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Simultaneous approximation of a set
JOURNAL
OF APPROXIMATION
THEORY
Simultaneous
21, 205-208
Approximation
CHARLES Computer
(1977)
Science Department, London, Ontario, Communicated
of a Set
B. DUNHAM University of W’esrcwOntorio, Canada N6A 5B9 by R. Bojanic
Received September 30, 1974
.A family in a linear space is to be simultaneously approximated by a finitedimensional linear subspace with respect to a norm. Existence of a best approximation is established. Characterization and uniqueness results are obtained. Discretization of the family can be employed to get a best approximation.
Let F be a linear space with norm i ~. Let {& ,..., $,I be a linearly independent subset of .F and define
The problem of approximation find a parameter A to minimize
is: For a given nonempty
subset F of .F’,
Such a parameter A is called best and L(A) is called a best approximation to F. This is a generalization of the classical problem of linear approximation with respect to a norm, in which F consists of a single element5 The omitted proofs of results in this paper use arguments similar to the classical ones for the case of F a single element. One of the first to consider this general approximation problem was Golomb [7, pp. 264ff]. Tt has also been considered by Laurent and Tuan [S] and for F a pair by Goel et al. [6]. A related approximation problem is that of approximation by all of 9, the problem of centers [9]. A case of special interest has been the case in which iI ~!is the Chebyshev norm. The first widely distributed result on this was by the author [5]. Laurent and Tuan cite two neglected earlier papers by Golomb and Remez. Diaz and McLaughlin extended the author’s results in [2] and studied complex linear Chebyshev approximation in [3]. Nonlinear simultaneous Chebyshev approximation has been studied by Blatt [I]. 205 Copyright All rights
C 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSS
0021-9045
The
problem
a function let
of
simultaneous
(weakly)
II be a function
on
on
.Y
F
set
the
value
much it
better
seems
case
than value
we
have
unknown, approximation This
of
wish
of
is the
only
and
F be as in the
for
138:
It can there
be shown
is no
of
ol example.
1
1 experimentally
we may by
family
know
and
the
a function
Family of
P simultaneously.
the
In
the
was taken
measurement
function
F‘
Y only.
,g of X, and
i\
\imultaneou>
for
Chebyshev
functions of
X
preceding
this
of
approximation
fewer
Y by
variables.
a function
paragraph.
problem
is
of
Suppose of .I’. Let
a \ii~e
s be
an
263
tf:
then
also
given
in
17. pp.
9, p. X3]. loss
3 routine
computing
c>(F, ./I)
that
of generality
c(E‘,
A).
where
F is the
closure
of F. Hence
F closed.
in considering
CJ(F. .) is a c.on~:c’.r,funt.tion.
COROLLARY. Use
studying
For
approach.
by
approximant
Motivation
the when
suitable
/I, a function
.r(-.
h is approximated
possible
approximate
approslmation
/I is measured
a multivalued
variables
in
parameters.
to measure,
parameter
to
8, p.
that
approximate the
is particularly
several
\-
case
case
effectively
approach
function
the
the to
the
.I%).
~1 is difficult
11. In
appropriate
that
In
parameter
or
us define
h(.v.
1 ,:‘,,: J’ F Yi. of the
arise\
a parameter
.L’. Let
,g,,(.u) and
approximation
dependent
best
for
minimizing
a convex
approximations.
In
function
some
cases
IS a general
approach
this
only
is
the
to known
approach. Let
M(F,
Proof. there is
;I
,4) :
j f’: if’
Sufficiency coefficient
L(A)1
CJ(F. A);.
is obvious. vector B
Suppose such that
1-l B)
f
.4 is e(M(F,
not
bsst to A)), B)
Hence f‘
L.( A )
/.
$I( F, .-l).
.21(F, A). c( ,4f(F.
Then 4).
,!I).
SIMULTANEOUS
By continuity
there is an open cover CJof M(F, A) such that
(9 llf(ii)
207
APPROXIMATION
W)II < Ilf-
L(A)II,
fE lJ
for 0 < h < I and suchf,
If U contains all of F, A is not best to F. Otherwise let V = F nonempty. As U is open, V is a closed set. Let
U be
71= SUP{llf - woII:f E VI, then as V is compact, there is f E V for which the supremum is attained and since M(F, A) n V is empty, 77 < e(F, A). There exists p > 0 such that u II W - 4 < 4F, -4 - rl. We have for f E V, 0 < X < p,
/If - L(hB i- (1 - h)A)Il < IIf - L(A)lI + II WB -- A)ll (17 + e(F, A) - 7 = e(F, A). Combining
this with (I) we have for all h in (0, min{p, I}] ~if -
L(AB + (1 - $W
< e(F, A),
f c F,
and by compactness, e(F, hB -t (1 - A)A) < e(F, A), proving the theorem. It is a consequence of the theorem that if M(F, A) consists of a single element f and L(A) is a unique best approximation to f, L(A) is a unique best approximation to F. THEOREM.
Let 11jl be strict and F compact. Then a best approximation
is
unique. Let us now consider some special norms. The L, norms are strict for I < p < cc and are covered by the above theorem. The Chebyshev norm is not strict and nonuniqueness can occur. The papers [l; 51 should be consulted. In the case of the L, norm, it appears that nonuniqueness can occur easily. THEOREM.
(fi ,fi}, fi < fi . If = J-G - -WN = JCh -.fXL then
Let J be the integral on X. Let F =
fl G L(A) Gfi> and .f(W) -fJ L(A) is a best L, approximation
to F.
In the case where fi and fi are different constants and L is a polynomial approximating function of degree > 1, there are many such A’s. Carroll [4]
208
CHARLES
for best simultaneous L, approximations
b
B. UU\l1A\I
to be
A case of particular interest is where F,, is a sequence of subset\ of /: \uch that for all f E F, there is a sequence ;,f;,; -h,f; ,f;, (~F,, In this case it ib seen that [F,,; ---f F and the theorem can be applied. The case where if;,; is a sequence of finite sets is of practical interest, as computing a best approximation relative to a set F of functions is easier if F is finite. The process of replacing approximation of infinite F by approximation of finite F-, is ;I process of discwtixtion, which is used extensively in many other approximation problems. KEFERENCES i.
7. 3. 1. 5. 6.
7.
P. BLATT, “Nicht-Lineal-c Gleichmlssigc Sini~ilranapproximation.” IXahcv approximation of c~ set of bounded complex-valued functions, J. App,pl.o.vim~fio~~ T/~~wr~ 2 (1969). 4 19-432. M. P. CARROLL, Simultaneous f., approximation of ;I compact set of real-\alucd functions, NNIJI~. Mar/r. 19 (1972), 1 IO-1 15. C. B. DUUHAM. Simultaneous Chebyshev approximation of functions on an intw\:il. Pwc~. Anw. Math. Sm. 18 (1967). 472~~477. D. S. C&FL, A. Horr.~~un, C. N4SIM, AND B. F4. SAH\I~L, On best simultancou\ &ii>proximation in normed linear spaces, Cut~rcrrl. M~rir. Brrll. 17 (1974). 523 -527. h’l. GOLOMD, Lectures on Theory of Approximation, Argonne National Laborator), t-l.
1962.
8. P. J. LA~~RENT AXO P. D. TuA~\, Global approximation 01‘ a compact >et by clemcn;\ of a convex set in a normed space. Numw. Mar/~. 15 (I 9701. 137-l 50. 9. J. I). WARD, Chebyshev centcls in spaces of continuou\ functions. Pati;ic J. .\lrri/l. 52 (1974), 283-287. F. DF.I;TSCH, revieu
of 6, #I 3556, .Mntl?. fircieir,.~ 51 ( I970), 1901. I I. A. S. HOLLAND, B. N. SAHK~.Y. ASD J. TZIMBALARIO. On best ~im~~ltaneous tion, J. Apptwximafio/r T/7cor~~ 17 (1976), 187-181;. 11. A. BRONDSTED, A note on best simultaneous approximation in narlwd C‘uwtrrl. ,Wnf/r. B/d/. 19 (1976). ?S’,- 360.
10.
appro\inw linear
space,,
OF APPROXIMATION
THEORY
Simultaneous
21, 205-208
Approximation
CHARLES Computer
(1977)
Science Department, London, Ontario, Communicated
of a Set
B. DUNHAM University of W’esrcwOntorio, Canada N6A 5B9 by R. Bojanic
Received September 30, 1974
.A family in a linear space is to be simultaneously approximated by a finitedimensional linear subspace with respect to a norm. Existence of a best approximation is established. Characterization and uniqueness results are obtained. Discretization of the family can be employed to get a best approximation.
Let F be a linear space with norm i ~. Let {& ,..., $,I be a linearly independent subset of .F and define
The problem of approximation find a parameter A to minimize
is: For a given nonempty
subset F of .F’,
Such a parameter A is called best and L(A) is called a best approximation to F. This is a generalization of the classical problem of linear approximation with respect to a norm, in which F consists of a single element5 The omitted proofs of results in this paper use arguments similar to the classical ones for the case of F a single element. One of the first to consider this general approximation problem was Golomb [7, pp. 264ff]. Tt has also been considered by Laurent and Tuan [S] and for F a pair by Goel et al. [6]. A related approximation problem is that of approximation by all of 9, the problem of centers [9]. A case of special interest has been the case in which iI ~!is the Chebyshev norm. The first widely distributed result on this was by the author [5]. Laurent and Tuan cite two neglected earlier papers by Golomb and Remez. Diaz and McLaughlin extended the author’s results in [2] and studied complex linear Chebyshev approximation in [3]. Nonlinear simultaneous Chebyshev approximation has been studied by Blatt [I]. 205 Copyright All rights
C 1977 by Academic Press, Inc. of reproduction in any form reserved.
ISSS
0021-9045
The
problem
a function let
of
simultaneous
(weakly)
II be a function
on
on
.Y
F
set
the
value
much it
better
seems
case
than value
we
have
unknown, approximation This
of
wish
of
is the
only
and
F be as in the
for
138:
It can there
be shown
is no
of
ol example.
1
1 experimentally
we may by
family
know
and
the
a function
Family of
P simultaneously.
the
In
the
was taken
measurement
function
F‘
Y only.
,g of X, and
i\
\imultaneou>
for
Chebyshev
functions of
X
preceding
this
of
approximation
fewer
Y by
variables.
a function
paragraph.
problem
is
of
Suppose of .I’. Let
a \ii~e
s be
an
263
tf:
then
also
given
in
17. pp.
9, p. X3]. loss
3 routine
computing
c>(F, ./I)
that
of generality
c(E‘,
A).
where
F is the
closure
of F. Hence
F closed.
in considering
CJ(F. .) is a c.on~:c’.r,funt.tion.
COROLLARY. Use
studying
For
approach.
by
approximant
Motivation
the when
suitable
/I, a function
.r(-.
h is approximated
possible
approximate
approslmation
/I is measured
a multivalued
variables
in
parameters.
to measure,
parameter
to
8, p.
that
approximate the
is particularly
several
\-
case
case
effectively
approach
function
the
the to
the
.I%).
~1 is difficult
11. In
appropriate
that
In
parameter
or
us define
h(.v.
1 ,:‘,,: J’ F Yi. of the
arise\
a parameter
.L’. Let
,g,,(.u) and
approximation
dependent
best
for
minimizing
a convex
approximations.
In
function
some
cases
IS a general
approach
this
only
is
the
to known
approach. Let
M(F,
Proof. there is
;I
,4) :
j f’: if’
Sufficiency coefficient
L(A)1
CJ(F. A);.
is obvious. vector B
Suppose such that
1-l B)
f
.4 is e(M(F,
not
bsst to A)), B)
Hence f‘
L.( A )
/.
$I( F, .-l).
.21(F, A). c( ,4f(F.
Then 4).
,!I).
SIMULTANEOUS
By continuity
there is an open cover CJof M(F, A) such that
(9 llf(ii)
207
APPROXIMATION
W)II < Ilf-
L(A)II,
fE lJ
for 0 < h < I and suchf,
If U contains all of F, A is not best to F. Otherwise let V = F nonempty. As U is open, V is a closed set. Let
U be
71= SUP{llf - woII:f E VI, then as V is compact, there is f E V for which the supremum is attained and since M(F, A) n V is empty, 77 < e(F, A). There exists p > 0 such that u II W - 4 < 4F, -4 - rl. We have for f E V, 0 < X < p,
/If - L(hB i- (1 - h)A)Il < IIf - L(A)lI + II WB -- A)ll (17 + e(F, A) - 7 = e(F, A). Combining
this with (I) we have for all h in (0, min{p, I}] ~if -
L(AB + (1 - $W
< e(F, A),
f c F,
and by compactness, e(F, hB -t (1 - A)A) < e(F, A), proving the theorem. It is a consequence of the theorem that if M(F, A) consists of a single element f and L(A) is a unique best approximation to f, L(A) is a unique best approximation to F. THEOREM.
Let 11jl be strict and F compact. Then a best approximation
is
unique. Let us now consider some special norms. The L, norms are strict for I < p < cc and are covered by the above theorem. The Chebyshev norm is not strict and nonuniqueness can occur. The papers [l; 51 should be consulted. In the case of the L, norm, it appears that nonuniqueness can occur easily. THEOREM.
(fi ,fi}, fi < fi . If = J-G - -WN = JCh -.fXL then
Let J be the integral on X. Let F =
fl G L(A) Gfi> and .f(W) -fJ L(A) is a best L, approximation
to F.
In the case where fi and fi are different constants and L is a polynomial approximating function of degree > 1, there are many such A’s. Carroll [4]
208
CHARLES
for best simultaneous L, approximations
b
B. UU\l1A\I
to be
A case of particular interest is where F,, is a sequence of subset\ of /: \uch that for all f E F, there is a sequence ;,f;,; -h,f; ,f;, (~F,, In this case it ib seen that [F,,; ---f F and the theorem can be applied. The case where if;,; is a sequence of finite sets is of practical interest, as computing a best approximation relative to a set F of functions is easier if F is finite. The process of replacing approximation of infinite F by approximation of finite F-, is ;I process of discwtixtion, which is used extensively in many other approximation problems. KEFERENCES i.
7. 3. 1. 5. 6.
7.
P. BLATT, “Nicht-Lineal-c Gleichmlssigc Sini~ilranapproximation.” IX
1962.
8. P. J. LA~~RENT AXO P. D. TuA~\, Global approximation 01‘ a compact >et by clemcn;\ of a convex set in a normed space. Numw. Mar/~. 15 (I 9701. 137-l 50. 9. J. I). WARD, Chebyshev centcls in spaces of continuou\ functions. Pati;ic J. .\lrri/l. 52 (1974), 283-287. F. DF.I;TSCH, revieu
of 6, #I 3556, .Mntl?. fircieir,.~ 51 ( I970), 1901. I I. A. S. HOLLAND, B. N. SAHK~.Y. ASD J. TZIMBALARIO. On best ~im~~ltaneous tion, J. Apptwximafio/r T/7cor~~ 17 (1976), 187-181;. 11. A. BRONDSTED, A note on best simultaneous approximation in narlwd C‘uwtrrl. ,Wnf/r. B/d/. 19 (1976). ?S’,- 360.
10.
appro\inw linear
space,,