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A note on some convergence properties of spline functions
Camp. & Maths. with Apvlr. Vol. 4. pp. 27-J-279 @ Pergamon Press Ltd.. 1978. Printed in Greal Britain
A NOTE ON SOME CONVERGENCE PROPERTIES OF SPLINE FUNCTIONS S. E. SIMS Department of Mathematics, Louisiana Tech University, Ruston, Louisiana 71270 U.S.A. Communicated by P. L. Ode11 (ReceivedJuly l!lT’) Abstract-Convergence properties for type 1, type 2’ and type k splines will be proved using the minimum norm property. Similar properties for the type 2 and modified type k splines will be established using the fundamental identity. 1. INTRODUCTION
Using either the minimum * norm property or the fundamental identity convergence properties for several types of spline functions will be established. In what follows the notation and definitions will be in accordance with[l]. Definition 1.1. Let A: x0 < x1 < . . * c xN be a mesh on [a, b]. A polynomial spline of degree 2n - 1 of deficiency k(0 < k 5 n) on [a, b] is a function S,, in C2”-‘-‘[a, b] such that S*@‘)(x)= 0 for all X in (Xi-r, xi), i = 1,. . . , N. Given a function f on [a, b] satisfying appropriate differentiability criteria a spline of deficiency k which interpolates to f on the mesh A must satisfy the condition
spfj x.) = f”‘(x.) 9
,
t
k-l,
j=O,l,...,
3
i=l,...,
N-l.
(1.1)
Additional interpolation constraints at the endpoints a and b can be used to define the following different types of splines. In each case i = 0, N. Type 1: Sa”‘Cf,x,) I = f”‘(x,)I ,
j = 0, 1, . . . , n - 1.
Type 2: SaU)Cf I x.) I =f”‘(x,)I ,
J. - n, . . . ,2n - 2.
Type 2’: S*“cf,Xi)=O, Type k:
j=n ,...,
Sa”‘Cf, xi) = f”‘(x.)I ,
j=O,l,...,
k-l.
j = n, . . . ,2n - k - 1.
S~“‘cf,xi) = 0, Modified Type k:
2n-2.
Sb”‘cf, Xi) = f”‘(xi),
j=O,l,...,
Sb”‘cf, Xi) = f”‘(X’)I 1
j=n,...,2n-k-1.
2. CONVERGENCE
k-l.
PROPERTIES
In order to obtain the desired properties the following lemma whose proof follows from Rolle’s Theorem is necessary. LEMMA 2.1. There exist points 5Zjin (a, b) such that s*“‘cf
2.) 9
I
=
f”‘(T.)
I
j=O,l,...,n-1,
3
if N z n - k for the type 1, type k and modified type k splines, and if N 2 n - k + 2 for the type 2 and type 2’ splines. Existence and uniqueness theorems can be established for each of these splines provided a sufficient number of points is chosen in the mesh A. Specifically the number of points N must satisfy k(N - 1) 2 n - 2 for the type 2 and type 2’ splines, and k(N + 1) 5 n for the type k and modified type k splines while N > 0 is the only restriction for the type 1 spline. .
2n
s. E.
278
SIMS
In the following theorems it will be assumed that the number of mesh points satisfies the condition, N z 0 for the type 1 spline, N L max {n - k + 2,1+ (n - 2)/k] for the type 2 and type 2’ splines, and N 2 max nik for the type k and modified type k splines. THEOREM2.1.
Let the sequence of functions cf,,,}and the function f be in C’“[a, b]. Suppose that lim Irmti’- f”‘l]== 0,
ml+-
j=O,l,...,
n.
(2.1)
If S,&, x) represents either the type 1, type 2’ or type k splines then, Jim I(Sau’Cfm, .) - SAti’cf, .))I= = 0 Illforj=O,l,...
,n-l.Furthermore,forj=O.l,...,
n-l
((S&(i’(fmt *)- SA”‘cf*‘)((a5
(2.2)
Mj
where Mi = 2
Ikm’n-i) _f’n-i’ll_cb _ a)“-‘-‘.
Proof. In this proof the linearity of the spline operator of each of the above types will be to simplify the notation. It is easy to see that, _
used
Sd”-“Cfm
-f,x)=
[_,
Sh(“‘Cf,,, -f,
t)dt +Sh(“-‘V-f,,,-.f,in-I).
Thus. ISA’“-“(f,,,-f, x,1
S16 JSa'"'&, -f,t)ldt + If,,,‘“-“(&_,) I
-f’“-‘)(fp,)l
where Lemma 2.1 was applied to obtain the last term in this inequality. Now the CauchySchwartz inequality yields,
Jab ISi”)(fm -f, t)l
dt 5 (I,*
(&?J,,
-f,
t),2dt)“2(h
- ,)“z.
But the minimum norm property implies, I* [Sa’“‘cf,,,- f, t)]’ dt 5 I’ Lf,‘“‘(t) -f’“‘(t)]2 dt. (I L7 A combination of these three inequalities along with taking the supremum over [a, b] gives
Hence (2.1) and (2.2) hold for j = n - 1. Again the above lemma gives, Sd”-2’cf, -f,x,
= [_2 S’“-“cf, -f, t) dt +f,,,‘“-2)(fn_Z)-f’“-2’(&-2), n
and
thus, (“-2’Cfm - f, *)I(-s (]P”Cf, - f, .)(I,@ - a) + (v,(“-*’ - f’“-*‘((= (ISA 5 ]bm(“’_f’“‘]]& - a)2 + ]~m’“-”-f’“-” _f’“-‘)]lm(b - a) + I~m(n-2’ _94’1],.
Continuing in this manner gives (2.1) and (2.2).
(2.3)
A note on some convergence properties of spiine functions
219
Property (2.1) of this theorem could have been proven using the best approximation property along with the continuity of the best approximation operator on a linear subspace. The advantage of the proof given here is that inequality (2.2) is obtained. However, the type 2 and modified type k splines do not satisfy a minimum norm property, and thus the fundamental identity is necessary to prove properties (2.1) and (2.2). Using the interpolation condition (1.1) the fundamental identity[l] for these two splines reduces to, I
0bIf’“‘(x) - S,(?j, x)]’ dx + 1 b [S*(“)u, x)12dx 0
where in the summation p = I for the type 2 spline, and p = n -k for the modified type k spline. Taking into account the remaining interpolation constraints for these two splines the following . inequality is ultimately obtained, 1’0 [Sd”‘(jy X)]’dX 5 (”0
Lo”‘]
dx - 2 2 i=p Ipi
- Sb”‘cf, Xi)]]f(2n-i-‘)(Xi)]
(4.2)
where again p = 1 or n - k depending on the type of spline and i = 0, N. THEOREM2.2.
In addition to the hypothesis of the previous theorem suppose that for m = 1,2, . . ., fm’)(Xi)
=
fo”(xi),
i = 0, N,
j=n,...,
2n-p-1,
(2.5)
where p = 1 or k depending on the type spline being used. If S&f, x) represents either the type 2 or modified type k spline then (2.1) and (2.2) hold. Proof. The proof is exactly as in the previous theorem with inequality (2.3) following from (2.4) by an application of (2.5). In addition to the hypothesis of Theorem 2.1 suppose that, for p = 1 or n - k, the sequences {Sao’cf,,,,xi)}, i = 0, N, j = p, . . ., n - 1 are bounded, and that
THEOREM2.3.
lim
fm(2n-i-I)(xi)
= f(2n-i-11(Xi)r
WI-
i=O,N,
j=p ,...,
n-1.
Again if Sa(f, x) represents either the type 2 or modified type k spline then (2.1) holds. Proof. The proof proceeds as in Theorem 2.1 until inequality (2.3) is reached. In the present case (2.3) is replaced by (2.4) to ultimately yield for i = 0, N and p = 1 or n - k, (n-‘)CfmIISA
f, .)I115 [ 1” Lfm(“)(x)- p”‘(x)]’ dx + 2 $: ]fm’)(Xi)- fo”(Xi)- SA,“‘cf,- f, xi)] 0 x Ifm(2n-j-yXi)
_pn-i-1)(4,1]1’2~b
_
a)112 + 1p~n-1
_ f”-ym.
lim ]]Sb(n-lY.f,- f, .)(I_= 0, mand the remainder of the proof is as before. REFERENCES I. J. H. Ahlberg, E. N. Nilson and J. L. Walsh. The Theory of Splines and Their Applications. Ac’ademicPress, New York (1967). 2. D. E. Wulbert, Continuity of Metric Projections, Trans. Am. math. Sot. 134, 335-341 (1968).
A NOTE ON SOME CONVERGENCE PROPERTIES OF SPLINE FUNCTIONS S. E. SIMS Department of Mathematics, Louisiana Tech University, Ruston, Louisiana 71270 U.S.A. Communicated by P. L. Ode11 (ReceivedJuly l!lT’) Abstract-Convergence properties for type 1, type 2’ and type k splines will be proved using the minimum norm property. Similar properties for the type 2 and modified type k splines will be established using the fundamental identity. 1. INTRODUCTION
Using either the minimum * norm property or the fundamental identity convergence properties for several types of spline functions will be established. In what follows the notation and definitions will be in accordance with[l]. Definition 1.1. Let A: x0 < x1 < . . * c xN be a mesh on [a, b]. A polynomial spline of degree 2n - 1 of deficiency k(0 < k 5 n) on [a, b] is a function S,, in C2”-‘-‘[a, b] such that S*@‘)(x)= 0 for all X in (Xi-r, xi), i = 1,. . . , N. Given a function f on [a, b] satisfying appropriate differentiability criteria a spline of deficiency k which interpolates to f on the mesh A must satisfy the condition
spfj x.) = f”‘(x.) 9
,
t
k-l,
j=O,l,...,
3
i=l,...,
N-l.
(1.1)
Additional interpolation constraints at the endpoints a and b can be used to define the following different types of splines. In each case i = 0, N. Type 1: Sa”‘Cf,x,) I = f”‘(x,)I ,
j = 0, 1, . . . , n - 1.
Type 2: SaU)Cf I x.) I =f”‘(x,)I ,
J. - n, . . . ,2n - 2.
Type 2’: S*“cf,Xi)=O, Type k:
j=n ,...,
Sa”‘Cf, xi) = f”‘(x.)I ,
j=O,l,...,
k-l.
j = n, . . . ,2n - k - 1.
S~“‘cf,xi) = 0, Modified Type k:
2n-2.
Sb”‘cf, Xi) = f”‘(xi),
j=O,l,...,
Sb”‘cf, Xi) = f”‘(X’)I 1
j=n,...,2n-k-1.
2. CONVERGENCE
k-l.
PROPERTIES
In order to obtain the desired properties the following lemma whose proof follows from Rolle’s Theorem is necessary. LEMMA 2.1. There exist points 5Zjin (a, b) such that s*“‘cf
2.) 9
I
=
f”‘(T.)
I
j=O,l,...,n-1,
3
if N z n - k for the type 1, type k and modified type k splines, and if N 2 n - k + 2 for the type 2 and type 2’ splines. Existence and uniqueness theorems can be established for each of these splines provided a sufficient number of points is chosen in the mesh A. Specifically the number of points N must satisfy k(N - 1) 2 n - 2 for the type 2 and type 2’ splines, and k(N + 1) 5 n for the type k and modified type k splines while N > 0 is the only restriction for the type 1 spline. .
2n
s. E.
278
SIMS
In the following theorems it will be assumed that the number of mesh points satisfies the condition, N z 0 for the type 1 spline, N L max {n - k + 2,1+ (n - 2)/k] for the type 2 and type 2’ splines, and N 2 max nik for the type k and modified type k splines. THEOREM2.1.
Let the sequence of functions cf,,,}and the function f be in C’“[a, b]. Suppose that lim Irmti’- f”‘l]== 0,
ml+-
j=O,l,...,
n.
(2.1)
If S,&, x) represents either the type 1, type 2’ or type k splines then, Jim I(Sau’Cfm, .) - SAti’cf, .))I= = 0 Illforj=O,l,...
,n-l.Furthermore,forj=O.l,...,
n-l
((S&(i’(fmt *)- SA”‘cf*‘)((a5
(2.2)
Mj
where Mi = 2
Ikm’n-i) _f’n-i’ll_cb _ a)“-‘-‘.
Proof. In this proof the linearity of the spline operator of each of the above types will be to simplify the notation. It is easy to see that, _
used
Sd”-“Cfm
-f,x)=
[_,
Sh(“‘Cf,,, -f,
t)dt +Sh(“-‘V-f,,,-.f,in-I).
Thus. ISA’“-“(f,,,-f, x,1
S16 JSa'"'&, -f,t)ldt + If,,,‘“-“(&_,) I
-f’“-‘)(fp,)l
where Lemma 2.1 was applied to obtain the last term in this inequality. Now the CauchySchwartz inequality yields,
Jab ISi”)(fm -f, t)l
dt 5 (I,*
(&?J,,
-f,
t),2dt)“2(h
- ,)“z.
But the minimum norm property implies, I* [Sa’“‘cf,,,- f, t)]’ dt 5 I’ Lf,‘“‘(t) -f’“‘(t)]2 dt. (I L7 A combination of these three inequalities along with taking the supremum over [a, b] gives
Hence (2.1) and (2.2) hold for j = n - 1. Again the above lemma gives, Sd”-2’cf, -f,x,
= [_2 S’“-“cf, -f, t) dt +f,,,‘“-2)(fn_Z)-f’“-2’(&-2), n
and
thus, (“-2’Cfm - f, *)I(-s (]P”Cf, - f, .)(I,@ - a) + (v,(“-*’ - f’“-*‘((= (ISA 5 ]bm(“’_f’“‘]]& - a)2 + ]~m’“-”-f’“-” _f’“-‘)]lm(b - a) + I~m(n-2’ _94’1],.
Continuing in this manner gives (2.1) and (2.2).
(2.3)
A note on some convergence properties of spiine functions
219
Property (2.1) of this theorem could have been proven using the best approximation property along with the continuity of the best approximation operator on a linear subspace. The advantage of the proof given here is that inequality (2.2) is obtained. However, the type 2 and modified type k splines do not satisfy a minimum norm property, and thus the fundamental identity is necessary to prove properties (2.1) and (2.2). Using the interpolation condition (1.1) the fundamental identity[l] for these two splines reduces to, I
0bIf’“‘(x) - S,(?j, x)]’ dx + 1 b [S*(“)u, x)12dx 0
where in the summation p = I for the type 2 spline, and p = n -k for the modified type k spline. Taking into account the remaining interpolation constraints for these two splines the following . inequality is ultimately obtained, 1’0 [Sd”‘(jy X)]’dX 5 (”0
Lo”‘]
dx - 2 2 i=p Ipi
- Sb”‘cf, Xi)]]f(2n-i-‘)(Xi)]
(4.2)
where again p = 1 or n - k depending on the type of spline and i = 0, N. THEOREM2.2.
In addition to the hypothesis of the previous theorem suppose that for m = 1,2, . . ., fm’)(Xi)
=
fo”(xi),
i = 0, N,
j=n,...,
2n-p-1,
(2.5)
where p = 1 or k depending on the type spline being used. If S&f, x) represents either the type 2 or modified type k spline then (2.1) and (2.2) hold. Proof. The proof is exactly as in the previous theorem with inequality (2.3) following from (2.4) by an application of (2.5). In addition to the hypothesis of Theorem 2.1 suppose that, for p = 1 or n - k, the sequences {Sao’cf,,,,xi)}, i = 0, N, j = p, . . ., n - 1 are bounded, and that
THEOREM2.3.
lim
fm(2n-i-I)(xi)
= f(2n-i-11(Xi)r
WI-
i=O,N,
j=p ,...,
n-1.
Again if Sa(f, x) represents either the type 2 or modified type k spline then (2.1) holds. Proof. The proof proceeds as in Theorem 2.1 until inequality (2.3) is reached. In the present case (2.3) is replaced by (2.4) to ultimately yield for i = 0, N and p = 1 or n - k, (n-‘)CfmIISA
f, .)I115 [ 1” Lfm(“)(x)- p”‘(x)]’ dx + 2 $: ]fm’)(Xi)- fo”(Xi)- SA,“‘cf,- f, xi)] 0 x Ifm(2n-j-yXi)
_pn-i-1)(4,1]1’2~b
_
a)112 + 1p~n-1
_ f”-ym.
lim ]]Sb(n-lY.f,- f, .)(I_= 0, mand the remainder of the proof is as before. REFERENCES I. J. H. Ahlberg, E. N. Nilson and J. L. Walsh. The Theory of Splines and Their Applications. Ac’ademicPress, New York (1967). 2. D. E. Wulbert, Continuity of Metric Projections, Trans. Am. math. Sot. 134, 335-341 (1968).