denote the set of of the a.p. functions by /B (gothic,
please! :-)).
Set ____
<f,g> = M{f(t)g(t)} and ||f|| = <f,f>^1/2
t
_
( z is the conjugate of the complex number z ; almost periodicity extends
naturally to functions R-->C ).
This way /B becomes an incomplete Hilbert space. Its dimension is the
power of the continuum.
Consider the system of functions (e ) defined by
nu nu /in R
e (x) = exp(i*nu*x)
nu
Then ||e ||^2 = M({|e |^2} = M {1} = 1
nu t nu t
<e ,e > = 0 for nu != mu
nu mu
For the a.p. function f:R-->R the "generalized Fourier
coefficients" are defined as follows:
c = <f,e >
nu nu
From Bessel's inequality
r
/SIGMA |c |^2 <= ||f||^2 for any r and 0 <= k <= r
k=0 nu
k
we deduce that for every positive p , the number of values of nu for
which |c | >= p , can not exceed (1/p^2)||f||^2 . In particular c = 0
nu nu
for all real nu except at most a denumerable set {nu(1),nu(2),...} .
The series
/SIGMA c * e (x)
k nu(k) nu(k)
is called the "generalized Fourier series of f(x) .
*Theorem T2 (The Fundamental Theorem of A.P. Functions)*. The generalized
Fourier series associated with the continuous a.p. function f:R-->R
converges in the metric of the space /B to the function f .
*Definition D2*. A distribution function (in the probabilistic sense) is a
non-decreasing real-valued function F:R-->R such that
(i) F is right continuous, i.e. lim F(y) = F(x)
y->x
+
(ii) lim F(y) = 0 and lim F(y) = 1
y->-oo y->oo
*Definition D3*. For the distribution function F , its characteristic
function is defined as being the Fourier-Stieltjes Transform of f :
oo
phi(t) = INT exp(i*t*x) dF(x)
-oo
*Theorem T3*. A distribution function is discrete (i.e. its range is
discrete) if and only if its characteristic function is almost periodic.
Books:
-----
[1] Frigyes Riesz and Bela Sz.-Nagy, Functional Analysis, Dover
Publications, Inc. New York,1990, ISBN 0-486-66289-6
[2] John Knopfmacher, Abstract Analytic Number Theory, Dover Publications,
Inc. New York, 1990, ISBN 0-486-66344-2
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1999.1 | another reference | WRKSYS::ROTH | Geometry is the real life! | Tue Sep 19 1995 06:10 | 7 |
I think I've seen a book by Bohr on the topic in the Chelsea catalog. I'm not sure what motivated the investigation of this subject but had origionally guessed that it may have come from noninear oscilations or limit cycles - maybe I'm mistaken though. I never read up on it. - Jim | |||||
| 1999.2 | More properties and generalization | EVTSG8::ESANU | Au temps pour moi | Tue Sep 19 1995 08:01 | 60 |
*Definition D4*. Let f:R-->C and let epsilon > 0 . An
epsilon-almost-period of f is a number tau such that
sup |f(x-tau) - f(x)| < epsilon
x /in R
*Notation*. AP(R) = the set of the almost periodic functions R-->C .
oo
*Theorem T4*. AP(R) is a closed subalgebra of L (R) .
oo
*Theorem T5 (Bohr)*. Let h /in L (R) be differentiable and assume that
h' is almost periodic. Then h itself is almost periodic.
*Proposition P7*. If f is almost periodic and differentiable and if f'
is uniformly continuous, then f' is itself almost periodic.
*Notation*. For X = Hausdorff space, C (X) = the set of the continuous
b
bounded functions X-->R (or C ). This is a Banach space.
*Notation*. If G is a locally compact group and f /in C (G) , let
b
f : G-->R (C) , f (s) = f(sx) (the right translations of f )
x x
O(f) = the closure of the set {f / x /in G} in C (G) .
x b
*Definition D5*. The set of the almost periodic functions G--> R (C) is
AP(G) = { f /in C (G) / O(f) is compact}
b
*Proposition P8*. The definition D5 coincides with the classical definition
of almost periodicity (definition D1, note 1999.0) if G = R .
* Proposition P9*. If the topological group G is compact, then
AP(G) = C(G) (= {Continuous functions G-->R (C) })
Books:
------
[3] John B. Conway, A Course in Functional Analysis, Springer-Verlag, 1990,
New York, Heidelberg, Berlin, ISBN 0-387-97245-5 or 3-540-972245-5
[4] Yitzhak Katznelson, An Introduction to Harmonic Analysis, Dover
Publications, Inc. New York, 1976, ISBN 0-486-63331-4
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