ETNA
Electronic Transactions on Numerical Analysis.
Volume 24, pp. 103-107, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
WEIERSTRASS’ THEOREM IN WEIGHTED SOBOLEV SPACES
WITH DERIVATIVES: ANNOUNCEMENT OF RESULTS ANA PORTILLA , YAMILET QUINTANA , JOSE M. RODRIGUEZ , AND EVA TOURIS Abstract. We characterize the set of functions which can be approximated by smooth functions and by polynomials with the norm
!"$#
for a wide range of (even non-bounded) weights % ’s. We allow a great deal of independence among the weights
% ’s.
Key words. Weierstrass’ theorem, weight, Sobolev spaces, weighted Sobolev spaces
AMS subject classifications. 41A10, 46E35, 46G10
1. Introduction. If & is any compact interval, Weierstrass’ Theorem says that
the largest set of functions which can be approximated by polynomials in the norm
if we identify, as usual, functions which are equal almost everywhere.
In [29] and [24] we study the same problem with the norm ,2-/(43+ defined by
5768579 2:<;>=@?A
ess supBDCFEHG
6
')(*&+
is
,
,.-/(0&1+
(4IJ+7G 3K(4IJ+JL
where 3 is a (bounded or unbounded) weight, i.e. a non-negative measurable function.
Considering weighted norms ,M-/(*3+ , has been proved to be interesting mainly because
of two reasons: on the one hand, it allows to wider the set of approximable functions (since
the functions in ,M-/(*3+ can have singularities where the weight
tends to zero); and, on the
6
other hand , it is possible
to
find
functions
which
approximate
whose
qualitative behaviour
6
is similar to the one of at those points where the weight tends to infinity.
A
If 3
(432N1L7OPO7O7LQ3.RS+ is a vectorial weight, we study this approximation problem with
the Sobolev norm T RVU -W(43+ defined by
57685PX
" :Y;>= ?ZA
[
R
\Q] N
^
^ 6
: =
\ ^ 9
^ :<;
=
O
The papers [27], [28], [29], [30], [31], [1], [32], [24] and [25] are the beginning of a
theory of Sobolev spaces with respect to general measures for _a`cbd`fe . This theory
plays an important role in the location of the zeroes of the Sobolev orthogonal polynomials
(see [19], [20], [28] and [30]). The location of these zeroes allows to prove results on the
asymptotic behaviour of Sobolev orthogonal polynomials (see [19]).
g
Received November 30, 2004. Accepted for publication February 3, 2005. Recommended by J. Arvesú.
Departamento de Matemáticas, Universidad Carlos III de Madrid, 30 Avenida de la Universidad, 28911 Leganés
(Madrid), SPAIN (apferrei, jomaro, etouris@math.uc3m.es). Research partially supported by a
grant from DGI(BFM 2003-04870), Spain. Ana Portilla and Jose M. Rodriguez are also partially supported by a
grant from DGI(BFM 2003-06335-C03-02), Spain.
Departamento de Matemáticas Puras y Aplicadas, Edificio Matemáticas y Sistemas (MYS), Apartado Postal:
89000, Caracas 1080 A, Universidad Simón Bolı́var, Caracas, Venezuela (yquintana@usb.ve).
103
ETNA
Kent State University
etna@mcs.kent.edu
104
A. PORTILLA, Y. QUINTANA, J. M. RODRIGUEZ, AND E. TOURIS
6
2. Main results. The fundamental results of this paper guarantee that a function be: = of polynomials (respectively, smooth functions) in the norm
longs to the closure of the space
6 \
RVU
T
-W(*3+ ,if and only if,
belongs to the closure of polynomials (respectively, smooth
functions) in the norm ,2-/(43 \ + , for every hi`kjl`nm .
This article is an abridged version of the paper [26].
A
(43 N L7OPO7OPL3 R + on o p@LQq>r satisfying:
T HEOREM 2.1. Let be a vectorial weight 3
(i) suv t _FwF3.RKxye .
\
\ † F‡
(ii) 3 \{z ,|}<- ~Q (o p@LQq>r€ƒ‚F„ L7OPO7OPLQ„
+L for every h)`ˆjlxnm .
\“
BŒ
3 \ (4I‰+8Š s ‹ _wŽ(_@a3 \Q‘ 
+ Š `c’ , a.e. in some neighborhood of „ , for every ”
_ `
Š
ŠŠ
“
B Œ"Ÿ1 Š
•
V
R
ž
`—– \ , h˜`™jš`ym›œ ,… and 3.RVž (*I‰H
+ Šs ‹
_FwF3.R‰Š `¡’ , a.e. in some neighborhood of „
Š
Š
•
Š
Š
for every _{`
`¢–WRVž .
Then the closure of the space of polynomials in T RVU -/(*3+ is
9 :Y; =7¨
: =
£ … ?A¥¤ 6 z
? 6 R z ¦l§
RFU
- (43+
T
, - (*3 R +
O
(iii)
,
R EMARK 1.
(i) We observe that this theorem characterizes the closure of ¦©§ T RVU -ª(43+ in T RVU -/(43+ ,
in terms of the similar problem in , - (*3ƒRD+ . This question of approximation in , - (43.RD+ is
solved in [24].
A
e
(ii) The hypothesis (ii) is not restrictive at all, since if ess lim sup BF« ‹ 3 \ (4IJ+
for
n
z
¬
an infinite number of points „
, for some hk`­j™x®m , then h is the only polynomial in
RVU
,|-/(*3 \ + , and it is trivial to find the closure of the space of polynomials in T
-W(*3+ .
BŒ
\ (4I‰+8Š s ‹ _Fw(_|a3 \Q‘ S
+ Š `n’ , is much weaker than
Š
Š
BŒ
Š
Š
3 \ (4IJ+ Š s ‹ _wF3 \Q‘
are allowed to be h .
Š `¡’ , since some 3 \Q‘
Š
Š
Š
Š
(iv)
The
possibility
of some 3 \ to be bounded is, naturally, allowed. That is to say,
\
\†
‡
‚V„ L7O7OPO¯LQ„
might be the empty set.
Sketch of the …proof. It is obvious that the closure of the space of polynomials in T RFU -W(43+
£
is contained in
.
£
(iii) Notice that hypothesis, 3
Then, it suffices to prove that every function
in … can be approximated by polynomials
6
‡
z £
= (43+ . Let us consider then,
and ‚Qb‰° ° a sequence of polynomials
in the norm T 6 RV: U -W
‡
R
converging to
in the norm ,M-/6 (43.RD+ . From the sequence ‚b‰° ° we will construct another
RVU
-W(*3+ .
one of polynomials converging to in the norm T
The key idea in order to carry out such a process, is to find, from b±° , a polynomial ²P°U R in
³
³
, where is the space of polynomials which have a primitive of order m in T RFU -W(43+ . If ¦
³
were a Hilbert space ³ and
a closed subspace, it would suffice to take as ²°U R the orthogonal
projection of b ° on . However, since our norms do not come from an inner product, the
problem is much more complicated; fortunately, we can find a finite set of polynomials ´ in
, - (*3 R + , such that ² °U R can be expressed as a linear combination of b ° and elements of ´ .
A
D EFINITION
2.2. We say that a vectorial
weight 3
(43.N1L7O7OPO7LQ3ƒRD+ in o „‰Lµr is of type _
if _wF3 R z , (o „‰Lµr4+ and 3 N L7OPO7OPL3 RVž z ,M-ª(o „¶L"µr4+ .
In the following theorems we describe the closure of smooth functions in Sobolev spaces
with weights.
A
T HEOREM 2.3. Let us consider a vectorial weight 3
(*3.N!LPO7OPO¯L3.RD+ of type _ in a
A
RVU
RFU
compact interval &
o „‰Lµr . Then the closure of ¦”§ T
-/(0&LQ3+ , '
-/( ¬ + § T
-W(*&¶L3+
R ¬
RVU
RFU
§
and ' ( +
T
-W(*&¶L3+ in T
-W(*&¶L3+ are, respectively,
ETNA
Kent State University
etna@mcs.kent.edu
105
WEIERSTRASS’ THEOREM IN WEIGHTED SOBOLEV SPACES WITH · DERIVATIVES
£
?ZA
¤ 6
z
¤ 6
£)¹{?ZA
T
T
z
-
RVU
z
£)º{?ZA®¤ 6
RVU
: =
? 6 R z
(*&¶L3+
: =
? 6 R z
(*&¶L3+
-
RVU
T
(*&¶L3+
: =
R z
? 6
-
¦l§
, -
'
(0&1+ §
-
')(0&1+ §
(0&LQ3 R +
, , -
9 :<¸ ; = ¨
U
L
9 :Y¸ ; = ¨
U
(*&¶L3.RD+
9 :Y¸ ; = ¨
U
(*&¶L3.RS+
L
O
R EMARK 2.
(i) We observe that Theorem 2.3 characterizes the closure of ' R ( ¬ + §
RVU
RVU
RVU
RVU
T
-W(0&LQ3+ , '
-W( ¬ + § T
-W(*&¶L3+ and ¦ª§ T
-/(*&¶L3+ in T
-W(0&LQ3+ , in terms of the
similar problem in ,M-/(*&¶L3.R!+ . This question of approximation in ,2-ª(*&¶L3.R!+ is solved in
[24].
R
(ii) If 3 R z ,M-ª(*&+ , then the closure of ' ( ¬ + , ¦ and '
-W( ¬ + are the same. This is a
consequence of Bernstein’s proof of Weierstrass’ Theorem (see e.g. [5, p. 113]), which gives
a sequence of polynomials converging uniformly up to the m -th derivative for any function in
R
'
(*&+ .
Sketch of the proof. The inclusion
X
R
'
( ¬ + §
T
RFU
(*&¶L3+
:Y¸ >
; =¼»
U
-
6
£
£iº
º
is obvious. Let6 : us= now consider a function z
, and let ½ z ')(
R
approximates
in ,|-/(0&LQ3 R + norm. If we consider the function
¾
RFž
[
?A
(*I‰+
: =
6 \
\Q] N
(*I¿›k„Ž+
(*„Ž+
\
B
—Á
jÀ
‹
½>(4Â+
(*Il›ˆÂ+
¬ +
, be a function which
RVž
(0mi›¡_V+ÀÄÃ
Â|L
6
¾
it is possible to show that approximates in T RFU -/(*&¶L3+ norm.
D EFINITION 2.4. We say that… Å©LƉL are comparable functions in the set Ç if there exists
ž
Ŕ`¡Æl`n’Å a.e. in Ç .
a positive constant ’ such that ’
A
D EFINITION 2.5. We say that
weight 3
(43.N!LPO7OPO7L3.RS+ in o „‰Lµr is of type œ
a vectorial
¹
º
if there exist real numbers
xn„
x¡„
xn„!… È
`yµ such that
„¿`—„
(i) _FwF3 R z ,… (Qo „ LQ„ È r4+ , and 3 N L7OPO7O7LQ3 RVž z ,M-É(Qo „‰L"µrÊ+ ,
¹
(ii) if „kxĄ , then 3 \ is comparable to a finite non-decreasing weight in o „‰LQ„ rËL for
hi`kjl`nm ,
º
(iii) if „1Ȕx̵ , then 3 \ is comparable to a finite non-increasing weight in o „ L"µrËL for
hi`kjl`nm .
A
T HEOREM 2.6.A Let us consider a vectorial weight 3
(*3.N!LPO7O7O7L3.RD+ of type œ in a
R ¬
RFU
RVU
o „¶L"µr . Then the closure of '
( + § T
-W(*&¶L3+ in T
-”(*&¶L3+ is
compact interval &
£
È
?A
¤ 6
z
T
RVU
-
(*&¶L3+
? 6
: =
\
z
')(0&1+ §
6
£
, -
(*&¶L3 \ +
9 :Y¸ ;
U
¨
=
for
h˜`™jš`nm
O
Sketch of the proof. Given a function z
È , we can split
it as a sum of three functions
by using an appropriate partition of unity. The function in o „ LQ„1È7r can be approximated with
similar arguments than the measures of type _ . In the approximation of the functions in
¹
º
o „‰L„ r and o „ Lµ"r we use shift arguments that allow us to construct a convolution with an
approximation of the identity.
ETNA
Kent State University
etna@mcs.kent.edu
106
A. PORTILLA, Y. QUINTANA, J. M. RODRIGUEZ, AND E. TOURIS
The next theorem makes the results of this paper more valuable since it allows to deal
with weights which can be obtained by “gluing” simpler ones.
‡
T HEOREM 2.7. Let us consider strictly
increasing sequences
of real numbers ‚V„¶° ,
‘
‡
ž
‚Fµ¯°
(Í belonging to a finite set, to Î , Î
or Î ) with µ°ž xτ° ‘ xе° for every Í . Let
A
?AÒÑ
3
(*3MN1L7OPO7OPL3.RS+ be a vectorial weight … in the interval
°>o „°JLµ°1r . Assume that for
&
each Í there exists an interval &P°™ÓÌo „° ‘ Lµ°1r with 3 L7O7OPO7LQ3ƒR z ,M-/(*&¯°¶+ . Let us assume
also that for each Í we have either 3 is of type _ in o „°>L"µ°!r , or _FwF3.R z ,M-/(o „1°JL"µ°1r4+ . Then
the closure of ' R (0&1+ § T RVU -/(0&LQ3+ in T RVU -/(0&LQ3+ is
?A¥¤ 6
£
&
T
º
RFU
z
T
-
(*&¶L3+
? 6
: =
R z
')(0&1+ §
, -
(*&¶L3 R +
9 :Y¸ ; =±¨
U
We can deduce the following consequence.
A
(43 N L7OPO7O7LQ3
T HEOREM 2.8. Let us consider … a vectorial weight 3
R
, with 3 z ,M}Y- ~ (0&1+ and _wF3.R z , }Y~ (0&1+ . Then the closure of ' (*&+
RVU
-W(0&LQ3+ is
£
º
?A
¤ 6
z
T
RFU
-
(*&¶L3+
? 6
: =
R z
')(0&1+ §
, -
(*&¶L3 R +
§
R +
T
9 :Y¸ ; = ¨
U
O
in the interval
RVU
-”(*&¶L3+ in
O
Proof. This theorem is a direct consequence of Theorems 2.3 and 2.7. It is enough
to
‘
ž
o
„
L
µ
r
Í
Î
Î
Î
split & as … a union of
compact
intervals
(
belonging
to
a
finite
set,
to
,
or
),
°
°
xԄ ° ‘
xÕµ ° for every
Í . We have that 3 is of type _ in each o „ ° … Lµ ° r , since
with µ °ž
?ZA
3 z ,|-/(o „ ° Lµ ° r4+ and _wF3 R z ,
(Qo „ ° Lµ ° rÊ+ for every Í . If we choose & °
o „ ° ‘ Lµ ° r , then
we can apply Theorems 2.3 and 2.7.
REFERENCES
[1] V. A LVAREZ , D. P ESTANA , J. M. R ODR ÍGUEZ , AND E. R OMERA , Weighted Sobolev spaces on curves, J.
Approx. Theory, 119 (2002), pp. 41–85.
[2] A. B RANQUINHO , A. F OULQUI É , AND F. M ARCELL ÁN , Asymptotic behavior of Sobolev type orthogonal
polynomials on a rectifiable Jordan curve or arc, Constr. Approx., 18 (2002), pp. 161–182.
[3] R. C. B ROWN AND B. O PIC , Embeddings of weighted Sobolev spaces into spaces of continuous functions,
Proc. R. Soc. Lond. A, 439 (1992), pp. 279–296.
[4] A. C ACHAFEIRO AND F. M ARCELL ÁN , Orthogonal polynomials of Sobolev type on the unit circle, J. Approx.
Theory, 78 (1994), pp. 127–146.
[5] P. J. D AVIS , Interpolation and Approximation, Dover, New York, 1975.
[6] B. D ÖELLA
×8Ø # Ø$ÙV ECCHIA , G. M ASTROIANNI , AND J. S ZABADOS , Approximation with exponential weights in
, J. Math. Anal. Appl., 272 (2002), pp. 1–18.
[7] W. N. E VERITT AND L. L. L ITTLEJOHN , The density of polynomials in a weighted Sobolev space, Rendiconti
di Matematica, Serie VII, 10 (1990), pp. 835–852.
[8] W. N. E VERITT, L. L. L ITTLEJOHN , AND S. C. W ILLIAMS , Orthogonal polynomials in weighted Sobolev
spaces, in Lecture Notes in Pure and Applied Mathematics, 117, Marcel Dekker, 1989, pp. 53–72.
[9] W. N. E VERITT, L. L. L ITTLEJOHN , AND S. C. W ILLIAMS , Orthogonal polynomials and approximation in
Sobolev spaces, J. Comput. Appl. Math., 48 (1993), pp. 69–90.
[10] A. F OULQUI É , F. M ARCELL ÁN , AND K. PAN , Asymptotic behavior of Sobolev-type orthogonal polynomials
on the unit circle, J. Approx. Theory, 100 (1999), pp. 345–363.
[11] J. H EINONEN , T. K ILPEL ÄINEN , AND O. M ARTIO , Nonlinear potential theory of degenerate elliptic equations, Oxford Science Publ., Clarendon Press, 1993.
[12] A. I SERLES , P. E. K OCH , S. P. N ORSETT, AND J. M. S ANZ -S ERNA , Orthogonality and approximation in
a Sobolev space, in Algorithms for Approximation, J. C. Mason and M. G. Cox, eds., Chapman & Hall,
London, 1990.
[13] A. I SERLES , P. E. K OCH , S. P. N ORSETT, AND J. M. S ANZ -S ERNA , Weighted Sobolev spaces and capacity,
J. Approx. Theory, 65 (1991), pp. 151–175.
[14] T. K ILPEL ÄINEN , Weighted Sobolev spaces and capacity, Ann. Acad. Sci. Fennicae, Series A. I. Math., 19
(1994), pp. 95–113.
ETNA
Kent State University
etna@mcs.kent.edu
WEIERSTRASS’ THEOREM IN WEIGHTED SOBOLEV SPACES WITH · DERIVATIVES
107
[15] A. K UFNER , Weighted Sobolev Spaces, Teubner Verlagsgesellschaft, Teubner-Texte zur Mathematik (Band
31), Leipzig, 1980, also published by John Wiley & Sons, New York, 1985.
[16] A. K UFNER AND B. O PIC , How to define reasonably weighted Sobolev spaces, Commentationes Mathematicae Universitatis Caroline, 25 (1984), pp. 537–554.
[17] A. K UFNER AND A. M. S ÄNDIG , Some Applications of Weighted Sobolev Spaces, Teubner Verlagsgesellschaft, Teubner-Texte zur Mathematik (Band 100), Leipzig, 1987.
[18] D. S. L UBINSKY , Weierstrass’ theorem in the twentieth century: a selection, Quaestiones Mathematicae, 18
(1995), pp. 91–130.
[19] G. L ÓPEZ L AGOMASINO AND H. P IJEIRA , Zero location and Ú -th root asymptotics of Sobolev orthogonal
polynomials, J. Approx. Theory, 99 (1999), pp. 30–43.
[20] G. L ÓPEZ L AGOMASINO , H. P IJEIRA , AND I. P ÉREZ , Sobolev orthogonal polynomials in the complex plane,
J. Comp. Appl. Math., 127 (2001), pp. 219–230.
[21] V. G. M AZ ’ JA , Sobolev spaces, Springer-Verlag, New York, 1985.
[22] B. M UCKENHOUPT , Hardy’s inequality with weights, Studia Math., 44 (1972), pp. 31–38.
[23] A. P INKUS , Weierstrass and approximation theory, J. Approx. Theory, 107 (2000), pp. 1–66.
[24] A. P ORTILLA , Y. Q UINTANA , J. M. R ODR ÍGUEZ , AND E. T OUR ÍS , Weierstrass’ Theorem with weights, J.
Approx. Theory, 127 (2004), pp. 83–107.
[25] A. P ORTILLA , Y. Q UINTANA , J. M. R ODR ÍGUEZ , AND E. T OUR ÍS , Weighted Weierstrass’ theorem with first
derivatives, preprint.
[26] A. P ORTILLA , Y. Q UINTANA , J. M. R ODR ÍGUEZ , AND E. T OUR ÍS , Weierstrass’ theorem in weighted
sobolev spaces with · derivatives, Rocky Mt. J. Math., to appear.
[27] J. M. R ODR ÍGUEZ , V. A LVAREZ , E. R OMERA , AND D. P ESTANA , Generalized weighted Sobolev spaces
and applications to Sobolev orthogonal polynomials I, Acta Appl. Math., 80 (2004), pp. 273–308.
[28] J. M. R ODR ÍGUEZ , V. A LVAREZ , E. R OMERA , AND D. P ESTANA , Generalized weighted Sobolev spaces and
applications to Sobolev orthogonal polynomials II, Approx. Theory and its Appl., 18 (2002), pp. 1–32.
[29] J. M. R ODR ÍGUEZ , Weierstrass’ theorem in weighted Sobolev spaces, J. Approx. Theory, 108 (2001),
pp. 119–160.
[30] J. M. R ODR ÍGUEZ , The multiplication operator in Sobolev spaces with respect to measures, J. Approx. Theory, 109 (2001), pp. 157–197.
[31] J. M. R ODR ÍGUEZ , Approximation by polynomials and smooth functions in Sobolev spaces with respect to
measures, J. Approx. Theory, 120 (2003), pp. 185–216.
[32] J. M. R ODR ÍGUEZ AND V. A. YAKUBOVICH , A Kolmogorov-Szegö-Krein type condition for weighted
Sobolev spaces, Indiana U. Math. J., 54 (2005), pp. 575–598.
[33] W. R UDIN , Real and Complex Analysis, second edition, McGraw-Hill, 1974.