Diagonal Matrix Sequences and their Spectral Symbols
aa r X i v : . [ m a t h . NA ] O c t Diagonal Matrix Sequences and their SpectralSymbols
Giovanni Barbarino
Abstract
The spectral symbols are useful tools to analyse the eigenvalue dis-tribution when dealing with high dimensional linear systems. Given amatrix sequence with an asymptotic symbol, the last one depends onlyon the spectra of the individual matrices, seen as a not ordered set. Wecan then focus only on diagonal sequences and sort the eigenvalues sothat they become an approximation of the symbol sampling. We showthat this is linked to the concept of diagonal Generalized Locally Toeplitz(GLT) sequences, and in particular we prove that any diagonal sequencewith a real valued symbol can be permuted in order to obtain a diagonalGLT sequence with the same symbol. A Matrix Sequence is an ordered collection of complex valued matrices withincreasing size, and is usually denoted as { A n } n , where A n ∈ C n × n .Matrix sequences naturally arise in several contexts. For example, the dis-cretization of a linear differential or integral equation by a linear numericalmethod (such as the finite difference method, the finite element method, themodern isogeometric analysis, etc.) leads to linear systems whose sizes are di-rectly proportional to the accuracy of the method. The speed of convergenceof the solvers (Conjugate Gradient, Preconditioned Krylov methods, Multigridtechniques, etc.) depends on the spectra of the matrices, so the knowledge ofthe eigenvalue distribution is a strong tool which we can use to choose or todesign the best solver and method of discretization.It is often observed in practice that the matrix sequences { A n } n generated bydiscretization methods possess a spectral symbol , that is a measurable functiondescribing the asymptotic distribution of the eigenvalues of A n . We recall thata spectral symbol associated with a sequence { A n } n is a measurable functions k : D ⊆ R n → C satisfying lim n →∞ n n X i =1 F ( λ i ( A n )) = 1 µ ( D ) Z D F ( k ( x )) dx for every continuous function F : C → C with compact support, where D is ameasurable set with finite Lebesgue measure µ ( D ) > . In this case we write { A n } n ∼ λ k ( x ) .
1e can also consider the singular values of the matrices instead of the eigenval-ues. In this case we have that k : D ⊆ R n → C is a singular value symbol of thesequence { A n } n if it satisfies lim n →∞ n n X i =1 F ( σ i ( A n )) = 1 µ ( D ) Z D F ( | k ( x ) | ) dx for every continuous function F : R → C with compact support, where D is ameasurable set with finite Lebesgue measure µ ( D ) > . In this case we write { A n } n ∼ σ k ( x ) . If { A n } n ∼ λ k ( x ) , then we can consider the diagonal matrices D n ∈ C n × n that contain the eigenvalues of A n . We get again that { D n } n ∼ λ k ( x ) , so wecan focus only on diagonal sequences. Moreover, we will consider only spectralsymbol with domain D = [0 , , so from now on k : [0 , → C . Given a diagonal matrix D n ∈ C n × n , with diagonal entries d ( n ) i = [ D n ] i,i wecan consider the piecewise linear function k n : [0 , → C such that interpolatethe values d ( n ) i on the nodes i/n and is linear in between. k n (0) = 0 k n (cid:18) in (cid:19) = d ( n ) i ∀ i = 1 , , . . . , n We say that D n converges piecewise to a function k : [0 , → C if the interpo-lations k n ( x ) converge in measure to k ( x ) . In this case, we write { D n } n ⇀ k ( x ) . We will give a more rigorous definition later. The first important result regardingthis convergence is expressed by the following Theorem that will be proved inSection 3.
Theorem.
Given a diagonal sequence { D n } n and a measurable function k :[0 , → C , then { D n } n ⇀ k ( x ) = ⇒ { D n } n ∼ λ k ( x ) The opposite implication does not hold, since the piecewise convergencedepends on the order of the eigenvalues, whereas the spectral symbol dependsonly on their values.The space of matrix sequences that admit a singular value symbol on a fixeddomain D has been shown to be closed with respect to a notion of convergencecalled the Approximating Classes of Sequences (a.c.s.) convergence. This notionand result are due to Serra-Capizzano [8], but were actually inspired by Tilli’spioneering paper on LT sequences [10]. Given a sequence of matrix sequences {{ B n,m } n } m , it is said to be a.c.s. convergent to { A n } n if there exists a sequence2 N n,m } n,m of "small norm" matrices and a sequence { R n,m } n,m of "small rank"matrices such that for every m there exists n m with A n = B n,m + N n,m + R n,m , k N n,m k ≤ ω ( m ) , rk( R n,m ) ≤ nc ( m ) for every n > n m , and ω ( m ) m →∞ −−−−→ , c ( m ) m →∞ −−−−→ . In this case, we will use the notation {{ B n,m } n } m a.c.s. −−−→ { A n } n .It has been observed that the sequences { A n } n arising from differential equa-tions can often be obtained through an a.c.s. limit of sums of products of specialdiagonal and Toeplitz sequences, for which we can easily deduce the spectralsymbol. This justifies the interest in the space of Generalized Locally Toeplitz(GLT) sequences, which contains both the before-mentioned class of sequences,gains the structure of a C -algebra, and is closed with respect to the a.c.s. conver-gence. We will report only a few properties of this space, but for a detailed pre-sentation of the GLT sequences and their applications refer to [2],[9],[10],[4],[6]and references therein.For every GLT matrix sequence { A n } n one of its singular value symbol k ( x ) is chosen, called GLT symbol , and denoted as { A n } n ∼ GLT k ( x ) . Key examplesof diagonal GLT sequences are { D n ( a ) } n , where a : [0 , → C is an almosteverywhere (a.e.) continuous function, and D n ( a ) = diag i =1 ,...,n a (cid:18) in (cid:19) = a (cid:0) n (cid:1) a (cid:0) n (cid:1) . . . a (1) It is easy to verify that { D n ( a ) } n ∼ σ a ( x ) ⊗ , where a ( x ) ⊗ , × [ − π, π ] → C is a two-variable function that is constant with respect to the second variable.This function is chosen as the GLT symbol of { D n ( a ) } n .In the case of diagonal matrix sequences, the choice of one symbol can beseen as a particular sorting of their eigenvalues, as expressed in the followingtheorems, proved in Section 3 and 4. Theorem.
Given a diagonal sequence { D n } n and a measurable function k :[0 , → C , then { D n } n ⇀ k ( x ) ⇐⇒ { D n } n ∼ GLT k ( x ) ⊗ Theorem.
Given a real diagonal sequence { D n } n and one of its spectral symbols k : [0 , → R , then { P n D n P Tn } ∼ GLT k ( x ) ⊗ where P n are permutation matrices. The theory of piecewise convergence is not enough in order to prove thecomplex version of the last result, since the field of complex numbers does nothave a natural order. The result holds nonetheless, and we will prove it in aseparate document. 3
A.c.s. and GLT
In this section, we introduce some technical results that will be used in the othersections. Moreover, we recall some known lemmas and theorems about spectralsymbols, acs convergence and GLT sequences.
Given a matrix A ∈ C n × n , we can define the function p ( A ) := min i =1 ,...,n +1 (cid:26) i − n + σ i ( A ) (cid:27) where σ ( A ) ≥ σ ( A ) ≥ · · · ≥ σ n ( A ) are the singular values of A , and byconvention σ n +1 ( A ) = 0 . Another way to compute this quantity makes use ofthe singular value decomposition (svd) of A . In fact, if A = U Σ V where Σ is adiagonal matrix containing the singular vales of A sorted in a decreasing order,we can decompose it into the sum of two diagonal matrices Σ = σ σ . . . σ n = e Σ ( i ) + b Σ ( i ) e Σ ( i ) = σ . . . σ i − . . . b Σ ( i ) = . . . σ i . . . σ n . So we get that for all i = 1 , . . . , n +1 it holds A = e A ( i ) + b A ( i ) := U e Σ ( i ) V + U b Σ ( i ) V and p ( A ) = min i =1 ,...,n +1 (cid:26) i − n + σ i ( A ) (cid:27) = min i =1 ,...,n +1 ( rk( e A ( i ) ) n + k b A ( i ) k ) When { A n } n is a sequence with singular value symbol zero, and in this case wecall it zero-distributed , Theorem 3.2 in [6] says something more on the sequences e A ( i n ) n and b A ( i n ) n , where i n are the indexes that minimize rk( e A ( i ) n ) n + k b A ( i ) n k . Lemma 2.1.
Let { A n } n be a matrix sequence, and let e A n := e A ( i n ) n , ¯ A n := ¯ A ( i n ) n be the sequences that minimize rk( e A ( i ) n ) n + k b A ( i ) n k . The following statement are equivalent. { A n } n ∼ σ • rk( e A n ) = o ( n ) , k b A n k = o (1) • lim n →∞ { i : σ i ( A n ) > ε } n = 0 ∀ ε > The function p ( A ) is subadditive, so we can introduce the pseudometric d acs on the space of matrix sequences d acs ( { A n } n , { B n } n ) = lim sup n →∞ p ( A n − B n ) . It has been proved ([3],[5]) that this distance induces the a.c.s. convergencealready introduced. In other words, d acs ( { A n } n , {{ B n,m } n } m ) m →∞ −−−−→ ⇐⇒ {{ B n,m } n } m a.c.s. −−−→ { A n } n . This result holds since the function p embodies the concept of acs convergence.In fact it is equivalent to p ( A ) = inf (cid:26) rk( R ) n + k N k : A = R + N (cid:27) . This justifies the following result:
Lemma 2.2.
Let {{ B n,m } n } m and { A n } n be matrix sequences and let R n,m := ^ A n − B n,m ( i n ) , N n,m := \ B n,m − A n ( i n ) , where i n are the indexes that minimize rk( ^ A n − B n,m ( i ) ) n + k \ A n − B n,m ( i ) k . Then {{ B n,m } n } m a.c.s. −−−→ { A n } n if and only if for every m there exists n m with A n = B n,m + N n,m + R n,m , k N n,m k ≤ ω ( m ) , rk( R n,m ) ≤ nc ( m ) for every n > n m , and ω ( m ) m →∞ −−−−→ , c ( m ) m →∞ −−−−→ . Proof.
Notice that {{ B n,m } n } m a.c.s. −−−→ { A n } n ⇐⇒ d acs ( { A n } n , {{ B n,m } n } m ) m →∞ −−−−→ ⇐⇒ lim sup n →∞ p ( A n − B n,m ) m →∞ −−−−→ ⇐⇒ lim sup n →∞ p ( N n,m + R n,m ) m →∞ −−−−→ ⇐⇒ lim sup n →∞ k N n,m k + rk( R n,m ) n m →∞ −−−−→ ⇐⇒ lim sup n →∞ k N n,m k m →∞ −−−−→ , lim sup n →∞ rk( R n,m ) n m →∞ −−−−→ . m there exists n m with k N n,m k ≤ ω ( m ) , rk( R n,m ) ≤ nc ( m ) ∀ n > n m where ω ( m ) m →∞ −−−−→ , c ( m ) m →∞ −−−−→ . Notice that if A n is a diagonal matrix, then e A ( i n ) n and b A ( i n ) n are also di-agonal matrices. If a diagonal sequence { D n } n is zero distributed, thanks toLemma 2.1, we can find e D ( i n ) n and b D ( i n ) n respectively low rank and low normdiagonal matrices that sums up to D n . Moreover, using Lemma 2.2, given {{ D n,m } n } m a.c.s. −−−→ { D n } n diagonal sequences we can find R n,m and N n,m re-spectively low rank and low norm diagonal matrices such that D n − D n,m = N n,m + R n,m .In [1], has been proved that the pseudometric d acs on the space of matrix se-quences is complete. Using the same arguments, we can prove the followingmore detailed statement. Theorem 2.1.
Let {{ B n,m } n } m be a sequence of matrix sequences that is aCauchy sequence with respect to the pseudometric d acs . There exists a crescentmap m : N → N with lim n →∞ m ( n ) = ∞ such that for every crescent map m ′ : N → N that respects • m ′ ( n ) ≤ m ( n ) ∀ n • lim n →∞ m ′ ( n ) = ∞ we get {{ B n,m } n } m a.c.s. −−−→ { B n,m ′ ( n ) } n . In particular, the pseudometric d acs is complete.Proof. By definition of Cauchy sequence, for every integer k > , there existsan index M k such that d acs ( { B n,s } n , { B n,t } n ) ≤ − k ∀ s, t ≥ M k . We can suppose that M k are strictly increasing accordingly to k . Let us fix k > and consider all the couple of distinct sequences ( { B n,s } n , { B n,t } n ) with M k +1 ≥ s, t ≥ M k . Their is less then − k , so we can use the definition ofdistance d acs ( { B n,s } n , { B n,t } n ) = lim sup n →∞ p ( B n,s − B n,t ) ≤ − k and obtain that there exists an index N s,t such that p ( B n,s − B n,t ) ≤ − k ∀ n ≥ N s,t . Since the number of couples ( s, t ) between M k and M k +1 is finite, we can definethe minimum of N s,t as e N k := min M k +1 ≥ s,t ≥ M k N s,t N k by the following recursive definition N = e N , N k = max { e N k , N k − + 1 } ∀ k > . Notice that, given k > , n ≥ N k and M k +1 ≥ s, t ≥ M k we have n ≥ N k ≥ e N k ≥ N s,t = ⇒ p ( B n,s − B n,t ) ≤ − k . (1)We can now define the function m ( n ) := ( N > nM k N k +1 > n ≥ N k ∀ k > The map m ( n ) is surely increasing, and it diverges to infinite as M k are strictlyincreasing. Let m ′ : N → N be a crescent map with • m ′ ( n ) ≤ m ( n ) ∀ n • lim n →∞ m ′ ( n ) = ∞ and set A n := { B n,m ′ ( n ) } n . We can now prove that {{ B n,m } n } m converges acsto { A n } n . Let m be a fixed index with M k ≤ m < M k +1 . Let n be a fixedindex with N k +1 ≤ N s ≤ n < N s +1 and since m ′ ( n ) is not bounded, we can alsoassume m ′ ( n ) ≥ m . There also exists a t such that M k ≤ M t ≤ m ′ ( n ) < M t +1 ,and M t ≤ m ′ ( n ) ≤ m ( n ) = M s = ⇒ t ≤ s. Since the function p is subadditive, we get p ( A n − B n,m ) = p ( B n,m ′ ( n ) − B n,m ) ≤≤ p ( B n,m ′ ( n ) − B n,M t ) + t − X r = k +1 p ( B n,M r +1 − B n,M r ) + p ( B n,M k +1 − B n,m ) . Using (1), we know that n ≥ N s ≥ N t , M t +1 > m ′ ( n ) ≥ M t = ⇒ p ( B n,M k +1 − B n,m ) ≤ − t n ≥ N s ≥ N t > N r = ⇒ p ( B n,M r +1 − B n,M r ) ≤ − r ∀ k < r < tn ≥ N k +1 > N k , M k +1 > m ≥ M k = ⇒ p ( B n,M k +1 − B n,m ) ≤ − k so we get p ( B n,m ′ ( n ) − B n,M t ) + t − X r = k +1 p ( B n,M r +1 − B n,M r ) + p ( B n,M k +1 − B n,m ) ≤ − t + t − X r = k +1 − r + 2 − k = t X r = k − r ≤ ∞ X r = k − r = 4 · − k . This means that for all m with M k ≤ m < M k +1 and for all n with N k +1 ≤ n and such that m ′ ( n ) ≥ m , we get p ( A n − B n,m ) ≤ · − k d acs ( { A n } n , { B n,m } n ) = lim sup n →∞ p ( A n − B n,m ) ≤ · − k ∀ M k ≤ m < M k +1 = ⇒ lim sup m →∞ d acs ( { A n } n , { B n,m } n ) ≤ lim sup k →∞ · − k = 0 . This entails that {{ B n,m } n } m a.c.s. −−−→ { A n } n . The acs convergence has nice properties linked to the singular value symbolsand GLT symbols of the sequences, as we will see in the next subsection. Thespectral symbols behave nicely only on the sequences of hermitian matrices, butwhen we work with diagonal matrices, not necessarily with real entries, we canalso regain some properties.
Lemma 2.3.
Given any almost everywhere continuous function a : [0 , → C ,then D n ( a ) ∼ λ a ( x ) . Proof.
Given any G ∈ C c ( C ) , we know that lim n →∞ n n X i =1 G (cid:18) a (cid:18) in (cid:19)(cid:19) = Z G ( a ( x )) dx since G ◦ a : [0 , → C is a bounded and almost everywhere continuous function,thus Riemann Integrable. Lemma 2.4.
Let { D n } n ∼ λ f ( x ) and { Z n } n ∼ σ be diagonal matrices se-quences, where f is any measurable function on a domain D ⊆ R n with finitenon-zero measure. Then { D n } n + { Z n } n ∼ λ f ( x ) Proof.
Using Lemma 2.1, we know that given any ε > we have lim n →∞ { i : σ i ( Z n ) > ε } n = 0 . Let the singular values of D n and Z n be sorted such that the i -th singular valueis the absolute value of the i -th entry on their diagonal. σ i ( D n ) = (cid:12)(cid:12)(cid:12) [ D n ] i,i (cid:12)(cid:12)(cid:12) , σ i ( Z n ) = (cid:12)(cid:12)(cid:12) [ Z n ] i,i (cid:12)(cid:12)(cid:12) , σ i ( D n + Z n ) = (cid:12)(cid:12)(cid:12) [ D n + Z n ] i,i (cid:12)(cid:12)(cid:12) . We thus have σ i ( D n + Z n ) − σ i ( D n ) ≤ σ i ( Z n ) . Given any G ∈ C c ( C ) , we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( σ i ( D n + Z n )) − Z D G ( f ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( σ i ( D n + Z n )) − n n X i =1 G ( σ i ( D n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( σ i ( D n )) − Z D G ( f ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( σ i ( D n + Z n )) − n n X i =1 G ( σ i ( D n )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X σ i ( Z n ) >ε G ( σ i ( D n + Z n )) − n n X i =1 G ( σ i ( D n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X σ i ( Z n ) ≤ ε G ( σ i ( D n + Z n )) − n n X i =1 G ( σ i ( D n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k G k ∞ { i : σ i ( Z n ) > ε } n + ω G ( ε ) and the second term of the sum tends to zero when n goes to infinity. This leadsto lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( σ i ( D n + Z n )) − Z D G ( f ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ω G ( ε ) ∀ ε > but ω G ( ε ) is the modulus of continuity of G and tends to zero as ε goes to zero,so we get the thesis lim n →∞ n n X i =1 G ( σ i ( D n + Z n )) = Z D G ( f ( x )) = ⇒ { D n } n + { Z n } n ∼ λ f ( x ) The space of diagonal sequences that admit a spectral symbol possess alsoa closure property with respect to the acs convergence of sequences and theconvergence in measure of measurable functions.
Lemma 2.5.
Let { D n } n and {{ D n,m } n } m be diagonal matrices sequences, andlet a : D → C , a m : D → C be measurable functions defined on D ⊆ R n that isa measurable set with non zero finite measure. If • { D n,m } n,m ∼ λ a m ( x ) • { D n,m } n,m a.c.s. −−−→ { D n } n • a m ( x ) µ −→ a ( x ) then { D n } n ∼ λ a ( x ) .Proof. Let G ∈ C c ( C ) , and let the eigenvalues of the diagonal matrices be sortedso that the i -th eigenvalue corresponds to the i -th diagonal element. λ i ( D n ) = [ D n ] i,i , λ i ( D n,m ) = [ D n,m ] i,i .
9e know that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( λ i ( D n )) − Z D G ( a ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( λ i ( D n )) − n n X i =1 G ( λ i ( D n,m )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( λ i ( D n,m )) − Z D G ( a m ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z D G ( a m ( x )) dx − Z D G ( a ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12) . Using Lemma 2.2, D n = D n,m + R n,m + N n,m where R n,m and N n,m are diagonalmatrices and for every m there exists n m with rk( R n,m ) ≤ c ( m ) n, k N n,m k ≤ s ( m ) ∀ n > n m lim m →∞ s ( m ) = lim m →∞ c ( m ) = 0 . Moreover, there exists a function t ( m ) such that (cid:12)(cid:12)(cid:12)(cid:12)Z D G ( a m ( x )) dx − Z D G ( a ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ t ( m ) lim m →∞ t ( m ) = 0 So we can fix ε > and find an index M such that c ( m ) ≤ ε, s ( m ) ≤ ε, t ( m ) ≤ ε ∀ m > M. We know that λ i ( D n ) − λ i ( D n,m ) = λ i ( D n − D n,m ) = λ i ( N n,m + R n,m ) = λ i ( N n,m )+ λ i ( R n,m ) but if n > n m and m > M , then ε < | λ i ( D n − D n,m ) | = ⇒ ε < | λ i ( N n,m ) | + | λ i ( R n,m ) | ≤ ε + | λ i ( R n,m ) | = ⇒ = λ i ( R n,m ) . This means that { i : ε < λ i ( D n − D n,m ) } ≤ rk( R n,m ) so we can estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( λ i ( D n )) − n n X i =1 G ( λ i ( D n,m )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i : λ i ( D n − D n,m ) ≤ ε G ( λ i ( D n )) − G ( λ i ( D n,m )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i : λ i ( D n − D n,m ) >ε G ( λ i ( D n )) − G ( λ i ( D n,m )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ω G ( ε ) + 2 rk ( R n,m ) n k G k ∞ ≤ ω G ( ε ) + 2 ε k G k ∞ ∀ n > n m , ∀ m > M.
10e obtain lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( λ i ( D n )) − Z D G ( a ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( λ i ( D n )) − n n X i =1 G ( λ i ( D n,m )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 G ( λ i ( D n,m )) − Z D G ( a m ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)Z D G ( a m ( x )) dx − Z D G ( a ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ ω G ( ε ) + 2 ε k G k ∞ + ε ∀ ε > and this concludes lim n →∞ n n X i =1 G ( λ i ( D n )) = Z D G ( a ( x )) = ⇒ { D n } n ∼ λ a ( x ) . A matrix sequence { A n } n may have several different singular values symbols,even on the same domain. For specific sequences we can choose one of theirsymbols, and denote it as GLT symbol of the sequence { A n } n ∼ GLT k ( x, θ ) . If we call G the set of sequences that own a GLT symbol, we know that • G is a C algebra. In fact, given { A n } n , { B n } n ∈ G and c ∈ C , we havethat { A n + B n } n ∈ G { A n B n } n ∈ G { cA n } n ∈ G . • G is closed with respect to the acs pseudometric. If { B n,m } m ∈ G for all m , then {{ B n,m } n } m a.c.s. −−−→ { A n } n = ⇒ { A n } n ∈ G The chosen symbols have all the same domain D = [0 , × [ − π, π ] . Let M D be the set of measurable functions on D , where we identify two functions if theyare equal almost everywhere. The choice of the symbol can be seen as a map S : G → M D . if we endow M D with the metric d M that induces the convergence in measure,then S can be seen as a map of algebras and complete pseudometric spaces,with the following properties that are proved in [6] or in [1]. Theorem 2.2. . S is an homomorphism of algebras. Given { A n } n , { B n } n ∈ G and c ∈ C ,we have that S ( { A n + B n } n ) = S ( { A n } n + { B n } n ) ,S ( { A n B n } n ) = S ( { A n } n ) S ( { B n } n ) ,S ( { cA n } n ) = cS ( { A n } n ) .
2. The kernel of S are exactly the zero-distributed sequences.3. S preserves the distances. Given { A n } n , { B n } n ∈ G we have d acs ( { A n } n , { B n } n ) = d m ( S ( { A n } n ) , S ( { B n } n )) . S is onto. All measurable functions are GLT symbols.5. GLT symbols are spectral symbols: { A n } n ∈ G = ⇒ { A n } n ∼ σ S ( { A n } n )
6. The graph of S is closed in G × M D . If {{ B n,m } n } m are sequences in G that converge acs to { A n } n , and their symbols converge in measure to k ( x, θ ) , then S ( { A n } n ) = k ( x, θ ) . Let us formalize the concept of piecewise convergence shown in the introduction.Given a vector v ∈ C n , let f : [0 , → C be the piecewise linear function thatinterpolates v i on the points i/n of the interval [0 , , where v := 0 . It respects f (cid:18) in (cid:19) = v i ∀ i = 0 , , . . . , n and its analytical expression is f ( x ) = nx ( v ( n ) i − v ( n ) i − ) + i ( v ( n ) i − − v ( n ) i ) + v ( n ) i x ∈ (cid:20) i − n , in (cid:21) Definition 3.1.
Given a sequence { D n } n of diagonal matrices, let d n = diag ( D n ) ,and f n the maps associated to d n . We say that D n converges piecewise to a func-tion f : [0 , → C if the sequence f n converges to f in measure. In this case wewrite { D n } n ⇀ f ( x ) . This definition copes well with the concept of zero distributed sequences anddiagonal sampling of continuous functions.12 .1 Zero Distributed and Diagonal Sampling Sequences
We recall here that a sequence is called zero distributed if the function zero is asingular value symbol. Moreover, they are all GLT sequences with symbol zero,and coincide with the kernel of the map S in Theorem 2.2, that is, are the onlyGLT sequences with symbol zero. In the case of diagonal matrices, we can provethat a diagonal sequence is zero-distributed if and only if it converges piecewiseto the function zero. Lemma 3.2.
Given a sequence of diagonal matrices { Z n } n , the following state-ment are equivalent.1. { Z n } n ⇀ { Z n } n ∼ GLT { Z n } n ∼ λ Proof.
A matrix sequence is a GLT sequence with symbol zero if and only if itis a zero distributed sequence, so we can substitute the second statement with { Z n } n ∼ σ . ⇒ Using Lemma 2.1, we can split the eigenvalues of Z n into two diagonalmatrices Z n = b Z n + e Z n with rk( b Z n ) = o ( n ) and k e Z n k = o (1) . This means thatgiven ε , there exists N such that rk( b Z n ) < εn, k e Z n k < ε ∀ n ≥ N or also said as { i : | λ i ( Z n ) | ≥ ε } < εn ∀ n ≥ N. We can now use the definition of spectral symbol to prove the thesis. Given any F ∈ C c ( C ) , we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 F ( λ i ( Z n )) − F (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X | λ i ( Z n ) |≥ ε F ( λ i ( Z n )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X | λ i ( Z n ) | <ε F ( λ i ( Z n )) − F (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ { i : | λ i ( Z n ) | ≥ ε } n k F k ∞ + ω F ( ε ) ≤ ε k F k ∞ + ω F ( ε ) ∀ n ≥ N where ω F ( ε ) is the modulus of continuity of the function F . We thus get lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n X i =1 F ( λ i ( Z n )) − F (0) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε k F k ∞ + ω F ( ε ) for every ε > . So we get to the conclusion lim n →∞ n n X i =1 F ( λ i ( Z n )) = F (0) = Z F (0) dx = ⇒ { Z n } n ∼ λ ⇐ = 3) Let F ∈ C c ( R ) . If G : C → C is defined as G ( z ) := F ( | z | ) , we getthat this is still a continuous and compact supported function G ∈ C c ( C ) . Wealso know that the singular values of a diagonal matrix are the absolute valuesof the eigenvalues. With this we can conclude lim n →∞ n n X i =1 F ( σ i ( Z n )) = lim n →∞ n n X i =1 F ( | λ i ( Z n ) | ) = lim n →∞ n n X i =1 G ( λ i ( Z n ))= Z G (0) dx = Z F (0) dx = ⇒ { Z n } n ∼ σ
01 = ⇒ Let f n ( x ) be the piecewise linear functions associated with Z n . Wefix ε > and study the quantity E i,n := n · µ (cid:26) x ∈ (cid:20) i − n , in (cid:21) : | f n ( x ) | > ε (cid:27) that we can divide into E i,n = E + i,n + E − i,n where E + i,n := n · µ (cid:26) x ∈ (cid:20) i − n , in (cid:21) : f n ( x ) > ε (cid:27) E − i,n := n · µ (cid:26) x ∈ (cid:20) i − n , in (cid:21) : f n ( x ) < − ε (cid:27) . Suppose now that f n (cid:0) in (cid:1) > ε . Recall that f n ( x ) = nx (cid:20) f n (cid:18) in (cid:19) − f n (cid:18) i − n (cid:19)(cid:21) + i (cid:20) f n (cid:18) i − n (cid:19) − f n (cid:18) in (cid:19)(cid:21) + f n (cid:18) in (cid:19) . Let us divide the analysis in cases. • If f n (cid:0) i − n (cid:1) ≥ ε , then E i,n = 1 . • If | f n (cid:0) i − n (cid:1) | < ε , then E − i,n = 0 , so E + i,n = E i,n . Given x ∈ (cid:2) i − n , in (cid:3) wehave f n ( x ) > ε ⇐⇒ in > x > ε − i ( f n (cid:0) i − n (cid:1) − f n (cid:0) in (cid:1) ) − f n (cid:0) in (cid:1) n ( f n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) ) so E + i,n = n " in − ε − i ( f n (cid:0) i − n (cid:1) − f n (cid:0) in (cid:1) ) − f n (cid:0) in (cid:1) n ( f n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) ) = f n (cid:0) in (cid:1) − εf n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) . Using the hypothesis, we get E i,n = f n (cid:0) in (cid:1) − εf n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) ≥ f n (cid:0) in (cid:1) − εf n (cid:0) in (cid:1) + ε = 1 − εf n (cid:0) in (cid:1) + ε ≥ − ε ε + ε = 13 . • If f n (cid:0) i − n (cid:1) ≤ − ε , then given x ∈ (cid:2) i − n , in (cid:3) we have f n ( x ) > ε ⇐⇒ in > x > ε − i ( f n (cid:0) i − n (cid:1) − f n (cid:0) in (cid:1) ) − f n (cid:0) in (cid:1) n ( f n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) ) , n ( x ) < − ε ⇐⇒ i − n < x < − ε − i ( f n (cid:0) i − n (cid:1) − f n (cid:0) in (cid:1) ) − f n (cid:0) in (cid:1) n ( f n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) ) , so E + i,n = n " in − ε − i ( f n (cid:0) i − n (cid:1) − f n (cid:0) in (cid:1) ) − f n (cid:0) in (cid:1) n ( f n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) ) = f n (cid:0) in (cid:1) − εf n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) ,E − i,n = n " − ε − i ( f n (cid:0) i − n (cid:1) − f n (cid:0) in (cid:1) ) − f n (cid:0) in (cid:1) n ( f n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) ) − i − n = − ε − f n (cid:0) i − n (cid:1) f n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) ,E i,n = f n (cid:0) in (cid:1) − εf n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) + − ε − f n (cid:0) i − n (cid:1) f n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) = 1 − εf n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) . Using the hypothesis, we get E i,n = 1 − εf n (cid:0) in (cid:1) − f n (cid:0) i − n (cid:1) ≥ − ε ε + ε = 13 The same would happen if f n ( i/n ) < − ε , so µ { x : | f n ( x ) | > ε } = n X i =1 µ (cid:26) x : | f n ( x ) | > ε, x ∈ (cid:20) i − n , in (cid:21)(cid:27) ≥ X i : | f n ( i/n ) | > ε µ (cid:26) x : | f n ( x ) | > ε, x ∈ (cid:20) i − n , in (cid:21)(cid:27) = X i : | f n ( i/n ) | > ε n E i,n ≥ n (cid:26) i : (cid:12)(cid:12)(cid:12)(cid:12) f n (cid:18) in (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:27) . We also know that f n µ −→ , so lim n →∞ µ { x : | f n ( x ) | > ε } = 0 ∀ ε > that leads to lim n →∞ n (cid:26) i : (cid:12)(cid:12)(cid:12)(cid:12) f n (cid:18) in (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) > ε (cid:27) = 0 ∀ ε > The values (cid:12)(cid:12) f n (cid:0) in (cid:1)(cid:12)(cid:12) are the singular values of Z n for every i > , so we get also lim n →∞ n { i : σ i ( Z n ) > ε } = 0 ∀ ε > and thanks to Lemma 2.1, we can conclude that { Z n } n ∼ σ . ⇐ = 2) Let ε > be a fixed value. We notice that | f n ( x ) | ≥ ε = ⇒ max (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) f n (cid:18) i − n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) f n (cid:18) in (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:27) ≥ ε ∀ x ∈ (cid:20) i − n , in (cid:21)
15o we deduce µ { x : | f n ( x ) | > ε } = n X i =1 µ (cid:26) x : | f n ( x ) | > ε, x ∈ (cid:20) i − n , in (cid:21)(cid:27) = X i :max {| f n ( i − n ) | , | f n ( in ) |} ≥ ε µ (cid:26) x : | f n ( x ) | > ε, x ∈ (cid:20) i − n , in (cid:21)(cid:27) ≤ n (cid:26) i : max (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) f n (cid:18) i − n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) f n (cid:18) in (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:27) ≥ ε (cid:27) ≤ n (cid:26) i : (cid:12)(cid:12)(cid:12)(cid:12) f n (cid:18) in (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ε (cid:27) . The values (cid:12)(cid:12) f n (cid:0) in (cid:1)(cid:12)(cid:12) are the singular values of Z n for every i > , so we can useLemma 2.1, and obtain that the last quantity converges to zero when n tendsto infinity. Therefore, lim n →∞ µ { x : | f n ( x ) | > ε } = 0 ∀ ε > meaning that f n µ −→ .Given the sampling diagonal matrices { D n ( a ) } n , where a : [0 , → C is acontinuous function, and D n ( a ) := diag i =1 ,...,n a (cid:18) in (cid:19) = a (cid:0) n (cid:1) a (cid:0) n (cid:1) . . . a (1) we can prove that { D n ( a ) } n ⇀ a ( x ) . Lemma 3.3.
Given any continuous map a : [0 , → C , the functions f n asso-ciated to D n ( a ) converge in measure to a . In particular, { D n ( a ) } n ⇀ a ( x ) . Proof.
Any continuous function on [0 , is uniformly continuous thanks toHeine-Cantor theorem, so given any ε > , there exists δ > such that | x − y | ≤ δ = ⇒ | a ( x ) − a ( y ) | ≤ ε. Let n be a natural number with n − < δ . If we denote a k := a ( k/n ) , then | a ( x ) − a i | ≤ ε ∀ x ∈ (cid:20) i − n , in (cid:21) ∀ i. On the same interval, the function f n is a segment in the complex space from a i − to a i for every i > . We know that | a i − a i − | ≤ ε , so | f n ( x ) − a i | ≤ ε ∀ x ∈ (cid:20) i − n , in (cid:21) ∀ i > . | f n ( x ) − a ( x ) | ≤ ε ∀ x ∈ (cid:20) n , (cid:21) ∀ n > δ − . So the functions f n ( x ) converge in measure to a ( x ) and { D n ( a ) } n ⇀ a ( x ) The set of continuous functions is dense in the space of measurable func-tions, so we can prove some approximation results on convergent sequences ofcontinuous functions.
Lemma 3.4.
Given any a : [0 , → C measurable function, and a m ∈ C ([0 , continuous functions that converge in measure to a ( x ) , there exists a crescentand unbounded map m ( n ) such that { D n ( a m ( n ) ) } n ∼ GLT a ( x ) ⊗ { D n ( a m ( n ) ) } n ⇀ a ( x ) Proof.
The functions a m ( x ) are continuous, so { D n ( a m ) } n ∼ GLT a m ( x ) ⊗ ,and a m ( x ) ⊗ converges in measure to a ( x ) ⊗ . This means that the sequence a m ( x ) ⊗ is a Cauchy sequence in M D , but the map S of Theorem 2.2 preservesthe distances, so {{ D n ( a m ) } n } m is also a Cauchy sequence. Thanks to Theorem2.1, we know that there exists an unbounded crescent map m ( n ) such that {{ D n ( a m ) } n } m a.c.s. −−−→ { D n ( a m ( n ) ) } n . The map S has also a closed graph, so we get { D n ( a m ( n ) ) } n ∼ GLT a ( x ) ⊗ . Thanks to Lemma 3.3, we also know that { D n ( a m ) } n ⇀ a m ( x ) for every m ,meaning that the piecewise linear functions a n,m ( x ) associated to D n ( a m ) con-verge to a m ( x ) . The measure convergence is metrizable through the distance d M , so we can define the following indexes. • a m ( x ) converges to a ( x ) in measure, so for every k > there exists anindex M k such that M k < M k +1 , d M ( a m ( x ) , a ( x )) ≤ − k ∀ m ≥ M k ∀ k > • a n,m ( x ) converges to a m ( x ) in measure, so there exists an index e N m suchthat d M ( a n,m ( x ) , a m ( x )) ≤ − k ∀ n ≥ e N m and we can define a strictly increasing set of indexes N k with the followingrecursive procedure. N = max m ≤ M n e N m o N k = max (cid:26) max M k ≤ m ≤ M k +1 { e N m } , N k − + 1 (cid:27) ∀ k >
17e can now define a new crescent unbounded map m ( n ) m ( n ) := ( N > nM k N k +1 > n ≥ N k ∀ k > and prove that the sequence a n,m ′ ( n ) ( x ) converges to a ( x ) for any crescent un-bounded map m ′ ( n ) such that m ′ ( n ) ≤ m ( n ) . In fact, suppose that n ≥ N k and m ′ ( n ) ≥ M k . We get d M ( a n,m ′ ( n ) ( x ) , a ( x )) ≤ d M ( a n,m ′ ( n ) ( x ) , a m ′ ( n ) ( x )) + d M ( a m ′ ( n ) ( x ) , a ( x )) and m ′ ( n ) ≥ M k = ⇒ d M ( a m ′ ( n ) ( x ) , a ( x )) ≤ − k M k ≤ M s ≤ m ′ ( n ) < M s +1 , m ′ ( n ) ≤ m ( n )= ⇒ M k ≤ M s ≤ m ( n ) , k ≤ s, n ≥ N k = ⇒ d M ( a n,m ′ ( n ) ( x ) , a m ′ ( n ) ( x )) ≤ − s ≤ − k . Consequentially, d M ( a n,m ′ ( n ) ( x ) , a ( x )) ≤ − k so the distance goes to zero when n tends to infinity,and this implies that { D n ( a m ′ ( n ) ) } n ⇀ a ( x ) If we consider m ( n ) := min { m ( n ) , m ( n ) } we find that it is an unlimitedcrescent function and it is bounded by both maps m ( n ) and m ( n ) . Usingagain Lemma 2.1 and what we’ve shown before, we prove the thesis. We are now ready to see how the piecewise convergence, the concept of spec-tral symbol and the GLT symbols are connected when dealing with diagonalsequences.
Theorem 3.1.
Given any diagonal sequence { D n } n and a measurable function f : [0 , → C , then { D n } n ⇀ f ( x ) = ⇒ { D n } n ∼ λ f ( x ) . Proof.
Let f m ( x ) be continuous functions that converge in measure to f ( x ) .Using Lemma 3.4, we can find a map m ( n ) such that { D n ( f m ( n ) ) } n ∼ GLT f ( x ) ⊗ , { D n ( f m ( n ) ) } n ⇀ f ( x ) . We know that { D n ( f m ) } n ∼ GLT f m ( x ) ⊗ , but the map S in Theorem 2.2preserves the distances, so d M ( f m ( x ) ⊗ , f ( x ) ⊗ → ⇒ d acs ( {{ D n ( f m ) } n } m , { D n ( f m ( n ) ) } n ) → ⇒ {{ D n ( f m ) } n } m a.c.s. −−−→ { D n ( f m ( n ) ) } n . { D n ( f m ) } n ∼ λ f m ( x ) , so all the hypothesis of Lemma2.5 are satisfied and we obtain { D n ( f m ( n ) ) } n ∼ λ f ( x ) . Both the sequences { D n } n and { D n ( f m ( n ) ) } n converge piecewise to f ( x ) . Thismeans that both the sequences of piecewise linear functions g n ( x ) and h n ( x ) associated to { D n } n and { D n ( f m ( n ) ) } n converge in measure to f ( x ) . The dif-ference of the two functions converges in measure to , but g n ( x ) − h n ( x ) is thepiecewise linear function associated to D n − D n ( f m ( n ) ) so we get that { D n } n − { D n ( f m ( n ) ) } n ⇀ and using Lemma 3.2, we get { D n } n − { D n ( f m ( n ) ) } n ∼ σ . Eventually, using Lemma 2.4 we conclude { D n } n = { D n ( f m ( n ) ) } n + ( { D n } n − { D n ( f m ( n ) ) } n ) ∼ λ f ( x ) Theorem 3.2.
Given { D n } n a sequence of diagonal matrices, and k : [0 , → C any measurable function, then the following are equivalent • D n ⇀ k ( x ) , • D n ∼ GLT k ( x ) ⊗ .Proof. Given k ( x ) a measurable function, we can always find a sequence ofcontinuous functions k m ( x ) that converges to k ( x ) in measure. Using Lemma3.4, there exists a map m ( n ) such that { D n ( k m ( n ) ) } n ∼ GLT k ( x ) ⊗ { D n ( k m ( n ) ) } n ⇀ k ( x ) . Using the algebra properties of the GLT space, we get { D n } n ∼ GLT k ( x ) ⊗ ⇐⇒ { D n ( k m ( n ) ) − D n } n ∼ GLT and thanks to Lemma 3.2 we have { D n ( k m ( n ) ) − D n } n ∼ GLT ⇐⇒ { D n ( k m ( n ) ) − D n } n ⇀ . The piecewise convergence is linear: if { A n } n ⇀ a ( x ) and { B n } n ⇀ b ( x ) , it iseasy to see that { A n } n + { B n } n ⇀ a ( x ) + b ( x ) . We can thus conclude that { D n ( k m ( n ) ) − D n } n ⇀ ⇐⇒ { D n } n ⇀ k ( x ) . A sequence { A n } n may have several different spectral or singular values symbols.When dealing with real valued symbols, though, there is a preferred one, called decreasing rearrangement . 19 .1 Decreasing Rearrangement Any measurable function f : D → R has a decreasing rearrangement that is adecreasing function g : [0 , → R with the same distribution. We can define itas g ( y ) := inf (cid:26) z : µ { x : f ( x ) > z } µ ( D ) ≤ y (cid:27) . It is easy to check that it is a decreasing function, and moreover the followingproperty hold: µ { x : f ( x ) > z } µ ( D ) = µ { y : g ( y ) > z } ∀ z ∈ R . This result can be found in any graduate book of analysis, like [7]. If f ( x ) isa spectral symbol for a sequence { A n } n , then we can prove that also g ( y ) is aspectral symbol for the same sequence. Lemma 4.1.
Given { A n } n a matrix sequence, f : D → R a measurable functionwith D ⊆ R n a set with finite non-zero measure, and g : [0 , → R is itsdeacrising rearrangement, then { A n } n ∼ λ f ( x ) = ⇒ { A n } n ∼ λ g ( x ) . Proof.
Let χ ( a,b ] be the indicator function of the real interval ( a, b ] . We knowthat µ ( D ) Z D χ ( a,b ] ( f ( x )) dx = µ { x : f ( x ) > a } − µ { x : f ( x ) > b } µ ( D )= µ { y : g ( y ) > a } − µ { y : g ( y ) > b } = Z χ ( a,b ] ( g ( y )) dy. This implies that given any step function G , obtained as a linear combinationsof indicator functions of real intervals ( a, b ] , we have that µ ( D ) Z D G ( f ( x )) dx = Z G ( g ( y )) dy. Any real valued, compact supported and continuous function can be approxi-mated in infinity norm arbitrarily well by step functions, so given F ∈ C c ( R ) and ε > , we can take a step function G such that k F ( x ) − G ( x ) k ∞ ≤ ε. This means that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ ( D ) Z D F ( f ( x )) dx − Z F ( g ( y )) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) µ ( D ) Z D F ( f ( x )) dx − µ ( D ) Z D G ( f ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) µ ( D ) Z D G ( f ( x )) dx − Z G ( g ( y )) dy (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z G ( g ( y )) dy − Z F ( g ( y )) dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε + 0 + ε ε > , so µ ( D ) Z D F ( f ( x )) dx = Z F ( g ( y )) dy. This concludes that for every F ∈ C c ( C ) , lim n →∞ n n X i =1 F ( λ i ( A n )) = 1 µ ( D ) Z D F ( f ( x )) dx = Z F ( g ( y )) dy. This means that we can work only with decreasing functions. In particu-lar, for these functions it is possible to establish a connection between spectralsymbols and piecewise convergence. The next lemma shows that the converseimplication of Theorem 3.1 holds with additional hypothesis.
Lemma 4.2.
Let { D n } n be a sequence of diagonal hermitian matrices, and let f n be the maps associated to d n = diag ( D n ) . If f n and f are decreasing realvalued and measurable functions on [0 , , then { D n } n ∼ λ f ( x ) = ⇒ { D n } n ⇀ f ( x ) Proof. f is a decreasing function, so the set of its discontinuity points havemeasure zero. If we suppose that f n do not converge to f in measure, then inparticular they do not converge punctually almost everywhere, so there is a set S ⊆ [0 , with non-zero measure such that lim n →∞ f n ( x ) = f ( x ) ∀ x ∈ S. Let x ∈ S be a point where f is continuous and x = 0 . There exists ε > such that | f n ( x ) − f ( x ) | > ε frequently. Surely, there exist infinite index m such that f m ( x ) − f ( x ) > ε or there exist infinite index m such that f m ( x ) − f ( x ) < − ε . Suppose without loss of generality that the first caseholds, and let g n be the subsequence of g n := f k n such that g n ( x ) − f ( x ) > ε ∀ n ∈ N . Using the continuity of f in x , we can find δ > such that f ( x − z ) < f ( x ) + ε ∀ ≤ z ≤ δ and x > δ . Let M := f ( x ) + 2 ε , and divide in cases.Case 1.) Suppose the sequence g n ( δ ) has a limit point z .Take T > max { M + ε, z } and let G ∈ C c ( R ) such that ≤ G ( x ) ≤ for every x ∈ R , and G ( x ) = ( x ∈ [ M, T ] , x ( M − ε, T + ε ) . Notice that g n ( x ) > M , δ < x , and g n are decreasing, so g n ( δ ) > M , meaningthat z ≥ M . Moreover, T > z implies that g n ( x ) ∈ [ M, T ] ∀ x ∈ [ δ, x ] frequently in n.
21f we consider only the points i/n with i = 1 , . . . , n , then k n k n X i =1 G (cid:18) g n (cid:18) ik n (cid:19)(cid:19) ≥ n i : ik n ∈ [ δ, x ] o k n ≥ x − δ frequently when k n > δ . Moreover, we have x > x − δ = ⇒ f ( x ) ≤ f ( x − δ ) < f ( x ) + ε = M − ε so Z G ( f ( x )) dx ≤ x − δ. Finally, using the definition of spectral symbol, we obtain x − δ ≤ lim n →∞ n n X i =1 G (cid:18) f n (cid:18) in (cid:19)(cid:19) = lim n →∞ n n X i =1 G ( λ i ( D n ))= Z G ( f ( x )) dx ≤ x − δ, that is an absurd.Case 2.) Suppose the sequence g n ( δ ) has not a limit point.This means that any subsequence of g n ( δ ) is not bounded, but g n ( δ ) > M forevery n , so they must diverge to + ∞ . Let γ > such that < γ < δ , anddenote R = f ( γ ) , T = f (1 − γ ) . We can find a function G ∈ C c ( R ) such that ≤ G ( x ) ≤ for every x ∈ R , and G ( x ) = ( x ∈ [ T, R ] , x ( T − ε, R + ε ) . Since g n ( δ ) diverges to infinity, then it will definitively be greater than R + ε ,and consequently x < δ = ⇒ g n ( x ) ≥ g n ( δ ) > R + ε. This means that k n k n X i =1 G (cid:18) g n (cid:18) ik n (cid:19)(cid:19) ≤ n i : ik n ∈ [ δ, o k n ≤ − δ frequently when k n > δ . Moreover, we have − γ > x > γ = ⇒ T = f (1 − γ ) ≤ f ( x ) ≤ f ( γ ) = R, so Z G ( f ( x )) dx ≥ − γ. − γ > − δ ≥ lim n →∞ n n X i =1 G (cid:18) f n (cid:18) in (cid:19)(cid:19) = lim n →∞ n n X i =1 G ( λ i ( D n ))= Z G ( f ( x )) dx ≥ − γ that is an absurd.Using this result on decreasing functions, we can use the natural order of R to obtain results regarding permutated diagonal matrices. Here we can use all the results proved in the other sections, to show that if asequence of diagonal matrices has a spectral symbol, then we can reorder theelements on the diagonal to produce a GLT sequence with the same symbol.
Theorem 4.1.
Given { D n } n a sequence of diagonal matrices with real entriessuch that { D n } n ∼ λ f ( x ) , with f : [0 , → R , then there exist permutationmatrices P n such that { P n D n P Tn } ∼ GLT f ( x ) ⊗ . Proof.
Notice that the definition of spectral symbol depends only on the eigen-values, and not on their order, so any permutation of the elements on the diag-onal of D n does not change the spectral symbol.Let g ( x ) be the decreasing rearrangement of f ( x ) , and let Q n be permutationmatrices such that Q n D n Q Tn have the eigenvalues sorted in decreasing order onthe diagonal, for every n . Using Lemma 4.1, we know that { Q n D n Q Tn } n ∼ λ g ( x ) but now g ( x ) and the functions associated to Q n D n Q Tn are all decreasing, so wecan apply Lemma 4.2 and obtain { Q n D n Q Tn } n ⇀ g ( x ) . Thanks to Theorem 3.2, we also know that { Q n D n Q Tn } n ∼ GLT g ( x ) ⊗ . Since f ( x ) is a measurable functions, then there exist a sequence f n ( x ) of con-tinuous function that converge to f ( x ) in measure, and thanks to Lemma 3.4,we can find a sequence of diagonal matrices { D ′ n } n such that { D ′ n } n ∼ GLT f ( x ) ⊗ { D ′ n } n ∼ λ f ( x ) . We can then find again permutation matrices S n that sort the elements of D ′ n in decreasing order, so that Lemma 4.1, Lemma 4.2 and Theorem 3.2 lead to { S n D ′ n S Tn } n ∼ GLT g ( x ) ⊗ . Using the fact that the GLT space is an algebra, we obtain { S n D ′ n S Tn − Q n D n Q Tn } n ∼ GLT that, thanks to Lemma 3.2, leads to { S n D ′ n S Tn − Q n D n Q Tn } n ∼ λ . Permuting again the entries, we have { D ′ n − S Tn Q n D n Q Tn S n } n ∼ λ so we can use again Lemma 3.2, that leads to { D ′ n − S Tn Q n D n Q Tn S n } n ∼ GLT but GLT is an algebra, so { S Tn Q n D n Q Tn S n } n ∼ GLT f ( x ) ⊗ . If S Tn Q n = P n , we conclude that { P n D n P Tn } n ∼ GLT f ( x ) ⊗ . References [1] G. Barbarino. Equivalence between GLT sequences and measurable func-tions.
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