- Email: [email protected]
On saturation with splines
JOURNAL
OF
APPROXIMATION
THEORY
13, 491494 (1975)
Note On Saturation 0. Department
of Mathematics,
University
DEDICATED
TO
OCCASION
with
Splines
SHISHA
of Rhode Island, Kingston,
PROFESSOR OF HIS
G.
G.
SIXTH-FIFTH
There is a whole class of results stating function very closely by splines of degree essentially to a polynomial of degree .< k. theorem below for which we give a proof [ll], [12] and [13].
LORENTZ
Rhode Island 02881
ON
THE
BIRTHDAY
that if one can approximate a -G/C, then the function reduces One variant of this class is the using an argument employed in
2. Let k be an integer 30, xy = a + j(b - a) n-l,j = are II congruent subintervals real functions defined on j-1,2 ,...> n, coincide with
and let - a3 < a < b C. co. For n = 11,2,... let 0, I)...) n, SO that I,,j = (x$!!, x$“)),j = I, 2 ,..., n, of (a, b). For n = 1, Z,... let S, be the set of all [a, b] - {-YA’~), xi”),..., xl:‘> which in each In,j , a polynomial of degree 5; k.
THEOREM. Letfbe CIrealfunction definedon (a, b). Suppose,for n := 1, Z,..., there is an s, E S, such that, as n + co,
sup
.&,hl-L$‘,.r:“)
,n,) 1f(x)
. . . ..S.&
-~ s,(x)! = o(l/n~‘l).
Then f coincides on (a, b) with a polynomial of degree -.
We assume, as we may, a = 0, b = 1. Let E n=
flkil s~rn.ll-l0,lln,?ln
sup
,...,
Copyright (( 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
640/12/4-II
1)
I .fW - sn(-x>i,
491
II -~ I, 2,...,
492
0.
SHISHA
so that E, -+ 0. Then, for II -y 1. 2. .. .. m (with s,(O) :- lim.r-o+o .rJx))
1. 2. .... HI - n, we have
as n, m ---f co. Hence s,(O) has a finite limit which we denotef(0). Then, for n = 1, 2 ,..., If(O) - s,(O)! -~:lim,,,_, ~S,,,(O)--~.x,,(O)I : ’ E,,/M~‘~~.We define similarly s,(l), n = 1, 2,..., and f( 1) and have, for II x 2, 3. ....
We prove now thatfis continuous in [0, I]. Let 0 -I c < 1 and 0 <: E < 3. Choose an integer n ‘I (3/~)ll(~‘-l) such that SUP .dErO.ll
, f(x)
- s,(s); ‘:: 1/W’ 1
~(l.ln.?~lJ,....(n-l)/sl
and such that c is not of the form j/n, 0 <.j < n,j an integer. Let jl be the integer satisfying (j, - 1)/n < c :$ .jJn, 1 X j, :S n. There is a 8 > 0 such thatif/ h / < Gandc -C h E ((j, -- I)in,.j,/n), then / s,(c t h) - s,(c)1 < c/3. For such an I?, 1f(c -L h) - f(c)1 ::. / f(~ - II) - S,(C c 17);-t j S,(C -t h) - s,(c)/
-‘- I .s,,(c) --f(c)]
i (2/w1)
+ (E/3) < E.
< co and let g be a real function defined and bounded Let-cooom<( on [o(, p]. For x and x f (k t I)h in [OI,/3] consider the (k + I)th difference
It is 0 if g coincides in [a, ,8] with a polynomial t 2 0 set
A well-known
of degree -
tool we shall use is the fact [14, p. 1041 that
implies that w,c+l(g; cr, p; t) = 0 for every t > 0.
ON SATURATION
WITH
493
SPLINES
Let 0 < j h j < t, 0 < x < 1, 0 < x + (k + 1)h < 1. Following [Ill, [12] and [13], let n, be the largest positive integer 12for which the closed line-segment L joing x to x + (k + 1)h lies in some interval [(j - 1)/n, j/n], 1 (6n&l. For otherwise if say 1: C I, = [Go- II/no,jo/qJ, 1 < j, < no, then L would lie either in one of the two closed halves of 1, or in its (open) middle third. Tn each case the maximality of no is contradicted. Extend the definition of s,~ (if no :, 1) so as to be continuous in pro. Since So: coincides there with a polynomial of degree Gk, we have Am%,,, = 0 and hence
!O~+“f(x)l= / dPl[f(x) - s,,(x)]1< “f (” 1: ‘) j f(s t j/z) - sn,(x+ j/2)1 j=O < 2”+1 (j,~l)y~
I f(v)
- &z,(u>l g 2ki~1%*l~~+1
< E,J12(k t 1) I h \I”“’ -5 q(t) tk+l, where
70) = W(k + l)lkil n>16mr)t,-I en- 0,
as
t-+0+0.
Thus for every t > 0, wk+l(f, 0, 1; t)/tk+l .< T)(t) and therefore w k+l(f; 0, 1; t) = 0 for every t > 0. Hence Ak”‘f(x) whenever 0 < x < 1, 0 < x + Ck + 1)k <, 1 and therefore Ahif
= 0
whenever
= 0
j > k + 1, 0 SZ~x -g 1, 0 < x -t-j/z < 1. (1)
This together with the continuity offin [0, l] imply that f coincides there with a polynomial of degree
f
(+)(I
) xj(1 - q-j
[L&J(O)]
(‘i’ ) xi,
II :~= 1, 2 )...)
j=O
converges uniformly
tof(x)
in [0, 11. By (1)
B,(x) E $ [&J(O)]
(‘,
xj,
II = k -i- I, k + 2 ,..,r
60
and hencef(x) degree
= lim,,,
B,(x) must coincide in [0, l] with a polynomial
of
494
0. SHISHA REFERENCES
1. J. H. AHLBERG, E. N. NILSO> AND J. L. WALSH, Math. Sot. 129 (1967), 391-413. 2. J. H. AIILBERG,
Applications,”
Complex cubic splints,
Twrs.
Anw.
E. Iv. NELSON AND J. L. WALSH, “The Theory of Splines and Theit Academic Press, New York and London, 1967. Local spline approximation by moments, J. Moth. M&I. 16 (1967),
3. Cr. BIRKHOFF, 987-990. 4. R. DEVOR~ AND F. RICHARDS, Saturation and inverse theorems for spline approximation, i~r “Spline Functions and Approximation Theory.” pp. 73 -82, Birkhluser Verlag. Base1 and Stuttgart, 1973. 5. R. DEVOKE AND F. RICHAKDS, The degree of approximation by C‘hebyshevian splines. Tram. Amrr. Math. Sot. 181 (19731, 401-41X. 6. D. GAIER, Saturation bei Spline-Approximation und Quadratut. Nunrrr. .Cfut/t. 16 (1970), I29 -140. University of Toronto Press, Toronto. 7. G. G. LORE~~L. “Bernstein Polynomials,” 1953. 8. F. RICIIARDS, On the saturation class for spiine functions, PWC. ~It~rer.. Math. SW. 33 (1972). 471476. 9. K. SCI-IERER, On the best approximation of continuous function:; by splines. SIAM J. Nwner. Anal. 7 (1970). 418323. 10. K. SCHEIWX, Characterization of generalized Lipschitz classes by best approximation with splines, SIAM J. Numer. Am/. 11 (1974), 283-304. I I. 0. SHISHA. A characterization of functions having Zygmund’s property, .I. Approsirrru-
tiorl Theor), 9 (1973), 395-397. 12. 0. SHISHA, Characterization of smoothness properties of functions by means of their degree of approximation by splines, J. Appro.wziatio/~ Theory 12 (I 974). 365-371, 13. 0. SHISHA, On the degree of approximation by step functions. J. Approximutio~~ Theory 12 (1974), 435-436. of Functions of a Real Variable,” Macmillan, 14. A. F. TIMA&, “Theory of Approximation New York, 1963.
OF
APPROXIMATION
THEORY
13, 491494 (1975)
Note On Saturation 0. Department
of Mathematics,
University
DEDICATED
TO
OCCASION
with
Splines
SHISHA
of Rhode Island, Kingston,
PROFESSOR OF HIS
G.
G.
SIXTH-FIFTH
There is a whole class of results stating function very closely by splines of degree essentially to a polynomial of degree .< k. theorem below for which we give a proof [ll], [12] and [13].
LORENTZ
Rhode Island 02881
ON
THE
BIRTHDAY
that if one can approximate a -G/C, then the function reduces One variant of this class is the using an argument employed in
2. Let k be an integer 30, xy = a + j(b - a) n-l,j = are II congruent subintervals real functions defined on j-1,2 ,...> n, coincide with
and let - a3 < a < b C. co. For n = 11,2,... let 0, I)...) n, SO that I,,j = (x$!!, x$“)),j = I, 2 ,..., n, of (a, b). For n = 1, Z,... let S, be the set of all [a, b] - {-YA’~), xi”),..., xl:‘> which in each In,j , a polynomial of degree 5; k.
THEOREM. Letfbe CIrealfunction definedon (a, b). Suppose,for n := 1, Z,..., there is an s, E S, such that, as n + co,
sup
.&,hl-L$‘,.r:“)
,n,) 1f(x)
. . . ..S.&
-~ s,(x)! = o(l/n~‘l).
Then f coincides on (a, b) with a polynomial of degree -.
We assume, as we may, a = 0, b = 1. Let E n=
flkil s~rn.ll-l0,lln,?ln
sup
,...,
Copyright (( 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.
640/12/4-II
1)
I .fW - sn(-x>i,
491
II -~ I, 2,...,
492
0.
SHISHA
so that E, -+ 0. Then, for II -y 1. 2. .. .. m (with s,(O) :- lim.r-o+o .rJx))
1. 2. .... HI - n, we have
as n, m ---f co. Hence s,(O) has a finite limit which we denotef(0). Then, for n = 1, 2 ,..., If(O) - s,(O)! -~:lim,,,_, ~S,,,(O)--~.x,,(O)I : ’ E,,/M~‘~~.We define similarly s,(l), n = 1, 2,..., and f( 1) and have, for II x 2, 3. ....
We prove now thatfis continuous in [0, I]. Let 0 -I c < 1 and 0 <: E < 3. Choose an integer n ‘I (3/~)ll(~‘-l) such that SUP .dErO.ll
, f(x)
- s,(s); ‘:: 1/W’ 1
~(l.ln.?~lJ,....(n-l)/sl
and such that c is not of the form j/n, 0 <.j < n,j an integer. Let jl be the integer satisfying (j, - 1)/n < c :$ .jJn, 1 X j, :S n. There is a 8 > 0 such thatif/ h / < Gandc -C h E ((j, -- I)in,.j,/n), then / s,(c t h) - s,(c)1 < c/3. For such an I?, 1f(c -L h) - f(c)1 ::. / f(~ - II) - S,(C c 17);-t j S,(C -t h) - s,(c)/
-‘- I .s,,(c) --f(c)]
i (2/w1)
+ (E/3) < E.
< co and let g be a real function defined and bounded Let-cooom<( on [o(, p]. For x and x f (k t I)h in [OI,/3] consider the (k + I)th difference
It is 0 if g coincides in [a, ,8] with a polynomial t 2 0 set
A well-known
of degree -
tool we shall use is the fact [14, p. 1041 that
implies that w,c+l(g; cr, p; t) = 0 for every t > 0.
ON SATURATION
WITH
493
SPLINES
Let 0 < j h j < t, 0 < x < 1, 0 < x + (k + 1)h < 1. Following [Ill, [12] and [13], let n, be the largest positive integer 12for which the closed line-segment L joing x to x + (k + 1)h lies in some interval [(j - 1)/n, j/n], 1
!O~+“f(x)l= / dPl[f(x) - s,,(x)]1< “f (” 1: ‘) j f(s t j/z) - sn,(x+ j/2)1 j=O < 2”+1 (j,~l)y~
I f(v)
- &z,(u>l g 2ki~1%*l~~+1
< E,J12(k t 1) I h \I”“’ -5 q(t) tk+l, where
70) = W(k + l)lkil n>16mr)t,-I en- 0,
as
t-+0+0.
Thus for every t > 0, wk+l(f, 0, 1; t)/tk+l .< T)(t) and therefore w k+l(f; 0, 1; t) = 0 for every t > 0. Hence Ak”‘f(x) whenever 0 < x < 1, 0 < x + Ck + 1)k <, 1 and therefore Ahif
= 0
whenever
= 0
j > k + 1, 0 SZ~x -g 1, 0 < x -t-j/z < 1. (1)
This together with the continuity offin [0, l] imply that f coincides there with a polynomial of degree
f
(+)(I
) xj(1 - q-j
[L&J(O)]
(‘i’ ) xi,
II :~= 1, 2 )...)
j=O
converges uniformly
tof(x)
in [0, 11. By (1)
B,(x) E $ [&J(O)]
(‘,
xj,
II = k -i- I, k + 2 ,..,r
60
and hencef(x) degree
= lim,,,
B,(x) must coincide in [0, l] with a polynomial
of
494
0. SHISHA REFERENCES
1. J. H. AHLBERG, E. N. NILSO> AND J. L. WALSH, Math. Sot. 129 (1967), 391-413. 2. J. H. AIILBERG,
Applications,”
Complex cubic splints,
Twrs.
Anw.
E. Iv. NELSON AND J. L. WALSH, “The Theory of Splines and Theit Academic Press, New York and London, 1967. Local spline approximation by moments, J. Moth. M&I. 16 (1967),
3. Cr. BIRKHOFF, 987-990. 4. R. DEVOR~ AND F. RICHARDS, Saturation and inverse theorems for spline approximation, i~r “Spline Functions and Approximation Theory.” pp. 73 -82, Birkhluser Verlag. Base1 and Stuttgart, 1973. 5. R. DEVOKE AND F. RICHAKDS, The degree of approximation by C‘hebyshevian splines. Tram. Amrr. Math. Sot. 181 (19731, 401-41X. 6. D. GAIER, Saturation bei Spline-Approximation und Quadratut. Nunrrr. .Cfut/t. 16 (1970), I29 -140. University of Toronto Press, Toronto. 7. G. G. LORE~~L. “Bernstein Polynomials,” 1953. 8. F. RICIIARDS, On the saturation class for spiine functions, PWC. ~It~rer.. Math. SW. 33 (1972). 471476. 9. K. SCI-IERER, On the best approximation of continuous function:; by splines. SIAM J. Nwner. Anal. 7 (1970). 418323. 10. K. SCHEIWX, Characterization of generalized Lipschitz classes by best approximation with splines, SIAM J. Numer. Am/. 11 (1974), 283-304. I I. 0. SHISHA. A characterization of functions having Zygmund’s property, .I. Approsirrru-
tiorl Theor), 9 (1973), 395-397. 12. 0. SHISHA, Characterization of smoothness properties of functions by means of their degree of approximation by splines, J. Appro.wziatio/~ Theory 12 (I 974). 365-371, 13. 0. SHISHA, On the degree of approximation by step functions. J. Approximutio~~ Theory 12 (1974), 435-436. of Functions of a Real Variable,” Macmillan, 14. A. F. TIMA&, “Theory of Approximation New York, 1963.