Generalized hyperbolic functions, circulant matrices and functional equations
There is a contrast between the two sets of functional equations f 0 ( x + y ) = f 0 ( x ) f 0 ( y ) + f 1 ( x ) f 1 ( y ) , f 1 ( x + y ) = f 1 ( x ) f 0 ( y ) + f 0 ( x ) f 1 ( y ) , and f 0 ( x - y ) = f 0 ( x ) f 0 ( y ) - f 1 ( x ) f 1 ( y ) , f 1 ( x - y ) = f 1 ( x ) f 0 ( y ) - f 0 ( x ) f 1...
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| Published in: | Linear algebra and its applications Vol. 406; pp. 272 - 284 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York, NY
Elsevier Inc
01.09.2005
Elsevier Science |
| Subjects: | |
| ISSN: | 0024-3795, 1873-1856 |
| Online Access: | Get full text |
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| Summary: | There is a contrast between the two sets of functional equations
f
0
(
x
+
y
)
=
f
0
(
x
)
f
0
(
y
)
+
f
1
(
x
)
f
1
(
y
)
,
f
1
(
x
+
y
)
=
f
1
(
x
)
f
0
(
y
)
+
f
0
(
x
)
f
1
(
y
)
,
and
f
0
(
x
-
y
)
=
f
0
(
x
)
f
0
(
y
)
-
f
1
(
x
)
f
1
(
y
)
,
f
1
(
x
-
y
)
=
f
1
(
x
)
f
0
(
y
)
-
f
0
(
x
)
f
1
(
y
)
satisfied by the even and odd components of a solution of
f(
x
+
y)
=
f(
x)
f(
y). Schwaiger and, later, Förg-Rob and Schwaiger considered the extension of these ideas to the case where
f is sum of
n components. Here we shorten and simplify the statements and proofs of some of these results by a more systematic use of matrix notation. |
|---|---|
| ISSN: | 0024-3795 1873-1856 |
| DOI: | 10.1016/j.laa.2005.04.011 |