- Source: Oscillation (mathematics)
In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point. As is the case with limits, there are several definitions that put the intuitive concept into a form suitable for a mathematical treatment: oscillation of a sequence of real numbers, oscillation of a real-valued function at a point, and oscillation of a function on an interval (or open set).
Definitions
= Oscillation of a sequence
=Let
(
a
n
)
{\displaystyle (a_{n})}
be a sequence of real numbers. The oscillation
ω
(
a
n
)
{\displaystyle \omega (a_{n})}
of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of
(
a
n
)
{\displaystyle (a_{n})}
:
ω
(
a
n
)
=
lim sup
n
→
∞
a
n
−
lim inf
n
→
∞
a
n
{\displaystyle \omega (a_{n})=\limsup _{n\to \infty }a_{n}-\liminf _{n\to \infty }a_{n}}
.
The oscillation is zero if and only if the sequence converges. It is undefined if
lim sup
n
→
∞
{\displaystyle \limsup _{n\to \infty }}
and
lim inf
n
→
∞
{\displaystyle \liminf _{n\to \infty }}
are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞.
= Oscillation of a function on an open set
=Let
f
{\displaystyle f}
be a real-valued function of a real variable. The oscillation of
f
{\displaystyle f}
on an interval
I
{\displaystyle I}
in its domain is the difference between the supremum and infimum of
f
{\displaystyle f}
:
ω
f
(
I
)
=
sup
x
∈
I
f
(
x
)
−
inf
x
∈
I
f
(
x
)
.
{\displaystyle \omega _{f}(I)=\sup _{x\in I}f(x)-\inf _{x\in I}f(x).}
More generally, if
f
:
X
→
R
{\displaystyle f:X\to \mathbb {R} }
is a function on a topological space
X
{\displaystyle X}
(such as a metric space), then the oscillation of
f
{\displaystyle f}
on an open set
U
{\displaystyle U}
is
ω
f
(
U
)
=
sup
x
∈
U
f
(
x
)
−
inf
x
∈
U
f
(
x
)
.
{\displaystyle \omega _{f}(U)=\sup _{x\in U}f(x)-\inf _{x\in U}f(x).}
= Oscillation of a function at a point
=The oscillation of a function
f
{\displaystyle f}
of a real variable at a point
x
0
{\displaystyle x_{0}}
is defined as the limit as
ϵ
→
0
{\displaystyle \epsilon \to 0}
of the oscillation of
f
{\displaystyle f}
on an
ϵ
{\displaystyle \epsilon }
-neighborhood of
x
0
{\displaystyle x_{0}}
:
ω
f
(
x
0
)
=
lim
ϵ
→
0
ω
f
(
x
0
−
ϵ
,
x
0
+
ϵ
)
.
{\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(x_{0}-\epsilon ,x_{0}+\epsilon ).}
This is the same as the difference between the limit superior and limit inferior of the function at
x
0
{\displaystyle x_{0}}
, provided the point
x
0
{\displaystyle x_{0}}
is not excluded from the limits.
More generally, if
f
:
X
→
R
{\displaystyle f:X\to \mathbb {R} }
is a real-valued function on a metric space, then the oscillation is
ω
f
(
x
0
)
=
lim
ϵ
→
0
ω
f
(
B
ϵ
(
x
0
)
)
.
{\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(B_{\epsilon }(x_{0})).}
Examples
1
x
{\displaystyle {\frac {1}{x}}}
has oscillation ∞ at
x
{\displaystyle x}
= 0, and oscillation 0 at other finite
x
{\displaystyle x}
and at −∞ and +∞.
sin
1
x
{\displaystyle \sin {\frac {1}{x}}}
(the topologist's sine curve) has oscillation 2 at
x
{\displaystyle x}
= 0, and 0 elsewhere.
sin
x
{\displaystyle \sin x}
has oscillation 0 at every finite
x
{\displaystyle x}
, and 2 at −∞ and +∞.
(
−
1
)
x
{\displaystyle (-1)^{x}}
or 1, −1, 1, −1, 1, −1... has oscillation 2.
In the last example the sequence is periodic, and any sequence that is periodic without being constant will have non-zero oscillation. However, non-zero oscillation does not usually indicate periodicity.
Geometrically, the graph of an oscillating function on the real numbers follows some path in the xy-plane, without settling into ever-smaller regions. In well-behaved cases the path might look like a loop coming back on itself, that is, periodic behaviour; in the worst cases quite irregular movement covering a whole region.
Continuity
Oscillation can be used to define continuity of a function, and is easily equivalent to the usual ε-δ definition (in the case of functions defined everywhere on the real line): a function ƒ is continuous at a point x0 if and only if the oscillation is zero; in symbols,
ω
f
(
x
0
)
=
0.
{\displaystyle \omega _{f}(x_{0})=0.}
A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.
For example, in the classification of discontinuities:
in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides);
in an essential discontinuity, oscillation measures the failure of a limit to exist.
This definition is useful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε (hence a Gδ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition.
The oscillation is equivalent to the ε-δ definition by a simple re-arrangement, and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given ε0 there is no δ that satisfies the ε-δ definition, then the oscillation is at least ε0, and conversely if for every ε there is a desired δ, the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.
Generalizations
More generally, if f : X → Y is a function from a topological space X into a metric space Y, then the oscillation of f is defined at each x ∈ X by
ω
(
x
)
=
inf
{
d
i
a
m
(
f
(
U
)
)
∣
U
i
s
a
n
e
i
g
h
b
o
r
h
o
o
d
o
f
x
}
{\displaystyle \omega (x)=\inf \left\{\mathrm {diam} (f(U))\mid U\mathrm {\ is\ a\ neighborhood\ of\ } x\right\}}
See also
Wave equation
Wave envelope
Grandi's series
Bounded mean oscillation
References
Further reading
Kata Kunci Pencarian:
- Oscillation (mathematics)
- Oscillation
- Cis (mathematics)
- Oscillator (disambiguation)
- Self-oscillation
- Classification of discontinuities
- Neural oscillation
- Bounded mean oscillation
- Plasma oscillation
- Oscillation theory
