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Fourier series of B-splines
Journal of Computational North-Holland
Fourier
and Applied
Mathematics
125
21 (1988) 125-127
series of B-splines
B.C. SOH School of Engineering, Received
Keywords:
Science University of Malaysia, Jalan Bandaraya,
30000 Ipoh, Malaysia
27 March 1987
B-splines,
Fourier
series
I feel it is interesting to note that the Fourier series of Schoenberg’s B-splines on the unit circle obtained in [3] by convolution can be derived via difference operators as follows. First, we denote U:= {z E C: 1z 1 = l} and for any z E U, z = e’“; i = m, x E [0, 271) and k a positive integer. Next, let s and v be two positive integers write w = eih where h = 27/k, c k and define the Gaussian binomial coefficients (see [1,2]) by
- l)(tP- 1).** ws“+l-1) s :=(0” [ 1 ( -l)( ‘-1) (( --1) V
w
. . .
w
(J
>
and in particular [G] = 1 and [z] = 0 for v < 0 or v > s. The following basic identities involving the Gaussian binomial coefficients will be useful: (1 -,f)(l-
&$).
. . (I-
Jg
=
y(_l)q n t; 1],.(.-1)/2y
0)
v=o
(for a proof, see [2, p. 188]), ~~~(-1)*[“:Ijwy(ll)/2wi’=o
po(l-w'-')=
vs=o,
l,...,n,
(2) s=o, l,...,n,
~1(-1)'-'["~l],.~~-1~/2vw-'v;
(3)
w=o
P#S A;+*f(z)=
~~'(-l)'["tl]~~~~-"/'f(;~~+l-~),
ZE
,y,
(4)
v=o
where A,f(z)
:=f(ZQ)
-f(Z)>
We define the B-splines on the unit
M;(z):= 0377-0427/87/$3.50
A;f(z) circle
$A;+l+,J~~-(i+n+l)),
0 1987, Elsevier Science Publishers
:= A;-‘f(zw)
- w”-‘A;-‘f(z),
by j=O,
l,.__,k_l,
B.V. (North-Holland)
n = 1,2,.
.. .
B. C. Soh / Fourier series of B-splines
126
where &(zW-“)
I
:=
arc[ 6Y-n-1,
0,
z E
( ZCC” - 1) n,
elsewhere.
w’) >
Then, by (4), we easily obtain
Theorem (Schoenberg
[3]). For any z E U,
(5)
MO”(z) = f c#, -CC where 1
cp =
1 -(/P
n
IL@J n(
/L--s
1 lz 2ai s=o ni
1 - Qs-I*
Proof. For PLE (0, l,..., 27
1 5=2a
=-
/A--s
1’
epipx
1<(_1).[
2’, 5
pE
(0, L...,n},
pe
{O,l)...,
?I}.
n},
MO” (elx)
1o
i ’
?‘I t
dx
l],“(~-l,/z/‘“+l)h(ei(i.-uh) _ 1)”
e--iPX
vh
1
1
n
(by noting P-S
=-nik s=o S#P For ~4 (0, l,...,
(J-P
-
n}, we have
1 CP= -2~ /a-vM:(e’x) _
(2) and (3)).
i
2’, ,‘,
ni’ [n i
edipx dx l]
WV(Y-l)/2~(hn+1’h(ei(x-~h)
_
1)”
ediPx
. v=o
=Further,
1 n 27Ti s=. ni
1 - &P (by noting (1) and (2)). P--s
if we follow Schoenberg b,, = (1 - w-‘)//A
[3], and let
and in particular
we can write (5) in the form MO”(z) = &
E bpbp-1 . . . bp-,,z’Y -cc
b, = 2 IA/k,
0
dx
dx
B.C. Soh / Fourier series of B-splines
127
References [l] S. Karlin, Total Positiuity, Vol. I (Stanford University Press, Stanford, CA, 1968). [2] G. Pblya and Szegii, Problems and Theorems in Analysis I (Springer, Berlin, 1972). [3] I.J. Schoenberg, On polynomial spline functions on the circle I, II, in: Proc. Confer. Functions, Budapest, 1972, pp. 403-433.
Constructive
Theory
of
Fourier
and Applied
Mathematics
125
21 (1988) 125-127
series of B-splines
B.C. SOH School of Engineering, Received
Keywords:
Science University of Malaysia, Jalan Bandaraya,
30000 Ipoh, Malaysia
27 March 1987
B-splines,
Fourier
series
I feel it is interesting to note that the Fourier series of Schoenberg’s B-splines on the unit circle obtained in [3] by convolution can be derived via difference operators as follows. First, we denote U:= {z E C: 1z 1 = l} and for any z E U, z = e’“; i = m, x E [0, 271) and k a positive integer. Next, let s and v be two positive integers write w = eih where h = 27/k, c k and define the Gaussian binomial coefficients (see [1,2]) by
- l)(tP- 1).** ws“+l-1) s :=(0” [ 1 ( -l)( ‘-1) (( --1) V
w
. . .
w
(J
>
and in particular [G] = 1 and [z] = 0 for v < 0 or v > s. The following basic identities involving the Gaussian binomial coefficients will be useful: (1 -,f)(l-
&$).
. . (I-
Jg
=
y(_l)q n t; 1],.(.-1)/2y
0)
v=o
(for a proof, see [2, p. 188]), ~~~(-1)*[“:Ijwy(ll)/2wi’=o
po(l-w'-')=
vs=o,
l,...,n,
(2) s=o, l,...,n,
~1(-1)'-'["~l],.~~-1~/2vw-'v;
(3)
w=o
P#S A;+*f(z)=
~~'(-l)'["tl]~~~~-"/'f(;~~+l-~),
ZE
,y,
(4)
v=o
where A,f(z)
:=f(ZQ)
-f(Z)>
We define the B-splines on the unit
M;(z):= 0377-0427/87/$3.50
A;f(z) circle
$A;+l+,J~~-(i+n+l)),
0 1987, Elsevier Science Publishers
:= A;-‘f(zw)
- w”-‘A;-‘f(z),
by j=O,
l,.__,k_l,
B.V. (North-Holland)
n = 1,2,.
.. .
B. C. Soh / Fourier series of B-splines
126
where &(zW-“)
I
:=
arc[ 6Y-n-1,
0,
z E
( ZCC” - 1) n,
elsewhere.
w’) >
Then, by (4), we easily obtain
Theorem (Schoenberg
[3]). For any z E U,
(5)
MO”(z) = f c#, -CC where 1
cp =
1 -(/P
n
IL@J n(
/L--s
1 lz 2ai s=o ni
1 - Qs-I*
Proof. For PLE (0, l,..., 27
1 5=2a
=-
/A--s
1’
epipx
1<(_1).[
2’, 5
pE
(0, L...,n},
pe
{O,l)...,
?I}.
n},
MO” (elx)
1o
i ’
?‘I t
dx
l],“(~-l,/z/‘“+l)h(ei(i.-uh) _ 1)”
e--iPX
vh
1
1
n
(by noting P-S
=-nik s=o S#P For ~4 (0, l,...,
(J-P
-
n}, we have
1 CP= -2~ /a-vM:(e’x) _
(2) and (3)).
i
2’, ,‘,
ni’ [n i
edipx dx l]
WV(Y-l)/2~(hn+1’h(ei(x-~h)
_
1)”
ediPx
. v=o
=Further,
1 n 27Ti s=. ni
1 - &P (by noting (1) and (2)). P--s
if we follow Schoenberg b,, = (1 - w-‘)//A
[3], and let
and in particular
we can write (5) in the form MO”(z) = &
E bpbp-1 . . . bp-,,z’Y -cc
b, = 2 IA/k,
0
dx
dx
B.C. Soh / Fourier series of B-splines
127
References [l] S. Karlin, Total Positiuity, Vol. I (Stanford University Press, Stanford, CA, 1968). [2] G. Pblya and Szegii, Problems and Theorems in Analysis I (Springer, Berlin, 1972). [3] I.J. Schoenberg, On polynomial spline functions on the circle I, II, in: Proc. Confer. Functions, Budapest, 1972, pp. 403-433.
Constructive
Theory
of