On the stability of the polynomial L 2 -projection on triangles and tetrahedra

Transcription

1 ASC Reot No. 5/0 On the stability of the olynomial L -ojection on tiangles and tetaheda J.M. Melenk und T. Wuze Institute fo Analysis and Scientific Comuting Vienna Univesity of Technology TU Wien ISBN

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3 On the stability of the olynomial L -ojection on tiangles and tetaheda J.M. Melenk T. Wuze July 3, 0 Abstact Fo the efeence tiangle o tetahedon T, we study the stability oeties of the L T )-ojection Π N onto the sace of olynomials of degee N. We show Π N u L T ) C u L T ) u H T ) and Π N u H T ) CN + ) / u H T ). This imlies otimal convegence ates fo the aoximation eo u Π N uu L T ) fo all u H k T ), k > /. Intoduction and main esults The study of olynomials and thei oeties as the olynomial degee tends to infinity has a vey long histoy in numeical mathematics. Concening aoximation and stability oeties of vaious high ode aoximation oeatos, the univaiate case is easonably well undestood in the way of examles, we mention the monogahs [6] fo othogonal olynomials and [8] fo issues concening aoximation); by tenso oduct aguments, also fo the case of olynomial aoximation on d- dimensional hye cubes a significant numbe of esults is available. The situation is less develoed fo simlices, and it is the uose of this note to contibute in the aea by studying the stability oeties of the olynomial L -ojection on tiangles o tetaheda. Ou main esults ae Theoems. and.3 below. These two theoems genealize known esults fo tenso oduct domains: Theoem. is the analog of [, Lemma 4.] and coesondingly, Co.. is the analog of [, Lemma 4.4] and [5, Lemma 3.5]) and Theoem.3 coesonds to [4, Thm..]. Indeendently, closely elated esults have ecently been obtained in [5]. The novelty of the esent wok ove [5] is twofold: Fistly, in the language of Coollay. below, we extend the aoximation esult of [5] fom s to s > /. Secondly, we study the H -stability of the L ojection. Although the esults of the esent note ae of indeendent inteest, the stability esult of Theoem. has alications in the analysis of the h-vesion of discontinuous Galekin methods h-dgfem) as demonstated in [4]. Moe geneally, simlicial elements ae, due to thei geate geometic flexibility as comaed to tenso oduct elements, commonly used in high ode finite element codes so that an undestanding of stability and aoximation oeties of olynomial oeatos defined on simlices could be useful in othe alications of high ode finite element methods h-fem) as well. We efe to [6, 7, 6,, 3] fo vaious asects of h-fem. To fix the notation, we intoduce the efeence tiangle T, the efeence tetahedon T 3 as well Institut fü Analysis und Scientific Comuting, Technische Univesität Wien, Austia melenk@tuwien.ac.at).

4 as the efeence cube by T : {x, y) R : < x <, < y < x}, T 3 : {x, y, z) R 3 : < x, y, z, x + y + z < },.a).b) S d :, ) d, d {,, 3}..c) Thoughout, we will denote by P N the sace of olynomials of total) degee N. We then have: Theoem.. Let T be the efeence tiangle o tetahedon and denote by Π N : L T ) P N the L T )-ojection onto the sace of olynomials of degee N. Then thee exists a constant C > 0 indeendent of N such that In aticula, theefoe, Π N u L T ) C u L T ) u H T ) u H T )..) Π N u L T ) C u / B, T ) u B /, T ),.3) whee the Besov sace B /, T ) is defined by B/, T ) L T ), H T )) /, and we used the eal method of inteolation see, e.g., [7, 8] fo details). Coollay.. Let T be the efeence tiangle o tetahedon. Then fo evey s > / thee exists a constant C s > 0 such that u Π N u L T ) C s N + ) s /) u H s T ) u H s T )..4) Theoem.3. Let T be the efeence tiangle o tetahedon. Then thee exists a constant C > 0 such that fo all N N Π N u H T ) CN + ) / u H T ) u H T )..5) We will only show the oofs fo the 3D case in Theoem 5.3 this theoem combines Theoems. and Coollay.) and in Theoem 6., which fomulates Theoem.3. The D case is teated with simila ideas. Details of the oof of Theoem. in D can be found in the Bachelo Thesis [9]. Numeical esults In this section, we illustate the shaness of Theoems.,.3 fo the D and the D case. esent the best constants in the following D and D situations: Π N u)) C D mult u L I) u H I), Π N u) L Γ) CD mult u L T ) u H T ), We Π N u H I) CH D N + u H I) u P N.) Π N u H T ) CH D N + u H T ) u P N,.) whee I, ) and Γ, ) { } T. The best constants C D constained maximization oblem. Fo examle, C D mult max{ Π Nu L Γ) u L T ) u H T ), mult, CD mult u P N}, ae solutions of which can be solved using the techniue of Lagange multilies. The constant CH D is moe eadily accessible as the solution of an eigenvalue oblem since C D H su u P N u L Γ) u. H T ) The esult of the D situation ae esented in Table wheeas the outcome of the D calculations ae shown in Table. The D calculations ae in ageement with the esults of Theoem.,.3 wheeas the D esults illustate [, Lemma 4.] and [4, Thm..].

5 Π N su N u)) Π u PN su N u)) u L I) u H u PN I) u H I) Π N su N u)) Π u PN su N u)) u L I) u H u PN I) u H I) Table : D maximization oblems N su u PN Π N u L Γ) Π N su u L Γ) u L T ) u H T u PN ) u H T ) su u PN Π N u H T ) u H T ) N+) Table : D maximization oblems 3

6 3 One-dimensional esults In the tenso-oduct setting of suaes and hexaheda, the aguments leading to Theoems.,.3 can be educed to a one-dimensional setting. Most of this eduction to a one-dimensional setting is also ossible in the esent case of simlices, and the esent section ovides the necessay one-dimensional esults. Ou basic tool fo this dimension eduction is the so-called Duffy tansfomation see 4.) below), which mas the simlex into a hye cube. Moe imotantly, as noted aleady by [9, 6, 7] othogonal olynomials on the simlex can be defined in the tansfomed vaiables though oducts of univaiate Jacobi olynomials, which exessed the desied eduction to one-dimensional settings. Although the situation is technically moe comlicated than the tenso-oduct setting since Jacobi olynomials aise the tenso-oduct setting of suaes and hexaheda ultimately leads to the study of the moe common Gegenbaue/ultasheical olynomials see, e.g., [, Sec. 3.3.], [,3,]) the Jacobi olynomials ae classical othogonal olynomials and one can daw on a lethoa of known oeties fo the uose of both analysis and design of algoithms. This undelies, fo examle, the woks [, 5, 6, 8] and is also at the heat of the esent analysis. We denote by P n α,β), α, β >, n N, the Jacobi olynomials, [6]. Fom [6, 4.3.3)] we have the following othogonality elation fo Jacobi olynomials and, N 0 : x) α x) β P α,β) hee, δ, eesent the Konecke symbol and γ α,β) : α+β+ + α + β + x)p α,β) x)dx γ α,β) δ, 3.) Γ + α + )Γ + β + ). 3.)!Γ + α + β + ) Futhemoe, we abbeviate factos that will aea natually in ou comutations: + ) h, α) : + α + ) + α + ), g, α) : α h, α) : + α + ) + α), g, α) : + α + α ) + α), α + α ) + α), 3.3) + α) h 3, α) : + α + ) + α), g 3, α) : + α ) + α ). By diect calculation we can establish elations between h i and g i. Lemma 3.. Let h, h, h 3 and g, g, g 3 be defined in 3.3). Then thee holds fo any and α N 0 g +, α) γ α,0) ) h 3 +, α), + h, α) + ) + Futhemoe, fo any 0 + g +, α) h, α) γ α,0), h +, α) + ) + + g 3 +, α) h, α), 3.4) h 3 +, α) ) h, α) h, α) h 3, α). 3.6) Poof. This follows diectly by simle calculation and the definition of the tems. Details can be found in Aendix B. 4

7 We will denote by P α,0) the antideivative of P α,0), i.e., x P α,0) x) : P α,0) t) dt. 3.7) The following lemma states imotant elations between Jacobi olynomials, thei deivatives, and thei antideivatives. Lemma 3.. Let α N 0 and h i, g i, i {,, 3} be given by 3.3) and γ α,β) i) fo x ii) fo iii) fo by 3.). Then we have t) α P α,0) t) dt x) α h, α)p α,0) + x) + h, α)p α,0) x) + h 3, α)p α,0) ), x) P α,0) P α,0) x) g, α)p α,0) x) + g, α)p α,0) x) + g 3, α)p α,0) x), x) h, α) P α,0) ) x) + h, α) P α,0) ) x) + h 3 +, α) + α,0)) x). P + Poof. The oof of i) elies on elations satisfied by Jacobi olynomials; see Aendix B fo details. ii) is taken fom []; iii) is obtained by diffeentiating ii) and using Lemma 3.. The next lemma will be used in the oof of the ensuing Lemma 3.5, which is the D vesion of Theoem.3. Lemma 3.3. Thee exists K > 0 such that fo all α, N 0 x) α ) P α,0) x) dx K + + α) γ α,0). Poof. The assetion in case of 0 is tivial. Fo α 0 see [, 5.3)]. A diect calculation shows I 0 0, I α + ) 4 α+ α +, I 3 + α) α + ) α + 4α + 3 α+4 ; fo suitable K, the assetion of the lemma is theefoe tue fo {0,, } and all α. Thus, we may assume α,. We abbeviate P : P α,0) and I : x)α P x) dx. Fom Lemma 3. ii) with + and thee we get P + g +, α) P g +, α) g +, α)g, α) P + ε )P ε P, 3.8) whee ε : g +, α)g 3, α) g +, α)g, α) α + + α) ) + + α) + α ) + α). We note that 0 ε. Futhemoe, we calculate ε )P ε P ) ε ) P ) + ε P ) ε ε )P P 5

8 so that by integation, Cauchy-Schwaz, and 0 ε : x) α ε )P x) ε P x) ) dx ε )I + ε I ). By the othogonality oeties of the Jacobi olynomials, we conclude in view of 3.8) I+ g +, α) + g +, α) ) γ α,0) + g +, α) g +, α)g, α) ) x) α ε )P x) ε P x) ) dx ) γ α,0) + g +, α) g +, α)g, α) ) + ε )I + ε I ). We oceed now by an induction agument. We note that fo any K the claimed fomula is monotone inceasing in. Hence, we can estimate ε )I + ε I ) K ) + α + ) ) and obtain by some tedious estimates fo the othe tems see Aendix B) : I+ g +, α) ) γ α,0) + K + ) + + α) + ) + [ K + ) + + α) + [ K + ) + + α) + ) g +, α) g +, α)g, α) + K + α ) ) [ K + ) + K + ) + + α ) α,0) ] )γ + + α) + ) + K + ) + K + ) + + α ] ) ) + α + 3) α) + ) + α ) ]. K + ) + K + ) + + ) The oof is comlete by ensuing that K so that the exession in backets is bounded by. The essential ingedient of the one-dimensional analysis in [4, Thm..], [5, Lemma 3.5], [, Lemma 4.] is the ability to elate the exansion coefficients u n ) n0 of the Legende exansion u n u np n 0,0) to the exansion coefficients b n ) n0 of the Legende exansion u n b np n 0,0). This elation genealizes to the case of exansions in Jacobi olynomials. A fist esult in this diection is see also [5, Lemma.] and [, Lemma.]): Lemma 3.4. Let α N 0. Let U C, ) and let x) α Ux) as well as x) α+ U x) be integable. Futhemoe, assume lim x) +α Ux) 0 and lim + x)ux) 0. Then the x x exansion coefficients u : b : satisfy the following connection fomula fo : x) α Ux)P α,0) x)dx, x) α U x)p α,0) x)dx. u h, α)b + + h, α)b + h 3, α)b. 6

9 Poof. Follows fom an integation by ats and the eesentation of antideivatives of Jacobi olynomials in tems of Jacobi olynomials given in Lemma 3. i). We efe to Aendix B fo details. The following esult is the genealization of [4, Thm..] on the H -stability of the L -ojection. While [4, Thm..] studied the H -stability of the tuncated Legende exansion, we study hee the effect of tuncating a Jacobi exansion. Lemma 3.5. Let α N 0. Let u and b be defined as in Lemma 3.4. Then thee exists a constant C > 0 indeendent of α and N such that fo evey N N we have x) α N ) u P α,0) x) dx CN b. 0 Poof. We abbeviate P : P α,0). We comute N+ u P N+ N b + + N+ N+ + b [ N+ 0 [h, α)b + + h, α)b + h 3, α)b ] P h 3 +, α)p + + b With Lemma 3. iii) we theefoe conclude N+ u P N+ N+ h, α)p h, α)p + b N+ h 3 N +, α)p N+b N + b P + N+ N h, α)p h, α)p + h, α)p b. N+ h 3 N +, α)p N+b N + N + h 3 +, α)p + h, α)p b. ] Inseting the esult of Lemma 3.3 gives x) α u P N+ dx N+ + CN CN N 3 N+ N b + CN h 3N +, α) N + α) b N b. N N + α) h, α) b This allows us to conclude the agument since P α,0) ) x) 0 0. Lemma 3.6. Fo β > and U C 0, ) C0, ]) thee holds ) x β Ux) dx x β+ U x) dx + 0 β + 0 β + U). 7

10 Poof. This vaiant of the Hady ineuality can be shown using [4, Thm. 330]. See Aendix B fo details. Lemma 3.7. Let U C, ) and assume Ux) x) α dx <, U x) x) α dx <. Let u and b be defined as in Lemma 3.4. Then thee exist constants C, C > 0 indeendent of α and U such that + α) u C 0 b C U x) x) α dx. Poof. The esult follows fom the elation between u and b given in Lemma 3.4 and fom bounds fo h, h, h 3. Next, we show a shot lemma that will be useful in the oof of Lemma 3.9. Lemma 3.8. Let α N 0 and. Then thee exists a constant C > 0 indeendent of and α such that N α j+α j α + α N + α. Poof. The oof follows by the standad agument of majoizing the sum by an integal. While Lemma 3.4 shows that the coefficients u can be exessed as a shot linea combination of the coefficients b a maximum of 3 coefficients suffices), the convese is not so easy. The following lemma may be egaded as a weak convese of Lemma 3.4 since it allows us to bound the coefficients b in tems of the coefficients u and weighted sums of the coefficients b. This is the main esult of this section and the key ingedient of the oof of Theoem. as it is esonsible fo the multilicative stuctue of the bound in Theoem.. Lemma 3.9. Assume the hyotheses of Lemma 3.4. Let α N 0. Let u and b be defined as in Lemma 3.4. Then fo thee exists a constant C > 0 indeendent of and α such that b + b C α+ j j u j / j j Poof. We may assume that the ight-hand side of the estimate in the lemma is finite. In view of the sign oeties of h, h, h 3 and 3.6) we have We intoduce the abbeviations h, α) + h, α) h 3, α). 3.9) α : h, α) h 3, α) α + α + ) + α + ) + α), ε : α α + ) b j α + ) + α + ) + α + 4) + + α) + α). By eaanging tems in Lemma 3.4 and using the tiangle ineuality we get h 3, α) b u + h, α) b + h, α) b +. /. 8

11 We set and by alying 3.9) we aive at Iteating 3.0) once gives z : u h 3, α) b z + α b + α ) b ) b z + α z+ + α + b + + α + ) b + ) + α ) b + Suaing and Cauchy-Schwaz yields z + α z + + α α + ) ) b + + α α + ) b + z + α z + + ε ) b + + ε b +. b z + α z + ) + z + α z + ) ε ) b + + ε b + ) If we abbeviate fo the fist two addends we obtain which we ewite as + ε ) b + + ε b + + ε ε ) b + b + z + α z + ) + z + α z + ) ε ) b + + ε b + ) + ε ) + ε ε ) ) b + + ε + ε ε ) ) b +. f : z + α z + ) + z + α z + ) ε ) b + + ε b + ) 3.) b f + ε )b + + ε b +, b b + f + ε b + b +). Next, we want to emloy a telescoing sum. Since we assume that the sums in the ight side of the statement of this lemma ae finite, i.e. u j <, b j <, 3.) j j j j and since j + α) α we have b 0 fo. Hence, we can wite j b + b b +j b +j+ + b +j b +j+ f +j + ε +j b ++j b ++j) + f++j + ε ++j b +3+j b ) ++j f +j ε +j b ++j + ε +j ε +j+ ) b ++j + ε ++j b +3+j f +j ε b + ε ++j b +3+j + ε +j ε +j+ ) b ++j + ε ++j b +3+j f +j ε b + + ε ++j ε ++j ) b +3+j + ε +j ε +j+ ) b ++j f +j ε b + + ε +j ε +j+ ) b ++j. 9

12 We conclude, noting that ε 0, b + b b + b + ε b + F + S +, 3.3) whee F : f j, j 3.4) S : ε jb j j with ε j : ε j ε j. 3.5) By ositivity of ε j and f j we have S + S as well as F + F. Theefoe, we get fom 3.3) and the definition of S Alying the notation we have Iteating 3.6) N times leads to A calculation shows S ε b + ε +b + + S + S + + max{ε, ε +}S +3 + max{ε, ε +}F + + max{ε, ε +})S + + max{ε, ε +}F. N S S +N+ + ε ε : max{ε, ε +} S + ε )S + + ε F. 3.6) +j) + N j ε +jf +j i0 + ε +i). 3.7) ε j αα + j)3 α + j) 5 α α + j). 3.8) Fom the definition of S in 3.5), 3.), and 3.8) it follows that lim S 0. Futhemoe, we can bound the oduct unifomly in N : N N N + ε +j) ex ln + ε +j) ex ε +j, 3.9) whee in the last estimate we used the fact that ln + x) x fo x 0. Fom 3.8) we get N ε +j N α N α + + j) α α + j) α α + j N, 3.0) whee we have used Lemma 3.8 in the last ste. Since α α+ <, inseting 3.0) in 3.9) gives N + ε +j) C. 3.) 0

13 Now, by assing to the limit N in 3.7), we obtain a closed fom bound fo S : S j ε +jf +j i0 + ε +i). Alying 3.0), 3.), 3.8) and the definition of F we can simlify S ε +jf +j j Inseting this estimate in 3.3) and using i j b + b F + f i α α + j) i i f i j α α++ <, we aive at α α + j) α α + F. α α + + F + F + F + F. 3.) We ae left with estimating F. By the definition of F in 3.4) and the definition of f in 3.) we have F z j + α j z j+ ) + j + α j z j+ ) j j z ε j ) b j+ + ε j b j+ ). 3.3) Now we estimate both sums seaately stating with the fist one: j + α j z j+ ) j z zj + αj zj+ zj. 3.4) }{{} j j Next, we use the elation between u and b fom Lemma 3.4. Futhemoe, we note that h 3, α) α+) γ α,0). Hence, we obtain z u h 3, α) u u α+ γ α,0) h 3, α) α+ u γ α,0) h 3, α) h, α)b + + h, α)b + h 3, α)b α+ u α γ α,0) ) b + + α b + b ). Inseting this in the bound 3.4), we get by alying the Cauchy-Schwaz ineuality fo sums j + α j z j+ ) j z α+ j α+ j j j u j α j ) b j+ + α j b j + b j ) u j / j j b j /.

14 We continue by estimating the second sum in 3.3). Using again z α+ u /γ α,0) we get z j + α j z j+ ) ε j ) b j+ + ε j b j+ ) j α+ j α+ j α+ j α+ j j j j j uj + α j u j+ ) ε j ) b j+ + ε j b j+ ) }{{}}{{}}{{} u j + u j+ ) b j+ + b j+ ) u j u j / / j j j+ In view of 3.) the last two estimates conclude the oof. 4 Exansions j b j+ b j To save sace we will sometimes denote oints in R 3 by just one lette, i.e. ξ ξ, ξ, ξ 3 ) fo oints in T 3 and η η, η, η 3 ) fo oints in S Duffy-Tansfomation We ecall the definition of the efeence tiangle, tetahedon, and the d-dimensional hye cube in.). The 3D-Duffy tansfomation D : S 3 T 3, [0], is given by + η ) η ) η 3 ) Dη, η, η 3 ) : ξ, ξ, ξ 3 ), + η ) ) η 3 ), η 3 4.) 4 / /. with invese D ξ, ξ, ξ 3 ) η, η, η 3 ) + ξ, + ξ ), ξ 3. ξ + ξ 3 ξ 3 Lemma 4.. The Duffy tansfomation is a bijection between the oen) cube S 3 and the oen) tetahedon T 3. Additionally, [ D ξi η) : η j ] 3 i,j D η) ) η ) η 3 ) 4 η ) η 3 ) η ) η 3 ) η 3) η ) η ) + η ) 4 + η ) + η ) 0 η ) η 0 0 η ) η 3 ),, ) ) det D η η3.

15 Poof. See, fo examle, [6]. Lemma 4.. Let D be the Duffy tansfomation and Γ : T { }. Then DΓ) Γ and D is an isometic isomohism with esect to the L Γ)-nom, i.e., fo sufficiently smooth functions u, we have fo ũ u D the elation u L Γ) ũ L Γ). Poof. Follows by insection. 4. Othogonal olynomials on tetaheda In tems of Jacobi olynomials P n α,β) we intoduce othogonal olynomials on the efeence tetahedon T 3 often associated with the names of Dubine o Koonwinde, [9, 6, 7]. Lemma 4.3. Let,, N 0 and set ψ,, : ψ,, D, whee ψ,, is defined by ψ,, η) : P 0,0) η )P +,0) η )P ++,0) η 3 ) ) ) η + η3. Then the functions ψ,, ae L T 3 ) othogonal and satisfy ψ,, P ++ T 3 ) and ψ,, ξ)ψ,, ξ) dξ δ, δ, δ, T δ, δ, δ, γ 0,0) γ +,0) γ ++,0) Poof. The oof can be found in Aendix B. 4.3 Exansion in tems of ψ,, Since ψ,, ),, N0 fom a set of othogonal olynomials we have that any u L T 3 ) can be exanded as whee u,,0 ψ,,, u L T 3 ) ψ,, ψ,, L T 3 ),,0 γ 0,0) + γ +,0) ++ γ ++,0) u,, ψ,,, 4.) u,, : ψ,,, u L T 3 ). 4.3) A basic ingedient of the oofs of Theoems. and.3 is the eduction of the analysis to onedimensional settings, fo which we have ovided the necessay esults in Section 3. The vaiable that will lay a secial ole is η 3 as will become aaent in Definition 4.4 below. Fo a function u defined in T 3 we intoduce the tansfomed function ũ : u D and get ) ) u,, uξ)ψ,, ξ) dξ ũη) ψ η η3,, η) dη T 3 S 3 ) + ) ũη)p 0,0) η )P +,0) η )P ++,0) η ++ η3 η 3 ) dη. 4.4) S 3 3

16 Definition 4.4. Let, N 0 and u L T 3 ). We define the functions U, : R R and Ũ, : R R as well as the coefficients ũ,, and ũ,, by ) + U, η 3 ) : ũη)p 0,0) η )P +,0) η η ) dη dη, 4.5) Ũ, η 3 ) : U,η 3 ), 4.6) η 3 ) + ũ,, : ũ,, : With this notation, we have by comaing 4.4) with 4.7) u,, ++ η 3 ) ++ Ũ, η 3 )P ++,0) η 3 ) dη 3, 4.7) η 3 ) ++ Ũ,η 3 )P ++,0) η 3 ) dη ) η 3 ) ++ Ũ, η 3 )P ++,0) η 3 ) dη 3 ++ ũ,,. 4.9) Since fo sufficiently smooth functions u the tansfomed function ũ is constant on η 3, othogonalities of the Jacobi olynomials give us U, ) 0 fo, ) 0, 0) 4.0) 4.4 Poeties of the univaiate functions U, and Ũ, We stat with some eliminay consideations egading estimates fo atial deivatives of the tansfomed function ũ. We have η ũη) η ) η 3 ) u) Dη), 4.) 4 η ũη) + η ) η 3 ) 4 η3 ũη) + η ) η ) u) Dη) + η 3) u) Dη), 4.) u) Dη) + η ) u) Dη) + 3 u) Dη), 4.3) 4 whee i is the atial deivative with esect to the i-th agument. In aticula, we get ) ) ) η ũη) dη u) Dη) η η3 dη S 3 η S 3 ξ u L T 3 ) u L T 3 ), 4.4) ) ) η ũη) η dη u) Dη) + ) ) η η η3 dη + ξ S 3 S 3 u L T 3 ) ξ u L T 3 ) + ξ u L T 3 ) u L T 3 ). 4.5) These estimates will be useful to ove the following lemmas. Lemma 4.5 oeties of U, ). Let u H T ) and U, be defined in Definition 4.4. Then thee exists a constant C > 0 indeendent of u such that ) U, η 3 ) η3 dη 3 u L T 3 ), 4.6),0,0 γ 0,0) γ 0,0), γ +,0) γ +,0) γ 0,0) γ +,0) ) U,η 3 ) η3 dη 3 C u L T 3 ), 4.7) + ) U, η 3 ) dη 3 C u L T 3 ). 4.8) 4

17 Futhemoe, we have fo Γ T { } 0 γ 0,0) γ +,0) U, ) u L Γ). 4.9) ) Poof. We fist ove 4.6) and 4.7). The fact that P 0,0) η )P +,0) η ) η ae othogonal olynomials in a weighted L -sace on S and the definition of U, imly fo fixed η 3 ) the eesentation ũη),0 γ 0,0) + γ +,0) which in tun gives ) ũη) η dη dη S ) η3 U, η 3 )P 0,0) η )P +,0) η ),0 γ 0,0) Since det D η Simila to the eesentation of ũ above, we get fo η3 ũ η3 ũη),0 γ +,0) ) η, U, η 3 ). 4.0) ), ) multilication with η3 and integation in η3 gives 4.6). γ 0,0) + γ +,0) Reasoning as in the case of 4.6) yields,0 γ 0,0) γ +,0) U,η 3 )P 0,0) η )P +,0) η ) ) η. U,η 3 ) ) η3 dη 3 u L T 3 ), which immediately leads to 4.7). We now tun to the oof of 4.8). By definition, we have U, η 3 ) + ũη)p 0,0) S We conside the integation in η. Integation by ats then yields ũη)p +,0) η ) η ) + dη ) ũη) η ) + +,0) P + η ) η )P +,0) η ) η ) + dη dη. 4.) η ũη) η ) +) P +,0) + η )dη η ũη) η ) + +,0) P + η ) + + )ũη) η ) +,0) P + η )dη. Hence, we obtain by inseting into 4.) U, η 3 ) η ũη)p 0,0) S ũη)p 0,0) S η ũη)p 0,0) S S η ũ)η) η ) η ) + P +,0) + η )dη dη η ) η ) P +,0) + η )dη dη η ) η ) + P +,0) + η )dη dη P 0,0) + η ) η ) P +,0) + η )dη dη, 5

18 whee in the last euation we used integation by ats in η and the fact that t)dt 0 fo. With the abbeviation g i : g i +, + ), i,, 3 we have by Lemma 3. ii) fo, the following elationshis: P +,0) P 0,0) η ) g P +,0) + η ) + g P +,0) η ) + g 3 P +,0) η ), 4.) ) P 0,0) + η ) P 0,0) η ). 4.3) P 0,0) η ) + Futhemoe, we intoduce two abbeviations ) + z, η 3 ) : η ũ)η)p 0,0) η η ) P +,0) η )dη dη 4.4) S ) + z, η 3 ) : u) D)η)P 0,0) η η ) P +,0) η )dη dη. 4.5) S Since we have 4.), using 4.), 4.3), 4.4) and 4.5) we get U, η 3 ) g z,+ η 3 + g z, η 3 ) + g 3 z, η 3 ) ) ) ) + η3 [ + g z+,+η 3 ) z,+η 3 ) ) + 4 We use g, g, g ) U, η 3 ) + g z+, η 3 ) z, η 3 ) ) + g 3 z+, η 3 ) z, η 3 ) )]. to aive at ) η3 z +,+ jη 3 ) + z,+ jη 3 ) ) + z,+ j η 3 ). 4.6) To estimate the tems on the ight side we note that the abbeviations z, and z, lead us to the eesentations η ũη) u) D)η),0,0 Since the olynomials P 0,0) η ) tiangle T we have γ 0,0) γ 0,0) η + γ +,0) + γ +,0) η ũη) η S ũ) D)η) η S z, η 3 )P 0,0) η ) z, η 3 )P 0,0) η ) η η ) P +,0) η ) ) P +,0) η ). ) +,0) P η ) ae othogonal olynomials on the efeence ) dη dη ) dη dη,0,0 γ 0,0) γ 0,0) γ +,0) γ +,0) Fomula 4.7) togethe with an integation in η 3 and an alication of 4.5) gives,0 γ 0,0) γ +,0) z, η 3 ) 4.7) z, η 3 ). 4.8) ) z, η 3 ) dη 3 η ũη) η dη u L S 3 T 3 ). 6

19 Fom the eesentation 4.8) we get by a multilication with use of 4.4) that,0 ) η3, an integation in η3 and the ) γ 0,0) γ +,0) z, η 3 ) η3 dη 3 ) ) u) D)η) η η3 dη u L S 3 T 3 ). The last two esults in combination with 4.6) conclude the agument. We finally ove 4.9). Fo the estimate 4.9), we use 4.0) with η 3. Noting Lemma 4. we have and theefoe the esult follows. u L Γ) ũ L Γ) S ũη, η, ) dη dη The estimate 4.8) does not ovide a bound fo the secial cases 0 o 0. The teatment of these two secial cases is the uose of the following lemma. Lemma 4.6. Assume the same hyotheses as in Lemma 4.5. Then thee exists a constant C > 0 indeendent of u such that 0 0 γ 0,0) γ 0,0) 0 γ,0) γ +,0) 0 U 0, η 3 ) dη 3 C u L T 3 ) 4.9) U,0 η 3 ) dη 3 C u L T 3 ). 4.30) Poof. The oof of 4.9) stat with the obsevation ) ) U 0, ũη)dη P,0) η η ) dη. We exand fo fixed η 3 the function η ũη, η, η 3 )dη as well as its deivative in tems of P,0). Lemma 3.7 then yields U 0, η 3 ) dη 3 + ) U 0, η 3 ) dη 3 0 γ 0,0) 0 γ,0) η ũη)dη η )dη dη 3 γ,0) η ũη) η S 3 ) dη u L T 3 ), whee we aealed to 4.5) in the last estimate. Analogously we deal with 4.30), whee we exand the ma η ) + ũη, η, η 3 ) η dη again fo fixed η 3 in tems of P 0,0) and conclude with Lemma 3.7 and 4.4). The estimates fo the functions U, imly easily coesonding bounds fo the functions Ũ,: Lemma 4.7 oeties of Ũ,). Let u H T ) and Ũ, be defined in Definition 4.4. Then thee exists a constant C > 0 indeendent of u such that η 3 ) ++ Ũ, η 3 ) dη 3 u L 4 T 3 ), 4.3) 4,0,0 γ 0,0) γ 0,0) γ +,0) γ +,0) η 3 ) ++ Ũ,η 3 ) dη 3 C u L T 3 ). 4.3) 7

20 Poof. We have Ũ,η 3 ) η 3 ) +) U, η 3 ) and theefoe Ũ,η 3 ) η 3 ) +) U,η 3 ) + + ) η 3 ) ++) U, η 3 ). Hence, 4 4 η 3 ) ++ Ũ, η 3 ) dη 3 η 3 ) ++ Ũ,η 3 ) dη 3 ) η3 U,η 3 ) dη 3 η3 Using the esults of Lemma 4.5 concludes the agument. ) U,η 3 ) dη ) U, η 3 ) dη 3. These esults allow us to get bounds fo weighted sums of the coefficients ũ,, and ũ,, given in Definition 4.4: Coollay 4.8. Assume the hyotheses of Lemma 4.7 and let ũ,,, ũ,, be given by Definition 4.4. Then thee exist constants C indeendent of u such that,,0,,0 γ 0,0) γ 0,0) γ +,0) γ +,0) ae othogonal olynomials in a weighted L -sace, exand- Poof. Since the olynomials P ++,0) ing Ũ, yields the eesentation Futhemoe, we have Ũ, η 3 ) 0 γ ++,0) γ ++,0) ũ,, C u L T 3 ), 4.33) ũ,, C u L T 3 ). 4.34) γ ++,0) η 3 ) ++ Ũ,η 3 ) dη 3 ũ,, P ++,0) η 3 ). 0 γ ++,0) ũ,,. The statement 4.33) now follows diectly fom 4.3) of Lemma 4.7. Analogously, we deal with 4.34), whee we exand Ũ, and conclude with 4.3) of Lemma Connections between ũ,, and ũ,, To obtain the multilicative stuctue in the estimation of Theoem. one key ingedient will be connections between ũ,, and ũ,,. This connection is essentially a one-dimensional effect and follows fom Lemma 3.4: Coollay 4.9. Let ũ,, and ũ,, be as in Definition 4.4 and h, h and h 3 be given by 3.3). Then fo and, 0 thee holds ũ,, h, + + )ũ,,+ + h, + + )ũ,, + h 3, + + )ũ,,. Poof. By density we may assume that u C T 3 ). Lemma 3.4 then imlies the esult. Fo a detailed vesion of this oof see Aendix B. 8

21 5 Tace stability Fist, we will establish a eesentation fo the tansfomed function ũ u D on the face Γ : T { }. Using A.4) and A.3) we get Theefoe we have fo ξ Γ P ++,0) ) ) P 0,++) ) ). ψ,, ξ) ψ,, D ξ, ξ, ) ) ) η Alying the exansion of u in 4.) we aive at ũη, η, ) uξ, ξ, ),,0 γ 0,0) + γ +,0) P 0,0) η ) η ) P 0,0) η )P +,0) η ). ) P +,0) η ) ) u,, ++ In view of Lemma 4. as well as the exansion 4.), we obtain fo the L Γ)-nom u L Γ) ũ L Γ),0 γ 0,0) γ +,0) ) u,, 0 ++ γ ++,0) γ ++,0). 5.). 5.) The next lemma will show that the infinite sum ove in 5.) can be exessed as a finite sum. Lemma 5.. Let N. Then thee holds ) N ++ γ ++,0) u,, ) N h N, + + ) + N N + γ ++,0) N ũ,,n ) + h 3 +, + + ) Poof. We abbeviate n : + +. In view of Coollay 4.9 we have ) N n γ n,0) ) N N+ + N+ 4.9) u,, + γ n,0) ) N + γ n,0) ũ,, + γ ++,0) + h, n )ũ,,+ + h, n )ũ,, + h 3, n )ũ,, ) h, n ) + ) h, n ) + N ) ũ,, + [ + ) N h N, n ) + γ n,0) γ n,0) ũ,, ũ,, + h, n ) γ n,0) γ n,0) N ũ,,n + N ) + h 3 +, n ) + + h, n ) N N γ n,0) h 3 +, n ) γ n,0) + ) + h 3 +, n ) + By 3.5), the exession in backets vanishes and that concludes the oof. 9 ) ũ,,. γ n,0) ũ,, + ] γ n,0) + ũ,,.

22 Since Lemma 5. assumes N, the tems coesonding to 0 in 5.) ae not included. We study this case now in Lemma 5.: Lemma 5.. Let u H T 3 ) and conside the eesentation of the noms in 5.). Fo 0 thee exists a constant C > 0 indeendent of, and u such that,0 γ 0,0) γ +,0) u,,0 ++ γ ++,0) 0 C u L T 3 ) u H T 3 ). Poof. We fist note that the coefficient u 0,0,0 u L T 3 ) so that we may focus on the sum with, ) 0, 0). Since γ 0,0) +, γ+,0) + ++ and γ++,0) , we get,0 γ 0,0) γ +,0) u,,0 ++ γ ++,0) ) 4 u,,0. To bound the sum on the ight-hand side, we note that an integation by ats and 4.0) give fo, ) 0, 0)) ++ u,,0,0 η 3 ) ++ U, η 3 )dη 3 5.3) U, ) + These two euations yield two estimates fo u,,0. ineuality ) η 3 ) ++3 U,η 3 )dη 3, 5.4) Fom 5.3), we get with the Cauchy-Schwaz ++ u,,0 η 3 ) ++ U, η 3 )dη 3 ) ) η 3 ) ++ 3 ) dη3 η 3 ) U, η 3 ) dη Fom 5.4), we obtain a second estimate as follows: ++ u,, ) ++6 U, ) + whee η 3 ) ++3 U,η 3 )dη 3 Inseting this in the bound befoe yields η 3 U, η 3 ) dη 3 5.5) η 3 ) u,, ) U, ) + η 3 ) ++3 U,η 3 )dη 3 ), ) ) ) dη3 η 3 ) U,η 3 ) dη 3 η 3 U,η 3 ). + + ) η 3 U,η 3 ) dη ) 0

23 Next, we abbeviate σ, : η 3 U, η 3 ) dη 3, τ, : Hence, alying 5.5) and 5.6) we have { ++ u,,0 ++4 min + + σ,, ) ) U, ) + min η 3 U,η 3 ) dη 3. U, ) + { σ,, ) U, ) ) σ,τ,, whee in the last ste we used that min{a, b } a b. This leads us to Hence, we conclude u,,0 + + ) 4 u,,0,0 )} + + τ, } ) 3 τ, + + ) U, ) ) σ,τ,. 5.7) + + ) U, ) +,0 + + ) U, ) +,0 u L Γ) + u L T 3 ) u L T 3 ) + + ) σ,,0 + + ) σ, τ,,0 / + + ) τ, whee, in the last ineuality, we aealed to Lemma 4.5. The statement of the lemma now fom the multilicative tace ineuality u L Γ) u L T 3 ) u H T 3 ) of, e.g., [3, Thm..6.6] and the tivial bound.,0 We ae now in osition to ove Theoem. as well as Coollay.: Theoem 5.3 tace stability of L -ojection, aoximation oeties). Fo N N 0 denote by Π N the L T 3 )-ojection onto P N T 3 ). Thee exists a constant C > 0 indeendent of N and u such that Futhemoe, Π N u L Γ) C u L T 3 ) u H T 3 ) u H T 3 ). 5.8) / Π N u L Γ) C u B /, T 3 ) u B /, T 3 ). 5.9) Additionally, fo each s > / thee is C s > 0 such that u Π N u L Γ) C s N + ) s /) u H s T 3 ) u H s T ). 5.0) Poof. In view of the multilicative tace ineuality u L Γ) u L T 3 ) u H T 3 ) see, e.g., [3, Thm..6.6]) we will only show the statement u Π N u L Γ) u L T 3 ) u H T 3 ). We abbeviate n : + + and c : γ 0,0) u Π N u L Γ),0 c γ +,0) max{0,n+ }. By 5.), we have to bound ) n γ n,0) u,,

24 + N N N c c N+ 0 N+ N+ ) ) c ) 0 n γ n,0) n γ n,0) n γ n,0) u,, u,, u,, N N c ) n 0 0 N+ γ n,0) u,, }{{} : S n + u,, + N+ 0 c ) γ n,0) }{{} : S 3 n c u,,0. N+ 0 γ n,0) 0 } {{ } : S 5 + N N+ 0 N+ N c ) 0 N+ 0 n γ n,0) c ) 0 c ) u,, n γ n,0) n γ n,0) u,, u,, } {{ } : S N 0 N+ c n γ n,0) 0 u,,0 } {{ } : S 4 Lemma 5. immediately gives S 4 +S 5 u L T 3 ) u H T 3 ). Fom Lemma 5. as well as the estimates and h N +, n ) + N N, + 0,..., N h 3 N +, n ), h 3 N +, n ) N + + N N, + 0,..., N γ n,0) N+ N , γ n,0) N+ N we obtain fo S S N N 0 0 Analogously, we get fo S and S 3 S S 3 N c ++3)ũ,,N ) ũ,,n ). 0 N+ N+ 0 c ++3)ũ,, + ++3) ũ,,0 ) c ++3)ũ,, + ++3) ũ,,0 ). Alying Lemma 3.9 the owes of two in the estimates above and in the lemma annihilate each othe.

25 Hence, S + S + S 3 gives S + S + S 3 N N N c 0 N+ N+ 0 N+ c c u L T 3 ) u L T 3 ), γ n,0) γ n,0) γ n,0) ũ,, ũ,, ũ,, / N / / 0 0 γ n,0) γ n,0) γ n,0) ũ,, ũ,, ũ,, whee in the last estimate, we have used the Cauchy-Schwaz ineuality fo sums and Coollay 4.8. Since u L T 3 ) u H T 3 ) this concludes the oof of 5.8). The estimate 5.9) follows diectly fom 5.8) in view of [7, Lemma 5.3]. Finally 5.0) is shown in the standad way. Fo abitay P N we have by the ojection oety of Π N as well as the continuity of the tace oeato γ 0 : B / /, T ) L T 3 ) cf., e.g., [7, Sec. 3]) u Π N u L T ) u L T ) + Π N u ) L T ) C u / B, T ). Hence, u Π N u L T ) C inf P N u B /, T ). Fix s > /. Let Π N : L T ) P N be an aoximation oeato with the oeties fo examle, one can combine the aoximation esults of [9, Aendix A] fo hye cubes with the well-known extension oeato of Stein, [5, Cha. VII]) u Π N u L T ) CN s u H s T ), u Π N u H s T ) C u H s T ) u H s T ). By inteolation theoy, we then have that Id Π N is a bounded linea oeato H s T ) B /, T ) L T ), H s T )) s /)/s, with nom Id Π N B /, T ) Hs T ) CN ss /)/s. 6 H -stability Ou ocedue to study the H -stability of the L -ojection Π N is to study the diectional deivative that coesonds to the deivative η3 in the tansfomed vaiable. The full gadient can be obtained fom this diectional deivative and aoiate affine tansfomations of the efeence tetahedon T 3. The key ste is theefoe to contol η3 ΠN u, whee we denote Π N u : Π N u) D. This is the uose of the ensuing lemma. Lemma 6.. Thee exists a constant C > 0 indeendent of N such that η ) η 3 ) η3 ΠN uη) dη CN u L T 3 ) u H T 3 ). S 3 Poof. We abbeviate n : + +. We see that Π N uη) N N 0 0 N 0 γ 0,0) + γ +,0) + γ n,0) ũ,, ψ,, η) / / / 3

26 by ecalling the elation between u,, and ũ,,. Diffeentiating with esect to η 3 shows us that we have to estimate the two tems I : η ) η 3 ) N N N + S γ 0,0) γ +,0) γ n,0) ũ,, ) P 0,0) η )P +,0) η η ) η 3 ) + P n,0) ) η3 ) dη 6.) I : η ) η 3 ) N N N + + S γ 0,0) γ +,0) γ n,0) ũ,, ) P 0,0) η )P +,0) η η ) η 3 ) + P n,0) η 3 ) dη 6.) Fist, we conside 6.). Fom Lemma 3.5 with α n we get I N N 0 0 N N 0 0 γ 0,0) γ 0,0) γ +,0) γ +,0) N η 3 ) n N ) 0 0 γ n,0) γ n,0) ũ,, P n,0) ) η3 ) ũ,, N u L T 3 ), whee in the last ste, we aealed to Coollay 4.8. Thus, we aive at the desied bound fo I. Next, we conside 6.). We have N N + ) N I η 3 ) + ũ,, P n,0) η 3 ) dη γ 0,0) γ 0,0) γ +,0) γ +,0) 0 γ n,0) Lemma 3.6 with β + and the nomalization convention fo Jacobi olynomials A.3) now yield N N + ) I N + ) η 3 ) n n ũ,, P,0) ) η3 ) dη 3 I + I N N 0 0 N I + N N 0 0 N N 0 γ 0,0) γ 0,0) N 0 0 γ n,0) + γ +,0) + γ +,0) γ 0,0) ũ,, P n,0) γ +,0) N 0 N 0 ) γ n,0) γ n,0) 0 N u L T 3 ) + N u L {η T 3 :η 3 }) γ n,0) 0 ũ,, P n,0) ũ,, ) ) ũ,, + N u L T 3 ) + N u L T 3 ) u H T 3 ) N u L T 3 ) γ n,0) ) N + ) γ n,0) whee we used the fact that I is bounded above and the multilicative tace ineuality [3, Thm..6.6] as in Theoem 5.3 but now alied to the bottom face {η T 3 : η 3 }. 4 ũ,, dη 3

27 We can now ove Theoem.3: Theoem 6. H -stability of L -ojection). Thee exists a constant C > 0 indeendent of N such that Π N u L T 3 ) C N u L T 3 ) u H T 3 ). Poof. Fo a function v and the tansfomed function ṽ v D, the fomula 4.3) ovides a elation between η3 ṽ and v. Reaanging tems yields η3 ṽ) D ξ) + ξ ξ 3 vξ) + ξ ξ 3 vξ) + 3 vξ). Theefoe, when tansfoming to T 3 in Lemma 6. we get + ξ Π N uξ) + ξ Π N uξ) + 3 Π N uξ) ξ 3 ξ 3 T 3 dξ N u L T 3 ). 6.3) By the symmety oeties of T 3, we see that also the following two othe emutations of indices ae valid estimates: + ξ Π N uξ) + ξ 3 3 Π N uξ) + Π N uξ) T 3 ξ ξ dξ N u L T 3 ), 6.4) + ξ 3 3 Π N uξ) + ξ Π N uξ) + Π N uξ) ξ ξ dξ N u L T 3 ). 6.5) T 3 We abbeviate ax, y) : +x y, a ij : aξ i, ξ j ) and + a 3 + a a 3 a 3 + a + a a 3 + a 3 + a 3 a Aξ, ξ, ξ 3 ) : sym + a 3 + a a 3 + a a 3 + a 3. sym sym + a 3 + a 3 Hence, we see that by adding 6.3), 6.4), and 6.5) we aive at T 3 Π N u) Aξ) Π N u dξ N u L T 3 ) Next, we obseve that nea the to vetex,, ), we have + ξ ξ 3 and + ξ ξ 3. This imlies that the functions a 3 and a 3 ae unifomly bounded on T 3. Analogously, we get bounds fo a, a 3 and a, a 3 by studying the vetices,, ) and,, ). Theefoe, we have su Aξ) L T 3 ) <. ξ T 3 By constuction, the matix Aξ) is ointwise) symmetic ositive semidefinite. Ou goal is to show that Aξ) is in fact ositive definite on the set that stays away fom the face F oosite the vetex,, ). This can be done with techniues as in [] by establishing lowe bounds fo the eigenvalues of Aξ). A diect calculation eveals det Aξ) 6 ξ + ξ ξ + ξ ξ 3 + ξ + ξ + + ξ 3 + ξ 3 ξ + ξ + ξ 3 + ξ ) + ξ 3 ) + ξ ) 6 ξ + ξ + ξ 3 ) + ξ + ξ + ξ 3 ) + + ξ + ξ + ξ 3 ) + ξ ) + ξ 3 ) + ξ ) 6 + ξ ) + ξ 3 ) + ξ ). 5

28 The face oosite the vetex,, ) contains the vetices, ),,, ),,, ) and is given by the euation ξ + ξ + ξ Futhemoe, we conclude that the signed distance of an abitay oint ξ fom this face F is given by distξ, F ) ξ + ξ + ξ 3 + ). 3 Let, fo abitay δ > 0, T δ : {ξ T 3 distξ, F ) < δ}. Then, since we stay away fom the face F, it is clea that thee exists C δ > 0 such that det Aξ) C δ ξ T δ. Combining the above findings, we have that on T δ the matix Aξ) is in fact symmetic ositive definite. Since the enties of Aξ) ae unifomly bounded in ξ, Geshgoin s cicle theoem ovides a constant C ue such that all eigenvalues of Aξ) ae bounded by C ue. A lowe bound fo the eigenvalues is obtained as follows: Denoting fo fixed ξ T δ the eigenvalues 0 < λ λ λ 3, we get fom det A λ λ λ 3 C δ det A λ λ λ 3 λ C ue. This ovides the desied lowe bound fo λ. Thus, we conclude that fo evey δ > 0 we can find c δ > 0 such that Aξ) c δ I on T δ. Hence, c δ Π N u T dξ Π N u) Aξ) Π N u dξ N u L T 3 ). δ T 3 Affine tansfomations allow us to get analogous estimates fo the sets that stay away fom the othe faces of T 3. We theefoe get the desied esult. A Poeties of Jacobi olynomials We have the following useful fomulas see [6,. 350 f], [6]): Recusion Relations with a np α,β) n+ x) a n + a 3 nx)p n α,β) x) a 4 np α,β) n x) a n : n + )n + α + β + )n + α + β) a n : n + α + β + )α β ) a 3 n : n + α + β)n + α + β + )n + α + β + ) a 4 n : n + α)n + β)n + α + β + ) A.) with b nx) d dx P n α,β) x) b nx)p n α,β) x) + b 3 nx)p α,β) b nx) : n + α + β) x ) b nx) : n α β n + α + β)x) b 3 nx) : n + α)n + β) n x) A.) 6

29 Secial Values P α,β) n ) ) n + α n A.3) P n α,β) x) ) n P n β,α) x) A.4) Secial Cases Fo the Legende Polynomial L n x) thee holds Miscellaneous Relations n x L n x) P 0,0) n x) A.5) d dx P n α,β) x) α+,β+) α + β + n + )P n x) A.6) t) α + t) β P n α,β) t) dt x) α+ + x) β+ P α+,β+) n x) A.7) B Selected oofs B. Selected oofs fo Section 3 Poof of Lemma 3. i). Using eaanged vesions of A.), A.6) and A.7) we obtain x t) α P α,0) t)dt A.7) + x) x)α+ P α+,) x) x ) x) α P α+,) x) x) α d x ) + α + dx P α,0) x) A.6) A.) x) α x) + α + α,0) αp α A.) allows us now to elace the tem xp α,0) Hence, we get x α + α)x)p α,0) x) + + α)p α,0) x) + α x) + + α)p α,0) + α + ) + α) x) by tems involving P α,0) + { t) α P α,0) t)dt x) α + α + ) + α) + ) + α + ) + α)p α,0) + + α + ) + α + ) x) + + α) + α + )P α,0) x) + α + )α P α,0) α,0) x) + α)xp x) α,0) x), P x) and P α,0) x). αp α,0) x) + + α)p α,0) x) ) } x) 7

30 Reaanging tems gives { x t) α P α,0) t)dt x) α + α + ) + α) + ) + α + ) + α) + α + ) + α + ) P α,0) + x) } + α + + α + α + P α,0) x) + + α) + α + + α + P α,0) x) { x) α + ) P α,0) + + α + ) + α + ) x) }{{} h, α) + α + α + ) + α) }{{} h, α) P α,0) x) + + α) + α + ) + α) } {{ } h 3, α) } P α,0) x) Poof of 3.5). By definition of γ α,β) we obtain in aticula α+ + α +, which leads in combination with the definition of h, h and h 3 to ) h, α) + ) + + h +, α) + ) + + h 3 +, α) + + α α + 3 α ) α+ + )+ + α + ) + α + ) α+ + α + 4) + α + ) + + α α + ) + ) α+ + α + 5) + α + 4) )+ + α + α α + 3)α + α + 4) + α + ) + α + ) + α + 4 ) )+ + ) + α + 4) + + α + 3)α + α + ) + α + ) α + α + 4) + α + ) Simly multilying out the numeato concludes the oof egading the fist euation. Inseting the definition of h, h and h 3 also leads in the case of the second euation to the conclusion h, α) h, α) α + α + ) + + ) + α) + α) + α + ) + α + ) α + α + 4α + α) + α + ) + α + ) + α + ) + α) + α) + α + ) + α + ) h 3, α). Lemma B. details of the oof of Lemma 3.3). Define fo α, N 0 Then, fo α, we have 0 ε. ε : g +, α)g 3, α) g +, α)g, α) α + + α) ) + + α) + α ) + α). 8

31 Poof. Clealy, ε 0. To see the estimate ε, we have to show α + + α) ) α ) + α ) + 3α ) + α ) α ) + α + ) + α)) + 3α ) }{{}? + α) + + α) + α )? + α ) + α) + α + )? 0 This last ineuality is cetainly tue if α ) + α + ) + α) + 3α ) 4α ) + α + ) + α) 4α ) + α) + α) 4α ) α α + α) α 4α α + α) 4α α) + α)? 0? 0? 0? 0? 0? 0 Lemma B. details of the oof of Lemma 3.3). Fo α N, we have g +, α) g +, α) g +, α)g, α) ) γ α,0) + ) + α + ) + + ) + ) + ) + α + ) + + ) + B.) B.) Poof. We stat by obseving that α+ + α +, 0 B.3) so that we obtain γ α,0) α α + + α α We stat with the bound B.). Then g +, α)) γα,0) γ α,0) g +, α)) α + ) + α + ) + α α + ) + α + γα,0) + + α + ) + α + 3) + + ) + α + ) + ) + γα,0) + 9

32 whee, in the last ste, we used, α. We now tun to the bound B.). Then ) g +, α) g +, α)g, α) ) α + α + ) + α + ) + α ) + α) + α) + α + ) + α + + α ) α + α + ) + α + ) + α ) + α) + α) + α + ) + α + + α + ) α + α + ) + α + ) + α ) + α) + α α) + α + ) + α + + α ) + α + )α + α + ) + α ) + α) + α α) + α + ) + α + + α + α ) + α + α + α α + 3) + α ) + α + + α + α + + α + 3) α γα,0) + Lemma B.3. Let U C, ) and let x) α Ux) as well as x) α+ U be integable. Futhemoe, let lim x x)+α Ux) 0 and lim + x)ux) 0. x Conside h, h and h 3 fom 3.3). We define u : b : x) α Ux)P α,0) x)dx, x) α U x)p α,0) x)dx. Then fo and α N 0 the following elationshi holds: Poof. Fom A.7) we have fo x and fo x u h, α)b + + h, α)b + h 3, α)b. x x t) α P α,0) t)dt O + x) t) α P α,0) t)dt O x) α+). Hence, using the stiulated behavio of U at the endoints, the following integation by ats can be justified: u x) α Ux)P α,0) x)dx Ux) x t) α P α,0) ) x) U x) 30 x t) α P α,0) t)dt dx.

33 In aticula, we note that b is well-defined. Futhemoe, u U x) x x) α U x) t) α P α,0) t)dt dx h, α)p α,0) h, α)b + + h, α)b + h 3, α)b, + x) + h, α)p α,0) whee in the second euation we aealed to Lemma 3. i). 0 x) + h 3, α)p α,0) x) ) dx Lemma B.4. Fo β > and U C 0, ) C0, ]) thee holds ) x β Ux) dx x β+ U x) dx + β + β + U). Poof. We define accoding to the notation of [4, Thm. 330] { U x) 0 < x < fx) : and F x) : ft) dt 0 x > Fo β >, [4, Thm. 330] states Hence, 0 0 x β F x) dx x β Ux) U)) dx 0 x ) x β+ fx) dx β + 0 { 0 x > ) x β+ U x) dx. β + 0 U) Ux) 0 < x <. By eaanging tems, we get ) x β Ux) dx x β+ U x) dx + U) x β dx 0 β ) x +β U x) dx + β + β + U). 0 B. Selected oofs fo Section 4 Poof of Lemma 4.3. With the definition of D and the abbeviation n : + + we get ψ,, ξ, ξ, ξ 3 ) ψ,, + ξ, + ξ ), ξ 3 ξ + ξ 3 ξ 3 P 0,0) + ξ ) P +,0) + ξ ) +ξ ) ) P n,0) ξ ξ 3 ) 3 + ξ3. ξ + ξ 3 ξ 3 Exanding P 0,0) ψ,, ξ, ξ, ξ 3 ) x )P +,0) y ) k0 l0 k0 l0 c kl k+l + ξ ξ + ξ 3 k0 l0 c klx k y l leads to ) k ) + l ξ P n,0) ξ 3 ) ξ 3 + ξ ξ 3 c kl k+l + + ξ ) k + ξ ) l ξ 3) l ξ ξ 3 ) ξ + ξ 3 ) k P n,0) ξ 3 ). 3 ) ξ3 ) +

34 Since P n,0) is a olynomial of degee, we see by the last tems in the sum above that ψ,, P ++ T 3 ). To see the othogonality oety, we tansfom to the cube S 3 and make use of 3.) thee times ψ,, ξ)ψ,, ξ)dξ T 3 ) ) ψ,, η) ψ,, η) η η3 dη P 0,0) η )P 0,0) η η ) η3 S 3 + δ +) η3 ) + + P +,0) η )P +,0) η ) ) P n,0) η 3 )P n,0) η 3 ) dη dη dη 3 η ) + P +,0) ) ++ + P n,0) η 3 )P n,0) η 3 ) dη dη 3 + δ + + δ n + δ + + δ δ. η )P +,0) η ) η 3 ) n P n,0) η 3 )P n,0) η 3 ) dη 3 Lemma B.5 details of Lemma 4.). Let D be the Duffy tansfomation and Γ : T { }. Then DΓ) Γ and D is an isometic isomohism with esect to the L Γ)-nom. Poof. Obviously D is an isomohism and by definition DΓ) Γ, so we will only show the isomety oety. Let u be a uadatic integable function on T 3 and conside the tansfomed function ũ u D. We have ũ L Γ) ũη, η, η 3 ) dη dη dη 3 Γ ) + u η ) + η ), η, dη dη uξ, ξ, ) dξ ξ u L Γ). T u Dη, η, ) ) dη dη Poof of Coollay 4.9. To ove this coollay we want to make use of Lemma 3.4. Theefoe, we have to claify that the conditions in the lemma ae satisfied. We oceed in two stes. Fist, we euie u C T 3 ) and show the statement in this case and then we ague by density to achieve esults in H T 3 ). Ste : By assuming that u C T 3 ) we get ũ C S 3 ). Hence, fo fixed and, if we ecall the definition of U,, we see that the ma η 3 U, η 3 ) is smooth on [, ]. Consideing the definition of Ũ, Ũ, η 3 ) U,η 3 ) η 3 ) +, we see that Ũ, C [, )) and that Ũ, has at most one ole of maximal ode + at the oint η 3. In view of these eliminay consideations we conclude that the following limits exist and 3

35 that the conditions in Lemma 3.4 ae satisfied: and lim η 3) ++3 Ũ, η 3 ) lim η η 3 η3 3) ++3 U, η 3 ) 0. lim + η 3)Ũ,η 3 ) 0. η 3 Now the statement follows diectly fom Lemma 3.4 when looking at the definition of ũ,, and ũ,, and conseuently elacing U with Ũ, and α with + +. Ste : Let u H T 3 ). Since C T 3 ) is dense in H T 3 ), thee exists a seuence u n ) n N C T 3 ) such that u n u in H T 3 ) fo n. Because we have aleady oved that u n, n N satisfies ou statement, ensuing that the seuences of coefficients ũ n,, and ũ n,, coesonding to u n convege fo fixed, and will conclude the oof: We have ũ n,, ++ u n,, ++ T 3 u n ξ, ξ, ξ 3 )ψ,, ξ, ξ, ξ 3 )dξ dξ dξ 3. Since ψ,, ),, N0 foms an othogonal basis fo L T 3 ) and since H T 3 ) L T 3 ), the mas F : u ũ,, ae continuous linea functionals on H T 3 ) and thus lim n F u n ) F u). In case of ũ,, we study the functionals F : u ũ,, that ma C T 3 ) into R. Since F is a linea functional that is continuous with esect to the H T 3 )-nom, we see by density of C T 3 ) in H T 3 ) that it is indeed a well-defined continuous linea functional on H T 3 ) and thus again lim n F un ) F u). C Extended vesions of the tables of Section 33

36 N su u PN Π N u)) u L u H su u PN Π N u)) u H Table 3: Comuted constants C fo D maximization oblems 34

37 N su u PN Π N u L Γ) u L T ) u H T ) su u PN Π N u L Γ) u H T ) su u PN Π N u H T ) u H T ) N+) Table 4: Extended vesion of Table