The main theorem

In this section we characterize the stability constants λa and λa,0,0 in the Fourier transform domain, see Theorem 4.51. We also state a similar characterization which enables us to efficiently computeλaandλa,0,0, see Theorem 4.54.

Recall Definition 2.61. SinceE⁰⁰(χGx0) is left-translation-invariant, see Remark 4.15(i), we can representE⁰⁰(χGx0) as a convolution operator.

Lemma 4.42. For allu, v ∈Uper we have

E⁰⁰(χ_Gx₀)(u, v) =hf_V ∗v₀, u₀i, whereu0=u(·⁻¹)andv0=v(·⁻¹).

Proof. Letu, v ∈Uper. LetN ∈M0 such thatuandv areT^N-periodic.

Letu0=u(·⁻¹) andv0=v(·⁻¹). By Lemma 4.14 we have E⁰⁰(χ_Gx₀)(u, v) = X

g,h∈CN

u(g)^T∂_gTN∂_hTNE⁰⁰(χ_Gx₀)v(h)

= 1

|CN| X

g,h∈CN

X

t∈T^N

u₀(g⁻¹)^Tf_V(g⁻¹ht)v₀(h⁻¹)

= 1

|CN| X

g∈CN

u₀(g⁻¹)^Tf_V ∗v₀(g⁻¹)

=hfV ∗v₀, u₀i,

where in the third step we used thatv0((ht)⁻¹) =v0(h⁻¹) for allh∈ CN

andt∈ T^N.

Letϕ:R → {0, . . . ,|R| −1} be a bijection. We define an isomorphism betweenC^(m|R|)×n and (C^m×n)^R by

(ai,j)i∈{1,...,m|R|};j∈{1,...,n}7→ (a_i+mϕ(g),j)i∈{1,...,m};j∈{1,...,n}

g∈R.

Definition 4.43. We define the functionsg_R, g_R,0,0 ∈L¹(G,R^(d|R|)×d) by

g_R(g) =P δg,hId

h∈R for allg∈ G

and

g_R,0,0(g) =P₀ δg,hId

h∈R for allg∈ G,

whereP (resp. P₀) is the square matrix of orderd|R|such that the map R^d|R|→R^d|R|, x7→P x

is the orthogonal projection with respect to the norm k · k with kernel Uiso(R) (resp. Uiso,0,0(R)).

Remark 4.44. The support of bothg_Randg_R,0,0 is equal toR. We have gR(g) =pϕ(g) for allg∈ R

and

g_R,0,0(g) =p_0,ϕ(g) for allg∈ R,

wherep0, . . . , p_|R|−1, p0,0, . . . , p_0,|R|−1∈R^(d|R|)×dsuch thatP = (p0, . . . , p_|R|−1) andP0= (p0,0, . . . , p_0,|R|−1) and bothP andP0 are as above.

Due to the left-translation-invariance,k · kR andk · kR,0,0 can be repre-sented by means of convolution operators.

Lemma 4.45. For all u ∈ Uper we have that kukR = kgR ∗u0k2 and kukR,0,0=kgR,0,0∗u0k2, whereu0=u(·⁻¹).

Proof. Let u ∈ Uper and N ∈ M0 such that u is T^N-periodic. Let u0=u(·⁻¹). LetP ∈R(d|R|)×(d|R|)such that the map

R^d|R|→R^d|R|, x7→P x

is the orthogonal projection with kernelU_iso(R). We have kuk²_R= 1

|CN| X

g∈CN

P(u(gh))h∈R

2. (4.36)

For allg∈ G we define the function

δ_g:G → {0,1}, h7→δ_h,g.

For allg∈ G we have

P(u(gh))_h∈R=P(u0(h⁻¹g⁻¹))_h∈R

=P((δhId)∗u0(g⁻¹))_h∈R

= (P(δ_hI_d)_h∈R)∗u₀(g⁻¹)

=gR∗u0(g⁻¹). (4.37) By (4.36) and (4.37) we have

kuk²_R= 1

|CN| X

g∈CN

kgR∗u0(g⁻¹)k²=kgR∗u0k²₂. Analogously we havekukR,0,0=kgR,0,0∗u₀k2.

Proposition 4.29 implies the following corollary.

Corollary 4.46. Suppose that E⁰(χ_Gx0) = 0. Then for all periodic representations ρ of G and a ∈ C^ddρ such that kg^V_R(ρ)ak = 0 we have hf_V

V

(ρ)a, ai= 0.

Proof. Suppose thatE⁰(χGx0) = 0. Letρbe a periodic representation of G and a∈C^ddρ such that kgR

V(ρ)ak = 0. Without loss of generality we assume thatρ∈ E. We defineu∈U_per,Cby

u

V

(ρ⁰) =

((a 0_ddρ,dρ−1) ifρ⁰ =ρ 0dd_ρ0,d_ρ˜ else for allρ⁰∈ E. We denoteu₀=u(·⁻¹). We have 0 =dρkg^V_R(ρ)ak²=dρkg^V_R(ρ)u^V(ρ)k²=dρkg_R∗u

V

(ρ)k²=kg_R∗uk²₂

=kg_R∗Re(u)k²₂+kg_R∗Im(u)k²₂=kRe(u0)k²_R+kIm(u0)k²_R, (4.38) where we used Proposition 2.56 in the third step and Lemma 4.45 in the last step. Thus we havekRe(u0)kR = 0 and kIm(u0)kR = 0 which is equivalent to Re(u0),Im(u0) ∈ Uiso,0,0 by Theorem 3.34. We have E⁰⁰(χ_Gx₀)(Re(u₀),Re(u₀)) = 0 and E⁰⁰(χ_Gx₀)(Im(u₀),Im(u₀)) = 0 by Proposition 4.29 and Remark 4.30(ii). Thus we have

d_ρhfV

V

(ρ)a, ai=E⁰⁰(χ_Gx₀)(Re(u₀),Re(u₀)) +E⁰⁰(χ_Gx₀)(Im(u₀),Im(u₀))

= 0,

where the first step follows analogously to (4.38) with Lemma 4.42 instead of Lemma 4.45.

The following lemma shows that we can consider complex-valued instead of real-valued functions.

Lemma 4.47. We have λa= sup

c∈R

∀u∈Uper,C:ckgR∗uk²₂≤ hfV ∗u, ui and

λ_a,0,0= sup c∈R

∀u∈U_per,C:ckgR,0,0∗uk²₂≤ hfV ∗u, ui . Proof. By Lemma 4.42, Lemma 4.45 and since Uper = {u(·⁻¹)|u ∈ Uper}, we have

λa= sup c∈R

∀u∈Uper:ckgR∗uk²₂≤ hfV ∗u, ui and hence,

λa≥sup c∈R

∀u∈Uper,C:ckg_R∗uk²₂≤ hfV ∗u, ui =: RHS.

Now we show thatλ_a≤RHS. For allu∈U_per,Cwe have hfV ∗u, ui=hfV ∗Re(u),Re(u)i −ihfV ∗Re(u),Im(u)i

+ ihfV ∗Im(u),Re(u)i+hfV ∗Im(u),Im(u)i

=hf_V ∗Re(u),Re(u)i −iE⁰⁰(χ_Gx₀)(Im(u),Re(u)) + iE⁰⁰(χGx0)(Re(u),Im(u)) +hfV ∗Im(u),Im(u)i

=hfV ∗Re(u),Re(u)i+hfV ∗Im(u),Im(u)i

≥λ_akgR∗Re(u)k²₂+λ_akgR∗Im(u)k²₂

=λakgR∗uk²₂,

where in the second step we used Lemma 4.42.

The proof of the characterization ofλa,0,0 is analogous.

Recall that by Definition 2.19 all representations are unitary. Schwarz’s theorem implies the following lemma.

Lemma 4.48. For all g ∈ G we have fV(g⁻¹) = fV(g)^T and for all representationsρof G the matrix fV

V

(ρ)is Hermitian.

Proof. For allg∈ G we have fV(g⁻¹) = X

h₁,h₂∈G\{id}

δ_g−1,h⁻¹₂ h1L(h2)^T∂h₂∂h₁V(y0)L(h1)

−δ_g−1,h⁻¹₂ L(h2)^T∂h₂∂h₁V(y0)−δ_g⁻¹_,h1∂h₂∂h₁V(y0)L(h1) +δ_g⁻¹_,id∂h₂,h₁V(y0)

= X

h₁,h₂∈G\{id}

δ_g,h⁻¹

1 h2L(h2)^T(∂h₁∂h₂V(y0))^TL(h1)

−δg,h₂L(h2)^T(∂h₁∂h₂V(y0))^T

−δ_g,h−1

1 (∂h₁∂h₂V(y0))^TL(h1) +δg,id(∂h₁∂h₂V(y0))^T

=fV(g)^T. (4.39)

For all representationsρofGwe have f_V

V

(ρ) =X

g∈G

f_V(g)⊗ρ(g)

=X

g∈G

f_V(g⁻¹)⊗ρ(g⁻¹)

=X

g∈G

f_V(g)^H⊗ρ(g)^H

=

X

g∈G

fV(g)⊗ρ(g) ^H

=f_V

V

(ρ)^H,

where in the third step we used (4.39) and thatρis unitary.

Definition 4.49. The Loewner order is the partial order on the set of all Hermitian matrices of C^n×n defined by A ≥B if A−B is positive semidefinite. We define

λ_min(A, B) := sup c∈R

cB^HB ≤A ∈R∪ {±∞}

for all Hermitian matricesA∈C^n×n and matricesB ∈C^m×n. Remark 4.50. (i) By means of the dual problem we have

λ_min(A, B) = inf x^HAx

x∈Cⁿ,kBxk= 1 and

λmin(A,0m,n) =

(∞ ifA is positive semidefinite

−∞ else

for all Hermitian matricesA∈C^n×n and matricesB∈C^m×n\ {0}.

The proof is analogous to the proof of Proposition 4.10.

(ii) Suppose thatB has in addition ranknand consider thegeneralized eigenvalue problem Av =λB^HBv, i. e. the problem of finding the eigenvaluesof thematrix pencilA−λB^HB. Then the eigenvalues of the generalized eigenvalue problem are real andλmin(A, B) is equal to the smallest eigenvalue of the generalized eigenvalue problem, see [34, Chapter X, Theorem 11]. The eigenvalues of the gener-alized eigenvalue problem are equal to the eigenvalues of the ma-trixA(B^HB)⁻¹, see [58, Proposition 6.1.1], but the eigenvalues of A(B^HB)⁻¹are ill-conditioned. There exist many numerically stable algorithms, see, e. g., [8, Chapter 5], and thus many programming languages have a function for this problem; e. g. for Python the subpackage linalg of the package SciPy has the function eigvalsh.

Due to the left-translation-invariance,E⁰⁰(χ_Gx₀),k · kRandk · kR,0,0can be represented by means of multiplier operators. Thus we have the fol-lowing representation ofλ_a andλ_a,0,0. Recall thatE is a representation set of{ρ∈G |b ρis periodic}.

By Lemma 4.47 we have λa= sup

Sincea∈C^ddρ was arbitrary, we haveλ_min(f_V

The proof of the characterization ofλa,0,0 is analogous.

For the remainder of this section, we fix a complete set of representa-tives of the cosets of T F in G such that Indρ is well-defined for all ρ by Definition 2.23. In the following we write Indρ for Ind^G_{T F}ρ for all representationsρofT F.

Lemma 4.52. For all representationsρof T F the functions R^d2 →C^{(ddρ)×(ddρ)}, k7→fV are continuous and the functions

R^d2 →R∪ {±∞}, k7→λmin

Proof. Letρbe a representation ofT Fandfi denote theith function of the lemma for everyi∈ {1, . . . ,5}. By Theorem D.7 the functionsf1,f2

andf3 are continuous.

Let (kn)_n∈Nbe a sequence inR^d2 andk∈R^d2 be such that lim_n→∞kn= k. Without loss of generality we assume that lim sup_n→∞f4(kn)>−∞

and lim sup_n→∞f4(kn) = limn→∞f4(kn). Letλ∈ Rbe such that λ <

lim sup_n→∞f4(kn). We have λf2(kn)^Hf2(kn) ≤ f1(kn) for all n ∈ N large enough. Since the Loewner order is closed, i. e. the set{(A, B) ∈ X²|A ≤ B} is closed, where X ={A ∈ C^{(ddρ)×(ddρ)}|Ais Hermitian}, we haveλf₂(k)^Hf₂(k)≤f₁(k). Thus we haveλ≤f₄(k).

Analogously the functionf₅ is upper semicontinuous.

Recall Definition 2.38, Proposition 2.39 and Definition 2.32.

Definition 4.53. For all ρ∈T Fd and representations ρ⁰ ∈ρ we define the space group

Gρ⁰ :=Gρ.

The following theorem generalizes Theorem 3.6(b) of [40] from lattices to general configurations.

Theorem 4.54. Let Rbe a representation set of a representation set of T F/∼. For alld ρ∈RletK_ρ be a representation set ofR^d2/G_ρ. Then we have

λa= infn λmin

fV

V

(Ind(χkρ)), g^V_R(Ind(χkρ))

ρ∈R, k∈Kρ

o

and

λ_a,0,0= infn λ_min

f_V

V

(Ind(χ_kρ)), g^V_R,0,0(Ind(χ_kρ))

ρ∈R, k∈K_ρo . Proof. LetRbe a representation set of a representation set ofT F/∼. Ford allρ∈R letKρ be a representation set ofR^d2/Gρ. Letm∈Nsuch that M0 = mN. By Lemma 2.36(i) there exists a representation set R⁰ of a representation set ofT Fd/∼such thatρisT^m-periodic for allρ∈R⁰. Due to the existence of fundamental domains, see, e. g., [49, Theorem 6.6.13], for allρ∈R⁰there exists a representation setK_ρ⁰ ofR^d2/Gρ such thatL⁰_ρ is a dense subset ofK_ρ⁰, where L⁰_ρ ={k ∈K_ρ⁰| ∃N ∈M0 :k∈ L^∗_S/N}.

By Theorem 2.43 applied toR andR⁰, there exist a bijection ϕ: G

ρ∈R⁰

K_ρ⁰ → G

ρ∈R

Kρ, (k, ρ)7→(ϕ1(k, ρ), ϕ2(k, ρ)) (4.40)

and for allρ∈R⁰ andk∈K_ρ⁰ some Tk,ρ ∈U(d_Ind(χkρ)) such that

By Lemma 4.52 for all ρ ∈ R the function fρ is upper semicontinuous and thus we have By Theorem 4.51 we have

λ_a= infn

By Lemma D.3(ii) there exists a permutation matrixPn,p₁,...,p_k∈O(n(p1

+· · ·+pk)) for alln, p1, . . . , pk∈Nsuch that A⊗(B₁⊕ · · · ⊕B_k) =P_m,p^T

1,...,p_k((A⊗B₁)⊕ · · · ⊕(A⊗B_k))P_n,p1,...,pk for allA∈C^m×n andB_i∈C^pi×pi,i∈ {1, . . . , k}.

Now we show thatλ_a≤RHS. Letρ∈R⁰,k∈L⁰_ρandρ⁰ = Ind(χkρ). Let

whereQis the unitary matrixP_d|R|,dρ

1,...,d_ρn(I_d|R|⊗T). By (4.45), (4.46)

Analogously to (4.45) and (4.46) we have

The proof of the characterization ofλa,0,0 is analogous.

Remark 4.55. (i) By Lemma 4.52 the above theorem is also true if for all ρ ∈ R we weaken the assumption on Kρ and only assume that the closure ofKρ contains a representation set of R^d2/Gρ. In particular the theorem is also true if for all ρ∈R the setKρ is a fundamental domain ofR^d2/Kρ.

(ii) An algorithm for the determination of a representation set ofT Fd/∼

with the aid of the finite group (T F)m is given by Lemma 2.36, wherem∈Nsuch thatM0=mN.

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