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Bounds for the norm of certain spline projections, II
JOURNAL
OF APPROXIMATION
THEORY
35, 299-310 (1982)
Bounds for the Norm of Certain Spline Projections, II * E. NEIJIUN Institute of Computer Science, University of Wroc/aw, pl. Grunwaldzki 214, SO-384 Wroclaw, Poland Communicated
by G. Meinardus
Received May 10, 1981
1. INTRODUCTION
AND NOTATION
Let II and q be given natural numbers such that n + 1 > q > 0 (n > 0). By I we denote the unit interval [0, 1] and A, is an arbitrary but fixed partition of the interval I: A,:O=xO
1.
By Sp(2q - 1, A,) we denote the space of polynomial splines of degree 2q - 1, deficiency 1, and knots xi (i = 0, l,.,., n). Thus s E Sp(2q - 1,A,) if and only if (i) in each interval [xi--(, xi] (i = 1, 2,..., n) s coincides with an algebraic polynomial of degree 2q - 1 or less, (ii) s E C2q-2(1). It is known that Sp(2q - 1, A,) is a linear subspaceof the space C(I) and dim Sp(2q - 1, A,) = n + 2q - 1 (cf. [ 11). In the sequel we will assumethat each element s from the space Sp(2q - 1, A,) satisfies additionally some boundary conditions P(O) = s”‘(1) = 0
(j = 1, 2)...) q - l),
(1.1)
,(jl(o) = p(l)
(j = 2, 4 ,..., 2q - 2).
(1.2)
or = 0
The conditions (1.2) are called Lidstone-type conditions (cf. [&I]). * This paper was completed while the author was visiting at the Gesellschaft fiir Mathematik und Datenverarbeitung at St. Augustin, Federal Republic of Germany.
299
002 I-9045/82/080299-1 All
ZSOZ.OO/O
Copyright c 1982 by Academic Press. Inc. rights of reproduction in any form reserved.
300
E. NEUMAN
It is known (see, e.g., [ 11) that for given real numbers f;: (i = 0, I,..., n) there exists exactly one s E Sp(24 - 1, A,) interpolating the datafi, i.e., S(Xi) =A
(i = 0, l,..., n),
(1.3)
jointly with the boundary conditions (1.1) or (1.2) (cf. [ 1 I). Every such spline function s may be written in the Lagrange form s(x) = -t j&(x) iY0
(x E 4,
where si E Sp(2q - 1, A,), si(xj) = 6, (i,j = 0, l...., n) and every function si satisfies the boundary conditions (1.1) or (1.2). The functions si are the socalled fundamental spline functions. They play an important role in our further considerations. Consider the operator Pf,‘-’ defined by
(‘fi4- !f)(x) = f fCxi) si(x)
U-E C(Q).
(1.4)
i=O
It is obvious that P’,“- ’ is a linear, bounded and idempotent map from C(I) onto Sp(2q - 1, A,); thus Pz-’ is a projection. Let 1). ]loc stand for the sup-norm in the interval I. The inequality
IIf-P?‘Iloc is well known way,
(herefE
< (1 + IIC?‘lO dWf, Sp(% - 1,A,,)) C(1)). The operator norm /I . ]I is defined in the usual
IIp~-‘ll = sup llJytllx,. llfll,
< 1
From this inequality we see that the knowledge on the size of the norm Pf,‘- ’ is important. In this paper we will give some results concerning the norms of the projections P,2q- ’ . We continue our earlier investigations from [22], where the natural boundary conditions were imposed on the spline function s = Pfi4-‘J For other results for the non-periodic boundary conditions see [2-4, 12, 291. In the case of the periodic boundary conditions (i.e., such that s”‘(0) = G(I) for j = 0, l,..., 2q - 2) many results are known up to date (see [6, 12-20, 24-281). In Section 3 the cubic case (q = 2) is treated assuming the boundary conditions (1.1). For the second type boundary conditions some results are given in the above-mentioned paper [22]. Estimations from above for I]Pi 11 (for arbitrary knots) and explicit formulae for these norms for equidistant knots are given. In the final section the uniform upper bounds for ]IPill are
SPLINE
301
PROJECTIONS
derived (in the case of equidistant knots). The interpolant Pjlf satisfies the boundary conditions (1.1) or (1.2). For the related results concerning the norm of some quadratic spline projections, see, [3, 7, 10, 11, 19, 20, 23-25, 291. 2.
LEMMAS
For our further aims we define the bi-infinity sequence (di} in the following manner: dmi = 0, d, = 1, d, = 4, di+ , = 4 di - dip, (i = 1,2 ,... ). LEMMA
2.1.
If the numbers di are defined as above, then
did,-di-ldl+,=dl-i
= 4-,-z
if
O
if
i>l+l.
1,
ProoJ Since d, = [(2 + (3)“2)m+’ - (2 - (3)“2)““]/(2(3)“2) -1, O,...), then the desired result follows by direct calculations.
Let ~j,-~=~/~=~j~=Pi,~+~=O,
(m =
1
and
Pij=(-l)“‘dj--ld,-i-,/d,-,
(j < 0,
= (-l)“‘di_,d,-j-~/d,-,
(j 2 9
(i,j=
1, 2 ,..., n - 1).
(2-l)
We have the following LEMMA
2.2.
Zf the numbers ml” are such that ml!!, + 4mj” + ml’:, = 3n(dj+ ,,i - 6j-,,i),
m. (0 =
m(i)n
-_ 0
(i = 0, l,..., n; j =
1,
2 ,..., n -
l),
(2.2)
then m;8 = (-l)i+i+l
3n dj-*(d,-i-d,-i-2)/d,-,
=3n(di-,d,-i-2-di-2dn-i-I)/dn-I
(j = 9,
=(-1)iCi3nd,_j-~(di-di_,)/d,_,
(j > 9
(i = 0, l,..., n; j = ProoJ
(j < 0,
1,
2 ,..., n - 1).
(2.3)
It is known (see, e.g., [21]) that the matrix of the above linear
302
E.NEUMAN
system (2.2) possesses an inverse with entries pu given by (2.1). By virtue of (2.2) we have
,I”=
3n(bj+j-1 -Pj,i+l)*
Hence and from (2.1) we obtain the desired result (2.3).
1
LEMMA 2.3. Let xi = i/n, st E Sp(3,dJ (i = 0, l,..., n) and let each fundamental spline function si satisfy the boundary conditions (1.1) for q = 2. If x E (xi-, , x,) (j = 1, 2 ,..., n), then
sgn s,(x) = (-l)i+i = (-l)i+j+
tj < 0, 1
(j > 0 (i = 0, l,..., n; j = 1, 2 ,..., n).
The proof of (2.4) follows immediately
(2.4)
from Theorem 2 (Part I) in
151.I 3. CUBIC CASE
For the sake of brevity we introduce more notation. Let hi = xi -xi-, (j = 1, 2,..., n), h = maxl,j,, hp a = maxli_/l =, hi/[h,(h, + hi)]. Our first result is contained in the following THEOREM 3.1. Let the knots x, be arbitrary, (i = 0, l,..., n) and (P”, f )‘(O) = (Pjlf )‘( 1) = 0. Then
(Pjl f)(xi)
llP;ll < 1 + $4
=f (x,)
(3.1)
where a and h are defined as above.
The proof is quite similar to that of [6, Theorem omitted. I
11. For this reason it is
From (3.1) we have the following COROLLARY 3.1.
For equidistant
knots we have 11Pi II < {.
Now we shall give an explicit formula for the norm of the projection Pi in the case of equidistant knots. Let /y(x)=
+ Isr(x)l IS
(x E I), ’
SPLINE
303
PROJECTIONS
denote the so-called Lebesgue function for the projection P?-‘. It is known that I(Pz-‘II = 1(/12-’ /)oo. From this equality it follows that for our aims we must have more information on the functions s,. Let ,i” = s;(xJ (i, 1 = 0, l,..., n). By virtue of our assumptions we have rnb’) = mf,” = 0 for all 1. If x E [xi-, , xi] (i = 1, 2 ,..., n) and if knots xi are equidistant, then each fundamental spline s,(x) may be written as ~~(~)=6~-,~,@~(~)+6~,@~(1~-x)+mj!!,@,(x)-m~”@,(1
-x)
(1= 0, l,..., n; x E [xi-,, xi]; i = 1, 2 ,..., n - I),
(3.2)
where @Jo(x)= (1 + 2t)(l - r)*, cPI(X) = t(1 - 2)*/n, t=n(x-Xi-,), ( see,
e.g.,
[I, 61).
@,(l -x)>O,
If
then
XE [Xi-l,Xi],
(3.3)
Qo(x),
Qo(l -x>,
Q,(x),
and Q,(x) + Qo( 1 - x) = 1, Q,(x) + @,(I -x)
< 1/4n.
With the help of Lemma 2.3 and (3.2) one can prove Aft(x)=
l + al,n@,(x)-Pi.n@lll (XE
-x>
[Xi-l,Xj];i=
1,2,...,n),
where i-l
a i.n -- iT (-l)i+‘+‘mj!, I=0
+ 2 (-1)‘+‘mi!,, I=i
i-1 Pi,n
=
C
(-I)‘+‘+’
ml”
+
I=0
+
(-l)‘+‘mi’).
ri
By virtue of (2.3) the above formulae simplify to
a,,”
-&dj&*(d~-j-, ” I
Bi,n=-d,-,
6d
+ d,-,),
(3.4) n-r ‘_ I (d’- I
2 fdj-1)
(i = 1, 2 ,..., n).
304
E.NEUMAN
Thus if x E [xi-, , xi], then the Lebesgue function /ii(x)
may be written as
n;(x) = 1 + ; n--l [a,(l-t)+bir]f(l-t) (t = tZ(X- Xi- 1); i = 1, 2,.**9n),
(3.5)
(i = 1, 2)...) n).
(3.6)
where
bi=d,-i-[(di-*
+di-1)
Let Ai = max,,_,cXGXl~~(x) (i = 1,2 ,..., n). From (3.5) and (3.6) we see that ni(x)=/iz(l -x). Hence &=An+,-i (i= 1, 2,..., n). THEOREM
(Pif)‘(O)
3.2.
Let
= (Pif)‘(
Xi = i/n,
(Pif)(Xi)
IIPs,II = 1 + 3dj-*(dj-l 2d,-,
+ dj)
= 1+ &
[2d3’* + (3 - B)(ZUj + l)]
Proof.
= f(xi)
(i = OV 19***7 n)T
1) = 0. Then ifn=2j+
n-1
1 (j=O,
if fI = 2j
l,...),
(j = 1, 2,...),
According to Lemma 2.1 we have, by virtue of (3.6), ai+,-Ui=d,-,i-,
+dn-2i
=+4-n-,
+ d>j-n-2)
if
2i< n,
if
2i + 1 > n.
Now we consider two cases. Let lo.
n = 2j + 1 (j = 0, l,... ). By virtue of (3.6) and (3.7) one gets
Ui+ * - Ui > 0
if
i= I,2 ,..., j,
if
i=j+
l,j+2
,..., n,
(3.7)
305
SPLINE PROJECTIONS
and
bi+,-bi>O
if
i= 1,2 ,..., j,
if
i=j+
l,j+
2 ,..., n.
Thus al
aj+,>Uj+z>‘*‘>a,,
b,
bj+l>bj+z>***>b,.
and
(3.8)
Moreover, aj,, = bj,, . By virtue of (3.5) and (3.8) we have lIPil
max
=
A~(x)=Ajl
(
xj
2”.
+-
n = 2j (j = 1,2,...). Similarly a,
Uj+[;
3
+
-dj-l(dj-1
1 =l+2d”+,
dj)*
to the previous case we can prove Uj+l
>
Uj+2
> *-* > Cl,,
and
(3.9) b,
bj>bj+,>***>b,.
Moreover, Uj = bj+ r and Uj+i = bj. Hence IIP;fll = maxXj-,G.,Gxj/i~(x) Ai( If x E [Xi-i, Xi] and n = 2j, then, from (3.6) and (3.5), one has Al(x)=
1+;
[aj
n-1
+ (dj- 1 -
dj-*dj)
l] t( 1 - t).
=
(3.10)
From Lemma 2.1 one gets dj-, - dj-,d, = 1. The cubic polynomial (aj + t) t( 1 - f) attains its single maximum in the interval [0, 1 ] in the point where
t*,
f* = (\/s’-
aj + 1)/3
(l/3 < I* < l/2),
and 6 is the same as above. With the help of (3.10) we obtain the desired result. The last statement of the thesis follows immediately from (3.5), (3.8) and (3.9). I In Table I we give values of IIPiII and e, :=max,(i~n~i-min,,i,.~ifor small values of n. COROLLARY
M
3.2.
If Pi is defined as in Theorem 3.2, then
< IP:ll
= 1.549038....
I
306
E. NEUMAN TABLE n
IICll
2 3 4 5 6 7 8 9 10
I.0 1.222222 1.5 1.522407 1.545455 1.547116 1.548780 1.548900 1.549020 1.549028
4.
QUMTIC
I e,
0.0 0.0 0.262963 0.284312 0.307284 0.308939 0.310603 0.310723 0.310843 0.310851
CASE
In this section we assume that the knots xi are equidistant, i.e., xi = i/n for all i = 0, l,..., n. We give below an upper bound for the norm of the projection Pi, under the assumption that the spline function s = Pi f satisfies the boundary conditions (1.1) or (1.2) for q = 3. Let us denote fj = s(xJ, m,= s’(x,), M,= s”(xi), S,= s’“(x,) (j= 0, l,..., n). The first theorem follows. THEOREM and
let
(Pj,
4.1. Let x, = i/n, (Pjtf)(xJ =f(x,) f)U’(0) = (Pif)U’( 1) = 0 for j = 1,2.
Proof. It is known that the first derivatives consistency relations:
(i = 0,
l,...,
n; fe
C(Z))
Then
m, satisfy the following
22lm, + 19m, + 3m, = n(-235f0 + 65f, + 155f2 + 15f3), mj-2+26mj-l+66mj+26mj+l+m,+2 = 5n(-..-* 3m,-3 + 19m,-,
- W-I + 22lm,-,
+ lo?, L +J;.+*) = n(-15f,-,
(j = 2, 3 ,..., n - 2), - 155f,-*
- 65f,-,
+ 235f,)
(see, e.g., [9]). Let A denote the matrix of the above system of linear equations with unknowns m, (j = 1,2 ,..., n - 1; m, = m, = 0). Using the standard diagonal dominance argument we obtain IIA -’ llco < l/12 (here 1).JJcostands for the infinity norm of the square matrix). Now we take a function f E C(Z) such that (If Ila, Q 1. Let b = (b,, b2,..., b,- ,)r, where bj
SPLINE
denotes the right-hand obvious that
307
PROJECTIONS
side in the jth equation of the above system. It is max ICj
] b,] < 470~2.
Hence (4.1)
Hoskins [9] proved that Mj-,-6Mj+Mj+,=
-2On*(f,-,
-2f,+fi+l)
+8n(mj+,
0’ = 1, 2,..., nSimilarly
-mj-l)
l;M,=M,=O).
to that above one can prove (4.2)
For x E [x1-,, xi] (i = 1,2,..., n) the quintic spline Pi $ may by written as (Pif)(x)=fi-l
@o(f) +
+.fi@O(l
[M,-,
Q*(f)
-I) +
+
M,@,(l
[mi-~@l(f)-mi@l(l -
-t)lln
(4.3)
t,]/n2,
where t = n(x - xi-,), @Jx) = (1 - x)“( 1 + 3x + 6x2), Q,(x) =x(1 -x)3(1 @*(x)=x2(1
+3x),
(4.4)
-x)‘.
From (4.4) we see that @J!(X), Di( 1 -x) We also have
> 0 for 0
Go(x) + @Jl - x) = 1,
G,(x)+ @,(I-x) < &, @2(x)
+
@*Cl
-x) < &
(4.5)
(O
1).
308
E.NEUMAN
TakingfE and (4.9,
C(I) and such that ]]f]l,
Hence the desired result follows.
< 1 we obtain, by virtue of (4.1)-(4.3)
I
In the case when the boundary conditions (1.2) are imposed on the spline function Pifthen the upper bound for the norm of this projection is given in the following THEOREM 4.2. and let (Pif)ci)(0)
Let xi = i/n, (Pif)(xi) =f(xi) (i = 0, l,..., n;fE = (Pjlf)‘-‘)( 1) = 0 fir j = 2,4. Then
C(I))
Proof. We only sketch the proof because it is quite similar to the proof of Theorem 4.1. The consistency relations for the fourth-order derivatives Sj = s’“(xj) are 65S, + 26S, + S, = 120n4(-2f,
+ 5f, - 4fz +f3),
Sj-2 + 26Sj-1 + 66Sj + 26Sj+ 1+ Sj+z (j = 2, 3,...,n - 2), = 120n4(fj-2-44fi-,+ 6&-4&+, +fi+J S.-,+26S,-2 + 65S,-,= 120n4(fn-, -4fn-z + Sf,-,- 2f,) (see, e.g., [30]). Hence if ]]f Iloc,< 1 then max (Sjl < 160n”.
(4.6)
- 2fi +fj+,) - (Sj_1 + 8Sj + Sj+,)/120n2
(see,
O
e.g., [30]). By virtue of (4.6) one gets (4.7) For x E [xi-r, xi] (i = 1,2 ,..., n) the quintic spline (Pi f )(x) may be written as (Pi
f)(X)=&-,
Yo(1 + [Si-
-t)+JyO(t)+ 1 yV,(l
["i-*
- l) + si y2(t)]/n47
yY,(l
-
f)+
Miyl(r)]ln2
(4.8)
309
SPLINE PROJECTIONS
where t = n(x - xi- ,),
Y,(x) = (x3 -x)/6, Y’,(x) = (x’ -x)/120
- Y,(x)/6.
We seethat
Hence and from (4.6)-(4.8) we obtain the desired result.
1
ACKNOWLEDGMENT The author wishes to thank Dr. A. Pokrzywa of the Mathematical Institute. Polish Academy of Sciences, for helpful remarks on an earlier draft of this paper.
REFERENCES I. J. H. AHLBERG, E. N. NILSON. AND J. L. WALSH, “The Theory of Splines and Their Applications,” Academic Press. New York, 1967. 2. C. DE BOOR. “The Method of Projections as Applied to the Numerical Solution of Two Point Boundary Value Problems Using Cubic Splines.” Ph.D. dissertation, University of Michigan. Ann Arbor. 1966. 3. C. DE BOOR. On bounding spline interpolation, J. Approx. Theory 14 (1975). 191-203. 4, C. DE BOOR, On cubic spline functions that vanish at all knots, Adoan. Math. 20( 1976). l-17. 5. C. DE BOOR AND I. J. SCHOENBERG,Cardinal interpolation and spline functions. VIII. The Budan-Fourier theorem for splines and applications, in “Spline Functions. Karslruhe 1975” (K. Biihmer, G. Meinardus, and W. Schempp. Eds.). pp. l-79, Lecture Notes in Mathematics Vol. 501. Springer-Verlag, Berlin, 1976. 6. E. W. CHENEY .\ND F. SCHURER,A note on the operators arising in spline approximation. J. Approx. Theory 1 (1968), 94-102. 7. S. DEMKO. Interpolation by quadratic splines. J. Approx. Theory 23 (1978). 392400. 8. C. A. HALL AND W. W. MEYER. Optimal error bounds for cubic spline interpolation. .f. Approx. Theory 16 (1976). 105-122. 9. W. D. HOSK~NS, Algorithm 62. Interpolating quintic splines on equidistant knots. Comput.
J. 13( 1970),
437-438.
10. W. J. KAMMERER, G. W. REDDIEN. AND R. S. V&RCA. Quadratic interpolating splines. Numer.
Math.
22 (1974).
24 l-259.
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E. NEUMAN
I I. M. MARSDEN, Quadratic spline interpolation, Bull. Amer. Marh. Sm. 80 (1974). 903-906. 12. M. MARSDEN, Cubic spline interpolation of continuous functions, J. Approx. Theory IO (1974), 103-111. 13. G. MEINARDUS, ijber die Norm des Operators der Kardinalen Spline-Interpolation, J. Approx. Theory 16 (1976), 289-298. 14. G. MEMARDUS, Period&he Spline-Funktionen, in “Spline Functions, Karlsruhe 1975” (K. Bohmer, G. Meinardus, and W. Schempp, &is.), pp. 177-199, Lecture Notes in Mathematics Vol. 501, Springer-Verlag, Berlin, 1976. 15. G. MEMARDLJS, Computation of the norms of some spline interpolation operators, in “Polynomial and Spline Approximation” (B. N. Sahney, Ed.), pp. 155-161, NATO Advanced Study Institute Series, Riedel, Dordrecht, 1979. 16. G. MEINARDUS AND G. MERZ, Zur periodischen Spline-Interpolation, in “SplineFuntionen” (K. Bohmer, G. Meinardus, and W. Schempp, Eds.), pp. 177-195. Bibliographisches Institut, Mannheim, 1974. 17. G. ME~NARDUS AND G. MERZ, Zur periodischen Spline-Interpolation, II, in “Numerische Methoden der Approximationstheorie” (L. Collatz, G. Meinardus, and H. Werner, Eds.), Band 4, pp. 204-221, ISNM 42, Birkhauser-Verlag, Basel, 1978. 18. G. MEMARDUS AND G. D. TAYLOR, Periodic quadratic spline interpolant of minimal norm, J. Approx. Theor?, 23 (1978), 137-141. 19. G. MERZ, Normen von Spline-Interpolationsoperatoren, in “Approximation in Theorie und Praxis” (G. Meinardus, Ed.), pp. 183-208, Bibliographisches Institut. Mannheim. 1979. 20. G. MERZ, Interpolation mit periodischen Spline-Funktionen 11, J. Approx. Theocv, in press. 21. E. NEUMAN, The inversion of certain band matrices, (in Polish). Mnr. Sfos. 9 (1977), 15-24. 22. E. NEUMAN, Bounds for the norm of certain spline projections, J. Approx. Theory 27 (1979). 135-145. 23. E. NEUMAN, Quadratic splines and histospline projections, J. Approx. Theory 29 (1980). 297-304. 24. F. B. RICHARDS: Best bounds for the uniform periodic spiine operator. J. Approx. Theory 7 (1973), 302-317. 25. F. B. RICHARDS, The Lebesgue constants for cardinal spline interpolation, J. Approx. Theory 14 (1975), 83-92. 26. F. SCHURER AND E. W. CHENEY, On interpolating cubic splines with equally-spaced nodes, Nederl. Akad. Wetensch. Proc. Ser. A 71 (1968), 5 17-524. 27. F. SCHURER,A note on interpolating periodic quintic splines with equally spaced nodes, .I. Approx. Theory 1 (1968), 493-500. 28. F. SCHURER,A note on interpolating quintic spline fuctions, in “Approximation Theory” (A. Talbot, Ed.), pp. 71-81, Academic Press, London/New York, 1970. 29. J. TZIMBALARIO. On a class of interpolatory splines, J. Approx. Theory 23 (1978). 142-145. 30. R. A. USMANI AND S. A. WARSI, Smooth spline solutions for boundary value problems in plate deflection theory, Comput. Math. Appl. 6 (1980), 205-211.
OF APPROXIMATION
THEORY
35, 299-310 (1982)
Bounds for the Norm of Certain Spline Projections, II * E. NEIJIUN Institute of Computer Science, University of Wroc/aw, pl. Grunwaldzki 214, SO-384 Wroclaw, Poland Communicated
by G. Meinardus
Received May 10, 1981
1. INTRODUCTION
AND NOTATION
Let II and q be given natural numbers such that n + 1 > q > 0 (n > 0). By I we denote the unit interval [0, 1] and A, is an arbitrary but fixed partition of the interval I: A,:O=xO
1.
By Sp(2q - 1, A,) we denote the space of polynomial splines of degree 2q - 1, deficiency 1, and knots xi (i = 0, l,.,., n). Thus s E Sp(2q - 1,A,) if and only if (i) in each interval [xi--(, xi] (i = 1, 2,..., n) s coincides with an algebraic polynomial of degree 2q - 1 or less, (ii) s E C2q-2(1). It is known that Sp(2q - 1, A,) is a linear subspaceof the space C(I) and dim Sp(2q - 1, A,) = n + 2q - 1 (cf. [ 11). In the sequel we will assumethat each element s from the space Sp(2q - 1, A,) satisfies additionally some boundary conditions P(O) = s”‘(1) = 0
(j = 1, 2)...) q - l),
(1.1)
,(jl(o) = p(l)
(j = 2, 4 ,..., 2q - 2).
(1.2)
or = 0
The conditions (1.2) are called Lidstone-type conditions (cf. [&I]). * This paper was completed while the author was visiting at the Gesellschaft fiir Mathematik und Datenverarbeitung at St. Augustin, Federal Republic of Germany.
299
002 I-9045/82/080299-1 All
ZSOZ.OO/O
Copyright c 1982 by Academic Press. Inc. rights of reproduction in any form reserved.
300
E. NEUMAN
It is known (see, e.g., [ 11) that for given real numbers f;: (i = 0, I,..., n) there exists exactly one s E Sp(24 - 1, A,) interpolating the datafi, i.e., S(Xi) =A
(i = 0, l,..., n),
(1.3)
jointly with the boundary conditions (1.1) or (1.2) (cf. [ 1 I). Every such spline function s may be written in the Lagrange form s(x) = -t j&(x) iY0
(x E 4,
where si E Sp(2q - 1, A,), si(xj) = 6, (i,j = 0, l...., n) and every function si satisfies the boundary conditions (1.1) or (1.2). The functions si are the socalled fundamental spline functions. They play an important role in our further considerations. Consider the operator Pf,‘-’ defined by
(‘fi4- !f)(x) = f fCxi) si(x)
U-E C(Q).
(1.4)
i=O
It is obvious that P’,“- ’ is a linear, bounded and idempotent map from C(I) onto Sp(2q - 1, A,); thus Pz-’ is a projection. Let 1). ]loc stand for the sup-norm in the interval I. The inequality
IIf-P?‘Iloc is well known way,
(herefE
< (1 + IIC?‘lO dWf, Sp(% - 1,A,,)) C(1)). The operator norm /I . ]I is defined in the usual
IIp~-‘ll = sup llJytllx,. llfll,
< 1
From this inequality we see that the knowledge on the size of the norm Pf,‘- ’ is important. In this paper we will give some results concerning the norms of the projections P,2q- ’ . We continue our earlier investigations from [22], where the natural boundary conditions were imposed on the spline function s = Pfi4-‘J For other results for the non-periodic boundary conditions see [2-4, 12, 291. In the case of the periodic boundary conditions (i.e., such that s”‘(0) = G(I) for j = 0, l,..., 2q - 2) many results are known up to date (see [6, 12-20, 24-281). In Section 3 the cubic case (q = 2) is treated assuming the boundary conditions (1.1). For the second type boundary conditions some results are given in the above-mentioned paper [22]. Estimations from above for I]Pi 11 (for arbitrary knots) and explicit formulae for these norms for equidistant knots are given. In the final section the uniform upper bounds for ]IPill are
SPLINE
301
PROJECTIONS
derived (in the case of equidistant knots). The interpolant Pjlf satisfies the boundary conditions (1.1) or (1.2). For the related results concerning the norm of some quadratic spline projections, see, [3, 7, 10, 11, 19, 20, 23-25, 291. 2.
LEMMAS
For our further aims we define the bi-infinity sequence (di} in the following manner: dmi = 0, d, = 1, d, = 4, di+ , = 4 di - dip, (i = 1,2 ,... ). LEMMA
2.1.
If the numbers di are defined as above, then
did,-di-ldl+,=dl-i
= 4-,-z
if
O
if
i>l+l.
1,
ProoJ Since d, = [(2 + (3)“2)m+’ - (2 - (3)“2)““]/(2(3)“2) -1, O,...), then the desired result follows by direct calculations.
Let ~j,-~=~/~=~j~=Pi,~+~=O,
(m =
1
and
Pij=(-l)“‘dj--ld,-i-,/d,-,
(j < 0,
= (-l)“‘di_,d,-j-~/d,-,
(j 2 9
(i,j=
1, 2 ,..., n - 1).
(2-l)
We have the following LEMMA
2.2.
Zf the numbers ml” are such that ml!!, + 4mj” + ml’:, = 3n(dj+ ,,i - 6j-,,i),
m. (0 =
m(i)n
-_ 0
(i = 0, l,..., n; j =
1,
2 ,..., n -
l),
(2.2)
then m;8 = (-l)i+i+l
3n dj-*(d,-i-d,-i-2)/d,-,
=3n(di-,d,-i-2-di-2dn-i-I)/dn-I
(j = 9,
=(-1)iCi3nd,_j-~(di-di_,)/d,_,
(j > 9
(i = 0, l,..., n; j = ProoJ
(j < 0,
1,
2 ,..., n - 1).
(2.3)
It is known (see, e.g., [21]) that the matrix of the above linear
302
E.NEUMAN
system (2.2) possesses an inverse with entries pu given by (2.1). By virtue of (2.2) we have
,I”=
3n(bj+j-1 -Pj,i+l)*
Hence and from (2.1) we obtain the desired result (2.3).
1
LEMMA 2.3. Let xi = i/n, st E Sp(3,dJ (i = 0, l,..., n) and let each fundamental spline function si satisfy the boundary conditions (1.1) for q = 2. If x E (xi-, , x,) (j = 1, 2 ,..., n), then
sgn s,(x) = (-l)i+i = (-l)i+j+
tj < 0, 1
(j > 0 (i = 0, l,..., n; j = 1, 2 ,..., n).
The proof of (2.4) follows immediately
(2.4)
from Theorem 2 (Part I) in
151.I 3. CUBIC CASE
For the sake of brevity we introduce more notation. Let hi = xi -xi-, (j = 1, 2,..., n), h = maxl,j,, hp a = maxli_/l =, hi/[h,(h, + hi)]. Our first result is contained in the following THEOREM 3.1. Let the knots x, be arbitrary, (i = 0, l,..., n) and (P”, f )‘(O) = (Pjlf )‘( 1) = 0. Then
(Pjl f)(xi)
llP;ll < 1 + $4
=f (x,)
(3.1)
where a and h are defined as above.
The proof is quite similar to that of [6, Theorem omitted. I
11. For this reason it is
From (3.1) we have the following COROLLARY 3.1.
For equidistant
knots we have 11Pi II < {.
Now we shall give an explicit formula for the norm of the projection Pi in the case of equidistant knots. Let /y(x)=
+ Isr(x)l IS
(x E I), ’
SPLINE
303
PROJECTIONS
denote the so-called Lebesgue function for the projection P?-‘. It is known that I(Pz-‘II = 1(/12-’ /)oo. From this equality it follows that for our aims we must have more information on the functions s,. Let ,i” = s;(xJ (i, 1 = 0, l,..., n). By virtue of our assumptions we have rnb’) = mf,” = 0 for all 1. If x E [xi-, , xi] (i = 1, 2 ,..., n) and if knots xi are equidistant, then each fundamental spline s,(x) may be written as ~~(~)=6~-,~,@~(~)+6~,@~(1~-x)+mj!!,@,(x)-m~”@,(1
-x)
(1= 0, l,..., n; x E [xi-,, xi]; i = 1, 2 ,..., n - I),
(3.2)
where @Jo(x)= (1 + 2t)(l - r)*, cPI(X) = t(1 - 2)*/n, t=n(x-Xi-,), ( see,
e.g.,
[I, 61).
@,(l -x)>O,
If
then
XE [Xi-l,Xi],
(3.3)
Qo(x),
Qo(l -x>,
Q,(x),
and Q,(x) + Qo( 1 - x) = 1, Q,(x) + @,(I -x)
< 1/4n.
With the help of Lemma 2.3 and (3.2) one can prove Aft(x)=
l + al,n@,(x)-Pi.n@lll (XE
-x>
[Xi-l,Xj];i=
1,2,...,n),
where i-l
a i.n -- iT (-l)i+‘+‘mj!, I=0
+ 2 (-1)‘+‘mi!,, I=i
i-1 Pi,n
=
C
(-I)‘+‘+’
ml”
+
I=0
+
(-l)‘+‘mi’).
ri
By virtue of (2.3) the above formulae simplify to
a,,”
-&dj&*(d~-j-, ” I
Bi,n=-d,-,
6d
+ d,-,),
(3.4) n-r ‘_ I (d’- I
2 fdj-1)
(i = 1, 2 ,..., n).
304
E.NEUMAN
Thus if x E [xi-, , xi], then the Lebesgue function /ii(x)
may be written as
n;(x) = 1 + ; n--l [a,(l-t)+bir]f(l-t) (t = tZ(X- Xi- 1); i = 1, 2,.**9n),
(3.5)
(i = 1, 2)...) n).
(3.6)
where
bi=d,-i-[(di-*
+di-1)
Let Ai = max,,_,cXGXl~~(x) (i = 1,2 ,..., n). From (3.5) and (3.6) we see that ni(x)=/iz(l -x). Hence &=An+,-i (i= 1, 2,..., n). THEOREM
(Pif)‘(O)
3.2.
Let
= (Pif)‘(
Xi = i/n,
(Pif)(Xi)
IIPs,II = 1 + 3dj-*(dj-l 2d,-,
+ dj)
= 1+ &
[2d3’* + (3 - B)(ZUj + l)]
Proof.
= f(xi)
(i = OV 19***7 n)T
1) = 0. Then ifn=2j+
n-1
1 (j=O,
if fI = 2j
l,...),
(j = 1, 2,...),
According to Lemma 2.1 we have, by virtue of (3.6), ai+,-Ui=d,-,i-,
+dn-2i
=+4-n-,
+ d>j-n-2)
if
2i< n,
if
2i + 1 > n.
Now we consider two cases. Let lo.
n = 2j + 1 (j = 0, l,... ). By virtue of (3.6) and (3.7) one gets
Ui+ * - Ui > 0
if
i= I,2 ,..., j,
if
i=j+
l,j+2
,..., n,
(3.7)
305
SPLINE PROJECTIONS
and
bi+,-bi>O
if
i= 1,2 ,..., j,
if
i=j+
l,j+
2 ,..., n.
Thus al
aj+,>Uj+z>‘*‘>a,,
b,
bj+l>bj+z>***>b,.
and
(3.8)
Moreover, aj,, = bj,, . By virtue of (3.5) and (3.8) we have lIPil
max
=
A~(x)=Ajl
(
xj
2”.
+-
n = 2j (j = 1,2,...). Similarly a,
Uj+[;
3
+
-dj-l(dj-1
1 =l+2d”+,
dj)*
to the previous case we can prove Uj+l
>
Uj+2
> *-* > Cl,,
and
(3.9) b,
bj>bj+,>***>b,.
Moreover, Uj = bj+ r and Uj+i = bj. Hence IIP;fll = maxXj-,G.,Gxj/i~(x) Ai( If x E [Xi-i, Xi] and n = 2j, then, from (3.6) and (3.5), one has Al(x)=
1+;
[aj
n-1
+ (dj- 1 -
dj-*dj)
l] t( 1 - t).
=
(3.10)
From Lemma 2.1 one gets dj-, - dj-,d, = 1. The cubic polynomial (aj + t) t( 1 - f) attains its single maximum in the interval [0, 1 ] in the point where
t*,
f* = (\/s’-
aj + 1)/3
(l/3 < I* < l/2),
and 6 is the same as above. With the help of (3.10) we obtain the desired result. The last statement of the thesis follows immediately from (3.5), (3.8) and (3.9). I In Table I we give values of IIPiII and e, :=max,(i~n~i-min,,i,.~ifor small values of n. COROLLARY
M
3.2.
If Pi is defined as in Theorem 3.2, then
< IP:ll
= 1.549038....
I
306
E. NEUMAN TABLE n
IICll
2 3 4 5 6 7 8 9 10
I.0 1.222222 1.5 1.522407 1.545455 1.547116 1.548780 1.548900 1.549020 1.549028
4.
QUMTIC
I e,
0.0 0.0 0.262963 0.284312 0.307284 0.308939 0.310603 0.310723 0.310843 0.310851
CASE
In this section we assume that the knots xi are equidistant, i.e., xi = i/n for all i = 0, l,..., n. We give below an upper bound for the norm of the projection Pi, under the assumption that the spline function s = Pi f satisfies the boundary conditions (1.1) or (1.2) for q = 3. Let us denote fj = s(xJ, m,= s’(x,), M,= s”(xi), S,= s’“(x,) (j= 0, l,..., n). The first theorem follows. THEOREM and
let
(Pj,
4.1. Let x, = i/n, (Pjtf)(xJ =f(x,) f)U’(0) = (Pif)U’( 1) = 0 for j = 1,2.
Proof. It is known that the first derivatives consistency relations:
(i = 0,
l,...,
n; fe
C(Z))
Then
m, satisfy the following
22lm, + 19m, + 3m, = n(-235f0 + 65f, + 155f2 + 15f3), mj-2+26mj-l+66mj+26mj+l+m,+2 = 5n(-..-* 3m,-3 + 19m,-,
- W-I + 22lm,-,
+ lo?, L +J;.+*) = n(-15f,-,
(j = 2, 3 ,..., n - 2), - 155f,-*
- 65f,-,
+ 235f,)
(see, e.g., [9]). Let A denote the matrix of the above system of linear equations with unknowns m, (j = 1,2 ,..., n - 1; m, = m, = 0). Using the standard diagonal dominance argument we obtain IIA -’ llco < l/12 (here 1).JJcostands for the infinity norm of the square matrix). Now we take a function f E C(Z) such that (If Ila, Q 1. Let b = (b,, b2,..., b,- ,)r, where bj
SPLINE
denotes the right-hand obvious that
307
PROJECTIONS
side in the jth equation of the above system. It is max ICj
] b,] < 470~2.
Hence (4.1)
Hoskins [9] proved that Mj-,-6Mj+Mj+,=
-2On*(f,-,
-2f,+fi+l)
+8n(mj+,
0’ = 1, 2,..., nSimilarly
-mj-l)
l;M,=M,=O).
to that above one can prove (4.2)
For x E [x1-,, xi] (i = 1,2,..., n) the quintic spline Pi $ may by written as (Pif)(x)=fi-l
@o(f) +
+.fi@O(l
[M,-,
Q*(f)
-I) +
+
M,@,(l
[mi-~@l(f)-mi@l(l -
-t)lln
(4.3)
t,]/n2,
where t = n(x - xi-,), @Jx) = (1 - x)“( 1 + 3x + 6x2), Q,(x) =x(1 -x)3(1 @*(x)=x2(1
+3x),
(4.4)
-x)‘.
From (4.4) we see that @J!(X), Di( 1 -x) We also have
> 0 for 0
Go(x) + @Jl - x) = 1,
G,(x)+ @,(I-x) < &, @2(x)
+
@*Cl
-x) < &
(4.5)
(O
1).
308
E.NEUMAN
TakingfE and (4.9,
C(I) and such that ]]f]l,
Hence the desired result follows.
< 1 we obtain, by virtue of (4.1)-(4.3)
I
In the case when the boundary conditions (1.2) are imposed on the spline function Pifthen the upper bound for the norm of this projection is given in the following THEOREM 4.2. and let (Pif)ci)(0)
Let xi = i/n, (Pif)(xi) =f(xi) (i = 0, l,..., n;fE = (Pjlf)‘-‘)( 1) = 0 fir j = 2,4. Then
C(I))
Proof. We only sketch the proof because it is quite similar to the proof of Theorem 4.1. The consistency relations for the fourth-order derivatives Sj = s’“(xj) are 65S, + 26S, + S, = 120n4(-2f,
+ 5f, - 4fz +f3),
Sj-2 + 26Sj-1 + 66Sj + 26Sj+ 1+ Sj+z (j = 2, 3,...,n - 2), = 120n4(fj-2-44fi-,+ 6&-4&+, +fi+J S.-,+26S,-2 + 65S,-,= 120n4(fn-, -4fn-z + Sf,-,- 2f,) (see, e.g., [30]). Hence if ]]f Iloc,< 1 then max (Sjl < 160n”.
(4.6)
- 2fi +fj+,) - (Sj_1 + 8Sj + Sj+,)/120n2
(see,
O
e.g., [30]). By virtue of (4.6) one gets (4.7) For x E [xi-r, xi] (i = 1,2 ,..., n) the quintic spline (Pi f )(x) may be written as (Pi
f)(X)=&-,
Yo(1 + [Si-
-t)+JyO(t)+ 1 yV,(l
["i-*
- l) + si y2(t)]/n47
yY,(l
-
f)+
Miyl(r)]ln2
(4.8)
309
SPLINE PROJECTIONS
where t = n(x - xi- ,),
Y,(x) = (x3 -x)/6, Y’,(x) = (x’ -x)/120
- Y,(x)/6.
We seethat
Hence and from (4.6)-(4.8) we obtain the desired result.
1
ACKNOWLEDGMENT The author wishes to thank Dr. A. Pokrzywa of the Mathematical Institute. Polish Academy of Sciences, for helpful remarks on an earlier draft of this paper.
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