Intnat. J. Mh. & Math. Sci.
Vol.
407
(1978) 407-420
UNIFORM APPROXIMATION BY INCOMPLETE POLYNOMIALS
E. B. SAFF*
Department of Mathematics
University of South Florida
Tampa, Florida 33620 U.S.A.
R. S. VARGA* *
Department of Mathematics
Kent State University
Kent, Ohio 44242 U.S.A.
(Received June 5, 1978)
ABSTRACT.
For any
e
with 0
<
incomplete polynomials of type
e
< i,
,
it is known
that, for the set of all
n
i.e
[p(x)
> akxk
s
’n},
to have
k=s
the Weierstrass property on
[a@,
I],
2
--<
it is necessary that
ae --< 1.
In this paper, we show that the above inequalities are essentially sufficient
as wel I.
KEY WORDS AND PHRASES.
Incomplete polynomials, Weierstrass property, uniform
convergence.
AMS(MOS) SUBJECT CLASSIFICATION.
41AI0 primary; 41A30 secondary.
The research of this author was conducted as a Guggenheim Fellow, visiting at
the Oxford University Computing Laboratory, Oxford, England.
Research supported in part by the Air Force Office of Scientific Research
under Grant AFOSR-74-2729, and by the Department of Energy under Grant
EY-76-S-02-2075.
408
1.
E. B. SAFF & R.S. VARGA
INTRO JCTION.
At the Conference on Rational Approximation with Emphasis on Applications of
Pad
Approximants, held December 15-17, 1976 in Tampa, Florida,
Professor G. G. Lorentz introduced new results and open questions for
incomplete polynomials, defined as follows.
e
number with 0 <_
< I.
n
k=s
akxk,
.incomplete polynomial of type
set of all incomplete polynomials of type
degree, and that when
addition.
be any given real
Then, a real or complex polynomial of the form
p(x)
is said to be an
e
Let
if
s-> ’n.
Note that the
contains polynomials of arbitrary
> 0, this collection is not closed under ordinary
This set, however, is closed under ordinary multiplication.
For such incomplete polynomials, we have, combining recent results,
THEOREM I.I.
0 <
_< I, let
[Pn. (x)}i=l
respective types
[2],
(Lorentz
i’
and Saff-Varga
[4]).
For any fixed
e
with
be a sequence of incomplete polynomials of
where lim inf
11Pn. (x)
-<
Pno (x)
O, uniformly
l->t>0.
If
E [0,I], all i>l
M for all x
and lim deg
m’
Pn.
th en
on every closed subinterval of
[0,@2).
(1.2)
Furthermore, (1.2) is best possible in the sense that, for each @ with
0 <
type
<_
I, there
is a sequence
[n. (x)}i=l
satisfying (I.I) and a sequence
ln. (i)l
M for all i _> I.
of incomplete polynomials of
[i}i= I
with lim
Hence, the interval
[0,82)
zero in (1.2) cannot be replaced by any larger interval
i
2
for which
of convergence to
[0,
t2+)
for >0.
UNIFORM APPROXIMATION BY POLYNOMIALS
For generalizations of Theorem I.i, see [4] and
In Lorentz
(0 <
< I)
[2],
is said to have the Weierstrass
with
pni
[5].
the set of all incomplete polynomials of fixed type
every continuous function f defined on
Pn.l(x)}i=l’
409
[ate,l]
[a,l].
if, for
there exists a sequence
an incomplete polynomial of type
which converges uniformly to f on
[a,l]
property on
for all
I,
i----_
Evidently, from (1.2), a necessary
,
condition that the set of all incomplete polynomials of a fixed type
0 <
< I, has the Weierstrass property on
[a@,l]
is that
82 _< a@ < i.
(1.3)
The main purpose of this paper is to show that the condition (1.3) is
The outline of the paper is as follows.
essentially sufficient as well.
In
2,
we state our new results and comment on their sharpness and their
relation to known results in the literature.
results are then given in
2.
The proofs of these new
3.
STATEMENTS OF NEW RESULTS.
As our first result, we have
THEOREM 2.1.
[0, I]
For any fixed @ with 0 < 8 < I, let F be any continuous
which is not an incomplete polynomial of type 8.
Then,
a necessary and sufficient condition that F be the uniform limit on
[0, I]
function on
of a sequence of incomplete polynomials of type 8, is that
F(x)
0 for all 0 _< x _<
82.
(2.1)
As an application of Theorem 2.1, fix any @ with 0 < 8 < I and consider
any continuous function
on
[0,
2]
and on
[2+c,
on
I],
[0, i]
with
]IIIL=[0,1 ]
where 0 < c < I
2.
i and with
For
vanishing
> 0, there
410
E. B. SAFF & R. S. VARGA
n
exists, using l’neorem 2.1, an incomplete polynomial
lln IIL[O,I ] <
its maximum absolute value on
the sequence
[0, I]
e2.
portion (cf.
n
+ ].
assumes
Thus,
of incomplete polynomials, each of
type t, cannot tend uniformly to zero in
0 < e < 1-
[e2, 2
in the interval
[(n(X)/llnl]L=[0,1])J}j=l
with
sufficiently small, that
for
which implies
e
of type
[t 2
2
+]
for any
with
This observation then gives a different proof of the sharpness
[4])
of Theorem I.I.
We also remark that the sufficiency of
Theorem 2.1 improves a related result of Roulier
[3,
Theorem
4]
concerning
Bernstein polynomials.
From Theorem 2.1, the following is deduced.
real numbers such that 0 <
function f on
[
.
For any @ with 0 < @ < I, let
THEOREM 2.2.
,I],
< t for all
i
there exists a sequence
[@i}i= I
be any sequence of
1.
Then, for any continuous
Pn
(x)}i=l’
with each P
n.
an
incomplete polynomial of type t i, such that
P
n.
(x)
and such that the sequence
f(x), uniformly
[Pn. (x)}i=l
on
[@2
I]
(2.2)
is uniformly bounded on
[0,i].
In the case of major interest in Theorem 2.2, i.e., when t.
t as
’, we remark that the result of Theorem 2.2 is best possible in the
2
following sense
If [a,b D [t
1 with [a b [-t 2 l, then there are,
i
[a,b]
continuous functions on
[a,b]
by a sequence
of type
i’
which cannot be uniformly approximated on
[en. (x)}i=l,
where t.
e
as i
with each
.
en.
an incomplete polynomial
As other consequences of Theorems 2.1 and 2.2, we have
COROLLARY 2.3.
function f on
[@2,1].
For any @ with 0 <
< I, consider any
Then, for any q with i
q <
=,
continuous
there exists a
411
UNIFORM APPROXIMATION BY POLYNOMIALS
sequence [P
n.
(x)} i=l
with each P
n.
an incomplete polynomial of type
such that
llf-Pn.
e
@2 ,i]
and such that the sequence
[
1
If(t)-P n.
[Pn. (x)izl
is uniformly bounded on
of type @ is dense in the Banach space L
l)j
[0, i].
For any @ with 0 < @ < I, the set of incomplete polynomialB
COROLLARY 2.4.
If’liEq L[@2
(t)lqdt} I/q
for each q with i _< q <
q
"’[@2,1j
(with respect to the norm
=.
For any @ with 0 < @ < I, the set of incomplete
COROLLARY 2.5.
polynomials of type @ is dense in the space of continuous functions on
[@2
+ C, I] (with
0<<I
respect to the norm
e2
11.[1.==[2+,
3)
for every
The sharpness remarks following Theorem 2.2 similarly apply to the
results of Corollaries 2.3-2.5.
To conclude this section, we remark that Corollary 2.5 leaves as an
open question whether or not each continuous function f on
f(e
2)
# 0
[@2
I]
with
is the uniform limit of incomplete polynomials of type @.
attempting to settle this question, consider the special case of @
and f(x)
m
i on
[,I
inf[ii I
I].
m
x
In
I
Setting
i
gm (x)I L=[[,
I]
gm
i s a polynomial of degree
m]
a modified Remez algorithm was used to produce the following partial
numerical results, rounded to three decimal, where
alternation point in
[,i_.,
I] for
each
m--> I.
m
denotes the least
412
E. B. SAFF & R. S. VARGA
c
.220
.625
7
.304
.353
.i61
8
.3’07
.344
.279
.94
.43
.336
.402
9
-I0
.309
.289
.311
.330
’296
.380
’II
.313
.326
.’300
.365
-12
.314
.321
:13
.316
.317
e’s
are,
m
It is interesting to note that the
in this partial listing,
monotone increasing with m.
3.
PROOFS.
PROOF OF THEOREM 2.1.
_
[0,I]
Let F be any continuous function
which is
not an incomplete polynomial of type 8, and assume that F is the uniform
Then, (2.1)
limit of a sequence of incomplete polynomials of type 8.
follows from (1.2) of Theorem
I.I, establishing the necessity of (2.1).
For sufficiency, let n be any positive integer with n o
o
If
y
(I-
)-I.
denotes the integer part of the real number y, let
n-i
S (x)’=
n
k=nS
k
a x
k
fn >_ n o
(3.1)
be the (unique) least squares approximation to the constant function 1
on
[0, I],
i.e.,
I
n-I
n
0
(I-S n(t)
d
(i-
inf
a
k=n
k
tk)2d
a
k
is rea
Next, set
x
Qn(X)’=
S
0
n(t)
n-i
Sk
k=nS
(k+l)
at
k+l
x
fn >_ n O
(3.2)
413
UNIFORM APPROXIMATION BY POLYNOMIALS
Note that
Qn’
which is of degree at most n, is an incomplete polynomial
type 0 for all
n->
no,
(n0 +
since
From the bf6ntz theory of best
(cf. Cheney [i, p.
196])
i) _>
L2-approximation
on
[0,I],
it is known
that
n0
n-
n
qj
(I +qj)’
j=l
(3.3)
where
n0 +
qj
Since the
n0/n,
qj’s
I,
j
I, 2, "’’,
j
n0.
n
are consecutive integers, the product in (3.3) telescopes to
whence
I
n
0
(I -S n(t))
02
on the interval
[Qn(X)}n=n 0
[02,1].
arbitrary real number satisfying 0 <
02
x
-
Qn(X)
so that
_<
+
<
0
converges uniformly to the
02.
be an
From (3.2), we have
X
Qn(02
) +
02_
(I-S n(t))dt,
i
ISn(t)
dt
(3.4)
For this purpose, let
2_
Ix-02-Qn(X)
as n
0
n
We now show that the sequence
function x
.
g
2d
+
l-S
n(t) Idt,
fx
[02
E
I]
Applying the Cauchy-Schwarz inequality to the last integral, then
02-
llx-02"Qn(X)IIL[02
for all n > n
o
1
]_< +
Clearly, since
0
n
I
ISn(t) Idt+ (I+
III-SnlIL2[0,1]
1
I
0) 2
2_
0 as n-
(l-Sn)-dt
from
(3.4),
(3.5)
414
E. B. SAFF & R. S. VARGA
it follows that there is a constant M such that
IISnlIL2 [0,I] -<
Vn
M
n
o
Next, note that each S n (x) from (3.1) is an incomplete polynomial of
type nS/(n-l), and nS/(n-l)
8 as n -. Hence, using the more
general
of Theorem I.I (cf. Saff and Varga
L2-version
the discussion of
(2.4")])
S (x)
n
[4,
Thm. 2.2 and
gives that
0 uniformly on
[0,82-e],
-.
as n
(3.6)
Furthermore, on writing
i
2
82_e
i
(l-Sn(t))2dt
(I-S
0
n(t))2dt
(i -S
0
n(t))2dr
and applying (3.4) and (3.6), we obtain
(I-S
n(t))2dt 82
(02-e)
e.
(3.7)
Consequently, from (3.5)- (3.7).
I
lim sup
n-
llx-e2-Qn(X)llL.[82 i]
e + 0 +
(l+e-82) 2 /--,
and as e was arbitrary, then
lira
We next show that
0
x
82
it follos
I1=- e 2 -Qn(=)llL.[ e 2
Qn(X)
rom
0 uniformly on
the definition of
Cauchy-Schwrz inequality that
1]
O.
(3.8)
[0,82].
Qn
in
For any x with
(3.2) and the
UNIFORM APPROXIMATION BY POLYNOMIALS
415
2
.
0
Sn(t)dtl --<
S2(t)d
n
ISn(t) Idt -<
0
2
0
fx E
ISn(t) Idt
[0
whence
2
2])2 2.
_<
S
0
2(t)dt.
n
(3.9)
But
n
0
02
02
02
2
n
0
and as the last integral is just
0
0
2Qn(02)
S (t)dt,
n
from (3.2), then
2
S
0
Since
0
2
(l-S n(t)) d t
2(t)dt _<
n
(l-Sn)
0
2
0 as n
2Qn(02).
+
dt
from (3.4) and since
(3.to)
Qn ‘02,
from (3.8), it follows from (3.9) and (3.10) that
as n
lim
n-
llQnllq=E0 02]
Thus, on defining the continuous function L on
[0,I]
0<x<O
0
(3.11)
0.
by
2
e(x)
_02
2 < x < I,
we see from (3.8) and (3.11) that
lim
n
IlL(x)
qn(X)llLEO ]
o.
To extend (3.12), we nxt aert that any continuoea fumction G(x)
on
[0,I]
with
(3.12)
E. B. SAFF & R. S. VARGA
416
0,
O<_x<_8
2
(3.13)
G(x)
(x),
82
x <_
with
P(82)
p(82)
where P is any polynomial
0,
[0,I]
can be uniformly approximated on
Because
I,
by incomplete polynomials of type 8.
0, we can write
m
P(x)
bkxk(x 82).
(3.14)
Setting
en’= llx 2
Qn(X)llL[82,1]
Vn >_
n0,
it follows that
]Ixk(x-82" Qn(X))llL[82, I] <-en
Next, set B
max{Ibkl"
0 <_ k <
m}.
k=0’ I, 2,
Since the case B
0 of our assertion
is trivial, assume B > 0 and let 6 be an arbitrary positive number.
en
0 as n
from
(3.8), there
(3.15)
fn>n
0._
exists a positive integer N>_ n
O
Since
such
that
6
e n <_
(m+l) B
Vn > N.
(3.16)
Then, for the polynomial P(x) of (3.14), we have from (3.15) and 3.16) that
m
liP( x
k=0
bkxk QN+m-k(x) [I L[
2
I]
(3.17)
UNIFORM APPROXIMATION BY POLYNOMIALS
m
Next, we claim that R(x):
k--O
bkxkQN+m_k(X)
is an incomplete polynomial of
Indeed, its degree is at most N + m, and as
type 8.
incomplete polynomial of type 8, then each product
0 of order at least k
has a zero at x
k
+ (N+m-k)8.
is an
in this sum
But as
is an incomplete
Thus, as 6 > 0 was arbitrary, it follows from (3.17)
polynomial of type 8.
P(82)
that any polynomial P(x) with
[8 2,1] by
QN+m_k(x)
XkQN+m_k(X)
(N+m)8 + k(l-8) >_ (N+m)8, then R(x)
+ (N+m-k)8
417
0 can be uniformly approximated on
a sequence of incomplete polynomials of type 8.
Next, as it is
evident from (3.11) that
m
I
lira
then
bkxk QN+m-k (x)llL [0,82]
5
G(x) of (3.13) can be uniformly approximated on [0,I] by a sequence
of incomplete polynomials of type 8.
Now, for an arbitrary function F(x), continuous on [0,i] with
F(x)
0 on
82],
[0,
let
Un(X)
[82,
uniform approximation to F on
E n -0 as n-
.
Clearly,
F(x)
Since
(un(x)
(Un(X)
n(82))
u
continuous extension to
be the polynomial of degree n of best
i].
..Un(82)l
En: llF-UnIIL[82, i]’ then
.fUn(8 2) F(82)I < En’ whence
If
Un(82))llL[82, I] <- 2En,
Vn_> 0.
(3.18)
82 call Un (x) its
on [0, 82 ] for all n_> 0.
is a polynomial vanishing at
[0, i]
with
Un(X)
0
Thus, from (3.18),
IIF UnlIL=[0,1] <_ 2En
Vn> 0.
The previous discussion shows that there is an incomplete polynomial Pn
of type 8, for every n > 0, such that
IIUn PnlILoo[O,I] I-,n
(3.19)
E. B. SAFF & R. S. VARGA
418
whence, with (3.19),
IIF- Pnl IL[0,1]
Since E
n
0 as n-
approximated on
,
0 I
by
[P n (x)} n=0
(3.20)
where each P (x) is an incomplete
n
I
PROOF OF THEORM 2.2.
@n}n__O
Vn> 0
this proves (cf. (2.1)) that F(x) can be uniformly
polynomial of type O.
Since
,
+I
<_ 2 E
n
2
f(x) on [8 I]
Consider any continuous function
< @ n < @ for all
is any sequence of real numbers with 0
n_> 0, extend f continuously for each n to [0,i], by means of
f(x)o,)(x-0 )/(O[82,1],
x 6
f (x)"
n
n
0
Note that
2
2
<f(@
x
n’llf!L[0,1]’ ’LIII02’ ,I]
2
-0n)
[n}n=0
with
n>
for all
0
f
n
n
n> 0, and that each
of type O
Pnll Loo[O
I]
n
<
fn
satisfies
Applying Theorem 2.1, for any
0 for all n _> 0 and lim
Pn(X)
incomplete polynomial
IOn,2 02],
6
[O,en2].
the hypotheses of Theorem 2.1 with 0
sequence
X
?In
0, there is an
such that
n
Vn > 0
(3.21)
which implies that
llf
PnlIL[0 2 I] <- !Ifn PnlIL [0 I] <- n
Consequently, (2.2) holds.
It also follows from (3.21) that
IlPnlL=,[O,:I.] <--tl fnll[O,].]
that
pn }n=O
Vn_> O.
+’r] n <_
are uniformly bounded on
tl [e
[0,I].
+
qn
Vn> 0,
UNIFORM APPROXIMATION BY POLYNOMIALS
[8 2,1],
[a,b]
Pn.(X)}i=l
I, and suppose there
f(x)
take
f(x) uniformly on [a,b].
(x)
no
[a,b.
uniformly bounded on
If 0 < a < 6)
Pn.(a)
by Theorem i.i,
i
2
n.
PROOF OF COROLLARY 2.3.
n_> 0 and lim
define f
+ 6 n <_ I
n
on
f (x)’=
n
n
x
I/q <
n
[82+6 n’
16)
2+6 n )" (x’6)2)/n
x6 [o,e2],
is continuous on
[0,I]
6
this
But then,
]n/2,
with
,
n>
0 ior all
choose
for every n
I],
x
[6)2,6)2+6 n
and satisfies the hypotheses of
Note, moreover, that
Theorem 2.1.
is
.
by means of
f(x),
so that f
__[n]n=0
For any sequence
211IL[2 I;.6
such that
[0,I]
where
6).
since 8.
for any fixed q with 1 <_ q <
O, and
ln
6)i
2
0 # f(6) ).
(8)
rescaling that P
sequence
Similarly, if b > I, we deduce by
f(a).
0
with
[ 5, Prop. i-,
then from
[0,I]
i]
fPn (x)}i=l
Clearly
sequence is necessarily uniformly bounded on
2
exists a
of incomplete polynomials of respective types
such that P
with 8
[6)2
a,b
To prove the sharpness of Theorem 2 2, let
419
fnlIL[0,1] <_ IL[8 2, I]"
Now,
+n
I f’fnll Lq[8 2, i]
f(t)-fn(t) lqd
_<
211I e [@2, i] "61/qn < n/2"
8
there is an incomplete polynomial P of type
n
Applying Theorem 2.1 to fn
8 such that
llfn enl]L[O,l] < ]n/2,
PnIILq[8 2,1] < ?In/2"
IIfn
proving (2.3).
]n/2
+
Moreover,
flaILs[8 2,1]’
bounded on
[0,i].
which also implies that
Thus, by the triangle inequality,
since
IIPnlIL[0,1] <IIfn-PnllL[0,1] +IIfnilL[0,! <
it is clear that the sequence
l
Ilf-PnIIeq[82,1<n,
{Pn(X)}n=0
is uniformly
420
E. B. SAFF & R. S. VARGA
PROOF OF COROLLARY 2.4.
As an abvious consequence of the fact that the
.[e2 ,I]
continuous functions are dense in L
q
for any q
i, Corollary 2.4
then follows directly from Theorem 2.1 and Corollary 2.3.
PROOF OF COROLLARY 2.5.
to any continuous function on
With
[2
@
e.’=
1
for all i
I, simply apply Theorem 2.2
+ ,I, where 0 <
ACKNOWLEDGMENT
We wish to thank Mr. M. Lachance (University of South Florida) for
having made the calculations which produced the numbers in the tables.
REFERENCES
I.
Cheney, E. W.
1966.
2.
Lorentz, G. G.
Introduction to Approximation Theory, McGraw-Hill, New York,
Approximation by incomplete polynomials (problems and
results), Pad and Rational Approximations" Theory and. Applications
(E. B. Saff and R. S. Varga, eds.), pp. 289-302, Academic Press, Inc.,
New York, 1977.
3.
Roulier, J. A.
4.
Saff, E. B. and R. S. Varga
5.
Saff, E. B. and R. S. Varga On incomplete polynomials, Proceedings of the
Oberwolfach Conference, Numerische Methoden der Approximationentheorie,
(L. Collatz, G. Meinardus, and H. Werner, eds.), held November 14-19,
1977 (to appear).
Permissible bounds on the coefficients of approximating
polynomials, J. Approximation Theory 3(1970), 117-122.
The sharpness of Lorentz’s theorem on
incomplete polynomials, Trans. Amer. Math. Soc. (to appear).