Weierstrass' Theorem in Weighted Sobolev Spaces

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1 Joural of Approximatio Theory 108, (2001) doi: jath , available olie at o Weierstrass' Theorem i Weighted Sobolev Spaces Jose M. Rodr@ guez 1 Departmet of Mathematics, Uiversidad Carlos III de Madrid, Aveida de la Uiversidad, 30, 28911, Legae s, Madrid, Spai jomaromath.uc3m.es Commuicated by Doro S. Lubisky Received February 10, 2000; accepted March 31, 2000; published olie Jauary 18, 2001 We characterize the set of fuctios which ca be approximated by polyomials with the followig orm k & f & W k, ([a, b], w) := : j=0 & f ( j) & L ([a, b], wj ), for a big class of weights w 0, w 1,..., w k 2001 Academic Press. Key Words: Weierstrass' theorem; Sobolev spaces; weights. 1. INTRODUCTION If I is ay compact iterval, Weierstrass' theorem says that C(I) is the biggest set of fuctios which ca be approximated by polyomials i the orm L (I), if we idetify, as usual, fuctios which are equal almost everywhere. There are may geeralizatios of this theorem (see, e.g., the moograph [L]). Here we study the same problem with the orm L (I, w) defied by & f & L (I, w) :=ess sup x # I f(x) w(x), (1.1) where w is a weight, i.e., a o-egative measurable fuctio, ad we use the covetio 0 } =0. Observe that (1.1) is ot the usual defiitio of the L orm i the cotext of measure theory, although it is the correct oe whe we work with weights (see, e.g., [BO]). If w=(w 0,..., w k )isa 1 Research partially supported by a grat from DGES (MEC), Spai Copyright 2001 by Academic Press All rights of reproductio i ay form reserved.

2 120 JOSE M. RODRI GUEZ vectorial weight, we also study this problem with the Sobolev orm W k, (2, w) defied by k & f & W k, (2, w) := : j=0 & f ( j ) & L (2, w j ), where 2 := k j=0 supp w j. It is obvious that W 0, (2, w)=l (2, w). Weighted Sobolev spaces is a iterestig topic i may fields of mathematics (see, e.g., [HKM, K, Ku, KO, KS, ad T]). I [ELW1], [EL], ad [ELW2] the authors study some examples of Sobolev spaces for p=2 with respect to geeral measures istead of weights, i relatio with ordiary differetial equatios ad Sobolev orthogoal polyomials. The papers [RARP1], [RARP2], [R1], ad [R2] are the begiig of a theory of Sobolev spaces with respect to geeral measures for 1p. This theory plays a importat role i the locatio of the zeroes of the Sobolev orthogoal polyomials (see [LP, RARP2, ad R1]). The locatio of these zeroes allows us to prove results o the asymptotic behaviour of Sobolev orthogoal polyomials (see [LP]). Now, let us state the mai results here. We refer to the defiitios i the ext sectio. Throughout the paper, the results are umbered accordig to the sectio where they are proved. We deote by P k, (2, w) (k0) the set of fuctios which ca be approximated by polyomials i the orm W k, (2, w), where we idetify, as usual, fuctios which are equal almost everywhere. We must remark that the symbol P k, (2, w) has a slightly differet meaig i [RARP1], [RARP2], [R1], ad [R2]. First, we have results for the case k=0. Theorem 2.1. Let us cosider a closed iterval I ad a weight w # L (I), loc such that the set S of sigular poits of w i I has zero Lebesgue measure. The we have C (R) & L (I, w)=c(r) & L (I, w)=h, with H=[f# C(I"S) & L (I, w): for each a # S, _l a # R such that ess lim f(x)&l a w(x)=0], x # I, x a where the closures are take i L (I, w). If a # S is of type 1, we ca take as l a ay real umber. If a # S is of type 2, l a =ess lim w(x)=, x a f(x) for =>0 small eough. Furthermore, if I is compact we also have P 0, (I, w)=h. If f # H & L 1 (I), I is compact, ad S is coutable, we ca approximate f by polyomials with the orm &}& L (I, w)+&}& L 1 (I). The followig are two of the mai results for k1.

3 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 121 Theorem 5.3. Let us cosider a compact iterval I ad a vectorial weight w=(w 0,..., w k )#L (I) such that w &1 k # L 1 (I). The we have P k, (I, w)=[f: I Rf (k&1) # AC(I) ad f (k) # P 0, (I, w k )]. Theorem 5.4. Let us cosider a compact iterval I ad a vectorial weight w=(w 0,..., w k )#L (I) such that the set of sigular poits for w k i I has zero Lebesgue measure. Assume that there exist a 0 # I, a iteger 0r<k, ad costats c, $>0 such that (1) w j+1 (x)c x&a 0 w j (x) i [a 0 &$, a 0 +$] & I, for r j<k, (2) I"[a0 &=, a 0 +=] w &1 k <, for every =>0, (3) if r>0, a 0 is (r&1)-regular. The we have P k, (I, w)=[f: I Rf (k&1) # AC loc (I"[a 0 ]), f (k) # P 0, (I, w k ), _l # R with ess lim f (r) (x)&l w r (x)=0, x # I, x a 0 ess lim f ( j ) (x) w j (x)=0, x # I, x a 0 for r j<k ifr<k&1, ad f (r&1) # AC(I) if r>0]. This result gives the characterizatio of P k, (I, w) for the case of Jacobi weights (see Corollary 5.1). There are other characterizatios of P k, (I, w) with weaker hypothesis o w k (see Theorems 5.5, 5.6, ad 5.7). There are also results for weights which ca be obtaied by ``gluig'' simpler oes (see Theorems 5.1 ad 5.2). We have results for o-bouded itervals (see Theorem 6.1 ad Propositios 6.1, 6.2, ad 6.3 i Sectio 6). These results deal with the case of Laguerre, Freud, ad fast decreasig degree weights. The aalogue of Weierstrass' theorem with the orms W k, p (2, +) (with 1p< ad + a vectorial measure) ca be fouded i [RARP2] ad [R2]. Now we preset the otatios we use. Notatios. Throughout the paper k0 deotes a fixed atural umber. Also, all the weights are o-egative Borel measurable fuctios defied o a subset of R; if a weight is defied i a proper subset E/R, we defie it i R"E as zero. If the weight does ot appear explicitly, we mea that we are usig the weight 1. Give 0<m<k, a vectorial weight w ad a closed set E, we deote by W k, (E, w) the space W k, (2 & E, w E ) ad

4 122 JOSE M. RODRI GUEZ by W k&m, (2, w) the space W k&m, (2, (w m,..., w k )). We deote by supp v the support of the measure v(x) dx, i.e., the itersectio of every closed set ER verifyig R"E v=0. If A is a Borel set, A, / A, it(a), ad A deote, respectively, the Lebesgue measure, the characteristic fuctio, the iterior ad the closure of A. IfI, U are subsets of R, the symbol I U deotes the relative boudary of U i I. Byf ( j ) we mea the jth distributioal derivative of f. P deotes the set of polyomials. We say that a -dimesioal vector satisfies a oe-dimesioal property if each coordiate satisfies this property. Fially, the costats i the formulae ca vary from lie to lie ad eve i the same lie. The outlie of the paper is as follows. Sectio 2 is dedicated to the proof of the theorems i the case k=0. Sectio 3 presets most of the defiitios we eed to state the results i Sectios 5 ad 6. I Sectio 4 we collect the techical results of [RARP1], [RARP2], ad [R1] that we eed. We prove the theorems for k1 i Sectios 5 ad 6. The results for o-bouded itervals are proved i Sectio APPROXIMATION IN L (I, W) Defiitio 2.1. Give a measurable set A, we defie the essetial closure of A as the set ess cl A :=[x # R: A & (x&$, x+$) >0, \$>0]. Defiitio 2.2. If A is a measurable set, f is a fuctio defied i A with real values ad a # ess cl A, we say that ess lim x # A, x a f(x)=l # R if for every =>0 there exists $>0 such that f(x)&l <= for almost every x # A & (a&$, a+$). I a similar way we ca defie ess lim x # A, x a f(x) = ad ess lim x # A, x a f(x)=&. We defie the essetial limit superior ad the essetial limit iferior i A as follows: ess lim sup x # A, x a ess lim if x # A, x a f(x) := if $>0 f(x) :=sup $>0 ess sup x # A & (a&$, a+$) ess if x # A & (a&$, a+$) f(x), f(x). If we do ot specify the set A we are assumig that A=R. Remarks. 1. The essetial limit iferior (or superior) of a fuctio f does ot chage if we modify f i a set of zero Lebesgue measure.

5 WEIERSTRASS' THEOREM IN SOBOLEV SPACES We have ess lim sup x # A, x a f(x)ess lim if x # A, x a f(x), ess lim x # A, x a f(x)=l if ad oly if ess lim sup x # A, x a f(x)=ess lim if x # A, x a f(x)=l. 3. We impose the coditio a # ess cl A i order to have the uicity of the essetial limit. If we do ot have this coditio, the every real umber is a essetial limit for ay fuctio f. Defiitio 2.3. Give a iterval I ad a weight w i I we say that a # I is a sigularity of w (or sigular for w) ii if ess lim if x # I, x a w(x)=0. We say that a sigularity a of w is of type 1 if ess lim w(x)=0. x # I, x a I other cases we say that a is a sigularity of type 2. Remark. The set of poits which are ot sigular for w i I is a relative ope set i I. Lemma 2.1. Let us cosider a iterval I, a weight w i I, ad a poit a # I which is ot sigular for w i I. The there exists $>0 such that every fuctio i the closure of C(R) with the orm L (I, w) belogs to C(I & [a&$, a+$]). Remark. Observe that every fuctio i C(I ) ca be exteded to a fuctio i C(R); therefore, the closures of C(R) ad C(I ) with the orm L (I, w) are the same. Recall that we idetify fuctios which are equal almost everywhere. Proof. We have that sup $>0 ess if w(x)=l>0. x # I & (a&$, a+$) Therefore there exists $>0 with ess if w(x)> l x # I & (a&$, a+$) 2 >0.

6 124 JOSE M. RODRI GUEZ Hece, we have &g& L (I & (a&$, a+$), w) l 2 &g& L (I & (a&$, a+$))= l 2 max g(x), x # I & [a&$, a+$] for every g # C(R). This iequality gives the lemma, sice if f is the limit of fuctios [g ]/C(R) with the orm i L (I & (a&$, a+$), w), it ca be modified i a set of zero Lebesgue measure i such a way that it is the uiform limit of [g ] i I & [a&$, a+$]. Lemma 2.2. Let us cosider a iterval I, a weight w i I ad a sigular poit a of w i I of type 1. The every fuctio f i the closure of C(R) with the orm L (I, w) verifies Proof. ess lim x # I, x a f(x) w(x)=0. (2.1) Let us assume that (2.1) is ot true, i.e., ess lim sup f(x) w(x)=l>0. x # I, x a Therefore for every $>0 we have Sice a is of type 1 we deduce ess sup f(x) w(x)l>0. x # I & (a&$, a+$) ess lim g(x) w(x)=0, x # I & (a&$, a+$) for every g # C(R). This implies that for each g # C(R) ad =>0 there exists $>0 with Cosequetly, for this $>0 we have ess sup g(x) w(x)=. x # I & (a&$, a+$) & f& g& L (I, w)& f& g& L (I & (a&$, a+$), w) & f & L (I & (a&$, a+$), w)&&g& L (I & (a&$, a+$), w)l&=, for every =>0 ad g # C(R). Hece we have & f& g& L (I, w)l>0,

7 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 125 for every g # C(R). This implies that f ca ot be approximated by fuctios i C(R) with the orm L (I, w). Lemma 2.3. Let us cosider a iterval I ad a weight w # L (I). Deote by S the set of sigular poits of w i I. Assume that a # S is of type 1 ad S =0. The, for ay fixed =>0 ad f # C(I"S) & L (I, w) with ess lim x # I, x a f(x) w(x)=0, there exist a relative ope iterval U i I with a # U ad I U/I"S (ad U/it(I) if a # it(i)) ad a fuctio g # L (I, w) & C(U ) such that g= f i I"U, & f& g& L (I, w)<= (ad & f& g& L 1 (I)<= iff# L 1 (I)). Furthermore, we ca choose g with the additioal coditio g(a)=0 or eve g(a)=* for ay fixed * # R. Proof. Without loss of geerality we ca assume that a is a iterior poit of I, sice the case a #I is simpler. Take such that [a&1, a+1]/it(i). Sice S =0, there exist y #(a, a+1)"s ad x #(a&1, a)"s verifyig f( y ) 2 & + ess if f(x), f(x ) 2 & + ess if f(x). x #[a, a+1] x #[a&1, a] Let us defie ow the fuctio f (which is cotiuous i a ope eighbourhood of [x, y ], sice x, y S) as f (x) :={x&a x &a f(x ) if x #[x, a], x&a y &a f( y ) if x #[a, y ], f(x) if x # I"[x, y ]. Observe that f (x) 2 & + f(x) for almost every x #[x, y ]. Hece & f& f & L (I, w)=& f& f & L ([x, y ], w)2 & f & L ([x, y ], w)+2 & &w& L (I), ad this last expressio goes to 0 as, sice ess lim x # I, x a f(x) w(x)=0. If f # L 1 (I), we also have & f& f & L 1 (I)=& f& f & L 1 ([x, y ])2 & f & L 1 ([x, y ])+2 & ( y &x ), ad this expressio goes to 0 as. Observe that f (a)=0; it is easy to modify f i a small eighbourhood of a i order to have f (a)=*, for fixed * # R. This fiishes the proof of the lemma.

8 126 JOSE M. RODRI GUEZ Lemma 2.4. If A is a measurable set, we have: (1) ess cl A is a closed set cotaied i A. (2) A"ess cl A =0. (3) If f is a measurable fuctio i A _ ess cl A, a # ess cl A ad there exists ess lim x # ess cl A, x a f(x), the there exists ess lim x # A, x a f(x) ad ess lim x # A, x a f(x)= ess lim x # ess cl A, x a f(x). (4) If A >0 ad f is a cotiuous fuctio i R we have Proof. (1) is direct. & f & L (A)= sup x # ess cl A f(x). (2) is a cosequece of the Lebesgue differetiatio theorem, sice we have lim $ 0 1 2$ x+$ / A =1, x&$ for almost every x # A, ad this implies A & (x&$, x+$) >0 for almost every x # A ad every $>0. Assume ow that ess lim x # ess cl A, x a f(x)=l # R. Cosequetly, for every =>0 there exists $>0 such that for almost every x # ess cl A & (a&$, a+$) we have f(x)&l <=. Sice A"ess cl A =0, we have f(x)&l <=, for almost every x # A & (a&$, a+$). This gives (3) if l # R. The case l=\ is similar. The statemet (2) gives & f & L (A)& f & L (ess cl A) sup x # ess cl A f(x). We have f(x) & f & L (A) for almost every x # A. The f(x) & f & L (A) for every x # ess cl A, sice f is cotiuous. Therefore These two iequalities give (4). sup f(x) & f & L (A). x # ess cl A

9 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 127 Lemma 2.5. Let us cosider a iterval I, a weight w i I, ad a # I. If ess lim sup x # I, x a w(x)=l>0, the for every fuctio f i the closure of C(R) & L (I, w) with the orm L (I, w) there exists the fiite limit ess lim w(x)=, x a Proof. We have for every $>0 ad the f(x), for every 0<=<l. ess sup w(x)l>0, x # I & (a&$, a+$) [x # I & (a&$, a+$) :w(x)=] >0, for every $>0 ad 0<=<l. This implies that a belogs to ess cl A =, where A = :=[x # I : w(x)=]. If g # C(R) & L (I, w), 0<=<l, ad $>0, we have = &g& L (A = & [a&$, a+$])&g& L (A = & [a&$, a+$], w). Sice ess cl (A = & [a&$, a+$]) is a compact set ad g # C(R) & L (I, w), Lemma 2.4 (4) gives = max g(x) &g& L (A = & [a&$, a+$], w). x # ess cl (A = & [a&$, a+$]) Cosequetly, if [g ]/C(R) & L (I, w) coverges to f i L (I, w), the [g ] coverges to f uiformly i ess cl (A = & [a&$, a+$]) ad f # C(ess cl (A = & [a&$, a+$])) for every $>0. Therefore f # C(ess cl A = ). This fact ad Lemma 2.4 (3) give that, for 0<=<l, there exists ess lim x # A =, x a f(x)= ess lim x # ess cl A =, x a f(x)= lim x # ess cl A =, x a f(x). Lemma 2.6. Let us cosider a iterval I, a weight w i I, ad a sigular poit a of w i I. The every fuctio f i the closure of C(R) & L (I, w) with the orm L (I, w) verifies if (ess lim sup f(x) w(x) )=0. (2.2) =>0 w(x)<=, x a Proof. Observe first that a # ess cl([x # I : w(x)<=]) for every =>0, sice a is sigular for w i I. Let us assume that (2.2) is ot true, i.e., ess lim sup x # A = c, x a f(x) w(x)l>0,

10 128 JOSE M. RODRI GUEZ for every =>0, where A = := [x # I : w(x)=] ad A c = :=I"A =. Therefore for every =, $>0 we have ess sup x # A = c & (a&$, a+$) f(x) w(x)l>0. For each g # C(R) & L (I, w), =>0, ad $>0, we have Cosequetly &g& L (A = c & (a&$, a+$), w) = &g& L (I & (a&$, a+$))<. & f& g& L (I, w)& f& g& L (A = c & (a&$, a+$), w) ad therefore & f & L (A = c & (a&$, a+$), w) &&g& L (A = c & (a&$, a+$), w), & f& g& L (I, w)l&= &g& L (I & (a&$, a+$)), for every g # C(R) & L (I, w) ad $, =>0. Hece we obtai & f& g& L (I, w)l>0, for every g # C(R) & L (I, w). This implies that f caot be approximated by fuctios i C(R) & L (I, w). Lemma 2.7. Let us cosider a iterval I, a weight w i I, ad a # I. If ess lim x # I, x a w(x)=0 ad if =>0 (ess lim sup w(x)<=, x a f(x) w(x) )=0, the we have ess lim x # I, x a f(x) w(x)=0. Proof. For each '>0 there exist =, $ 1 >0 such that ess sup f(x) w(x)<'. w(x)<=, x # I & (a&$ 1, a+$ 1 ) We also have that there exists $ 2 >0 such that w(x)<= for almost every x # I & (a&$ 2, a+$ 2 ). If we take $ :=mi($ 1, $ 2 ), we obtai ess sup f(x) w(x) ess sup f(x) w(x)<', x # I & (a&$, a+$) w(x)<=, x # I & (a&$ 1, a+$ 1 ) ad this fiishes the proof. Lemma 2.8. Let us cosider a iterval I ad a weight w # L (I). Deote by S the set of sigular poits of w i I. Assume that a # S ad S =0. The, for ay fixed '>0 ad f # C(I"S) & L (I, w) such that

11 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 129 (a) (b) eough, if =>0 (ess lim sup w(x)<=, x a f(x) w(x))=0, there exists the fiite limit ess lim w(x)=, x a f(x), for =>0 small there exist a relative ope iterval U i I with a # U ad I U/I"S (ad U/it(I) if a # it(i)) ad a fuctio g # L (I, w) & C(U ) with g= f i I"U, & f& g& L (I, w)<' (ad & f& g& L 1 (I)<' iff# L 1 (I)). Furthermore, we ca choose g with the additioal coditio g(a)=ess lim w(x)=, x a f(x), for =>0 small eough. Proof. If a is of type 1, Lemmas 2.3 ad 2.7 give the result. Assume ow that a is of type 2. Without loss of geerality we ca assume that a is a iterior poit of I, sice the case a #I is simpler. We cosider first the case ess lim sup xa +w(x)>0 ad ess lim sup xa & w(x) >0. For each atural umber, let us choose = >0 with lim = =0 ad ess lim sup f(x) w(x)< 1 w(x)<=, x a. Let us cosider ow $ >0 with lim $ =0 ad ess sup x #(a&$, a+$ ) & A c = f(x) w(x)< 1, (2.3) where A = := [x # I : w(x) =] ad A = c := I"A =. We defie l := ess lim x # A=, x a f(x), for ay =>0 small eough. We ca take $ with the additioal property f(x)&l <1 for almost every x #(a&$, a+$ ) & A =. Let us choose # #(a, a+$ )"S ad #$ #(a&$, a)"s with f(# )&l <1 ad f(#$ )&l <1. We defie the fuctios a (x) ad b (x) i[#$, # ] as follows: ad a (x) :={l+(x&a) b (x) :={l+(x&a) mi[l, f(# )]&l # &a l+(x&a) mi[l, f(#$ )]&l #$ &a max[l, f(# )]&l # &a l+(x&a) max[l, f(#$ )]&l #$ &a if x #[a, # ], if x #[#$, a], if x #[a, # ], if x #[#$, a].

12 130 JOSE M. RODRI GUEZ Now we ca defie the fuctios g # L (I, w) & C([#$, # ]) i the followig way: a (x) if x #[#$, # ] ad f(x)a (x), g (x) :={b (x) if x #[#$, # ] ad f(x)b (x), f(x) i other case. Observe that a (x)g (x)b (x), a (x)&l <1, ad b (x)&l <1, for every x #[#$, # ]. Therefore g (x)&l <1 for x #[#$, # ] ad & f& g & L ([#$, # ] & A =, w) 2 &w& L (I). (2.4) We prove ow lim & f& g & L ([#$, # ], w)=0. The facts &g & L ([#$, # ] & A c =, w) \ l +1 + =, ad (2.3) give & f& g & L ([#$, # ] & A c =, w) < 1 + \ l +1 + =. This iequality ad (2.4) give & f& g & L ([#$, # ], w)< 1 + \ l +1 + = + 2 &w& L (I). If f # L 1 (I), we also have & f& g & L 1 (I)=& f& g & L 1 ([#$, # ]) & f & L 1 ([#$, # ])+ \ l +1 + (# &#$ ). This fiishes the proof i this case. If ess lim sup x a + w(x)>0 ad ess lim sup x a & w(x)=0, we oly eed to cosider the fuctios g for x>a ad the fuctios f i the proof of Lemma 2.3 for x<a (recall that we ca choose f with f (a)=l). The case ess lim sup x a + w(x)=0 ad ess lim sup x a & w(x)>0 is symmetric. The followig result is direct. Propositio 2.1. Let us cosider a sequece of closed itervals [I ] # 4 such that for each # 4 there exists a ope eighbourhood of I which does

13 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 131 ot itersect m{ I m. Deote by J the uio J := I. Let us cosider a weight w i J. The we have C(J) & L (J, w)=, C(I ) & L (I, w), where the closures are take i L with respect to w, i the correspodig iterval. We also have a similar result for cotiguous itervals. Propositio 2.2. Let us cosider a iterval I ad a weight w # L (I). loc Let us cosider a icreasig sequece of real umbers [a ] # 4, where 4 is either Z +, Z &, Z, or [1, 2,..., N] for some N # N such that I= [a, a +1 ] ad a is ot sigular for w i I if a is i the iterior of I. The we have C (I) & L (I, w) =C(I) & L (I, w) = { f #, # 4 C([a, a +1 ]) : f is cotiuous i each a # it(i) = = { f #, # 4 C ([a, a +1 ]) : f is cotiuous i each a # it(i) =, where the closures are take i L with respect to w, i the correspodig iterval. Remark. We ca esure C (I) & L (I, w)=c (R) & L (I, w) if I is closed. The same is obviously true for C(I) istead of C (I). Proof. The third equality is true sice C ([a, a +1 ])=C([a, a +1 ]) is a direct cosequece of Weierstrass' theorem ad w # L ([a, a +1 ]). We are goig to see that the closure of C (I) & L (I, w) ad C(I) & L (I, w) with the orm L (I, w) is the same. It is eough to prove that every f # C(I) ca be approximated by fuctios i C (I) with the orm L (I, w). We ca assume that 4=Z, sice the argumet i the other cases is simpler. Give =>0 ad f # C(I), for each # Z, there exists a fuctio g # C (R) with & f& g & L ([a 2&1, a 2+2 ], w)<=2. Let us cosider fuctios % # C (R) with % =0 i (&, a 2&1 ], % =1 i [a 2, ) ad 0% 1.

14 132 JOSE M. RODRI GUEZ We defie ow a fuctio g # C (I) byg(x) :=(1&% (x)) g &1 (x)+ % (x) g (x), if x #[a 2&1, a 2+1 ]. We have & f& g& L ([a 2&1, a 2 ], w)&(1&% )( f &g &1 )& L ([a 2&1, a 2 ], w) +&% ( f &g )& L ([a 2&1, a 2 ], w)<=2+=2==, & f& g& L ([a 2, a 2+1 ], w)=& f& g & L ([a 2, a 2+1 ], w)<=2, ad this implies & f& g& L (I, w)=. I order to see the secod equality, observe that the ideas above give that the result is true if it is true for the set 4=[1, 2, 3]. Let us cosider f # C([a 1, a 2 ])& C([a 2, a 3 ]) ad cotiuous i a 2. Give m # N there exist fuctios g 1 m # C([a 1, a 2 ]) ad g 2 m # C([a 2, a 3 ]) with & f& g 1 m & L ([a 1, a 2 ], w)+ & f& g 2 & m L ([a 2, a 3 ], w)<1m. I order to fiish the proof it is eough to costruct a fuctio g m # C([a 1, a 3 ]) satisfyig the iequality & f& g m & L ([a 1, a 3 ], w)<cm, where c is a costat idepedet of m. We kow that there exist positive costats $, c 1, ad c 2 such that [a 2 &$, a 2 +$][a 1, a 3 ], f(x)&f(a 2 ) <1m if x&a 2 $ ad 0<c &1 1 w(x)c 2 for almost every x #[a 2 &$, a 2 +$]. Lemma 2.1 gives that f # C([a 2 &$, a 2 +$]) ad the ad cosequetly f(x)&g 1 m (x) <c 1m, for every x #[a 2 &$, a 2 ], f(x)&g 2 m (x) <c 1m, for every x #[a 2, a 2 +$], g 1 m (x)&f(a 2) <(c 1 +1)m, for every x #[a 2 &$, a 2 ], g 2 m (x)&f(a 2) <(c 1 +1)m, for every x #[a 2, a 2 +$]. Let us defie g 0 m as the fuctio whose graph is the segmet joiig the poits (a 2 &$, g 1 (a m 2&$)) ad (a 2 +$, g 2 (a m 2+$)). The we have g 0 m (x)&f(a 2) <(c 1 +1)m, for every x #[a 2 &$, a 2 +$], g 0 m (x)&f(x) <(c 1+2)m, for every x #[a 2 &$, a 2 +$], &g 0 m & f & L ([a 2 &$, a 2 +$], w)<c 2 (c 1 +2)m. If we defie the fuctio g m # C([a 1, a 3 ]) by g 1 m (x), if x #[a 1, a 2 &$], g m (x) :={g 0 m (x), if x #[a 2&$, a 2 +$], g 2 m (x), if x #[a 2+$, a 3 ],

15 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 133 we have & f& g m & L ([a 1, a 3 ], w)<(c 2 (c 1 +2)+1)m. This fiishes the proof of Propositio 2.2. Propositio 2.3. Let us cosider a closed iterval I ad a weight w # L loc(i) such that the set S of sigular poits of w i I has zero Lebesgue measure. The we have C (R) & L (I, w)=c(r) & L (I, w)=h, with H := [f# C(I"S) & L (I, w): for each a # S, if (ess lim sup f(x) w(x))=0 =>0 w(x)<=, x a ad, if a is of type 2, there exists the fiite limit for =>0 small eough], ess lim w(x)=, x a f(x), where the closures are take i L (I, w). Furthermore, if I is compact we also have P 0, (I, w)=h. If f # H & L 1 (I), I is compact, ad S is coutable, we ca approximate f by polyomials with the orm &}& L (I, w)+&}& L 1 (I). Proof. Lemmas 2.1, 2.2, 2.7, 2.5, ad 2.6 give that H cotais C(R) & L (I, w). I order to see that H is cotaied i C(R) & L (I, w), assume first that I is compact; the w # L (I). Fix =>0 ad f # H. Lemma 2.8 gives that for each a # S there exist a relative ope iterval U a i I with a # U a ad I U a /I"S (ad U a /it(i) if a # it(i)) ad a fuctio g a # L (I, w) & C(U a) such that g a = f i I"U a ad & f& g a & L (I, w)<=. The set S is compact sice it is a closed set cotaied i the compact iterval I. Therefore there exist a 1,..., a m # S such that S/U a1 _ }}} _ U am. Without loss of geerality we ca assume that U a1,..., U am is miimal i the followig sese: for each i=1,..., m the set j{i U aj does ot cotai to U ai. Defie [: i, ; i ]:=U a i. Assume that we have U ai & U aj {<, with : i <: j. The miimal property gives U a i & U a j =[: j, ; i ] ad [: j, ; i ] & U ak =< for every k{i, j. We defie the fuctios g aj, a i (x) :=g ai, a j (x) := ; i&x ; i &: j g ai (x)+ x&: j ; i &: j g aj (x).

16 134 JOSE M. RODRI GUEZ Observe that g ai, a j # C([: j, ; i ]) ad satisfies g ai, a j (: j )=g ai (: j ), g ai, a j (; i ) = g aj (; i ), ad &g aj, a i & f & L ([: j, ; i ], w) " ; i&x (g ai (x)&f(x)) ; i &: j "L ([: j, ; i ], w) If we defie the fuctio g # L (I, w) & C(I) as + " x&: j (g aj (x)&f(x)) <2=. ; i &: j "L ([: j, ; i ], w) f(x) if x # I". i U ai g(x) :={g ai (x) if x # U ai, x. g ai, a j (x) if x # U ai & U aj, we have & f& g& L (I, w)<2=. If f # L 1 (I) ad S is coutable, cosider [a 1, a 2,...]=S. If we take g a with & f& g a & L 1 (I)<2 & =, it is direct that & f& g& L 1 (I)<2=. This fiishes the proof i this case. If I is ot compact, we ca choose a icreasig sequece [a ] # 4 of real umbers, where 4 is either Z +, Z &,orz such that I= [a, a +1 ] ad a is ot sigular for w i I if a is i the iterior of I. We ca take [a ] # 4 with the followig additioal property: max # 4 a =max I if there exists max I ad mi # 4 a =mi I if there exists mi I. The first part of the proof ad Propositio 2.2 give the result. We ca reformulate this result as follows. Theorem 2.1. Let us cosider a closed iterval I ad a weight w # L (I) loc such that the set S of sigular poits of w i I has zero Lebesgue measure. The we have C (R) & L (I, w)=c(r) & L (I, w)=h, with H=[f# C(I"S) & L (I, w) : for each a # S, j{i U aj _l a # R such that ess lim f(x)&l a w(x)=0], x # I, x a where the closures are take i L (I, w). If a # S is of type 1, we ca take as l a ay real umber. If a # S is of type 2, l a =ess lim w(x)=, x a f(x) for =>0 small eough. Furthermore, if I is compact we also have P 0, (I, w)=h. If f # H & L 1 (I), I is compact ad S is coutable, we ca approximate f by polyomials with the orm &}& L (I, w)+&}& L 1 (I).

17 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 135 Remark. If S is the uio of a fiite umber of itervals ad a set of zero Lebesgue measure, Theorem 2.1 ad Propositio 2.1 give C (R) & L (I, w). Proof. We oly eed to show the equivalece of the followig coditios (a) ad (b): (a) for each a # S, (a.1) if =>0 (ess lim sup w(x)<=, x a f(x) w(x) )=0, (a.2) if a is of type 2, there exists the fiite limit l a := ess lim w(x)=, x a f(x), for =>0 small eough, (b) for each a # S, there exists l a # R such that ess lim x # I, x a f(x)&l a w(x)=0. It is clear that (b) implies (a). Hypothesis (a.1) gives that for each '>0, there exist =, $>0 with & f & L ([a&$, a+$] & [w(x)<=], w)<'3 ad l a =<'3. By hypothesis (a.2) we ca choose $ with the additioal coditio & f&l a & L ([a&$, a+$] & [w(x)=], w)<'3. These iequalities imply & f&l a & L ([a&$, a+$], w)& f & L ([a&$, a+$] & [w(x)<=], w) + l a =+& f&l a & L ([a&$, a+$] & [w(x)=], w)<'. Corollary 2.1. Let us cosider a closed iterval I ad a weight w # L loc (I) such that the set S of sigular poits of w i I has zero Lebesgue measure. If f, g # C(R) & L (I, w) ad. # C(I) & L (I), the we also have f, f +, f &, max( f, g), mi( f, g),.f # C(R) & L (I, w). Proof. The characterizatio of C(R) & L (I, w) give i Theorem 2.1 or Propositio 2.3 implies the result for f ad.f. This fact ad max( f, g)= f +g+ f &g 2, mi( f, g)= f +g& f &g 2 gives the result for max( f, g) ad mi( f, g). The facts f + =max( f, 0), f & =max(&f, 0) fiish the proof. 3. PREVIOUS DEFINITIONS FOR SOBOLEV SPACES Most of our results for k1 use tools of Sobolev spaces. We iclude here the defiitios that we eed i order to uderstad these tools. First of all, we explai the defiitio of geeralized Sobolev space i [RARP1] for the particular case p= (the defiitio i [RARP1] covers

18 136 JOSE M. RODRI GUEZ the cases 1p, eve if the weights are substituted by measures). Oe ca thik that the atural defiitio of weighted Sobolev space (the fuctios f with k weak derivatives satisfyig & f ( j ) & L (w j )< for 0 jk) is a good oe; however this is ot true (see [KO] or [RARP1]). We start with some previous defiitios. Defiitio 3.1. We say that two fuctios u, v are comparable o the set A if there are positive costats c 1, c 2 such that c 1 u(x)v(x)c 2 for almost every x # A. We say that two orms &}& 1, &}& 2 i the vectorial space X are comparable if there are positive costats c 1, c 2 such that c 1 &x& 1 &x& 2 c 2 for every x # X. We say that two vectorial weights are comparable if they are comparable o each compoet. (We use here the covetio that 00=1.) I what follows the symbol a b meas that a ad b are comparable for a ad b fuctios or orms. Obviously, the spaces L (A, w) ad L (A, v) are the same ad have comparable orms if w ad v are comparable o A. Therefore, i order to study Sobolev spaces we ca chage a weight w by ay comparable weight v. Next, we shall defie a class of weights which plays a importat role i our results. Defiitio 3.2. We say that a weight w belogs to B ([a, b]) if w &1 # L 1 ([a, b]). Also, if J is ay iterval we say that w # B (J) if w # B (I) for every compact iterval IJ. We say that a weight belogs to B (J), where J is a uio of disjoit itervals i # A J i, if it belogs to B (J i ), for i # A. Observe that if vw i J ad w # B (J), the v # B (J). Defiitio 3.3. We deote by AC([a, b]) the set of fuctios absolutely cotiuous i [a, b], i.e. the fuctios f # C([a, b]) such that f(x)&f(a)= x f $(t) dt for all x #[a, b]. If J is ay iterval, AC a loc (J) deotes the set of fuctios absolutely cotiuous i every compact subiterval of J. Defiitio 3.4. Let us cosider a vectorial weight w=(w 0,..., w k ). For 0 jk we defie the ope set 0 j := [x # R : _ a ope eighbourhood V of x with w j # B (V)]. Observe that we always have w j # B (0 j ) for ay 0 jk. I fact, 0 j is the greatest ope set U with w j # B (U). Obviously, 0 j depeds o w, although w does ot appear explicitly i the symbol 0 j. It is easy to

19 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 137 check that if f ( j ) # L (0 j, w j ) with 1 jk, the f ( j ) # L 1 loc (0 j) ad f ( j&1) # AC loc (0 j ). Hypothesis. From ow o we assume that w j is idetically 0 i every poit of the complemet of 0 j. We eed this hypothesis i order to have complete Sobolev spaces (see [KO] ad [RARP1]). Remark. This hypothesis is satisfied, for example, if we ca modify w j i a set of zero Lebesgue measure i such a way that there exists a sequece : z0 with w &1 j [(:, ]] ope for every. Ifw j is lower semicotiuous, the it satisfies this coditio. The followig defiitios also deped o w, although w does ot appear explicitly. Let us cosider w=(w 0,..., w k ) a vectorial weight ad y # 2. To obtai a greater regularity of the fuctios i a Sobolev space we costruct a modificatio of the weight w i a eighbourhood of y, usig the followig versio (see a proof i [RARP1, Lemma 3.2]) of the Muckehoupt iequality (see [Mu], [M, p. 44]). This modified weight is equivalet i some sese to the origial oe (see Theorem A below). Muckehoupt iequality I. Let us cosider w 0, w 1 weights i (a, b). The there exists a positive costat c such that " b x g(t) dt "L ([a, b], w 0 )c &g& L ([a, b], w 1 ) for ay measurable fuctio g i [a, b], if ad oly if ess sup a<r<b w 0 (r) b r w &1 1 <. Defiitio 3.5. A vectorial weight w =(w 0,..., w k) is a right completio of a vectorial weight w with respect to y if w k :=w k ad there is a =>0 such that w j :=w j i the complemet of [ y, y+=] ad w j :=w j +w~ j, i [y, y+=] for 0 j<k, where w~ j is ay weight satisfyig: (i) (ii) w~ j # L ([y, y+=]), 4 (w~ j, w j+1)<, with 4 (u, v) := ess sup u(r) y+= v &1. y<r< y+= r

20 138 JOSE M. RODRI GUEZ Muckehoupt iequality I guaratees that if f ( j ) # L (w j ) ad f ( j+1) # L (w j+1 ), the f ( j ) # L (w j). Example. It ca be show that the followig costructio is always a completio: we choose w~ j :=0 if w j+1 B ((y, y+=]); if w j+1 # B ([y, y+=]) we set w~ j (x):=1i[y, y+=]; ad if w j+1 # B ((y, y+=])" B ([y, y+=]) we take w~ j (x) :=1 for x #[y+=2, y+=], ad for x #(y, y+=2). Remarks. w~ j (x) :=mi {1, \ y+= x &1 &1 w j+1+ =, 1. We ca defie a left completio of w with respect to y i a symmetric way. 2. If w j+1 # B ([y, y+=]), the 4 (w~ j, w j+1)< for ay weight w~ j # L ([y, y+=]). I particular, 4 (1, w j+1)<. 3. If w, v are two vectorial weights such that w j cv j for 0 jk ad v is a right completio of v, the there is a right completio w of w, with w jcv j for 0 jk (it is eough to take w~ j =v~ j ). Also, if w, v are comparable measures, v is a right completio of v if ad oly if it is comparable to a right completio w of w. 4. We always have w k=w k ad w jw j for 0 j<k. Defiitio 3.6. If w is a vectorial weight, we say that a poit y # R is right j-regular (respectively, left j-regular), if there exist =>0, a right completio w (respectively, left completio) of w, ad j<ik such that w i # B ([y, y+=]) (respectively, B ([y&=, y])). Also, we say that a poit y # R is j-regular if it is right ad left j-regular. Remarks. 1. A poit y # R is right j-regular (respectively, left j-regular), if at least oe of the followig properties is verified: (a) There exist =>0 ad j<ik such that w i # B ([y, y+=]) (respectively, B ([y&=, y])). Here we have chose w~ j =0. (b) There exist =>0, j<ik, :>0, ad $<i& j&1 such that w i (x): x& y $, for almost every x #[y, y+=] (respectively, [ y&=, y]). See Lemma 3.4 i [RARP1].

21 WEIERSTRASS' THEOREM IN SOBOLEV SPACES If y is right j-regular (respectively, left), the it is also right i-regular (respectively, left) for each 0i j. 3. We ca take i= j+1 i this defiitio sice by the secod remark to Defiitio 3.5 we ca choose w l=w l +1 # B ([y, y+=]) for j<l<i, if j+1<i. 4. If y is ot sigular for w j, the y # 0 j ad y is ( j&1)-regular. Whe we use this defiitio we thik of a poit [b] as the uio of two half-poits [b + ] ad [b & ]. With this covetio, each oe of the followig sets (a, b) _ (b, c) _ [b + ]=(a, b) _ [b +, c){(a, c), (a, b) _ (b, c) _ [b & ]=(a, b & ] _ (b, c){(a, c), has two coected compoets, ad the set (a, b) _ (b, c) _ [b & ] _ [b + ]=(a, b) _ (b, c) _ [b]=(a, c) is coected. We oly use this covetio i order to study the sets of cotiuity of fuctios: we wat that if f # C(A) ad f # C(B), where A ad B are uio of itervals, the f # C(A _ B). With the usual defiitio of cotiuity i a iterval, if f # C([a, b))& C([b, c]) the we do ot have f # C([a, c]). Of course, we have f # C([a, c]) if ad oly if f # C([a, b & ])& C([b +, c]), where by defiitio, C([b +, c])=c([b, c]) ad C([a, b & ])=C([a, b]). This idea ca be formalized with a suitable topological space. Let us itroduce some otatio. We deote by 0 ( j ) the set of j-regular poits or half-poits, i.e., y # 0 ( j ) if ad oly if y is j-regular, we say that y + # 0 ( j ) if ad oly if y is right j-regular, ad we say that y & # 0 ( j ) if ad oly if y is left j-regular. Obviously, 0 (k) =< ad 0 j+1 _ }}} _ 0 k 0 ( j ). Remark. If 0j<k ad I is a iterval, I0 ( j ), the the set I"(0 j+1 _ }}} _ 0 k ) is discrete (see the remark before Defiitio 7 i [RARP1]). Defiitio 3.7. We say that a fuctio h belogs to the class AC loc (0 ( j ) )ifh # AC loc (I) for every coected compoet I of 0 ( j ). Defiitio 3.8 (Sobolev space). If w=(w 0,..., w k ) is a vectorial weight, we defie the Sobolev space W k, (2, w) as the space of equivalece classes of V k, (2, w) :=[f: 2 Rf ( j ) # AC loc (0 ( j ) ) for 0 j<k ad & f ( j ) & L (2, w j )< for 0 jk]

22 140 JOSE M. RODRI GUEZ with respect to the semiorm k & f & W k, (2, w) := : j=0 & f ( j ) & L (2, w j ). Remark. If we are iterested i fuctios f with complex values, we oly eed to apply the results i this paper to the real ad imagiary parts of f. Defiitio 3.9. K(2, w) as If w is a vectorial weight, let us defie the space K(2, w) :=[g: 0 (0) Rg # V k, (0 (0), w), &g& W k, (0 (0), w) =0]. K(2, w) is the equivalece class of 0 i W k, (0 (0), w). This cocept ad its aalogue for 1p< play a importat role i the geeral theory of Sobolev spaces ad i the study of the multiplicatio operator i Sobolev spaces i particular (see [RARP1], [RARP2], [R1], [R2], ad Theorems A ad B below). Defiitio If w is a vectorial weight, we say that (2, w) belogs to the class C 0 if there exist compact sets M, which are a fiite uio of compact itervals, such that (i) M itersects at most a fiite umber of coected compoets of 0 1 _ }}} _ 0 k, (ii) K(M, w)=[0], (iii) M M +1, (iv) M =0 (0). Remarks. 1. Coditio (2, w)#c 0 is ot very restrictive. I fact, the proof of Theorem A below (see [RARP1, Theorem 4.3]) gives that if 0 (0) " (0 1 _ }}} _ 0 k ) has oly a fiite umber of poits i each coected compoet of 0 (0), ad K(2, w)=[0], the (2, w)#c The proof of Theorem A below gives that if for every coected compoet 4 of 0 1 _ }}} _ 0 k we have K(4, w)=[0], the (2, w)#c 0. Coditio 4 w 0 >0 implies K(4, w)=[0]. 3. Sice the restrictio of a fuctio of K(2, w) to M is i K(M, w) for every, we have that (2, w)#c 0 implies K(2, w)=[0].

23 WEIERSTRASS' THEOREM IN SOBOLEV SPACES TECHNICAL RESULTS I this sectio we collect the theorems we eed i order to prove the results i Sectios 5 ad 6. The ext results, proved i [RARP1], [RARP2], ad [R1], play a cetral role i the theory of Sobolev spaces with respect to measures (see the proofs i [RARP1, Theorems 4.3 ad 5.1]). We preset here a weak versio of these theorems which are eough for our purposes. Theorem A. Let w=(w 0,..., w k ) be a vectorial weight. Let K j be a fiite uio of compact itervals cotaied i 0 ( j ), for 0 j<k, ad w a right (or left) completio of w. If (2, w)#c 0, the there exist positive costats c 1 =c 1 (K 0,..., K k&1 ) ad c 2 =c 2 (w, K 0,..., K k&1 ) such that k&1 c 1 : j=0 &g ( j ) & L (K j )&g& W k, (2, w), c 2 &g& W k, (2, w )&g& W k, (2, w), \g # V k, (2, w). Theorem B. Let us cosider a vectorial weight w=(w 0,..., w k ). Assume that we have either (i) (2, w)#c 0 or (ii) 0 1 _ }}} _ 0 k has oly a fiite umber of coected compoets. The the Sobolev space W k, (2, w) is complete. Remark. See Theorem 5.1 i [RARP1] for further results o completeess. Lemma 3.3 i [RARP1] gives the followig result. Propositio A. Let w=(w 0,..., w k ) be a vectorial weight i [a, b], with w k0 # B ((a, b ]) for some 0<k 0 k. If we costruct a right completio w of w with respect to the poit a takig ==b&a, ad w j=w j for k 0 jk, the there exist positive costats c j such that k 0 c j &g ( j ) & L ([a, b], w : j ) i= j k 0 &1 &g (i ) & L ([a, b], w i )+ : i= j g (i ) (b), for all 0 j<k 0 ad g # V k, ([a, b], w). I particular, there is a positive costat c such that k 0 &1 c &g& W k, ([a, b], w )&g& W k, ([a, b], w)+ : g ( j ) (b), for all g # V k, ([a, b], w). The followig is a particular case of Corollary 4.3 i [RARP1]. j=0

24 142 JOSE M. RODRI GUEZ Corollary A. Let us cosider a vectorial weight w=(w 0,..., w k ). Let K j be a fiite uio of compact itervals cotaied i 0 ( j ), for 0 j<k. If (2, w)#c 0, the there exists a positive costat c 1 =c 1 (K 0,..., K k&1 ) such that k&1 c 1 : j=0 &g ( j+1) & L 1 (K j )&g& W k, (2, w), \g # V k, (2, w). A simple modificatio i the proof of Corollary A gives Corollary B. Recall that we use W k&m, (2, w) to deote the Sobolev space W k&m, (2, (w m,..., w k )). Corollary B. Let us cosider a vectorial weight w=(w 0,..., w k ). For some 0<mk, assume that (2, (w m,..., w k )) # C 0. Let K be a fiite uio of compact itervals cotaied i 0 (m&1). The there exists a positive costat c 1 =c 1 (K) such that c 1 &g& L 1 (K)&g& W k&m, (2, w), \g # V k&m, (2, w). Theorem 3.1 i [RARP2] ad its remark give the followig result. Theorem C. Let us cosider a vectorial weight w=(w 0,..., w k ) with (2, w)#c 0. Assume that K is a fiite uio of compact itervals J 1,..., J ad that for every J m there is a iteger 0k m k verifyig J m 0 (k m &1), if k m >0, ad Jm w j =0 for k m < jk, if k m <k. If w j # L (K) for 0< jk, the there exists a positive costat c 0 such that c 0 & fg& W k, (2, w)& f & W k, (2, w) (sup g(x) +&g& W k, (2, w)), x # 2 for every f, g # V k, (2, w) with g$=g"= }} } =g (k) =0 i 2"K. Corollary 3.1 i [RARP2] implies the followig result. Corollary C. Let us cosider a vectorial weight w=(w 0,..., w k ) i (a, b) with w k # B ([a, b]), w # L ([a, b]) ad K([a, b], w)=[0]. The there exists a positive costat c 0 such that c 0 & fg& W k, ([a, b], w)& f & W k, ([a, b], w)&g& W k, ([a, b], w) for every f, g # V k, ([a, b], w). The followig result is easy to prove.

25 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 143 Lemma A. Let us cosider w=(w 0,..., w k ) a vectorial weight with w j+1 (x)c 1 x&a 0 w j (x), for 0 j<k, a 0 # R ad x i a iterval I. Let. # C k (R) be such that supp.$[*, *+t], with max[ *&a 0, *+t&a 0 ]c 2 t ad &. ( j ) & L (I) c 3 t &j for 0jk. The, there is a positive costat c 0 which is idepedet of I, a 0, *, t, w,., ad g such that &.g& W k, (2, w)c 0 &g& W k, (I, w), for every g # V k, (2, w) with supp(.g)i. Remark. The costat c 0 ca deped o c 1, c 2, c 3 ad k. The ext result is the versio for p= of Corollary 3.2 i [R1]. It ca be proved as i the case 1p<. Corollary D. Let us cosider a compact iterval I ad a vectorial weight w=(w 0,..., w k )#L (I). Assume that there exist a 0 # I, a iteger 0r<k, ad costats c, $>0 such that w j+1 (x)c x&a 0 w j (x) i [a 0 &$, a 0 +$] & I, for r j<k. The a 0 is either right or left r-regular. We defie ow the followig fuctios, log 1 x=&log x, log 2 x=log(log 1 x),..., log x=log(log &1 x). A computatio ivolvig Muckehoupt iequality gives the followig result. Propositio B. Let us cosider a compact iterval I ad a vectorial weight w=(w 0,..., w k )#L (I). Assume that there exist a 0 # I, a iteger 0r<k, # N, $, c i >0, = i 0, ad : i, # i,..., 1 #i # R for rik such that (i) w i (x) e &c i x&a 0 &= i x&a 0 : i log # i 1 x&a 1 0 } } } log # i x&a 0 for x # [a 0 &$, a 0 +$] & I ad rik, (ii) : i N if = i =0 ad r<ik. The there exists a completio w of w such that the Sobolev orms W k, p (I, w) ad W k, p (I, w ) are comparable ad there exists rr 0 k with w j+1(x)c x&a 0 w j(x) i [a 0 &$, a 0 +$] & I, for r 0 j<k if r 0 <k, ad w r0 # B ([a 0 &$, a 0 +$] & I). I particular, a 0 is (r 0 &1)-regular if r 0 >0.

26 144 JOSE M. RODRI GUEZ 5. APPROXIMATION IN W K, (I, W) First of all, the ext results allow us to deal with weights which ca be obtaied by ``gluig'' simpler oes. Theorem 5.1. Let us cosider &a<b<c<d. Let w= (w 0,..., w k ) be a vectorial weight i (a, d) ad assume that there exists a iterval I[b, c] with w # L (I) ad (I, w)#c 0. The f ca be approximated by fuctios of C (R) i W k, ([a, d], w) if ad oly if it ca be approximated by fuctios of C (R) i W k, ([a, c], w) ad W k, ([b, d], w). Remark. If a, d # R ad w # L ([a, d]), the result is also true with P istead of C (R). This is a cosequece of Berstei's proof of Weierstrass' theorem (see, e.g., [D, p. 113]), which gives a sequece of polyomials covergig uiformly up to the k th derivative for ay fuctio i C k ([a, d]). Proof. [:, ;]I prove the o-trivial implicatio. Let us cosider J= [:, ;]/I ad a iteger 0k 1 k, such that J/(b, c) k1 (b, c) (k 1 &1) if k 1 >0, ad J w j =0 for k 1 < jk if k 1 <k. Let us cosider f # V k, ([a, d], w) ad. 1,. 2 # C (R) such that. 1 approximates f i W k, ([a, c], w) ad. 2 approximates f i W k, ([b, d], w). Set % # C (R) a fixed fuctio with 0%1, %=0 i (&, :] ad %=1 i [;, ). It is eough to see that %. 2 +(1&%). 1 approximates f i W k, ([a, d], w) or, equivaletly, i W k, (I, w). Theorem C with 2=I ad K=J gives & f&%. 2 &(1&%). 1 & W k, (I, w) &%( f &. 2 )& W k, (I, w)+&(1&%)( f &. 1 )& W k, (I, w) c(& f&. 2 & W k, (I, w)+& f&. 1 & W k, (I, w)), ad this fiishes the proof of the theorem. Theorem 5.2. Let us cosider strictly icreasig sequeces of real umbers [a ], [b ] ( belogig to a fiite set, to Z, Z +, or Z & ) with a +1 <b for every. Let w=(w 0,..., w k ) be a vectorial weight i (:, ;) := (a, b ) with &:<;. Assume that for each there exists a iterval I [a +1, b ] with w # L (I ) ad (I, w)#c 0. The f ca be approximated by fuctios of C (R) i W k, ([:, ;], w) if ad oly if it ca be approximated by fuctios of C (R) i W k, ([a, b ], w) for each.

27 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 145 Proof. We prove the o-trivial implicatio. Let us cosider. # C (R) which approximates f i W k, ([a, b ], w). By the proof of Theorem 5.1 we kow that there are % # C (R) ad positive costats c such that & f&%. +1 &(1&% ). & W k, (I, w) c (& f&. & W k, (I, w)+& f&. +1 & W k, (I, w)). Now, give =>0, it is eough to approximate f i [a, b ] with error less tha = mi[1, c &1, c &1 ]2. &1 Theorem 5.3. Let us cosider a compact iterval I ad a vectorial weight w=(w 0,..., w k )#L (I) such that w k # B (I). The we have P k, (I, w)=h 3 :=[f# V k, (I, w)f (k) # P 0, (I, w k )] =H 0 :=[f# V k, (I, w)f ( j ) # P 0, (I, w j ), for 0 jk ] =[f: I Rf (k&1) # AC(I) ad f (k) # P 0, (I, w k )]. Proof. We prove first H 3 P k, (I, w). If f # H 3, let us cosider a sequece [q ] of polyomials which coverges to f (k) i L (I, w k ). Let us choose a # I. The the polyomials satisfy Q (x) :=f(a)+ f $(a)(x&a)+ }}} + f (k&1) (a) (x&a)k&1 (k&1)! + x a q (t) (x&t)k&1 (k&1)! Q ( j ) (x)= f ( j ) (a)+ }}} + f (k&1) (a) (x&a)k& j&1 (k& j&1)! + x a for 0 j<k. Therefore, for 0 j<k, dt q (t) (x&t)k& j&1 (k& j&1)! f ( j ) (x)&q ( j )(x) = } x j&1 ( f (k) (x&t)k& (t)&q (t)) dt } a (k& j&1)! c I f (k) (t)&q (t) w k (t) w k (t) &1 dt c & f (k) &q & L (I, w k ). dt,

28 146 JOSE M. RODRI GUEZ Hece, we have for 0 j<k, & f ( j ) &Q ( j ) & L (I, w j )c & f (k) &q & L (I, w k ), sice w j # L (I). The we have obtaied that f # P k, (I, w). Sice 0 k =it(i), 0 1 _ }}} _ 0 k =it(i) is coected ad Theorem B gives that W k, (I, w) is complete; therefore P k, (I, w)h 0. The cotet H 0 H 3 is direct. The last equality is also direct sice the fact w k # B (I) gives 0 (k&1) =I. The f (k&1) # AC(I) for every f # V k, (I, w). Theorem 5.4. Let us cosider a compact iterval I ad a vectorial weight w=(w 0,..., w k )#L (I), such that the set S of sigular poits for w k i I has zero Lebesgue measure. Assume that there exist a 0 # I, a iteger 0r<k, ad costats c, $>0 such that (1) w j+1 (x)c x&a 0 w j (x) i [a 0 &$, a 0 +$] & I, for r j<k, (2) w k # B (I"[a 0 ]), (3) if r>0, a 0 is (r&1)-regular. The we have P k, (I, w)=h 4 := [f# V k, (I, w)f (k) # P 0, (I, w k ), _l # R with ess lim f (r) (x)&l w r (x)=0, x # I, x a 0 ad ess lim x # I, x a 0 f ( j ) (x) w j (x)=0, for r< j<k ifr<k&1] =H 0 := [f# V k, (I, w)f ( j ) # P 0, (I, w j ), for 0 jk ] =[f: I Rf (k&1) # AC loc (I"[a 0 ]), f (k) # P 0, (I, w k ), _l # R with ess lim f (r) (x)&l w r (x)=0, x # I, x a 0 ess lim x # I, x a 0 f ( j ) (x) w j (x)=0, for r j<k ifr<k&1, ad f (r&1) # AC(I) if r>0]. Remark. Muckehoupt iequality I gives that coditio w j+1 (x) c 1 x&a 0 w j (x) is ot as restrictive as it seems, sice may weights ca be modified i order to satisfy it (see Propositio B). Proof. We prove first H 4 P k, (I, w). Let us take f # H 4. Without loss of geerality we ca assume that a 0 is a iterior poit of I, sice the argumet is simpler if a 0 #I. Without loss of geerality we ca assume also l=0, sice i other case we ca cosider f(x)&lx r r! istead of f(x)

29 WEIERSTRASS' THEOREM IN SOBOLEV SPACES 147 (recall that w # L (I)). Cosider ow a fuctio. # C c (R) with.=1 i [&1, 1],.=0 i R"(&2, 2), ad 0.1 i R. For each # N, let us defie. (x) :=.((x&a 0 )) ad h :=(1&. ) f (r). We have & f (r) &h & W k&r, (I, w)=&. f (r) & W k&r, (I, w) c 0 & f (r) & W k&r, ([a 0 &2, a 0 +2], w), sice we are i the hypotheses of Lemma A, where *=a 0 &2, t=4, ad we cosider the iterval [a 0 &2, a 0 +2]: observe that *&a 0 = *+t&a 0 =2=t2 ad &. ( j ) & L (R)= j &. ( j ) & L (R) 4 k max[&.& L (R), &.$& L (R),..., &. (k) & L (R)] t &j. Hece, we deduce that & f (r) & h & W k&r, (I, w) 0 as, sice ess lim x # I, x a0 f ( j ) (x) w j (x)=0, for each r jk (Lemma 2.2 gives the result for j=k sice hypotheses w r # L (I) ad (1) give that a 0 is a sigularity of type 1 for w k i I). Therefore, i order to see that f (r) ca be approximated by polyomials i W k&r, (I, w) it is eough to see that each h ca be approximated by polyomials i W k&r, (I, w). Cosider weights w :=(w 0,..., w k&1, w k, ) with w k, :=w k +/ [a0 &1, a 0 +1]w k.it is direct that w # L (I) ad w k, # B (I). Observe that Corollary 2.1 gives h (k&r) # P 0, (I, w k ), sice h (k&r) =(1&. ) f (k) +F, with F = & k&r i=1 ( k&r i ). (i ) f (k&i ) # C(I) ad 1&. # C(I). Hece Theorem 5.3 implies that each h ca be approximated by polyomials i W k&r, (I, w ) ad cosequetly i W k&r, (I, w). Therefore, f ca be approximated by polyomials i W k&r, (I, w). This fiishes the proof if r=0. I other case, hypotheses (2) ad (3) give 0 (r&1) =I ad cosequetly f (r&1) # AC(I). Without loss of geerality we ca assume that there exists =>0 such that [a 0 &=, a 0 +=] is cotaied i the iterior of I ad w r 1 i I"[a 0 &=, a 0 +=]. I the other case we ca chage w by w* with w*:=w j j if j{r ad w r *:=w r +/ I"[a0 &=, a 0 +=]. It is obvious that it is more complicated to approximate f i W k, (I, w*) tha i W k, (I, w). Therefore, we have K([a, b], (w r,..., w k ))=[0] ad ([a, b], (w r,..., w k )) # C 0 (see Remark 2 to Defiitio 3.10). Let us cosider a sequece [q ] of polyomials covergig to f (r) i W k&r, (I, w). Corollary B gives & f (r) &q & L 1 (I)c & f (r) &q & W k&r, (I, w).

30 148 JOSE M. RODRI GUEZ The polyomials defied by Q (x) :=f(a)+ f $(a)(x&a)+}}}+f (r&1) (a) (x&a)r&1 (r&1)! + x a q (t) (x&t)r&1 (r&1)! dt, satisfy & f&q & W k, (I, w)c & f (r) &q & L 1 (I)+& f (r) &q & W k&r, (I, w) c & f (r) &q & W k&r, (I, w), ad we coclude that the sequece of polyomials [Q ] coverges to f i W k, (I, w). Sice 0 k =it(i)"[a 0 ], 0 1 _ }}} _ 0 k has at most two coected compoets ad Theorem B gives that W k, (I, w) is complete; therefore P k, (I, w)h 0. Observe that hypotheses w r # L (I) ad (1) give that a 0 is a sigularity of type 1 for w j i I, for each r< jk. By Theorem 2.1 there exists l # R with ess lim x # I, x a0 f (r) (x)&l w r (x)=0, if a 0 is a sigularity for w r i I; i the other case, it is a direct cosequece of the cotiuity of f (r) i a 0. This fact ad Lemma 2.2 give H 0 H 4. The last equality is direct by the defiitio of V k, (I, w); it is eough to remark that Corollary D ad (2) give 0 (r) =0 (r+1) =}}}=0 (k&1) =I"[a 0 ], ad (2) ad (3) give 0 (r&1) =I if r>0. If we apply Theorem 5.1, Theorem 5.4, ad Propositio B, we obtai the ext result for Jacobi-type weights. Corollary 5.1. Cosider a vectorial weight w such that w j (x) (x&a) : j (b&x) ; j with : j, ; j 0 for 0 jk. Assume that there exist 0r 1, r 2 <k such that a is r 1 -right regular if r 1 >0, bisr 2 -left regular if r 2 >0, ad verifyig either (i) : j+1 : j +1 for r 1 j<k ad ; j+1 ; j +1 for r 2 j<k, (ii) : j #[0,)"Z + for r 1 < jk, ad ; j #[0,)"Z + for r 2 < jk. The P k, ([a, b], w) =[f# V k, ([a, b], w)f ( j ) # P 0, ([a, b], w j ), for 0 jk]. Remark. The same argumet gives a similar result if there are also a fiite umber of sigularities i (a, b), ad eve if the sigularities i [a, b] are of more geeral type (as i Propositio B).