Doubling Transformations and Definite Integrals
A doubling transformation ψ ( x ) = x - λ - b x - μ , b , λ , μ ∈ R , b > 0 , has the property that for any absolutely integral function F ( x ) on R we have ∫ - ∞ ∞ F ( ψ ( x ) ) d x = ∫ - ∞ ∞ F ( x ) d x . Compositions of doubling transformations also satisfy this integral invariance property....
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| Veröffentlicht in: | International journal of applied and computational mathematics 2015-03, Vol.1 (1), p.33-44 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | A doubling transformation
ψ
(
x
)
=
x
-
λ
-
b
x
-
μ
,
b
,
λ
,
μ
∈
R
,
b
>
0
, has the property that for any absolutely integral function
F
(
x
)
on
R
we have
∫
-
∞
∞
F
(
ψ
(
x
)
)
d
x
=
∫
-
∞
∞
F
(
x
)
d
x
.
Compositions of doubling transformations also satisfy this integral invariance property. In this paper we give criteria for determining when a given rational function is a composition of two or more doubling transformations, and use this criteria for giving explicit families of such transformations such as
(
x
-
a
)
(
x
+
a
2
)
(
x
-
a
3
)
(
x
+
a
4
)
x
(
x
-
a
2
)
(
x
+
a
3
)
,
for
a
>
1
;
(
x
2
-
a
2
)
(
x
2
-
a
8
)
x
(
x
+
a
2
)
(
x
-
a
3
)
, for
a
>
1
; and
(
x
2
-
a
2
)
(
x
2
-
b
2
)
x
(
x
2
-
a
b
)
,
for
0
<
a
<
b
. |
|---|---|
| ISSN: | 2349-5103 2199-5796 |
| DOI: | 10.1007/s40819-014-0017-3 |