On the stability of the boundary trace of the polynomial L^2-projection on triangles and tetrahedra (extended version)
aa r X i v : . [ m a t h . NA ] J a n On the stability of the boundary trace of the polynomial L -projection ontriangles and tetrahedra (extended version) J.M. Melenk, T. Wurzer
Vienna University of Technology, Wiedner Hauptstraße 8-10, A-1040 Vienna
Abstract
For the reference triangle or tetrahedron T , we study the stability properties of the L ( T )-projection Π N onto thespace of polynomials of degree N . We show k Π N u k L ( ∂ T ) ≤ C k u k L ( T ) k u k H ( T ) . This implies optimal convergencerates for the approximation error k u − Π N u k L ( ∂ T ) for all u ∈ H k ( T ), k > / Keywords: polynomials on triangles, polynomials on tetrehedra, polynomial L -projection
1. Introduction and main results
The study of polynomials and their properties as the polynomial degree tends to infinity has a very long historyin numerical mathematics. Concerning approximation and stability properties of various high order approximationoperators, the univariate case is reasonably well understood (in the way of examples, we mention the monographs[27] for orthogonal polynomials and [9] for issues concerning approximation). One-dimensional results can oftenbe generalized to the case of the d -dimensional hyper cube by tensor product arguments. Indeed, a significantnumber of results is available for the tensorial case, and we point the reader to the area of spectral methods [1, 4]and to [15] and references therein. The situation is less developed for simplices, possibly due to a presumed lackof product structure. It is the purpose of this note to contribute to this field by studying the stability propertiesof the polynomial L -projection on triangles and tetrahedra, paying special attention to its trace on the boundary.Our main result is Theorem 1.1 below on the stability of the boundary trace of the L -projection. It generalizesknown results for tensor product domains to the case of triangles/tetrahedra: Theorem 1.1 is the analog of [14,Lemma 4.2] (and correspondingly, Corollary 1.2 is the analog of [14, Rem. 4.3] and [17, Lem. 3.5]). We mention thatthe two-dimensional case of triangles was studied in [30], where Theorem 1.1, Corollary 1.2, and Corollary 1.3 areshown for triangles. Independently, closely related results have recently been obtained in [6] and [12]. The noveltyof the present work over [6] is that, in the language of Corollary 1.2 below, we extend the approximation result of[6] from s ≥ s > /
2. The elegant proof of Corollary 1.3 given in [12, Lem. 6.4] relies heavily on Corollary 1.2;the analysis presented here provides the tools for an alternative proof of Corollary 1.3, which we outline as well.The stability result of Theorem 1.1 has applications in the analysis of the hp -version of discontinuous Galerkinmethods ( hp -DGFEM) as demonstrated in [25]. More generally, simplicial elements are, due to their greater geometricflexibility as compared to tensor product elements, commonly used in high order finite element codes so that anunderstanding of stability and approximation properties of polynomial operators defined on simplices could be usefulin other applications of high order finite element methods ( hp -FEM) as well. We refer to [23, 18, 7, 8, 24] for variousaspects of hp -FEM.To fix the notation, we introduce the reference triangle T , the reference tetrahedron T as well as the referencecube S d by T := { ( x, y ) ∈ R : − < x < , − < y < − x } , (1.1a) T := { ( x, y, z ) ∈ R : − < x, y, z < , x + y + z < − } , (1.1b) S d := ( − , d , d ∈ { , , } . (1.1c) Email addresses: [email protected] (J.M. Melenk), [email protected] (T. Wurzer)
Preprint submitted to Elsevier August 17, 2018 hroughout, we will denote by P N the space of polynomials of (total) degree N . We then have: Theorem 1.1.
Let T be the reference triangle or tetrahedron and denote by Π N : L ( T ) → P N the L ( T ) -projectiononto the space of polynomials of degree N ∈ N . Then there exists a constant C > independent of N such that k Π N u k L ( ∂ T ) ≤ C k u k L ( T ) k u k H ( T ) ∀ u ∈ H ( T ) . (1.2) In particular, therefore, k Π N u k L ( ∂ T ) ≤ C k u k B / , ( T ) ∀ u ∈ B / , ( T ) , (1.3) where the Besov space B / , ( T ) = ( L ( T ) , H ( T )) / , is defined as an interpolation space using the so-called realmethod (see, e.g., [29, 28] for details). As already mentioned above, the following corollary extends the admissible range for the parameter s from s ≥ s > / Corollary 1.2.
Let T be the reference triangle or tetrahedron. Then for every s > / there exists a constant C s > such that k u − Π N u k L ( ∂ T ) ≤ C s ( N + 1) − ( s − / k u k H s ( T ) ∀ u ∈ H s ( T ) . (1.4)The following result, which generalizes the analogous result on hyper cubes of [5, Thm. 2.2], is derived in [12, Lem. 6.4]with an integration by parts argument, a polynomial inverse estimate, and Corollary 1.2 with s = 1. Corollary 1.3 ([12, Lem. 6.4]) . Let T be the reference triangle or tetrahedron. Then there exists a constant C > such that for all N ∈ N k Π N u k H ( T ) ≤ C ( N + 1) / k u k H ( T ) ∀ u ∈ H ( T ) . (1.5)We will only prove Theorem 1.1 and Corollary 1.2 for the 3D case at the end of Section 4. We refer to the BachelorThesis [30] for the 2D version. We will present the key steps for an alternative proof of Corollary 1.3 at the end ofSection 5. Remark 1.4.
The regularity requirement u ∈ B / , ( T ) in the stability estimate (1.3) is essentially the minimal one,if uniformity in N is sought. To see this, we first note that taking the limit N → ∞ reproduces the known result(see, e.g., [28, Chap. 32]) k γ u k L ( ∂ T ) ≤ C k u k B / , ( T ) , where γ is the trace operator. Next, we recall that for any ε > H / ε ( T ) ⊂ B / , ( T ) ⊂ H / − ε ( T ) but the trace operator γ cannot be extended to a bounded linearoperator H / − ε ( T ) → L ( ∂ T ).Numerical results indicating for the 1D and the 2D situation the sharpness of Theorem 1.1 and Corollary 1.3 aregiven in Section 6 below. In the tensor-product case of squares and hexahedra, the arguments leading to Theorem 1.1 and Corollary 1.3 canbe reduced to a one-dimensional setting and have been worked out in [14]—we recapitulate the key steps later in thissubsection. Most of this reduction to one-dimensional settings is also possible for simplices, where Jacobi polynomialscome into play instead of the simpler Legendre or Gegenbauer/ultraspherical polynomials. Let us highlight somereasons why a reduction to one-dimensional situations is possible and what the key ideas of the proof of Theorem 1.1are: (I)
The first basic tool for a dimension reduction is a classical one, the so-called Duffy transformation (see (3.1)below), which maps the simplex into a hyper cube. As noted already by [19, 10, 18] orthogonal polynomials onthe simplex (see Lemma 3.3) can be defined in the transformed variables through products of univariate Jacobipolynomials, which expresses the desired reduction to one-dimensional settings. These Jacobi polynomials arise sincethe transformation from the simplex to the cube changes the classical (unweighted) Lebesgue measure to a weightedmeasure. The situation is technically more complicated than the tensor-product setting: first, Jacobi polynomials2rise (due to the weights) in contrast to the more common Gegenbauer/ultraspherical polynomials for the tensor-product setting. Second, the weight in the Jacobi polynomials is not fixed so that the dependence on both thepolynomial degree and the weight needs to be tracked explicitly (cf. the definition of the orthogonal polynomial onthe simplex in Lemma 3.3). Nevertheless, the Jacobi polynomials are classical orthogonal polynomials, and one candraw on a plethora of known properties for the purpose of both analysis and design of algorithms. We mention inpassing that these observations have been made and exploited previously in different connections, for example, inthe works [2, 20, 6, 18]. (II)
A second ingredient to the reduction to one-dimensional problems arises from the fact that we aim at traceestimates in Theorem 1.1. Such estimates hark back to techniques associated with the names of Gagliardo andNirenberg and are closely connected to the 1D Sobolev embedding H ⊂ L ∞ . For example, for the half space R d + := { ( x ′ , x d ) | x ′ ∈ R d − , x d > } , the trace estimate can be cast in multiplicative form as k u ( · , k L ( R d − ) . k u k L ( R d + ) h k u k L ( R d + ) + k ∂ a u k L ( R d + ) i where one has some freedom to choose the vector (field) a as long as it is not tangential to the plane R d − × { } (the classical proofs usually take a to be the normal to R d − × { } ). This freedomto suitably choose the vector field a is exploited in the proof of Corollary 1.2 given in [6], where a is a vector fieldpointing from one face of the reference tetrahedron to the vertex opposite. Our analysis here and that of [30] effectsa similar thing: It performs the analysis on the reference cube S for the Duffy-transformed function e u and singlesout ∂ η e u , which is closely related to the directional derivative ∂ a u on the simplex selected by [6]. (III) Given that we study the L -projection, the observation (II) shows that we need a connection between theexpansion of u in orthogonal polynomials on the triangle/tetrahedron and the expansion of ∂ a u . This is at theheart of the analysis of [6]. Likewise it is the key step in [30] and the present article. Working in the transformedvariables on the cube S d and denoting the transformed function by e u , we have to relate the expansion of e u to thatof ∂ η e u . Ultimately, the issue is to understand the relation between the sequences (ˆ u q ) ∞ q =0 and (ˆ b q ) ∞ q =0 , where theterms ˆ u q and ˆ b q are the coefficients of the function u = P q ˆ u q P ( α, q and its derivative u ′ = P q ˆ b q P ( α, q in terms ofthe expansion in series of Jacobi polynomials P ( α, q . A technical complication over the tensor product case is that afamily of expansions (parametrized by α ) has to be considered and that the dependence on α has to be tracked. Itis easy to express the coefficients ˆ u q in terms of the coefficients ˆ b q —see Lemma 2.3 for the general case and (1.6) forthe special case α = 0. It is harder to control the coefficients ˆ b q in terms of the coefficients ˆ u q . In the present work,this is achieved in Lemma 2.6 and discussed in more detail in the following point (IV). (IV) The refinement of the present analysis and [30] over [6] is the multiplicative structure of the estimate. Thisresults from a refined connection between the expansion of a function and its derivative. Let us review the originof the multiplicate estimate (1.2) for tensor product domains as given in [14, Lem. 4.1]), since similar ideas underliethe arguments here. For the case N ≥ T := ( − , | (Π N u )(1) | . k u k L ( T ) k u k H ( T ) . Bythe 1D Sobolev embedding theorem already mentioned above we have k u k L ∞ ( T ) . k u k L ( T ) k u k H ( T ) . It thereforesuffices to establish the inequality for u − Π N u . Following [17, Lem. 3.5], we expand u and its derivative u ′ inLegendre series (we write, as is common, L q = P (0 , q ): u = ∞ X q =0 ˆ u q L q , u ′ = ∞ X q =0 ˆ b q L q , with coefficients ˆ u q and ˆ b q explicitly given by ˆ u q = q +12 R T u ( x ) L q ( x ) dx and ˆ b q = q +12 R T u ′ ( x ) L q ( x ) dx . Orthog-onality properties of the Legendre polynomials imply (see Lemma 2.3 below with α = 0)ˆ u q = ˆ b q − q − − ˆ b q +1 q + 3 , q ≥ . (1.6)Since L q (1) = 1 for all q ∈ N , we get( u − Π N u )(1) = ∞ X q = N +1 ˆ u q = ˆ b N N + 1 + ˆ b N +1 N + 3 . (1.7)3he terms in the last expression are now estimated using a telescoping sum: ˆ b N N + 1 ! = ∞ X r = N ˆ b r r + 1 ! − ˆ b r +2 r + 2) + 1 ! = ∞ X r = N ˆ b r r + 1 − ˆ b r +2 r + 2) + 1 ! ˆ b r r + 1 + ˆ b r +2 r + 2) + 1 ! = ∞ X r = N ˆ u r +1 ˆ b r r + 1 + ˆ b r +2 r + 2) + 1 ! . ∞ X r = N r + 1) + 1 | ˆ u r | ! / ∞ X r = N (2 r + 1) ˆ b r r + 1 ! / . k u k L ( − , k u ′ k L ( − , , where we have used k u k L ( T ) = P ∞ q =0 | ˆ u q | q +1 and k u ′ k L ( T ) = P ∞ q =0 | ˆ b q | q +1 . We therefore conclude | ( u − Π N u )(1) | . k u k L k u ′ k L . In particular, the above developments provide a simple proof of Lemma 2.6 below for thespecial case α = 0. This lemma is at the heart of the multiplicative structure of of Theorem 1.1.
2. One-dimensional results
As mentioned in Section 1.1, many aspects of the proof of Theorem 1.1 and Corollary 1.3 can be reduced to one-dimensional settings. In this section, we collect the univariate results for the proof of Theorem 1.1 in Section 2.2 andfor the alternative proof of Corollary 1.3 in Section 2.3.
We denote by P ( α,β ) n , α , β > − n ∈ N , the Jacobi polynomials, [27]. From [27, (4.3.3)] we have the followingorthogonality relation for Jacobi polynomials and p , q ∈ N : Z − (1 − x ) α (1 − x ) β P ( α,β ) p ( x ) P ( α,β ) q ( x ) dx = γ ( α,β ) p δ p,q ; (2.1)here, δ p,q represents the Kronecker symbol and γ ( α,β ) p := 2 α + β +1 p + α + β + 1 Γ( p + α + 1)Γ( p + β + 1) p !Γ( p + α + β + 1) . (2.2)Furthermore, we abbreviate factors that will appear naturally in our computations: h ( q, α ) := − q + 1)(2 q + α + 1)(2 q + α + 2) , g ( q, α ) := 2 q + 2 α (2 q + α − q + α ) ,h ( q, α ) := 2 α (2 q + α + 2)(2 q + α ) , g ( q, α ) := 2 α (2 q + α − q + α ) , (2.3) h ( q, α ) := 2( q + α )(2 q + α + 1)(2 q + α ) , g ( q, α ) := − q − q + α − q + α − . By a direct calculation, we can establish relations between the h i and g i . Lemma 2.1.
Let h i , g i , i ∈ { , , } , be defined in (2.3). Then there holds for any q ≥ and α ∈ N g ( q + 1 , α ) γ ( α, q = h ( q + 1 , α ) γ ( α, q +1 , g ( q + 1 , α ) γ ( α, q = h ( q, α ) γ ( α, q , g ( q + 1 , α ) γ ( α, q = h ( q − , α ) γ ( α, q − , (2.4)( − q γ ( α, q h ( q, α ) + ( − q +1 γ ( α, q +1 h ( q + 1 , α ) + ( − q +2 γ ( α, q +2 h ( q + 2 , α ) = 0 . (2.5)4 urthermore, for any q ≥ Proof.
This follows directly from the definitions and simple calculations. Details can be found in Appendix B.We will denote by b P ( α, q the antiderivative of P ( α, q − , i.e., b P ( α, q ( x ) := Z x − P ( α, q − ( t ) dt. (2.7)The following lemma states important relations between Jacobi polynomials, their derivatives, and their antideriva-tives. Lemma 2.2.
Let α ∈ N and h i , g i , i ∈ { , , } , be given by (2.3) and γ ( α,β ) p by (2.2). Then we have(i) for q ≥ Z x − (1 − t ) α P ( α, q ( t ) dt = − (1 − x ) α (cid:16) h ( q, α ) P ( α, q +1 ( x ) + h ( q, α ) P ( α, q ( x ) + h ( q, α ) P ( α, q − ( x ) (cid:17) , (ii) for q ≥ b P ( α, q ( x ) = g ( q, α ) P ( α, q ( x ) + g ( q, α ) P ( α, q − ( x ) + g ( q, α ) P ( α, q − ( x ) , (iii) for q ≥ γ ( α, q P ( α, q ( x ) = h ( q − , α ) γ ( α, q − (cid:0) P ( α, q − (cid:1) ′ ( x ) + h ( q, α ) γ ( α, q (cid:0) P ( α, q (cid:1) ′ ( x ) + h ( q + 1 , α ) γ ( α, q +1 (cid:0) P ( α, q +1 (cid:1) ′ ( x ) . Proof.
The proof of (i) relies on known relations satisfied by Jacobi polynomials (specifically, [18, (A.3), (A.4),(A.8), (A.9)]); see Appendix B for details. (ii) is taken from [2]; (iii) is obtained by differentiating (ii) and usingLemma 2.1.
The essential ingredient of the one-dimensional analysis in [5, Thm. 2.2], [17, Lem. 3.5], [14, Lem. 4.1] is the abilityto relate the expansion coefficients (ˆ u q ) ∞ q =0 of the Legendre expansion u = P q ˆ u q P (0 , q to the expansion coefficients(ˆ b q ) ∞ q =0 of the Legendre expansion u ′ = P q ˆ b q P (0 , q ; we illustrated this point already in (IV) of Section 1.1This relation generalizes to the case of expansions in Jacobi polynomials. A first result in this direction is (see also[6, Lem. 2.1] and [2, Lem. 2.2]): Lemma 2.3.
Let α ∈ N . Let U ∈ C ( − , and let (1 − x ) α U ( x ) as well as (1 − x ) α +1 U ′ ( x ) be integrable.Furthermore, assume lim x → (1 − x ) α U ( x ) = 0 and lim x →− (1 + x ) U ( x ) = 0 . Then the expansion coefficients u q := Z − (1 − x ) α U ( x ) P ( α, q ( x ) dx, b q := Z − (1 − x ) α U ′ ( x ) P ( α, q ( x ) dx. satisfy the following connection formula for q ≥ : u q = h ( q, α ) b q +1 + h ( q, α ) b q + h ( q, α ) b q − . (2.8)5 urthermore, we have the representations U = ∞ X q =0 γ ( α, q u q P ( α, q U ′ = ∞ X q =0 γ ( α, q b q P ( α, q and the equalities Z − | U ( x ) | (1 − x ) α dx = ∞ X q =0 γ ( α, q | u q | , Z − | U ′ ( x ) | (1 − x ) α dx = ∞ X q =0 γ ( α, q | b q | . Proof.
Follows from an integration by parts and the representation of antiderivatives of Jacobi polynomials in termsof Jacobi polynomials given in Lemma 2.2 (i). We refer to Appendix B for details.This connection formula between the coefficients u q and b q allows us to bound a weighted sum of the coefficients u q by a weighted sum of the coefficients b q : Lemma 2.4.
Let U ∈ C ( − , and assume R − | U ( x ) | (1 − x ) α dx < ∞ as well as R − | U ′ ( x ) | (1 − x ) α dx < ∞ .Let u q and b q be defined as in Lemma 2.3. Then there exist constants C , C > independent of α and U such that ∞ X q =1 γ ( α, q ( q + α ) | u q | ≤ C ∞ X q =0 γ ( α, q | b q | ≤ C Z − | U ′ ( x ) | (1 − x ) α dx. Proof.
The result follows from the relation between u q and b q given in Lemma 2.3 and from bounds for h , h , h .The following simple lemma is merely needed for the proof of Lemma 2.6 below. Lemma 2.5.
Let α ∈ N and q ≥ . Then there exists a constant C > independent of q and α such that α N X j = q + α j ≤ αq + α ∀ N ≥ q + α. Proof.
Follows by the standard argument of majorizing the sum by an integral.While Lemma 2.3 shows that the coefficients u q can be expressed as a short linear combination of the coefficients b q (a maximum of 3 coefficients suffices), the converse is not so easy. The following lemma may be regarded as a weakconverse of Lemma 2.3 since it allows us to bound the coefficients b q in terms of the coefficients u q and weightedsums of the coefficients b q . This is the main result of this section and the key ingredient of the proof of Theorem 1.1as it is responsible for the multiplicative structure of the bound in Theorem 1.1. In the proof of Lemma 2.6, thereader will recognize several arguments from (IV) of Section 1.1. Lemma 2.6.
Assume the hypotheses of Lemma 2.3 and let u q and b q be defined as in Lemma 2.3. Then there existsa constant C > independent of q ≥ and α such that | b q − | + | b q | ≤ C α +1 X j ≥ q γ ( α, j u j / X j ≥ q − γ ( α, j b j / . roof. We may assume that the right-hand side of the estimate in the lemma is finite. In view of the sign propertiesof h , h , h and (2.6) we have | h ( q, α ) | + | h ( q, α ) | = | h ( q, α ) | . (2.9)We introduce the abbreviations α q := h ( q, α ) h ( q, α ) = α (2 q + α + 1)(2 q + α + 2)( q + α ) ∈ [0 , ,ε q := α q (1 − α q +1 ) = α ( q + 2)(2 q + α + 1)(2 q + α + 4)( q + 1 + α )( q + α ) ∈ [0 , . By rearranging terms in Lemma 2.3 and using the triangle inequality we get | h ( q, α ) | | b q − | ≤ | u q | + | h ( q, α ) | | b q | + | h ( q, α ) | | b q +1 | . We set z q := | u q || h ( q, α ) | and by applying (2.9) we arrive at | b q − | ≤ z q + α q | b q | + (1 − α q ) | b q +1 | . (2.10)Iterating (2.10) once gives | b q − | ≤ z q + α q (cid:0) z q +1 + α q +1 | b q +1 | + (1 − α q +1 ) | b q +2 | (cid:1) + (1 − α q ) | b q +1 |≤ z q + α q z q +1 + (cid:0) − α q (1 − α q +1 ) (cid:1) | b q +1 | + α q (1 − α q +1 ) | b q +2 | = z q + α q z q +1 + (1 − ε q ) | b q +1 | + ε q | b q +2 | . Squaring and applying the Cauchy-Schwarz inequality yields b q − ≤ ( z q + α q z q +1 ) + 2( z q + α q z q +1 ) (cid:0) (1 − ε q ) | b q +1 | + ε q | b q +2 | (cid:1) + (1 − ε q ) b q +1 + ε q b q +2 + 2 ε q (1 − ε q ) | b q +1 | | b q +2 |≤ ( z q + α q z q +1 ) + 2( z q + α q z q +1 ) (cid:0) (1 − ε q ) | b q +1 | + ε q | b q +2 | (cid:1) + (cid:0) (1 − ε q ) + ε q (1 − ε q ) (cid:1) b q +1 + (cid:0) ε q + ε q (1 − ε q ) (cid:1) b q +2 . We abbreviate the first two addends by f q := ( z q + α q z q +1 ) + 2( z q + α q z q +1 ) (cid:0) (1 − ε q ) | b q +1 | + ε q | b q +2 | (cid:1) (2.11)and obtain b q − ≤ f q + (1 − ε q ) b q +1 + ε q b q +2 , which we rewrite as b q − − b q +1 ≤ f q + ε q (cid:0) b q +2 − b q +1 (cid:1) . (2.12)Next, as was done in (IV) of Section 1.1, we use a telescoping sum. As we assume that the sums on the right-handside of the statement of this lemma are finite, i.e., X j γ ( α, j u j < ∞ , X j γ ( α, j b j < ∞ , (2.13)7nd since, γ ( α, j = (2 j + α + 1)2 − ( α +1) we have √ q | b q | → q → ∞ . Hence, b q − + b q = ∞ X j =0 b q − j − b q − j +2 + b q +2 j − b q +2 j +2 ≤ ∞ X j =0 f q +2 j + ε q +2 j (cid:0) b q +2+2 j − b q +1+2 j (cid:1) + f q +1+2 j + ε q +1+2 j (cid:0) b q +3+2 j − b q +2+2 j (cid:1) = ∞ X j =0 f q + j − ∞ X j =0 ε q +2 j b q +1+2 j + ∞ X j =0 ( ε q +2 j − ε q +2 j +1 ) b q +2+2 j + ∞ X j =0 ε q +1+2 j b q +3+2 j = ∞ X j =0 f q + j − ε q b q +1 + ∞ X j =0 ( ε q +1+2 j − ε q +2+2 j ) b q +3+2 j + ∞ X j =0 ( ε q +2 j − ε q +2 j +1 ) b q +2+2 j = ∞ X j =0 f q + j − ε q b q +1 + ∞ X j =0 ( ε q + j − ε q + j +1 ) b q +2+ j . We conclude, noting that ε q ≥ b q − + b q ≤ b q − + b q + ε q b q +1 ≤ F q + S q +2 , (2.14)where F q := X j ≥ q f j , (2.15) S q := X j ≥ q ε ′ j b j with ε ′ j := | ε j − − ε j − | . (2.16)By positivity of ε ′ j and f j we have S q +1 ≤ S q as well as F q +1 ≤ F q . Therefore, we get from (2.14) and the definitionof S q S q = ε ′ q b q + ε ′ q +1 b q +1 + S q +2 ≤ S q +2 + max { ε ′ q , ε ′ q +1 } S q +3 + max { ε ′ q , ε ′ q +1 } F q +1 ≤ (1 + max { ε ′ q , ε ′ q +1 } ) S q +2 + max { ε ′ q , ε ′ q +1 } F q . Abbreviating ε ′′ q := max { ε ′ q , ε ′ q +1 } we therefore have S q ≤ (1 + ε ′′ q ) S q +2 + ε ′′ q F q . (2.17)Iterating (2.17) N times leads to S q ≤ S q +2 N +2 N Y j =0 (1 + ε ′′ q +2 j ) + N X j =0 ε ′′ q +2 j F q +2 j j − Y i =0 (1 + ε ′′ q +2 i ) . (2.18)A calculation shows ε ′ j . α ( α + j ) ( α + j ) = α ( α + j ) . (2.19)From the definition of S q in (2.16), (2.13), and (2.19) it follows that lim q →∞ S q = 0. Furthermore, we can bound theproduct appearing in (2.18) uniformly in N with the aid of the elementary fact ln(1 + x ) ≤ x for x ≥ N Y j =0 (1 + ε ′′ q +2 j ) = exp N X j =0 ln(1 + ε ′′ q +2 j ) ≤ exp N X j =0 ε ′′ q +2 j , (2.20)8rom (2.19) we get N X j =0 ε ′′ q +2 j . N X j =0 α ( α + q + 2 j ) . α N X j = q α + j ) . αα + q ∀ N ≥ q, (2.21)where we used Lemma 2.5 in the last step. Since αα + q <
1, inserting (2.21) in (2.20) gives N Y j =0 (1 + ε ′′ q +2 j ) ≤ C. (2.22)Now, by passing to the limit N → ∞ in (2.18), we obtain a closed form bound for S q : S q ≤ ∞ X j =0 ε ′′ q +2 j F q +2 j j − Y i =0 (1 + ε ′′ q +2 i ) . Applying (2.21), (2.22), (2.19), and the definition of F q we can simplify S q . ∞ X j =0 ε ′′ q +2 j F q +2 j . X j ≥ q X i ≥ j f i α ( α + j ) = X i ≥ q f i i X j = q α ( α + j ) . αα + q F q . Inserting this estimate in (2.14) and using αα + q +2 <
1, we arrive at b q − + b q . F q + αα + q + 2 F q +2 . F q + F q +2 . F q . (2.23)We are left with estimating F q . By the definition of F q in (2.15) and the definition of f q in (2.11) we have F q = X j ≥ q ( z j + α j z j +1 ) + 2 X j ≥ q ( z j + α j z j +1 ) (cid:0) (1 − ε j ) | b j +1 | + ε j | b j +2 | (cid:1) . (2.24)Now we estimate both sums separately starting with the first one: X j ≥ q ( z j + α j z j +1 ) . X j ≥ q z j + α j |{z} ≤ z j +1 . X j ≥ q z j . (2.25)To proceed further, we use the relation between u q and b q from Lemma 2.3. Also, we note that h ( q, α ) & − ( α +1) γ ( α, q . Hence, we obtain z q = | u q | | h ( q, α ) | . α +1 | u q | γ ( α, q | u q | h ( q, α )= 2 α +1 | u q | γ ( α, q h ( q, α ) | h ( q, α ) b q +1 + h ( q, α ) b q + h ( q, α ) b q − | . α +1 | u q | γ ( α, q (cid:0) (1 − α q ) | b q +1 | + α q | b q | + | b q − | (cid:1) . Inserting this in (2.25), we get by applying the Cauchy-Schwarz inequality for sums X j ≥ q ( z j + α j z j +1 ) . α +1 X j ≥ q γ ( α, j | u j | (cid:0) (1 − α j ) | b j +1 | + α j | b j | + | b j − | (cid:1) . α +1 X j ≥ q γ ( α, j | u j | / X j ≥ q − γ ( α, j | b j | / . (2.26)9e continue by estimating the second sum in (2.24). Using again z q . α +1 | u q | /γ ( α, q we get X j ≥ q ( z j + α j z j +1 ) (cid:0) (1 − ε j ) | b j +1 | + ε j | b j +2 | (cid:1) . α +1 X j ≥ q γ ( α, j (cid:0) | u j | + α j |{z} ≤ | u j +1 | (cid:1)(cid:0) (1 − ε j ) | {z } ≤ | b j +1 | + ε j |{z} ≤ | b j +2 | (cid:1) . α +1 X j ≥ q γ ( α, j ( | u j | + | u j +1 | ) ( | b j +1 | + | b j +2 | ) . α +1 X j ≥ q γ ( α, j | u j | / X j ≥ q γ ( α, j +1 | b j +1 | / . α +1 X j ≥ q γ ( α, j | u j | / X j ≥ q − γ ( α, j | b j | / . (2.27)In view of (2.23) (2.24), the bounds (2.26), (2.27) allow us to conclude the proof. In this section, we study the stability of the operator that effects the truncation of an expansion in Jacobi polynomials.We analyze this operator in the weighted H -norm. In other words: The main result of this section, Lemma 2.8,generalizes [5, Thm. 2.2], where the case α = 0 is studied, which corresponds to the analysis of the H -stability ofthe L -projection. This section is closely tied to an alternative proof of Corollary 1.3 and not immediately requiredfor the proof of Theorem 1.1. Lemma 2.7.
For all α , q ∈ N Z − (1 − x ) α (cid:12)(cid:12)(cid:12)(cid:0) P ( α, q (cid:1) ′ ( x ) (cid:12)(cid:12)(cid:12) dx ≤ q ( q + 1 + α ) γ ( α, q . (2.28) Proof.
We abbreviate P q := P ( α, q and I q := R − (1 − x ) α | P ′ q ( x ) | dx . The assertion is trivial for the case q = 0. For α = 0 see [1, (5.3)]. A direct calculation shows I = 0 , I = ( α + 2) α +1 α + 1 , I = (3 + α )( α + 2)2( α + 1) 2 α +1 ;the assertion of the lemma is therefore true for q ∈ { , , } and all α . Thus, we may assume α ≥ q ≥
2. FromLemma 2.2, (ii) with q + 1 and q there we get P ′ q +1 = 1 g ( q + 1 , α ) P q − g ( q + 1 , α ) g ( q + 1 , α ) g ( q, α ) P q − + (1 − ε q ) P ′ q − − ε q P ′ q − , (2.29)where ε q := − g ( q + 1 , α ) g ( q, α ) g ( q + 1 , α ) g ( q, α ) = α (2 q + 1 + α )( q − q + 1 + α )(2 q + α − q + α ) . We note that 0 ≤ ε q ≤
1. Furthermore, we calculate (cid:0) (1 − ε q ) P ′ q − − ε q P ′ q − (cid:1) = (1 − ε q ) ( P ′ q − ) + ε q ( P ′ q − ) − ε q (1 − ε q ) P ′ q − P ′ q − so that by integration, Cauchy-Schwarz, and 0 ≤ ε q ≤ Z − (1 − x ) α (cid:0) (1 − ε q ) P ′ q − ( x ) − ε q P ′ q − ( x ) (cid:1) dx ≤ ((1 − ε q ) I q − + ε q I q − ) .
10y the orthogonality properties of the Jacobi polynomials, we conclude in view of (2.29) I q +1 = (cid:18) g ( q + 1 , α ) (cid:19) γ ( α, q + (cid:18) g ( q + 1 , α ) g ( q + 1 , α ) g ( q, α ) (cid:19) γ ( α, q − + Z − (1 − x ) α (cid:0) (1 − ε q ) P ′ q − ( x ) − ε q P ′ q − ( x ) (cid:1) dx ≤ (cid:18) g ( q + 1 , α ) (cid:19) γ ( α, q + (cid:18) g ( q + 1 , α ) g ( q + 1 , α ) g ( q, α ) (cid:19) γ ( α, q − + ((1 − ε q ) I q − + ε q I q − ) . We proceed now by an induction argument on q for fixed α . The induction hypothesis and the fact that q q ( q + 1 + α ) γ ( α, q is monotone increasing in q provides((1 − ε q ) I q − + ε q I q − ) ≤ q − q −
1) + α + 1) γ ( α, q − , and we obtain by some tedious estimates for the other two terms appearing in the bound of I q +1 (see Appendix B): I q +1 ≤ (cid:18) g ( q + 1 , α ) (cid:19) γ ( α, q + (cid:18) g ( q + 1 , α ) g ( q + 1 , α ) g ( q, α ) (cid:19) γ ( α, q − + 4( q − q + α ) γ ( α, q − ≤ q + 1)(( q + 1) + 1 + α ) γ ( α, q +1 " q + 1) + 44( q + 1) + ( q + α ) ( q − γ ( α, q − ( q + 2 + α ) ( q + 1) γ ( α, q +1 = 4(( q + 1) + 1 + α ) ( q + 1) γ ( α, q +1 (cid:20) − q − q ( q + α + 1) − q + 1)(2 q + α − q + α + 2) (cid:21) . The proof is completed by observing that the expression in brackets is bounded by 1.We now study the stability of truncating a Jacobi expansion.
Lemma 2.8.
Let α ∈ N . Let u q and b q be defined as in Lemma 2.3. Then there exists a constant C > (which isexplicitly available from the proof ) independent of α and N such that for every N ∈ N we have Z − (1 − x ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X q =0 γ ( α, q u q (cid:0) P ( α, q (cid:1) ′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ CN ∞ X q =0 γ ( α, q | b q | . Proof.
We abbreviate P q := P ( α, q and compute with the connection formula (2.8) between the coefficients u q and b q X q ≥ N +1 γ ( α, q u q P ′ q = X q ≥ N +1 γ ( α, q [ h ( q, α ) b q +1 + h ( q, α ) b q + h ( q, α ) b q − ] P ′ q = X q ≥ N b q γ ( α, q +1 h ( q + 1 , α ) P ′ q +1 + X q ≥ N +1 b q γ ( α, q h ( q, α ) P ′ q + X q ≥ N +2 b q γ ( α, q − h ( q − , α ) P ′ q − = X q ≥ N +2 b q " γ ( α, q − h ( q − , α ) P ′ q − + 1 γ ( α, q h ( q, α ) P ′ q + 1 γ ( α, q +1 h ( q + 1 , α ) P ′ q +1 + 1 γ ( α, N +1 h ( N + 1 , α ) P ′ N +1 b N +1 + N +1 X q = N γ ( α, q h ( q + 1 , α ) P ′ q +1 b q . With Lemma 2.2, (iii) we therefore conclude X q ≥ N +1 γ ( α, q u q P ′ q = X q ≥ N +2 b q γ ( α, q P q + 1 γ ( α, N +1 h ( N + 1 , α ) P ′ N +1 b N +1 + N +1 X q = N γ ( α, q h ( q + 1 , α ) P ′ q +1 b q . Z − (1 − x ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X q ≥ N +1 γ ( α, q u q P ′ q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ X q ≥ N +2 γ ( α, q b q + CN h ( N + 1 , α ) γ ( α, N +1 ( N + α ) b N +1 + CN N +1 X q = N γ ( α, q ( N + α ) h ( q + 1 , α ) b q ≤ CN X q ≥ N γ ( α, q b q . This allows us to conclude the argument since (cid:0) P ( α, (cid:1) ′ ( x ) = 0. Lemma 2.9 (Hardy inequality) . For β > − and U ∈ C (0 , ∩ C ((0 , there holds Z x β | U ( x ) | dx ≤ (cid:18) β + 1 (cid:19) Z x β +2 | U ′ ( x ) | dx + 1 β + 1 | U (1) | . Proof.
This variant of the Hardy inequality can be shown using [16, Thm. 330]. See Appendix B for details.
3. Properties of expansions on the tetrahedron
To save space we will sometimes denote points in R by just one letter, i.e. ξ = ( ξ , ξ , ξ ) for points in T and η = ( η , η , η ) for points in S . We recall the definition of the reference triangle, tetrahedron, and the d -dimensional hyper cube in (1.1). The3D-Duffy transformation D : S → T , [11], is given by D ( η , η , η ) := ( ξ , ξ , ξ ) = (cid:18) (1 + η )(1 − η )(1 − η )4 − , (1 + η )(1 − η )2 − , η (cid:19) (3.1)with inverse D − ( ξ , ξ , ξ ) = ( η , η , η ) = (cid:18) − ξ ξ + ξ − , ξ − ξ − , ξ (cid:19) . Lemma 3.1.
The Duffy transformation is a bijection between the (open) cube S and the (open) tetrahedron T .Additionally, D ′ ( η ) := (cid:20) ∂ξ i ∂η j (cid:21) i,j =1 = (1 − η )(1 − η ) 0 0 − (1 + η )(1 − η ) (1 − η ) 0 − (1 + η )(1 − η ) − (1 + η ) 1 ⊤ , ( D ′ ( η )) − = 1(1 − η )(1 − η ) η ) 2(1 + η )0 2(1 − η ) 1 − η − η )(1 − η ) , det D ′ = (cid:18) − η (cid:19) (cid:18) − η (cid:19) . Proof.
See, for example, [18]. 12estricted to the face η = − S , the 3D Duffy transformation reduces to the 2D version of the Duffytransformation, and there holds: Lemma 3.2.
Let D be the Duffy transformation and Γ := T × {− } . Set ˜Γ := S × {− } . Then D (˜Γ) = Γ and D | ˜Γ : ˜Γ → Γ is a bijection. Furthermore, for ˜ u = u ◦ D we have Z ( ξ ,ξ , − ∈ Γ u dξ dξ = Z ( η ,η , − ∈ ˜Γ ˜ u − η dη dη . In fact, D | ˜Γ is the standard Duffy transformation from S to T .Proof. Follows by inspection.
In terms of the Jacobi polynomials P ( α,β ) n we introduce orthogonal polynomials on the reference tetrahedron T oftenassociated with the names of Dubiner or Koornwinder, [10, 19, 18]: Lemma 3.3 (orthogonal polynomials on T ) . For p , q , r ∈ N set ψ p,q,r := ˜ ψ p,q,r ◦ D − , where ˜ ψ p,q,r is defined by ˜ ψ p,q,r ( η ) := P (0 , p ( η ) P (2 p +1 , q ( η ) P (2 p +2 q +2 , r ( η ) (cid:18) − η (cid:19) p (cid:18) − η (cid:19) p + q . Then the functions ψ p,q,r are L ( T ) -orthogonal, satisfy ψ p,q,r ∈ P p + q + r ( T ) , and Z T ψ p,q,r ( ξ ) ψ p ′ ,q ′ ,r ′ ( ξ ) dξ = δ p,p ′ δ q,q ′ δ r,r ′ p + 1 22 p + 2 q + 2 22 p + 2 q + 2 r + 3= δ p,p ′ δ q,q ′ δ r,r ′ γ (0 , p γ (2 p +1 , q p +1 γ (2 p +2 q +2 , r p +2 q +2 . Furthermore, any u ∈ L ( T ) can be expanded as u = ∞ X p,q,r =0 h ψ p,q,r , u i L ( T ) k ψ p,q,r k L ( T ) ψ p,q,r = ∞ X p,q,r =0 γ (0 , p p +1 γ (2 p +1 , q p +2 q +2 γ (2 p +2 q +2 , r u p,q,r ψ p,q,r , (3.2) u p,q,r := h ψ p,q,r , u i L ( T ) . (3.3) Proof.
The proof can be found in Appendix B.
Remark 3.4 (orthogonal polynomials in 2D) . Lemma 3.2 stated that D | η = − reduces to the standard 2D versionof the Duffy transformation (see, e.g., [18, Sec. 3.2.1.1] or [21, (3.2.20)] for concrete formulas). Correspondingly, thepolynomials ˜ ψ Dp,q := ˜ ψ p,q, ( · , · , −
1) = P (0 , p ( η ) P (2 p +1 , q ( η ) (cid:18) − η (cid:19) p lead to orthogonal polynomials ψ Dp,q given by ˜ ψ Dp,q = ψ Dp,q ◦ D | η = − with orthogonality properties Z T ψ Dp,q ( ξ ) ψ Dp ′ ,q ′ ( ξ ) dξ dξ = δ p,p ′ δ q,q ′ p + 1 22 p + 2 q + 2 = δ p,p ′ δ q,q ′ γ (0 , p γ (2 p +1 , q p +1 . An expansion analogous to (3.2), (3.3) below is valid; u ∈ L ( T ) can be written as u = ∞ X p,q =0 h u, ψ Dp,q i L ( T ) k ψ Dp,q k L ( T ) ψ Dp,q = X p,q γ (0 , p p +1 γ (2 p +1 , q u p,q ψ Dp,q ,u p,q = h ψ Dp,q , u i L ( T ) . See Appendix B for details. 13 .3. Expansion in terms of ψ p,q,r A basic ingredient of the proofs of Theorem 1.1 and Corollary 1.3 is the reduction of the analysis to one-dimensionalsettings, for which we have provided the necessary results in Section 2. As already flagged in (III) of Section 1.1,the η -variable plays a special role. This is captured in Definition 3.5 below, where the functions e U p,q and e U ′ p,q (withexpansion coefficients ˜ u p,q,r and ˜ u ′ p,q,r ) are introduced. Before that, we introduce for a function u defined in T thetransformed function ˜ u := u ◦ D and get u p,q,r = Z T u ( ξ ) ψ p,q,r ( ξ ) dξ = Z S ˜ u ( η ) ˜ ψ p,q,r ( η ) (cid:18) − η (cid:19) (cid:18) − η (cid:19) dη (3.4)= Z S ˜ u ( η ) P (0 , p ( η ) P (2 p +1 , q ( η ) P (2 p +2 q +2 , r ( η ) (cid:18) − η (cid:19) p +1 (cid:18) − η (cid:19) p + q +2 dη. Definition 3.5.
Let p, q ∈ N and u ∈ L ( T ) . Define the functions U p,q : ( − , → R and e U p,q : ( − , → R aswell as the coefficients ˜ u p,q,r and ˜ u ′ p,q,r by U p,q ( η ) := Z − Z − ˜ u ( η ) P (0 , p ( η ) P (2 p +1 , q ( η ) (cid:18) − η (cid:19) p +1 dη dη , (3.5) e U p,q ( η ) := U p,q ( η )(1 − η ) p + q , (3.6)˜ u p,q,r := Z − (1 − η ) p +2 q +2 e U p,q ( η ) P (2 p +2 q +2 , r ( η ) dη , (3.7)˜ u ′ p,q,r := Z − (1 − η ) p +2 q +2 e U ′ p,q ( η ) P (2 p +2 q +2 , r ( η ) dη . (3.8)With this notation, we have by comparing (3.4) with (3.7) u p,q,r = 12 p + q +2 Z − (1 − η ) p +2 q +2 e U p,q ( η ) P (2 p +2 q +2 , r ( η ) dη = 12 p + q +2 ˜ u p,q,r . (3.9)Since for sufficiently smooth functions u the transformed function ˜ u is constant on η = 1, the orthogonality propertiesof the Jacobi polynomials give us U p,q (1) = 0 for ( p, q ) = (0 , U p,q and e U p,q We start with some preliminary considerations regarding estimates for partial derivatives of the transformed function˜ u . We have ∂ η ˜ u ( η ) = (1 − η )(1 − η )4 ( ∂ u ) ◦ D ( η ) , (3.11) ∂ η ˜ u ( η ) = − (1 + η )(1 − η )4 ( ∂ u ) ◦ D ( η ) + (1 − η )2 ( ∂ u ) ◦ D ( η ) , (3.12) ∂ η ˜ u ( η ) = − (1 + η )(1 − η )4 ( ∂ u ) ◦ D ( η ) − (1 + η )2 ( ∂ u ) ◦ D ( η ) + ( ∂ u ) ◦ D ( η ) , (3.13)where ∂ i denotes the partial derivative with respect to the i -th argument. In particular, we get Z S | ∂ η ˜ u ( η ) | (cid:18) − η (cid:19) dη = Z S | ( ∂ u ) ◦ D ( η ) | (cid:18) − η (cid:19) (cid:18) − η (cid:19) dη = k ∂ ξ u k L ( T ) ≤ k∇ u k L ( T ) , (3.14)14nd Z S | ∂ η ˜ u ( η ) | (cid:18) − η (cid:19) dη . Z S | ( ∂ u ) ◦ D ( η ) | (cid:18) η (cid:19) (cid:18) − η (cid:19) (cid:18) − η (cid:19) dη + k ∂ ξ u k L ( T ) . k ∂ ξ u k L ( T ) + k ∂ ξ u k L ( T ) . k∇ u k L ( T ) . (3.15)These estimates are useful to prove the following lemmas. Lemma 3.6 (properties of U p,q ) . Let u ∈ H ( T ) and U p,q be defined in Definition 3.5. Then there exists a constant C > independent of u such that ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q Z − | U p,q ( η ) | (cid:18) − η (cid:19) dη = k u k L ( T ) , (3.16) ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q Z − | U ′ p,q ( η ) | (cid:18) − η (cid:19) dη ≤ C k∇ u k L ( T ) , (3.17) ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q ( p + q ) Z − | U p,q ( η ) | dη ≤ C k∇ u k L ( T ) . (3.18) Furthermore, we have for
Γ = T × {− } ∞ X p =0 γ (0 , p p +1 γ (2 p +1 , q | U p,q ( − | = k u k L (Γ) . (3.19) Proof.
Proof of (3.16) and (3.17):
The fact that P (0 , p ( η ) P (2 p +1 , q ( η ) (cid:0) − η (cid:1) p are orthogonal polynomials in aweighted L -space on S and the definition of U p,q imply (for fixed η ) the representation (cf. (B.7) for details)˜ u ( η ) = ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q U p,q ( η ) P (0 , p ( η ) P (2 p +1 , q ( η ) (cid:18) − η (cid:19) p , which in turn gives Z S | ˜ u ( η ) | (cid:18) − η (cid:19) dη dη = ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q | U p,q ( η ) | . (3.20)Since det D ′ = (cid:0) − η (cid:1) (cid:0) − η (cid:1) , multiplication with (cid:0) − η (cid:1) and integration in η gives (3.16).Similar to the representation of ˜ u above, we get for ∂ η ˜ u∂ η ˜ u ( η ) = ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q U ′ p,q ( η ) P (0 , p ( η ) P (2 p +1 , q ( η ) (cid:18) − η (cid:19) p . Reasoning as in the case of (3.16) yields ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q Z − (cid:12)(cid:12) U ′ p,q ( η ) (cid:12)(cid:12) (cid:18) − η (cid:19) dη . k∇ u k L ( T ) , which immediately leads to (3.17). Proof of (3.18) for p = 0 : We restrict our attention for the double sum (3.18) to the special case p = 0. We startwith the observation U ,q = Z − (cid:18)Z − ˜ u ( η ) dη (cid:19) P (1 , q ( η ) (cid:18) − η (cid:19) dη . η the function η R − ˜ u ( η , η , η ) dη as well as its derivative in terms of the orthogonalpolynomials P (1 , q , we get with Lemma 2.4 ∞ X q =0 γ (0 , γ (1 , q q Z − | U ,q ( η ) | dη ≤ ∞ X q =1 γ (1 , q ( q + 1) Z − | U ,q ( η ) | dη . Z − Z − (cid:12)(cid:12)(cid:12)(cid:12)Z − ∂ η ˜ u ( η ) dη (cid:12)(cid:12)(cid:12)(cid:12) (1 − η ) dη dη . Z S | ∂ η ˜ u ( η ) | (cid:18) − η (cid:19) dη . k∇ u k L ( T ) , where we appealed to (3.15) in the last estimate. Proof of (3.18) for p ≥ and q ≥ : We restrict our attention in the double sum in (3.18) to the case p ≥ q ≥ U p,q ( η ) = 12 p +1 Z S ˜ u ( η ) P (0 , p ( η ) P (2 p +1 , q ( η )(1 − η ) p +1 dη dη . (3.21)In this double integral, we consider the integration in η . Integration by parts then yields Z − ˜ u ( η ) P (2 p +1 , q ( η )(1 − η ) p +1 dη = (cid:16) ˜ u ( η )(1 − η ) p +1 b P (2 p +1 , q +1 ( η ) (cid:17) (cid:12)(cid:12)(cid:12) − − Z − ∂ η (cid:0) ˜ u ( η )(1 − η ) p +1 (cid:1) b P (2 p +1 , q +1 ( η ) dη = − Z − ∂ η ˜ u ( η )(1 − η ) p +1 b P (2 p +1 , q +1 ( η ) − ( p + 1)˜ u ( η )(1 − η ) p b P (2 p +1 , q +1 ( η ) dη . Hence, we obtain by inserting into (3.21) U p,q ( η ) = − p +1 Z S ∂ η ˜ u ( η ) P (0 , p ( η )(1 − η ) p +1 b P (2 p +1 , q +1 ( η ) dη dη + p + 12 p +1 Z S ˜ u ( η ) P (0 , p ( η )(1 − η ) p b P (2 p +1 , q +1 ( η ) dη dη = − p +1 Z S ∂ η ˜ u ( η ) P (0 , p ( η )(1 − η ) p +1 b P (2 p +1 , q +1 ( η ) dη dη − p + 12 p +1 Z S ( ∂ η ˜ u )( η ) b P (0 , p +1 ( η )(1 − η ) p b P (2 p +1 , q +1 ( η ) dη dη , where in the last equation we used integration by parts in η and the orthogonality property R − P (0 , p ( t ) dt = 0 for p ≥
1. With the abbreviation g i := g i ( q + 1 , p + 1), i = 1, 2, 3, we have by Lemma 2.2, (ii) for p , q ≥ b P (2 p +1 , q +1 ( η ) = g P (2 p +1 , q +1 ( η ) + g P (2 p +1 , q ( η ) + g P (2 p +1 , q − ( η ) , (3.22) b P (0 , p +1 ( η ) = 12 p + 1 (cid:16) P (0 , p +1 ( η ) − P (0 , p − ( η ) (cid:17) . (3.23)Furthermore, we introduce two abbreviations z p,q ( η ) := Z S ( ∂ η ˜ u )( η ) P (0 , p ( η ) (cid:18) − η (cid:19) p +1 P (2 p +1 , q ( η ) dη dη , (3.24)˜ z p,q ( η ) := Z S (( ∂ u ) ◦ D )( η ) P (0 , p ( η ) (cid:18) − η (cid:19) p +1 P (2 p +1 , q ( η ) dη dη . (3.25)16ince we have (3.11), using (3.22), (3.23), (3.24), and (3.25) we get U p,q ( η ) = − (cid:0) g z p,q +1 ( η ) + g z p,q ( η ) + g z p,q − ( η ) (cid:1) − (cid:18) p + 12 p + 1 (cid:19) (cid:18) − η (cid:19) h g (cid:0) ˜ z p +1 ,q +1 ( η ) − ˜ z p − ,q +1 ( η ) (cid:1) + g (cid:0) ˜ z p +1 ,q ( η ) − ˜ z p − ,q ( η ) (cid:1) + g (cid:0) ˜ z p +1 ,q − ( η ) − ˜ z p − ,q − ( η ) (cid:1)i . We use g , g , g . p + q to arrive at( p + q ) | U p,q ( η ) | . X j =0 (cid:18) − η (cid:19) (cid:16) | ˜ z p +1 ,q +1 − j ( η ) | + | ˜ z p − ,q +1 − j ( η ) | (cid:17) + | z p,q +1 − j ( η ) | . (3.26)To estimate the terms on the right-hand side we note that the abbreviations z p,q and ˜ z p,q lead us to the representations ∂ η ˜ u ( η ) = ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q z p,q ( η ) P (0 , p ( η ) (cid:18) − η (cid:19) p P (2 p +1 , q ( η ) , (( ∂ u ) ◦ D )( η ) = ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q ˜ z p,q ( η ) P (0 , p ( η ) (cid:18) − η (cid:19) p P (2 p +1 , q ( η ) . Since the polynomials P (0 , p ( η ) (cid:0) − η (cid:1) p P (2 p +1 , q ( η ) are orthogonal polynomials on the reference triangle T wehave Z S | ∂ η ˜ u ( η ) | (cid:18) − η (cid:19) dη dη = ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q | z p,q ( η ) | , (3.27) Z S | (( ∂ u ) ◦ D )( η ) | (cid:18) − η (cid:19) dη dη = ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q | ˜ z p,q ( η ) | . (3.28)Formula (3.27) together with an integration in η and an application of (3.15) gives ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q Z − | z p,q ( η ) | dη = Z S | ∂ η ˜ u ( η ) | (cid:18) − η (cid:19) dη . k∇ u k L ( T ) . From the representation (3.28) we get by a multiplication with (cid:0) − η (cid:1) , an integration in η , and the use of (3.14)that ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q Z − | ˜ z p,q ( η ) | (cid:18) − η (cid:19) dη = Z S | (( ∂ u ) ◦ D )( η ) | (cid:18) − η (cid:19) (cid:18) − η (cid:19) dη . k∇ u k L ( T ) . Proof of (3.18) for p ≥ and q = 0 : The remaining case for the double sum in (3.18) is p ≥ q = 0. This no longer difficult since we have with q = 0 that the relationship (3.22) simplifies to b P (2 p +1 , q +1 ( η ) = g P (2 p +1 , q +1 ( η ) + g P (2 p +1 , q ( η ) , (3.29)i.e., the same relationship holds except that g is set to zero. Hence, we may proceed exactly as in in the previouscase of p ≥ q ≥ Proof of (3.19):
For the estimate (3.19), we use (3.20) with η = −
1. Noting Lemma 3.2 we have k u k L (Γ) = Z S | ˜ u ( η , η , − | (cid:18) − η (cid:19) dη dη = ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q | U p,q ( − | . U p,q imply corresponding bounds for the functions e U p,q : Lemma 3.7 (properties of e U p,q ) . Let u ∈ H ( T ) and let e U p,q be defined in Definition 3.5. Then there exists aconstant C > independent of u such that ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q Z − (1 − η ) p +2 q +2 (cid:12)(cid:12)(cid:12) e U p,q ( η ) (cid:12)(cid:12)(cid:12) dη = k u k L ( T ) , (3.30)14 ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q Z − (1 − η ) p +2 q +2 (cid:12)(cid:12)(cid:12) e U ′ p,q ( η ) (cid:12)(cid:12)(cid:12) dη ≤ C k∇ u k L ( T ) . (3.31) Proof.
We have e U p,q ( η ) = (1 − η ) − ( p + q ) U p,q ( η ) and therefore e U ′ p,q ( η ) = (1 − η ) − ( p + q ) U ′ p,q ( η ) + ( p + q )(1 − η ) − ( p + q +1) U p,q ( η ) . Hence, 14 Z − (1 − η ) p +2 q +2 (cid:12)(cid:12)(cid:12) e U p,q ( η ) (cid:12)(cid:12)(cid:12) dη = Z − (cid:18) − η (cid:19) | U p,q ( η ) | dη , Z − (1 − η ) p +2 q +2 (cid:12)(cid:12)(cid:12) e U ′ p,q ( η ) (cid:12)(cid:12)(cid:12) dη . Z − (cid:18) − η (cid:19) (cid:12)(cid:12) U ′ p,q ( η ) (cid:12)(cid:12) dη + ( p + q ) Z − | U p,q ( η ) | dη . Using the results of Lemma 3.6 concludes the argument.These results allow us to get bounds for weighted sums of the coefficients ˜ u p,q,r and ˜ u ′ p,q,r given in Definition 3.5: Corollary 3.8.
Assume the hypotheses of Lemma 3.7 and let ˜ u p,q,r , ˜ u ′ p,q,r be given by Definition 3.5. Then thereexist constants C independent of u such that ∞ X p,q,r =0 γ (0 , p p +1 γ (2 p +1 , q γ (2 p +2 q +2 , r | ˜ u p,q,r | ≤ C k u k L ( T ) , (3.32) ∞ X p,q,r =0 γ (0 , p p +1 γ (2 p +1 , q γ (2 p +2 q +2 , r | ˜ u ′ p,q,r | ≤ C k∇ u k L ( T ) . (3.33) Proof.
Since the polynomials P (2 p +2 q +2 , r are orthogonal polynomials in a weighted L -space, expanding e U p,q yieldsthe representation e U p,q ( η ) = ∞ X r =0 γ (2 p +2 q +2 , r ˜ u p,q,r P (2 p +2 q +2 , r ( η )and therefore Z − (1 − η ) p +2 q +2 | e U p,q ( η ) | dη = ∞ X r =0 γ (2 p +2 q +2 , r | ˜ u p,q,r | . The statement (3.32) now follows directly from (3.30) of Lemma 3.7. Analogously, we deal with (3.33), where weexpand e U ′ p,q and conclude with (3.31) of Lemma 3.7. ˜ u p,q,r and ˜ u ′ p,q,r As we have mentioned in (IV) of Section 1.1, the multiplicative structure of the estimate of Theorem 1.1 is basedon a relation between the coefficients ˜ u p,q,r and ˜ u ′ p,q,r . This connection is essentially a one-dimensional effect andfollows from Lemma 2.3: 18 orollary 3.9. Let ˜ u p,q,r and ˜ u ′ p,q,r be as in Definition 3.5 and h i , i ∈ { , , } be given by (2.3). Then for r ≥ and p , q ≥ there holds ˜ u p,q,r = h ( r, p + 2 q + 2)˜ u ′ p,q,r +1 + h ( r, p + 2 q + 2)˜ u ′ p,q,r + h ( r, p + 2 q + 2)˜ u ′ p,q,r − . Proof.
By density we may assume that u ∈ C ∞ ( T ). Lemma 2.3 then implies the result. For a detailed version ofthis proof see Appendix B.
4. Trace stability (Proof of Theorem 1.1)
We start with establishing a representation of the transformed function ˜ u = u ◦ D on the face Γ := T × {− } . Using(A.5) and (A.4) we get P (2 p +2 q +2 , r ( −
1) = ( − r P (0 , p +2 q +2) r (1) = ( − r . Therefore we have for ξ ∈ Γ ψ p,q,r ( ξ ) = ˜ ψ p,q,r (cid:0) D − ( ξ , ξ , − (cid:1) = ( − r (cid:18) − η (cid:19) p P (0 , p ( η ) P (2 p +1 , q ( η ) . Applying the expansion of u in (3.2) we arrive at˜ u ( η , η , −
1) = u ( ξ , ξ , − ∞ X p,q,r =0 γ (0 , p p +1 γ (2 p +1 , q P (0 , p ( η ) (cid:18) − η (cid:19) p P (2 p +1 , q ( η )( − r u p,q,r p +2 q +2 γ (2 p +2 q +2 , r . In view of Lemma 3.2 as well as the expansion (3.2), we obtain for the L (Γ)-norm k u k L (Γ) = Z S | ˜ u ( η , η , − | (cid:18) − η (cid:19) dη dη = ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =0 ( − r u p,q,r p +2 q +2 γ (2 p +2 q +2 , r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4.1)Before proceeding further, we recall (IV) of Section 1.1, where we studied ( u − Π N u )(1) in equation (1.7). There, weobserved that the infinite sum P ∞ q = N +1 ˆ u q reduced to a short sum with merely two terms. A very similar effect takesplace here in the multi-dimensional case in that the infinite sum over r in (4.1) can be expressed as a finite sum: Lemma 4.1.
Let N ≥ . Then there holds ∞ X r = N ( − r p +2 q +2 γ (2 p +2 q +2 , r u p,q,r = ( − N h ( N, p + 2 q + 2) 2 p + q γ (2 p +2 q +2 , N ˜ u ′ p,q,N + N X r = N − ( − r +1 h ( r + 1 , p + 2 q + 2) 2 p + q γ (2 p +2 q +2 , r +1 ˜ u ′ p,q,r . roof. We abbreviate n pq := 2 p + 2 q + 2. In view of Corollary 3.9 we have ∞ X r = N ( − r n pq γ ( n pq , r u p,q,r ( . ) = ∞ X r = N ( − r p + q γ ( n pq , r ˜ u p,q,r = ∞ X r = N ( − r p + q γ ( n pq , r (cid:16) h ( r, n pq )˜ u ′ p,q,r +1 + h ( r, n pq )˜ u ′ p,q,r + h ( r, n pq )˜ u ′ p,q,r − (cid:17) = ∞ X r = N +1 ( − r − h ( r − , n pq ) 2 p + q γ ( n pq , r − ˜ u ′ p,q,r + ∞ X r = N ( − r h ( r, n pq ) 2 p + q γ ( n pq , r ˜ u ′ p,q,r + ∞ X r = N − ( − r +1 h ( r + 1 , n pq ) 2 p + q γ ( n pq , r +1 ˜ u ′ p,q,r = ∞ X r = N +1 ( − r ˜ u ′ p,q,r p + q " − h ( r − , n pq ) γ ( n pq , r − + h ( r, n pq ) γ ( n pq , r − h ( r + 1 , n pq ) γ ( n pq , r +1 + ( − N h ( N, n pq ) 2 p + q γ ( n pq , N ˜ u ′ p,q,N + N X r = N − ( − r +1 h ( r + 1 , n pq ) 2 p + q γ ( n pq , r +1 ˜ u ′ p,q,r . By (2.5), the expression in brackets vanishes and that concludes the proof.Since Lemma 4.1 assumes N ≥
1, the terms corresponding to r = 0 in the norm in (4.1) are not included. We nowstudy this special case in the following Lemma 4.2: Lemma 4.2.
Let u ∈ H ( T ) . Then there exists a constant C > independent of p , q , and u such that ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u p,q, p +2 q +2 γ (2 p +2 q +2 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k u k L ( T ) k u k H ( T ) . Proof.
We first note that the coefficient | u , , | ≤ p |T |k u k L ( T ) so that we may focus on the sum with ( p, q ) =(0 , γ (2 p +2 q +2 , = p +2 q +3 p +2 q +3 , we get ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u p,q, p +2 q +2 γ (2 p +2 q +2 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q ( p + q + 1) | u p,q, | . To bound the sum on the right-hand side, we note that an integration by parts and (3.10) give (for ( p, q ) = (0 , p + q +2 u p,q, = Z − (1 − η ) p + q +2 U p,q ( η ) dη (4.2)= 1 p + q + 3 (cid:18) p + q +3 U p,q ( −
1) + Z − (1 − η ) p + q +3 U ′ p,q ( η ) dη (cid:19) , (4.3)These two equations (4.2), (4.3) yield two estimates for u p,q, . From (4.2), we get with the Cauchy-Schwarz inequality | p + q +2 u p,q, | = (cid:12)(cid:12)(cid:12)(cid:12)Z − (1 − η ) p + q +2 U p,q ( η ) dη (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z − (cid:0) (1 − η ) p + q +1 (cid:1) dη (cid:19) (cid:18)Z − (1 − η ) | U p,q ( η ) | dη (cid:19) = 2 p +2 q +5 p + 2 q + 3 Z − (cid:18) − η (cid:19) | U p,q ( η ) | dη . (4.4)20rom (4.3), we obtain a second estimate as follows: | p + q +2 u p,q, | ≤ p + q + 3) p +2 q +6 | U p,q ( − | + (cid:12)(cid:12)(cid:12)(cid:12)Z − (1 − η ) p + q +3 U ′ p,q ( η ) dη (cid:12)(cid:12)(cid:12)(cid:12) ! , where (cid:12)(cid:12)(cid:12)(cid:12)Z − (1 − η ) p + q +3 U ′ p,q ( η ) dη (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18)Z − (cid:0) (1 − η ) p + q +2 (cid:1) dη (cid:19)(cid:18)Z − (1 − η ) | U ′ p,q ( η ) | dη (cid:19) = 2 p +2 q +7 p + q ) + 5 Z − (cid:18) − η (cid:19) | U ′ p,q ( η ) | . Inserting this in the bound before yields | p + q +2 u p,q, | ≤ p +2 q +6 ( p + q + 3) | U p,q ( − | + 1 p + q + 2 Z − (cid:18) − η (cid:19) | U ′ p,q ( η ) | dη ! . (4.5)Next, we abbreviate σ p,q := Z − (cid:18) − η (cid:19) | U p,q ( η ) | dη , τ p,q := Z − (cid:18) − η (cid:19) | U ′ p,q ( η ) | dη . Hence, applying (4.4) and (4.5) we have in view of the elementary observation min { a , b + c } ≤ b + min { a , c } ≤ b + | a | | c | (for real a , b , c ): | p + q +2 u p,q, | ≤ min (cid:26) p +2 q +5 p + q + 3 σ p,q , p +2 q +6 ( p + q + 3) (cid:18) | U p,q ( − | + 1 p + q + 2 τ p,q (cid:19)(cid:27) ≤ p +2 q +5 ( p + q + 3) (cid:18) p + q + 3 | U p,q ( − | + 2 p + q + 2 σ p,q τ p,q (cid:19) . This leads us to | u p,q, | . p + q + 3) | U p,q ( − | + 1( p + q + 2) σ p,q τ p,q . (4.6)Hence, we conclude ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q ( p + q + 1) | u p,q, | . ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q | U p,q ( − | + ∞ X p,q =0 γ (0 , p p +1 γ (2 p +1 , q σ p,q τ p,q . k u k L (Γ) + k u k L ( T ) k∇ u k L ( T ) , where, in the last inequality, we used Cauchy-Schwarz for the sum involving σ pq and τ pq and appealed to Lemma 3.6.The proof of the lemma is now completed with the aid of the multiplicative trace inequality k u k L (Γ) . k u k L ( T ) k u k H ( T ) (see, e.g., [3, Thm. 1.6.6]).We conclude this section with the proof of Theorem 1.1 and Corollary 1.2. Proof of Theorem 1.1 and Corollary 1.2:
In view of the multiplicative trace inequality k u k L (Γ) . k u k L ( T ) k u k H ( T ) (see, e.g., [3, Thm. 1.6.6]) we will only show the statement k u − Π N u k L (Γ) . k u k L ( T ) k u k H ( T ) . We abbreviate n pq := 2 p + 2 q + 2 and c pq := γ (0 , p p +1 γ (2 p +1 , q . By (4.1), we have to bound k u − Π N u k L (Γ) = ∞ X p,q =0 c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =max { ,N +1 − p − q } ( − r n pq γ ( n pq , r u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X p + q ≤ N c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r = N +1 − p − q ( − r n pq γ ( n pq , r u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X p + q ≥ N +1 c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =0 ( − r n pq γ ( n pq , r u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = N X p =0 N − p X q =0 c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r = N +1 − p − q ( − r n pq γ ( n pq , r u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + N X p =0 ∞ X q = N +1 − p c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =0 ( − r n pq γ ( n pq , r u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ X p = N +1 ∞ X q =0 c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =0 ( − r n pq γ ( n pq , r u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . N X p =0 N − p X q =0 c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r = N +1 − p − q ( − r n pq γ ( n pq , r u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | {z } =: S + N X p =0 ∞ X q = N +1 − p c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =1 ( − r n pq γ ( n pq , r u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | {z } =: S + ∞ X p = N +1 ∞ X q =0 c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =1 ( − r n pq γ ( n pq , r u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | {z } =: S + N X p =0 ∞ X q = N +1 − p c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n pq γ ( n pq , u p,q, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | {z } =: S + ∞ X p = N +1 ∞ X q =0 c pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n pq γ ( n pq , u p,q, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | {z } =: S . Lemma 4.2 immediately gives S + S . k u k L ( T ) k u k H ( T ) . Noting the estimates h ( N + 1 − p − q, n pq ) . p + qN . N , p + q = 0 , . . . , N,h ( N + 1 − p − q, n pq ) , h ( N + 2 − p − q, n pq ) . N + p + qN . N , p + q = 0 , . . . , N, γ ( n pq , N +1 − p − q = 2 N + 52 p +2 q +3 , γ ( n pq , N +2 − p − q = 2 N + 72 p +2 q +3 , we obtain for S from Lemma 4.1 S . N X p =0 N − p X q =0 c pq (cid:18)(cid:12)(cid:12)(cid:12) − ( p + q +3) ˜ u ′ p,q,N +1 − p − q (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) − ( p + q +3) ˜ u ′ p,q,N − p − q (cid:12)(cid:12)(cid:12) (cid:19) . Analogously, we get for S and S S . N X p =0 ∞ X q = N +1 − p c pq (cid:18)(cid:12)(cid:12)(cid:12) − ( p + q +3) ˜ u ′ p,q, (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) − ( p + q +3) ˜ u ′ p,q, (cid:12)(cid:12)(cid:12) (cid:19) ,S . ∞ X p = N +1 ∞ X q =0 c pq (cid:18)(cid:12)(cid:12)(cid:12) − ( p + q +3) ˜ u ′ p,q, (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) − ( p + q +3) ˜ u ′ p,q, (cid:12)(cid:12)(cid:12) (cid:19) . S + S + S S + S + S . N X p =0 N − p X q =0 c pq X r ≥ N +1 − p − q γ ( n pq , r | ˜ u p,q,r | / X r ≥ N − p − q γ ( n pq , r | ˜ u ′ p,q,r | / + N X p =0 ∞ X q = N +1 − p c pq X r ≥ γ ( n pq , r | ˜ u p,q,r | / X r ≥ γ ( n pq , r | ˜ u ′ p,q,r | / + ∞ X p = N +1 ∞ X q =0 c pq X r ≥ γ ( n pq , r | ˜ u p,q,r | / X r ≥ γ ( n pq , r | ˜ u ′ p,q,r | / . k u k L ( T ) k∇ u k L ( T ) , where in the last estimate, we have used the Cauchy-Schwarz inequality for sums and Corollary 3.8. Since k∇ u k L ( T ) ≤k u k H ( T ) this concludes the proof of (1.2).The estimate (1.3) follows directly from (1.2) in view of [28, Lem. 25.3].Finally, (1.4) is obtained in a fairly routine way from the stability of Π N just shown. Specifically, for arbitrary v ∈ P N we have by the projection property of Π N as well as the continuity of the trace operator γ : B / / , ( T ) → L ( ∂ T )(cf., e.g., [28, Sec. 32]) k u − Π N u k L ( ∂ T ) ≤ k u − v k L ( ∂ T ) + k Π N ( u − v ) k L ( ∂ T ) ≤ C k u − v k B / , ( T ) . Hence, k u − Π N u k L ( ∂ T ) ≤ C inf v ∈P N k u − v k B / , ( T ) . Fix s > /
2. Let e Π N : L ( T ) → P N be an approximationoperator with simultaneous approximation properties in a scale of Sobolev spaces, viz., k u − e Π N u k L ( T ) ≤ CN − s k u k H s ( T ) , k u − e Π N u k H s ( T ) ≤ C k u k H s ( T ) ∀ u ∈ H s ( T ) . (This can be achieved, for example, by combining the approximation results of [22, Appendix A] for hyper cubes withthe well-known extension operator of Stein, [26, Chap. VII].) The reiteration theorem (cf., e.g., [28, Thm. 26.3]) gives B / , ( T ) = ( L ( T ) , H s ( T )) ( s − / /s, . Also, we have H s ( T ) = ( H s ( T ) , H s ( T )) ( s − / /s, . By interpolationtheory, we then have that Id − e Π N is a bounded linear operator( H s ( T ) , H s ( T )) ( s − / /s, = H s ( T ) → B / , ( T ) = ( L ( T ) , H s ( T )) ( s − / /s, with norm k Id − e Π N k B / , ( T ) ← H s ( T ) ≤ CN − s ( s − / /s . (cid:3) H -stability (Proof of Corollary 1.3) Our procedure to study the H -stability of the L -projection Π N is to consider on the hyper cube S the derivative ∂ η e v N of the transformed function e v N = v N ◦ D , where v N := Π N u . This provides information about a directionalderivative of v N on T . Through affine transformations of the tetrahedron T , information about the full gradientof v N can be inferred.The key step is therefore to control ∂ η e Π N u , where we denote e Π N u := (Π N u ) ◦ D . This is the purpose of the ensuinglemma. Lemma 5.1.
There exists a constant
C > independent of N such that Z S (1 − η )(1 − η ) (cid:12)(cid:12)(cid:12) ∂ η e Π N u ( η ) (cid:12)(cid:12)(cid:12) dη ≤ CN k∇ u k L ( T ) ∀ u ∈ H ( T ) . roof. Given that Π N reproduces constant functions, a Poincar´e inequality allows us to reduce the problem toshowing the weaker estimate Z S (1 − η )(1 − η ) (cid:12)(cid:12)(cid:12) ∂ η e Π N u ( η ) (cid:12)(cid:12)(cid:12) dη ≤ CN k u k H ( T ) ∀ u ∈ H ( T ) . (5.1)We abbreviate n pq := 2 p + 2 q + 2. We see that e Π N u ( η ) = N X p =0 N − p X q =0 N − p − q X r =0 γ (0 , p p +1 γ (2 p +1 , q p + q γ ( n pq , r ˜ u p,q,r ˜ ψ p,q,r ( η )by recalling the relation between u p,q,r and ˜ u p,q,r . Differentiating with respect to η shows us that we have to estimatethe two terms I := Z S (1 − η )(1 − η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X p =0 N − p X q =0 N − p − q X r =0 γ (0 , p p +1 γ (2 p +1 , q γ ( n pq , r ˜ u p,q,r × P (0 , p ( η ) P (2 p +1 , q ( η ) (cid:18) − η (cid:19) p (1 − η ) p + q (cid:0) P ( n pq , r (cid:1) ′ ( η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dη (5.2) I := Z S (1 − η )(1 − η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X p =0 N − p X q =0 N − p − q X r =0 γ (0 , p p +1 γ (2 p +1 , q p + qγ ( n pq , r ˜ u p,q,r × P (0 , p ( η ) P (2 p +1 , q ( η ) (cid:18) − η (cid:19) p (1 − η ) p + q − P ( n pq , r ( η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dη (5.3)First, we consider (5.2). From Lemma 2.8 with α = n pq we get I = N X p =0 N − p X q =0 γ (0 , p p +1 γ (2 p +1 , q Z − (1 − η ) n pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − p − q X r =0 γ ( n pq , r ˜ u p,q,r (cid:0) P ( n pq , r (cid:1) ′ ( η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dη . N X p =0 N − p X q =0 γ (0 , p p +1 γ (2 p +1 , q ( N − p − q ) ∞ X r =0 γ ( n pq , r | ˜ u ′ p,q,r | . N k∇ u k L ( T ) , where in the last step, we appealed to Corollary 3.8. Thus, we arrive at the desired bound for I . Next, we consider(5.3). We have I = N X p =0 N − p X q =0 ( p + q ) γ (0 , p p +1 γ (2 p +1 , q Z − (1 − η ) p +2 q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − p − q X r =0 γ ( n pq , r ˜ u p,q,r P ( n pq , r ( η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dη Lemma 2.9 with β = 2 p + 2 q and the normalization convention for Jacobi polynomials (A.4) now yield I ≤ N X p =0 N − p X q =0 ( p + q ) γ (0 , p p +1 γ (2 p +1 , q p + q ) Z − (1 − η ) n pq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − p − q X r =0 γ ( n pq , r ˜ u p,q,r (cid:0) P ( n pq , r (cid:1) ′ ( η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dη + 1 p + q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − p − q X r =0 γ ( n pq , r ˜ u p,q,r P ( n pq , r ( − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ I + N X p =0 N − p X q =0 p + qγ (0 , p p +1 γ (2 p +1 , q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − p − q X r =0 γ ( n pq , r ˜ u p,q,r P ( n pq , r ( − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = I + N X p =0 N − p X q =0 p + qγ (0 , p p +1 γ (2 p +1 , q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − p − q X r =0 γ ( n pq , r ˜ u p,q,r ( − r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I + 2 N N X p =0 N − p X q =0 γ (0 , p p +1 γ (2 p +1 , q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =0 ( − r γ ( n pq , r ˜ u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r = N − p − q +1 ( − r γ ( n pq , r ˜ u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ I + 2 N N X p =0 N − p X q =0 γ (0 , p p +1 γ (2 p +1 , q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =0 ( − r γ ( n pq , r p + q +2 u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r = N − p − q +1 ( − r γ ( n pq , r p + q +2 u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ I + 2 N N X p =0 N − p X q =0 γ (0 , p p +1 γ (2 p +1 , q − ( p + q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =0 ( − r γ ( n pq , r p + q )+2 u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r = N − p − q +1 ( − r γ ( n pq , r p + q )+2 u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We now recall I . N k∇ u k L ( T ) from above. The sums can be estimated very generously: Using 2 − ( p + q ) ≤
1, weget for the first sum in view of (4.1) N X p =0 N − p X q =0 γ (0 , p p +1 γ (2 p +1 , q − ( p + q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =0 ( − r γ ( n pq , r p + q )+2 u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N X p =0 N − p X q =0 γ (0 , p p +1 γ (2 p +1 , q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r =0 ( − r γ ( n pq , r p + q )+2 u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k u k L (Γ) . k u k L ( T ) k u k H ( T ) , where we set Γ = T × {− } and used the multiplicative trace inequality, [3, Thm. 1.6.6]. For the second sum, weestimate again generously 2 − ( p + q ) ≤ k u − Π N u k L (Γ) in the proof of Theorem 1.1: N X p =0 N − p X q =0 γ (0 , p p +1 γ (2 p +1 , q − ( p + q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r = N − p − q +1 ( − r γ ( n pq , r p + q )+2 u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N X p =0 N − p X q =0 γ (0 , p p +1 γ (2 p +1 , q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X r = N − p − q +1 ( − r γ ( n pq , r p + q )+2 u p,q,r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . k u k L ( T ) k u k H ( T ) , and this completes the proof. Proof of Corollary 1.3:
For a function v and the transformed function ˜ v = v ◦ D , the formula (3.13) provides arelation between ∂ η ˜ v and ∇ v . Rearranging terms yields( ∂ η ˜ v ) ◦ D − ( ξ ) = − ξ − ξ ∂ v ( ξ ) − ξ − ξ ∂ v ( ξ ) + ∂ v ( ξ ) . Therefore, when transforming to T in Lemma 5.1 we get Z T (cid:12)(cid:12)(cid:12)(cid:12) − ξ − ξ ∂ Π N u ( ξ ) − ξ − ξ ∂ Π N u ( ξ ) + ∂ Π N u ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) dξ . N k∇ u k L ( T ) . (5.4)By the symmetry properties of T , we see that also the following two other permutations of indices are valid estimates: Z T (cid:12)(cid:12)(cid:12)(cid:12) − ξ − ξ ∂ Π N u ( ξ ) − ξ − ξ ∂ Π N u ( ξ ) + ∂ Π N u ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) dξ . N k∇ u k L ( T ) , (5.5) Z T (cid:12)(cid:12)(cid:12)(cid:12) − ξ − ξ ∂ Π N u ( ξ ) − ξ − ξ ∂ Π N u ( ξ ) + ∂ Π N u ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) dξ . N k∇ u k L ( T ) . (5.6)25e abbreviate a ( x, y ) := − x − y , a ij := a ( ξ i , ξ j ) and A ( ξ , ξ , ξ ) := a + a a a + a + a a + a + a a sym a + a a + a a + a sym sym a + a . Hence, we see that by adding (5.4), (5.5), and (5.6) we arrive at Z T ( ∇ Π N u ) ⊤ A ( ξ ) ∇ Π N u dξ . N k∇ u k L ( T ) Next, we observe that near the top vertex ( − , − , (cid:12)(cid:12)(cid:12)(cid:12) ξ − ξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ξ − ξ (cid:12)(cid:12)(cid:12)(cid:12) ≤ . This implies that the functions a and a are uniformly bounded on T . Analogously, we get bounds for a , a and a , a by studying the vertices (1 , − , −
1) and ( − , , − ξ ∈T k A ( ξ ) k L ∞ ( T ) < ∞ . By construction, the matrix A ( ξ ) is (pointwise) symmetric positive semidefinite. Our goal is to show that A ( ξ ) is infact positive definite on the set that stays away from the face F opposite the vertex ( − , − , − A ( ξ ). A direct calculation revealsdet A ( ξ ) = 16 ξ + 2 ξ ξ + 2 ξ ξ + 2 ξ + ξ + 1 + 2 ξ + 2 ξ ξ + 2 ξ + ξ ( − ξ ) ( − ξ ) ( − ξ ) = 16 ( ξ + ξ + ξ ) + 2( ξ + ξ + ξ ) + 1( − ξ ) ( − ξ ) ( − ξ ) = 16 (1 + ξ + ξ + ξ ) ( − ξ ) ( − ξ ) ( − ξ ) . The face opposite the vertex ( − , − , −
1) contains the vertices ( − − , − , , − , − , −
1) and is given bythe equation ξ + ξ + ξ + 1 = 0. Furthermore, we conclude that the signed distance of an arbitrary point ξ fromthis face F is given by dist( ξ, F ) = 1 √ ξ + ξ + ξ + 1) . Let, for arbitrary δ > T δ := { ξ ∈ T | dist( ξ, F ) < − δ } . Then, since we stay away from the face F, it is clear that there exists C δ > A ( ξ ) ≥ C δ ∀ ξ ∈ T δ . Combining the above findings, we have that on T δ the matrix A ( ξ ) is in fact symmetric positive definite. Since theentries of A ( ξ ) are uniformly bounded in ξ , Gershgorin’s circle theorem provides a constant C upper such that alleigenvalues of A ( ξ ) are bounded by C upper .A lower bound for the eigenvalues is obtained as follows: Denoting for fixed ξ ∈ T δ the eigenvalues 0 < λ ≤ λ ≤ λ ,we get from det A = λ λ λ C δ ≤ det A = λ λ λ ≤ λ C upper . This provides the desired lower bound for λ . Thus, we conclude that for every δ > c δ > A ( ξ ) ≥ c δ I on T δ . Hence, c δ Z T δ |∇ Π N u | dξ ≤ Z T ( ∇ Π N u ) ⊤ A ( ξ ) ∇ Π N u dξ . N k∇ u k L ( T ) . Affine transformations allow us to get analogous estimates for the sets that stay away from the other faces of T .We therefore get the desired result. (cid:3) . Numerical results In this section, we illustrate the sharpness of Theorem 1.1 and Corollary 1.3 for the 1D and the 2D case. We presentthe best constants in the following 1D and 2D situations: | (Π N u )(1) | ≤ C Dmult k u k L ( I ) k u k H ( I ) ∀P N ( I ) , (6.1) k Π N u k H ( I ) ≤ C DH √ N + 1 k u k H ( I ) ∀ u ∈ P N ( I ) , (6.2) k (Π N u ) k L (Γ) ≤ C Dmult k u k L ( T ) k u k H ( T ) ∀ u ∈ P N ( T ) , (6.3) k Π N u k H ( T ) ≤ C DH √ N + 1 k u k H ( T ) ∀ u ∈ P N ( T ) , (6.4)where I = ( − ,
1) and Γ = ( − , × {− } ⊂ ∂ T . The best constants C Dmult , C Dmult are solutions of constrainedmaximization problem. For example, C Dmult = max {k Π N u k L (Γ) | k u k L ( T ) k u k H ( T ) = 1 , u ∈ P N } , which can be solved using the technique of Lagrange multipliers. The constant C DH is more readily accessible as thesolution of an eigenvalue problem since C DH = sup u ∈P N k u k L (Γ) k u k H ( T ) . The result of the 1D situation is presented in Table 1 whereas the outcome of the 2D calculations is shown inTable 2. The 2D calculations are in agreement with the results of Theorem 1.1 and Corollary 1.3 whereas the 1Dresults illustrate [14, Lem. 4.1] and [5, Thm. 2.2]. N sup u ∈P N | (Π N u )(1) | k u k L I ) k u k H I ) sup u ∈P N | (Π N u )(1) | k u k H I ) N sup u ∈P N | (Π N u )(1) | k u k L I ) k u k H I ) sup u ∈P N | (Π N u )(1) | k u k H I )
55 2.9680 1.046260 2.9719 1.045565 2.9750 1.044870 2.9776 1.044375 2.9798 1.043880 2.9817 1.043485 2.9833 1.043190 2.9847 1.042895 2.9859 1.0425100 2.9869 1.0422105 2.9879 1.0420110 2.9887 1.0418115 2.9895 1.0416120 2.9901 1.0414
Table 1: 1D maximization problems
A. Properties of Jacobi polynomials
We have the following useful formulas (see [18, p. 350 f], [27]):
Recursion Relations a n P ( α,β ) n +1 ( x ) = ( a n + a n x ) P ( α,β ) n ( x ) − a n P ( α,β ) n − ( x ) (A.1)27 sup u ∈P N k Π N u k L k u k L T k u k H T sup u ∈P N k Π N u k L k u k H T sup u ∈P N k Π N u k H T k u k H T ( N + 1) Table 2: 2D maximization problems with a n := 2( n + 1)( n + α + β + 1)(2 n + α + β ) a n := (2 n + α + β + 1)( α − β ) a n := (2 n + α + β )(2 n + α + β + 1)(2 n + α + β + 2) a n := 2( n + α )( n + β )(2 n + α + β + 2) b n ( x ) ddx P ( α,β ) n ( x ) = b n ( x ) P ( α,β ) n ( x ) + b n ( x ) P ( α,β ) n − ( x ) (A.2)with b n ( x ) := (2 n + α + β )(1 − x ) b n ( x ) := n ( α − β − (2 n + α + β ) x ) b n ( x ) := 2( n + α )( n + β )28 pecial Values P ( α,β )0 ≡ P ( α,β ) n (1) = (cid:18) n + αn (cid:19) (A.4) P ( α,β ) n ( − x ) = ( − n P ( β,α ) n ( x ) (A.5) P ( α, ( x ) = 1 + α + 12 (2 + α + β )( x −
1) (A.6) b P ( α, ( x ) = Z x − P ( α, p ( t ) dt = 1 + x (A.7) Special Cases
For the Legendre Polynomial L n ( x ) there holds L n ( x ) = P (0 , n ( x ) (A.8) Miscellaneous Relations ddx P ( α,β ) n ( x ) = 12 ( α + β + n + 1) P ( α +1 ,β +1) n − ( x ) (A.9)2 n Z x − (1 − t ) α (1 + t ) β P ( α,β ) n ( t ) dt = − (1 − x ) α +1 (1 + x ) β +1 P ( α +1 ,β +1) n − ( x ) (A.10) B. Details for selected proofs
B.1. Selected proofs for Section 2B.1.1. Extended proofs of Lemma 2.1Proof of (2.5).
By definition of γ ( α,β ) p we obtain in particular γ ( α, q = 2 α +1 q + α + 1 , which leads in combination with the definition of h , h and h to( − q γ ( α, q h ( q, α ) + ( − q +1 γ ( α, q +1 h ( q + 1 , α ) + ( − q +2 γ ( α, q +2 h ( q + 2 , α )= ( − q +1 q + α + 12 α +1 q + 2(2 q + α + 1)(2 q + α + 2) + ( − q +1 q + α + 32 α +1 α (2 q + α + 4)(2 q + α + 2)+ ( − q +2 q + α + 52 α +1 q + α + 2)(2 q + α + 5)(2 q + α + 4)= ( − q +1 α (cid:18) q + 12 q + α + 2 + (2 q + α + 3) α (2 q + α + 4)(2 q + α + 2) − q + α + 22 q + α + 4 (cid:19) = ( − q +1 α (cid:18) ( q + 1)(2 q + α + 4) + (2 q + α + 3) α − ( q + α + 2)(2 q + α + 2)(2 q + α + 4)(2 q + α + 2) (cid:19) Simply multiplying out the numerator concludes the proof regarding the first equation.Inserting the definition of h , h and h also leads in the case of the second equation to the conclusion h ( q, α ) − h ( q, α ) = 2 α (2 q + α + 1) + 2( q + 1)(2 q + α )(2 q + α )(2 q + α + 1)(2 q + α + 2)= 4 q + 4 q + 6 qα + 2 α + 4 α (2 q + α )(2 q + α + 1)(2 q + α + 2)= (2 q + α + 2)(2 q + 2 α )(2 q + α )(2 q + α + 1)(2 q + α + 2) = h ( q, α ) . .1.2. Extended proofs of Lemma 2.2Proof of Lemma 2.2 (i). Using rearranged versions of (A.2), (A.9) and (A.10) we obtain Z x − (1 − t ) α P ( α, q ( t ) dt ( A. ) = − q (1 + x )(1 − x ) α +1 P ( α +1 , q − ( x )= − q (1 − x )(1 − x ) α P ( α +1 , q − ( x ) ( A. ) = − (1 − x ) α q (1 − x ) 2 q + α + 1 ddx P ( α, q ( x ) ( A. ) = − (1 − x ) α q q + α + 1 q ( α − (2 q + α ) x ) P ( α, q ( x ) + 2 q ( q + α ) P ( α, q − ( x )2 q + α = − (1 − x ) α αP ( α, q ( x ) + 2( q + α ) P ( α, q − ( x ) − (2 q + α ) xP ( α, q ( x )( q + α + 1)(2 q + α )(A.1) allows us now to replace the term xP ( α, q ( x ) by terms involving P ( α, q +1 ( x ), P ( α, q ( x ) and P ( α, q − ( x ). Hence, weget Z x − (1 − t ) α P ( α, q ( t ) dt = − (1 − x ) α q + α + 1)(2 q + α ) ( αP ( α, q ( x ) + 2( q + α ) P ( α, q − ( x ) − q + α + 1)(2 q + α + 2) (cid:16) q + 1)( q + α + 1)(2 q + α ) P ( α, q +1 ( x )+ 2 q ( q + α )(2 q + α + 2) P ( α, q − ( x ) − (2 q + α + 1) α P ( α, q ( x ) (cid:17)) Rearranging terms gives Z x − (1 − t ) α P ( α, q ( t ) dt = − (1 − x ) α q + α + 1)(2 q + α ) ( − q + 1)( q + α + 1)(2 q + α )(2 q + α + 1)(2 q + α + 2) P ( α, q +1 ( x )+ α q + 2 α + 22 q + α + 2 P ( α, q ( x ) + 2( q + α ) q + α + 12 q + α + 1 P ( α, q − ( x ) ) = − (1 − x ) α ( − q + 1)(2 q + α + 1)(2 q + α + 2) | {z } h ( q, α ) P ( α, q +1 ( x )+ 2 α (2 q + α + 2)(2 q + α ) | {z } h ( q, α ) P ( α, q ( x ) + 2( q + α )(2 q + α + 1)(2 q + α ) | {z } h ( q, α ) P ( α, q − ( x ) ) B.1.3. Extended proofs of Lemma 2.7
Lemma B.1 (details of the proof of Lemma 2.7) . Write I q,α := Z − (1 − x ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) P ( α, q (cid:17) ′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) dx Then(i) for α = 0 we have I q, = q ( q + 1) ≤ K · q ( q + 1) γ (0 , q with K = 130 ii) for α ≥ we have I ,α = ( α + 2) α +1 α + 1) ≤ K · · (1 + 1 + α ) γ ( α, with K = 1 / I ,α = (3 + α )( α + 2)2 α +1 α + 1) ≤ K · · (2 + 1 + α ) γ ( α, with K = 9 /
16 (B.2)
Proof. Proof of (i):
We have I q, = q ( q + 1) by [1, (5.3)]. The statement therefore follows directly. Proof of (ii):
A direct calculation shows I ,α · (1 + 1 + α ) γ ( α, = ( α + 2) α +1 α + 1 (3 + α )2 α +1 (2 + α ) = ( α + 3)4( α + 1)this last function is monotone decreasing in α so that its maximum on [1 , ∞ ) is attained for α = 1, which has thevalue 1 /
2. Analogously, we proceed for the case q = 2. We have I ,α · (2 + 1 + α ) γ ( α, = ( α + 2)( α + 3)2( α + 1) 2 α +1 α α +1 α ) = ( α + 2)( α + 5)4( α + 1)( α + 3) . Again, this last function is monotone decreasing on (0 , ∞ ) so that its maximum on [1 , ∞ ) is attained for α = 1. Lemma B.2 (details of the proof of Lemma 2.7) . Define for α , q ∈ N ε q := − g ( q + 1 , α ) g ( q, α ) g ( q + 1 , α ) g ( q, α ) = α (2 q + 1 + α )( q − q + 1 + α )(2 q + α − q + α ) . Then, for α , q ≥ we have ≤ ε q ≤ .Proof. Clearly, ε q ≥
0. To see the estimate ε q ≤
1, we have to show α (2 q + 1 + α )( q − ? ≤ ( q + α )( q + 1 + α )(2 q + α − ⇐⇒ α ( q − q + α −
2) + 3 α ( q − ? ≤ (2 q + α − q + α )( q + α + 1) ⇐⇒ (2 q + α − | {z } ≥ ( α ( q − − ( q + α + 1)( q + α )) + 3 α ( q − ? ≤ α ( q − − ( q + α + 1)( q + α ) + 3 α ( q − ? ≤ ⇐⇒ α ( q − − ( q + α + 1)( q + α ) ? ≤ ⇐⇒ α ( q − − ( q + α ) − ( q + α ) ? ≤ ⇐⇒ α ( q − − q − αq − α − ( q + α ) ? ≤ ⇐⇒ αq − α − q − α − ( q + α ) ? ≤ ⇐⇒ − α − ( q − α ) − ( q + α ) ? ≤ , which is indeed the case. Lemma B.3 (details of the proof of Lemma 2.7) . For α ∈ N , q ≥ we have (cid:18) g ( q + 1 , α ) (cid:19) γ ( α, q ≤ q + 1 · ( q + 1)(( q + 1) + α + 1) γ ( α, q +1 (B.3) (cid:18) g ( q + 1 , α ) g ( q + 1 , α ) g ( q, α ) (cid:19) γ ( α, q − ≤ q + 1 · ( q + 1)(( q + 1) + α + 1) γ ( α, q +1 , (B.4)4( q − q + α ) γ ( α, q − = 4(( q + 1) + 1 + α ) ( q + 1) γ ( α, q +1 (cid:18) − q + α + 2 (cid:19) (cid:18) − α ( q + 1)(2 q + α − (cid:19) (B.5)31 urthermore, we have q + 1) + (cid:18) − q + α + 2 (cid:19) (cid:18) − α ( q + 1)(2 q + α − (cid:19) = 1 − − q − αq + q α + q ( q + 1)(2 q + α − q + α + 2) = 1 − q − + q ( q −
1) + αq ( q − q + 1)(2 q + α − q + α + 2) Proof.
We start with the bound (B.3). We compute1( g ( q + 1 , α )) γ ( α, q q + 1)( q + 2 + α ) γ ( α, q +1 = (2( q + 1) + α − (2( q + 1) + α ) (2( q + 1) + α + 1)(2 q + α + 1)(2( q + 1) + 2 α ) ( q + 2 + α ) ( q + 1)= 1 q + 1 (2 q + 1 + α )(2( q + 1) + α ) (2( q + 1) + α + 1)(2( q + 1) + 2 α ) ( q + 2 + α ) = 4 q + 1 (2 q + 1 + α )(2 q + α + 2) (2 q + α + 3)(2 q + 2 α + 2) (2 q + 2 α + 4) ≤ q + 1 . We now turn to the bound (B.4). We first compute g ( q + 1 , α ) g ( q + 1 , α ) g ( q, α ) = 2 α (2( q + 1) + α − q + 1) + α ) (2( q + 1) + α − q + 1) + α )2( q + 1) + 2 α (2 q + α − q + α )2 q + 2 α = 2 α (2 q + α + 1)(2 q + α − q + 2 α + 2)(2 q + 2 α ) . Then (cid:18) g ( q + 1 , α ) g ( q + 1 , α ) g ( q, α ) (cid:19) γ ( α, q − γ ( α, q +1 q + 1)( q + 2 + α ) = (cid:18) α (2 q + α + 1)(2 q + α − q + 2 α + 2)(2 q + 2 α ) (cid:19) q + 1) + α + 12( q −
1) + α + 1 1( q + 1)( q + 2 + α ) = 4 q + 1 (2 α ) (2 q + α + 1) (2 q + α − (2 q + 2 α + 2) (2 q + 2 α ) q + α + 32 q + α − q + 2 α + 4) ≤ q + 1 (2 α ) (2 q + 2 α ) ≤ q + 1 . Finally, we show (B.5).4( q − q + α ) γ ( α, q − = 4( q + 2 + α ) ( q + 1) γ ( α, q +1 ( q + α ) ( q − γ ( α, q − ( q + 2 + α ) ( q + 1) γ ( α, q +1 = 4( q + 2 + α ) ( q + 1) γ ( α, q +1 (cid:18) − q + α + 2 (cid:19) q − q + 1 2 q + α + 32 q + α −
1= 4( q + 2 + α ) ( q + 1) γ ( α, q +1 (cid:18) − q + α + 2 (cid:19) (cid:18) − q + 1 (cid:19) (cid:18) q + α − (cid:19) = 4( q + 2 + α ) ( q + 1) γ ( α, q +1 (cid:18) − q + α + 2 (cid:19) (cid:18) − q + 1 + 42 q + α − − q + 1 42 q + α − (cid:19) = 4( q + 2 + α ) ( q + 1) γ ( α, q +1 (cid:18) − q + α + 2 (cid:19) (cid:18) − α + 2( q + 1)(2 q + α − (cid:19) , which is the claimed statement. 32 emma B.4. Let U ∈ C ( − , and let (1 − x ) α U ( x ) as well as (1 − x ) α +1 U ′ be integrable. Furthermore, let lim x → (1 − x ) α U ( x ) = 0 and lim x →− (1 + x ) U ( x ) = 0 . Consider h , h and h from (2.3). We define u q := Z − (1 − x ) α U ( x ) P ( α, q ( x ) dx,b q := Z − (1 − x ) α U ′ ( x ) P ( α, q ( x ) dx. Then for q ≥ and α ∈ N the following relationship holds: u q = h ( q, α ) b q +1 + h ( q, α ) b q + h ( q, α ) b q − . Proof.
From (A.10) we have for x → − Z x − (1 − t ) α P ( α, q ( t ) dt = O (1 + x )and for x → Z x − (1 − t ) α P ( α, q ( t ) dt = O (cid:0) (1 − x ) α +1 (cid:1) . Hence, using the stipulated behavior of U at the endpoints, the following integration by parts can be justified: u q = Z − (1 − x ) α U ( x ) P ( α, q ( x ) dx = (cid:18) U ( x ) Z x − (1 − t ) α P ( α, q ( x ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) − − Z − U ′ ( x ) Z x − (1 − t ) α P ( α, q ( t ) dt dx. In particular, we note that b q is well-defined. Furthermore, u q = − Z − U ′ ( x ) Z x − (1 − t ) α P ( α, q ( t ) dt dx = Z − (1 − x ) α U ′ ( x ) (cid:16) h ( q, α ) P ( α, q +1 ( x ) + h ( q, α ) P ( α, q ( x ) + h ( q, α ) P ( α, q − ( x ) (cid:17) dx = h ( q, α ) b q +1 + h ( q, α ) b q + h ( q, α ) b q − , where in the second equation we appealed to Lemma 2.2 (i). Lemma B.5.
For β > − and U ∈ C (0 , ∩ C ((0 , there holds Z x β | U ( x ) | dx ≤ (cid:18) β + 1 (cid:19) Z x β +2 | U ′ ( x ) | dx + 1 β + 1 | U (1) | . Proof.
We define according to the notation of [16, Thm. 330] f ( x ) := ( U ′ ( x ) 0 < x < x > F ( x ) := Z ∞ x f ( t ) dt = ( x > U (1) − U ( x ) 0 < x < . For β > −
1, [16, Thm. 330] states Z ∞ x β | F ( x ) | dx ≤ (cid:18) β + 1 (cid:19) Z ∞ x β +2 | f ( x ) | dx Z x β ( U ( x ) − U (1)) dx ≤ (cid:18) β + 1 (cid:19) Z x β +2 | U ′ ( x ) | dx. By rearranging terms, we get Z x β | U ( x ) | dx ≤ (cid:18) β + 1 (cid:19) Z x β +2 | U ′ ( x ) | dx + | U (1) | Z x β dx ≤ (cid:18) β + 1 (cid:19) Z x β | U ′ ( x ) | dx + 1 β + 1 | U (1) | . B.2. Selected proofs for Section 3
We start by recalling the definition of the 2D version of the Duffy transformation D D : R → R given by D D : ( η , η ) ( ξ , ξ ) = (cid:18)
12 (1 + η )(1 − η ) − , η (cid:19) . It maps S onto T . Its inverse ( D D ) − : T → S is given by( D D ) − : ( ξ , ξ ) ( η , η ) = (cid:18) ξ − ξ − , ξ (cid:19) . A calculation shows ( D D ) ′ = (cid:18) − η − η (cid:19) , det( D D ) ′ = 1 − η T (this is well-known, see, e.g., [21, Sec. 3.2.3] fordetails): Lemma B.6. (i) The polynomials ψ Dp,q defined by ψ Dp,q ◦ D D := P (0 , p ( η ) (cid:18) − η (cid:19) p P (2 p +1 , q ( η ) are orthogonal polynomials on T and satisfy Z T ψ Dp,q ψ Dp ′ ,q ′ dξ dξ = δ p,p ′ δ q,q ′ p + 1 22 p + 2 q + 2 = δ p,p ′ δ q,q ′ γ (0 , p γ (2 p +1 , q p +1 (ii) Any u ∈ L ( T ) can be expanded as u = X p,q γ (0 , p p +1 γ (2 p +1 , q (cid:18)Z T uψ Dp,q dξ dξ (cid:19) ψ Dp,q , (B.6) or, written as integrals over S with ˜ u := u ◦ D D ˜ u = X p,q γ (0 , p p +1 γ (2 p +1 , q Z S uP (0 , q ( η ) (cid:18) − η (cid:19) p +1 P (2 p +1 , q ( η ) dη dη ! P (0 , p ( η ) (cid:18) − η (cid:19) p P (2 p +1 , q ( η )(B.7) Lemma B.7 (details of Lemma 3.2) . Let D be the Duffy transformation and Γ := T × {− } . Set ˜Γ := S × {− } .Then D (˜Γ) = Γ and with the notation ˜ u := u ◦ D we have Z ˜Γ | ˜ u ( η , η , − | − η dη dη = Z Γ | u ( ξ , ξ , − | dξ dξ roof. Obviously D is an isomorphism and by definition D (Γ) = Γ, so we will only show the isometry property.Let u be a quadratic integrable function on T and consider the transformed function ˜ u = u ◦ D . We have Z ˜Γ | ˜ u ( η , η , − | − η dη dη = Z − Z − (cid:12)(cid:12) u (cid:0) D ( η , η , − (cid:1)(cid:12)(cid:12) − η dη dη = Z − Z − (cid:12)(cid:12)(cid:12)(cid:12) u (cid:18) (1 + η )(1 − η )2 − , η , − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) − η dη dη = Z − Z − (cid:12)(cid:12) u (cid:0) D D ( η , η ) , − (cid:1)(cid:12)(cid:12) − η dη dη = Z T | u ( ξ , ξ , − | dξ ξ = k u k L (Γ) . Proof of Lemma 3.3.
With the definition of D − and the abbreviation n pq := 2 p + 2 q + 2 we get ψ p,q,r ( ξ , ξ , ξ ) = ˜ ψ p,q,r (cid:18) − ξ ξ + ξ − , ξ − ξ − , ξ (cid:19) = P (0 , p (cid:18) − ξ ξ + ξ − (cid:19) P (2 p +1 , q (cid:18) ξ − ξ − (cid:19) P ( n pq , r ( ξ ) − ξ − ξ − ! p (cid:18) − ξ (cid:19) p + q . Expanding P (0 , p ( x − P (2 p +1 , q ( y −
1) = P pk =0 P ql =0 c kl x k y l leads to ψ p,q,r ( ξ , ξ , ξ ) = p X k =0 q X l =0 c kl k + l (cid:18) ξ ξ + ξ (cid:19) k (cid:18) ξ − ξ (cid:19) l P ( n pq , r ( ξ ) (cid:18) − ξ − ξ (cid:19) p (cid:18) − ξ (cid:19) p + q = p X k =0 q X l =0 c kl k + l p + q (1 + ξ ) k (1 + ξ ) l (1 − ξ ) q − l ( ξ − ξ ) p ( ξ + ξ ) k P ( n pq , r ( ξ ) . Since P ( n pq , r is a polynomial of degree r , we see by the last 2 terms in the sum above that ψ p,q,r ∈ P p + q + r ( T ).To see the orthogonality property, we transform to the cube S and make use of (2.1) three times Z T ψ p,q,r ( ξ ) ψ p ′ ,q ′ ,r ′ ( ξ ) dξ = Z S ˜ ψ p,q,r ( η ) ˜ ψ p ′ ,q ′ ,r ′ ( η ) (cid:18) − η (cid:19) (cid:18) − η (cid:19) dη = Z − Z − Z − P (0 , p ( η ) P (0 , p ′ ( η ) (cid:18) − η (cid:19) p + p ′ +1 P (2 p +1 , q ( η ) P (2 p ′ +1 , q ′ ( η ) × (cid:18) − η (cid:19) p + q + p ′ + q ′ +2 P ( n pq , r ( η ) P ( n p ′ q ′ , r ′ ( η ) dη dη dη = 22 p + 1 δ pp ′ − (2 p +1) Z − Z − (1 − η ) p +1 P (2 p +1 , q ( η ) P (2 p +1 , q ′ ( η ) × (cid:18) − η (cid:19) p + q + q ′ +2 P ( n pq , r ( η ) P ( n pq ′ , r ′ ( η ) dη dη = 22 p + 1 δ pp ′ p + 2 q + 2 δ qq ′ − n pq Z − (1 − η ) n pq P ( n pq , r ( η ) P ( n pq , r ′ ( η ) dη = 22 p + 1 δ pp ′ p + 2 q + 2 δ qq ′ p + 2 q + 2 r + 3 δ rr ′ . roof of Corollary 3.9. To prove this corollary we want to make use of Lemma 2.3. Therefore, we have to clarifythat the conditions in the lemma are satisfied. We proceed in two steps. First, we require u ∈ C ∞ ( T ) and showthe statement in this case and then we argue by density to achieve results in H ( T ).Step 1: By assuming that u ∈ C ∞ ( T ) we get ˜ u ∈ C ( S ). Hence, for fixed p and q , if we recall the definition of U p,q , we see that the map η U p,q ( η ) is smooth on [ − , e U p,q e U p,q ( η ) = U p,q ( η )(1 − η ) p + q , we see that e U p,q ∈ C ([ − , e U p,q has at most one pole of maximal order p + q at the point η = 1. In viewof these preliminary considerations we conclude that the following limits exist and that the conditions in Lemma 2.3are satisfied: lim η → (1 − η ) p +2 q +3 e U p,q ( η ) = lim η → (1 − η ) p + q +3 U p,q ( η ) = 0 . and lim η →− (1 + η ) e U p,q ( η ) = 0 . Now the statement follows directly from Lemma 2.3 when looking at the definition of ˜ u p,q,r and ˜ u ′ p,q,r and consequentlyreplacing U with e U p,q and α with 2 p + 2 q + 1.Step 2: Let u ∈ H ( T ). Since C ∞ ( T ) is dense in H ( T ), there exists a sequence ( u n ) n ∈ N ⊂ C ∞ ( T ) such that u n → u in H ( T ) for n → ∞ . Because we have already proved that u n , n ∈ N satisfies our statement, ensuringthat the sequences of coefficients ˜ u np,q,r and ˜ u ′ n p,q,r corresponding to u n converge for fixed p , q and r will conclude theproof:We have ˜ u np,q,r = 2 p + q +2 u np,q,r = 2 p + q +2 Z T u n ( ξ , ξ , ξ ) ψ p,q,r ( ξ , ξ , ξ ) dξ dξ dξ . Since ( ψ p,q,r ) p,q,r ∈ N forms an orthogonal basis for L ( T ) and since H ( T ) ⊂ L ( T ), the maps F : u ˜ u p,q,r arecontinuous linear functionals on H ( T ) and thus lim n →∞ F ( u n ) = F ( u ).In case of ˜ u ′ p,q,r we study the functionals e F : u ˜ u ′ p,q,r that map C ∞ ( T ) into R . 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