On the existence of compactly supported reconstruction functions in a sampling problem

On the exist e nce of compactly su p p o rted reconst ruction functions in a sampling pro blem A. G. Garc ´ ıa, ∗ M. A. Hern ´ andez-Medin a, † and G. P´ erez-Villal´ on ‡ * Departamen to de Matem´ aticas, Univ ersidad Carlos I I I de Madrid, Avda. de la Univ ersidad 30, 28 911 Legan ´ es-Madrid, Spain. † Departamen to de Matem´ atica Aplicada, E.T.S.I.T., U.P .M., Avd a. Complutense s/n , 2 8040 Madrid, Spain. ‡ Departamen to de Mat em´ atica Ap licada, E.U.I.T.T., U.P .M., Carr et. V alencia km. 7, 28031 Madrid, Spain. Abstract Assume that sa mples of a filtered version of a function in a shift-inv aria n t space are av alaible. This work deals with the exis tence of a sa mpling for m ula involving these samples and having recons truction functions with compact supp ort. Thus, low computational complexity is inv olv ed a nd t runcation errors are avoided. This is done in the ligh t o f the gener alized sa mpling theory by using the ov ersampling technique: more samples than strictly necess ary ar e used. F or a suitable choice of the sa mpling per io d, a necessary a nd sufficient condition is giv en in terms of the Kro neck er canonical form of a ma tr ix p encil. Comparing with o ther characterizations in the mathematical literature, the given here has an imp or ta n t a dv ant age: it can b e reliable computed by using the GUPTRI form of the ma tr ix p encil. Fina lly , a practical metho d for computing the co mpa ctly supp orted reco ns truction functions is g iven for the imp ortant case where the ov ersampling rate is minimum. Keyw ords : Shift-in v ariant spaces; Generalized sampling; Ove rsampling; Matrix p encils; Kronec k er canonical form; GUPTRI form. AMS : 15A2 1; 15A22; 42C15; 42C40; 94A20. 1 Statemen t of the problem Let V ϕ b e a shift-inv arian t space in L 2 ( R ) with s table ge nerator ϕ ∈ L 2 ( R ), i.e., V ϕ := n f ( t ) = X n ∈ Z a n ϕ ( t − n ) : { a n } ∈ ℓ 2 ( Z ) o ⊂ L 2 ( R ) . ∗ E-mail: agarc ia@math.uc3 m.es † E-mail: mahm@ mat.upm.es ‡ E-mail: gpere z@euitt.upm .es 1 No w ada y s , s ampling theory in shift-inv arian t sp aces is a v ery activ e researc h topic (see, for instance, [1, 2, 3, 4 , 5, 10, 21, 22] and the r eferen ces therein) since an appropr iate c hoic e for th e generator ϕ (for instance, a B-spline) eliminates most of th e pr oblems asso ciated with the cla ssical Shannon’s sampling theory [18]. Supp ose that a linear time-in v ariant sys tem L is defined on V ϕ . Und er suitable condi- tions, Unser and Aldroub i [4, 17] hav e found sampling formulas allo wing the reco v ering of an y fun ction f ∈ V ϕ from the sequence of samp les {  L f  ( n ) } n ∈ Z . Concretely , th ey pr ov e d that for any f ∈ V ϕ , f ( t ) = X n ∈ Z L f ( n ) S L ( t − n ) , t ∈ R , (1) where th e s equence { S L ( t − n ) } n ∈ Z is a Riesz basis for V ϕ . Note that a reconstruction function S L with compact sup p ort imp lies lo w computational complexit y and av oids tru n- cation errors. Ev en when the generator ϕ has compact sup p ort, rarely the same o ccurs with the reconstru ction function S L in formula (1). A wa y to o vercome this difficulty is to u se the ov ersampling tec hn iqu e, i.e, to tak e samples {  L f  ( nT ) } n ∈ Z with a s ampling p erio d T < 1. This is th e main goal in this pap er: Assu ming that b oth the generator ϕ and L ϕ ha ve compact su pp ort, we study the existence of stable samp ling form ulas w ith compactly sup p orted reconstru ction functions, w hic h all o w us to reco ver an y f ∈ V ϕ from the samples {  L f  ( nT ) } n ∈ Z , where the samp ling p er io d is T := r /s < 1 for some p ositive in tegers r < s . T his is don e in the ligh t of the generalized sampling theory obtained in [12] by follo win g an id ea of Djok o vic and V aidy anathan in [10]. F o r th e sak e of notational simplicit y we hav e assu med th at only samples from one linear time-inv arian t system L are a v alaible. In so doing, the problem is connected with the searc h of p olynomial left in verses of a certain s × r p olynomial matrix G ( z ) in timately related to the sampling problem. T aking adv antage of the sp ecial structure of the m atrix G ( z ) w e giv e a necessary and s u fficien t condition whic h inv olv es the matrix p encils theory . Concretely , this condition uses some information cont ained in the Kr on eck er canonical form of a matrix p encil asso ciated with the matrix G ( z ). F rom a p r actical p oint of view, this information can b e r etriev ed fr om th e GUPTRI (General UPp er TRIangular) form of the matrix p encil. It is wo rth to men tion that the GUPTRI form can b e stably computed. The m athematica l problem of finding a p olynomial left inv erse of a p olynomial matrix G ( z ) has b een stu died in [7] by Cv etk o vi´ c and V et terli in the filter banks setting. It in volv es the S mith canonica l form S ( z ) of the matrix G ( z ). Roughly sp eaking, the Smith canonical form of G ( z ) must con tain monomials in its diagonal. F rom a practical p oin t of view, th e Smith canonical form has an imp ortant d ra wb ac k: there is not a stable metho d for its compu tation. Another algebraic approac h is the f ollo wing (see, f or instance, [16]): Assu me that G ( z ) is a s × r Laur en t p olynomial matrix ( r < s ); wheneve r the greatest common d ivisor of all minors of maxim um order r is a monomial, then its Smith canonical form S ( z ) has monomials in its diagonal. Without loss of generalit y we can assume that the γ :=  s r  minors of order r in G ( z ) are p olynomials with p ositive p o wers in z . Inv oking Euclides algorithm w e can obtain  s r  p olynomials, f 1 ( z ) , . . . , f γ ( z ), such that γ X n =1 f n ( z ) A n ( z ) = m ( z ) , for all z ∈ C 2 where A n , 1 ≤ n ≤ γ , are the minors of order r of G ( z ) and m ( z ) is a monomial. Denote by D ′ n ( z ) the adjoin t m atrix corresp onding to the minor A n and D n ( z ) the m atrix obtained from D ′ n ( z ) b y adding s − r zero column s. Thus, D n ( z ) G ( z ) = A n ( z ) I r , and consequently  γ X n =1 f ′ n ( z ) D n ( z )  G ( z ) = I r , where f ′ n ( z ) := f ( z ) /m ( z ) could b e a Laur en t p olynomial, 1 ≤ n ≤ γ . F rom a practical p oint of view the drawbac k h ere is the effectiv e calculation of th e  s r  minors of G ( z ) whenev er r b ecomes larger. The p ap er is organized as f ollo ws: In Section 2 w e includ e the n eeded preliminaries to understand the raised problem. The existence of reconstruction functions with compact supp ort dep end s on the ran k , for z ∈ C \ { 0 } , of a p olynomial matrix G ( z ), associated with the sampling p roblem. In Section 3, a su itable choic e of the sampling p erio d T = r /s reduces our problem to a matrix p encil problem. Thus, we giv e a necessary and sufficient condition for the existence of compactly su p p orted reconstruction fu nctions whic h inv olv es the Kr onec k er canonical form of a singular matrix p encil. Section 4 is dev oted to compute a p olynomial left inv erse of the matrix G ( z ) in the imp ortant case where th e o v ersamp ling rate is minimum. Finally , w e briefly remind, as an App endix, the canonical forms alluded in what follo ws . 2 Preliminaries on generalized sampling F rom no w on, th e function ϕ ∈ L 2 ( R ) is a stable generator for th e shif t-inv arian t space V ϕ := n f ( t ) = X n ∈ Z a n ϕ ( t − n ) : { a n } ∈ ℓ 2 ( Z ) o ⊂ L 2 ( R ) , i.e., the sequence { ϕ ( · − n ) } n ∈ Z is a Riesz basis for V ϕ . A R iesz b asis in a separable Hilb ert space is the image of an orthonormal basis by means of a b oun ded inv ertible op er ator. Recall that th e sequence { ϕ ( · − n ) } n ∈ Z is a Riesz basis for V ϕ if and only if 0 < k Φ k 0 ≤ k Φ k ∞ < ∞ , where k Φ k 0 denotes the essentia l infi m u m of the function Φ( w ) := P k ∈ Z | b ϕ ( w + k ) | 2 in (0 , 1), and k Φ k ∞ its essenti al sup r em um ( b ϕ denotes, as usu al, the F ourier transform of ϕ ). F urtherm ore, k Φ k 0 and k Φ k ∞ are the optimal Riesz b ound s [6, p. 143] . W e assume throughout the p ap er that the functions in the sh ift-in v arian t space V ϕ are con tinuous on R . Equ iv alent ly , the generator ϕ is con tin u ous on R and the function P n ∈ Z | ϕ ( t − n ) | 2 is uniformly b ounded on R (see [22]). Thus, an y f ∈ V ϕ is defined as the p oin twise sum f ( t ) = P n ∈ Z a n ϕ ( t − n ) on R . Be s ides, V ϕ is a repro du cing kernel Hilb ert space where con vergence in the L 2 ( R )-norm implies p oint wise con vergence which is uniform on R (see [12]). The space V ϕ is the image of L 2 (0 , 1) b y means of the isomorphism T ϕ : L 2 (0 , 1) → V ϕ whic h maps th e orthonormal basis { e − 2 π inw } n ∈ Z for L 2 (0 , 1) on to the Riesz basis { ϕ ( t − n ) } n ∈ Z for V ϕ . Namely , for eac h F ∈ L 2 (0 , 1) th e function T ϕ F ∈ V ϕ is giv en by ( T ϕ F )( t ) := X n ∈ Z  F ( · ) , e − 2 π in ·  L 2 (0 , 1) ϕ ( t − n ) , t ∈ R . (2) 3 Supp ose that L is a linear time-in v arian t system d efined on V ϕ of one of the follo wing t yp es (or a linear com bination of b oth): (a) Th e impu lse r esp onse h of L b elongs to L 1 ( R ) ∩ L 2 ( R ). Th u s, for any f ∈ V ϕ w e h a ve  L f  ( t ) := [ f ∗ h ]( t ) = Z ∞ −∞ f ( x ) h ( t − x ) dx , t ∈ R . (b) L in vol v es samples of the fun ction itself, i.e., ( L f )( t ) = f ( t + d ), t ∈ R , for some constan t d ∈ R . F or fixed p ositiv e integ ers s > r , consider the sampling p erio d T := r /s < 1. The first goal is to reco ve r an y fu nction f ∈ V ϕ b y u sing a frame expansion in volving the samples  ( L f )( r n/s )  n ∈ Z . This can b e done in the light of the generalized sampling theory devel op ed in [12]. Indeed, since the sampling p oin ts r n/s , n ∈ Z , can b e expressed as  r n/s  n ∈ Z =  r m + ( j − 1) r /s  m ∈ Z ,j =1 , 2 ,...,s , the initial p roblem is equ iv alent to the reco v ery of f ∈ V ϕ from the sequences of samples {L j f  r n  } n ∈ Z ,j =1 , 2 ,...,s where the linear time-in v ariant systems L j , j = 1 , 2 , . . . , s , are defined by ( L j f )( t ) := ( L f )  t + ( j − 1) r /s  , t ∈ R . F ollo wing the nota tion in tro duced in [12], consider the functions g j ∈ L 2 (0 , 1), j = 1 , 2 , . . . , s , defined as g j ( w ) := X n ∈ Z ( L ϕ )  n + ( j − 1) r /s  e − 2 π inw , (3) the s × r matrix G ( w ) :=      g 1 ( w ) g 1 ( w + 1 r ) · · · g 1 ( w + r − 1 r ) g 2 ( w ) g 2 ( w + 1 r ) · · · g 2 ( w + r − 1 r ) . . . . . . . . . g s ( w ) g s ( w + 1 r ) · · · g s ( w + r − 1 r )      =  g j  w + k − 1 r   j =1 , 2 ,...,s k =1 , 2 ,...,r , and its relat ed constan ts α G := ess inf w ∈ (0 , 1 /r ) λ min [ G ∗ ( w ) G ( w )] , β G := ess sup w ∈ (0 , 1 /r ) λ max [ G ∗ ( w ) G ( w )] , where G ∗ ( w ) denotes the transp ose conjugate of the matrix G ( w ), and λ min (resp ectiv ely λ max ) the smallest (resp ectiv ely the largest) eig en v alue of th e p ositiv e semidefin ite matrix G ∗ ( w ) G ( w ). Notice that in the definition of the matrix G ( w ) we are considering the 1-p erio dic extensions of the inv olv ed fu nctions g j , j = 1 , 2 , . . . , s . Th us, the generalized s ampling th eory in [12] (see Theorem 1, Theorem 2 and its pro of ) giv es the follo wing sampling result in V ϕ : 4 Theorem 1 A ssume that the fu nctions g j , j = 1 , 2 , . . . , s , define d in (3) b elong to L ∞ (0 , 1) (this i s e quivalent to β G < ∞ ). Then the f ol lowing statements ar e e quivalent: (i) α G > 0 (ii) Ther e exist fu nctions a j in L ∞ (0 , 1) , j = 1 , 2 , . . . , s , such that  a 1 ( w ) , . . . , a s ( w )  G ( w ) = [1 , 0 , . . . , 0] a.e. in (0 , 1) . (4) (iii) Ther e exists a fr ame for V ϕ having the form { S j ( · − r n ) } n ∈ Z ,j =1 , 2 ,...,s such that, for any f ∈ V ϕ , we have f = X n ∈ Z s X j =1 ( L f )  r n + ( j − 1) r /s  S j  · − r n  in L 2 ( R ) . (5) In c ase the e quivalent c onditions ar e satisfie d, the r e c onstruction functions in (5) ar e given by S j = r T ϕ a j , j = 1 , 2 , . . . , s , wher e the functions a j , j = 1 , 2 , . . . , s , satisfy (4) . The c onver genc e of the series in (5) is also absolute and uniform on R . Recall th at a sequ ence { f k } is a fr ame for a separable Hilb ert sp ace H if there exist t wo co nstan ts A, B > 0 (frame b ounds ) suc h that A k f k 2 ≤ X k |h f , f k i| 2 ≤ B k f k 2 for all f ∈ H . Giv en a f rame { f k } for H the repr esen tation prop ert y of an y vect or f ∈ H as a series f = P k c k f k is retained, b ut, unlike the case of Riesz bases, the uniqu eness of this rep- resen tation (for ov ercomplete fr ames) is sacrificed. S uitable frame co efficien ts c k whic h dep end con tinuously and linearly on f are obtained by usin g the d u al frames { g k } of { f k } , i.e., { g k } is another frame for H su ch th at f = P k h f , g k i f k = P k h f , f k i g k for eac h f ∈ H . F or more details on the frame th eory see the s up erb monograph [6] and the references therein. It is wo rth to mentio n th at w h enev er the 1-p erio d ic fun ctions g j , j = 1 , 2 , . . . , s , are con tinuous on R , the conditions in Theorem 1 are also equiv alen t to the condition recent ly in tro duced in [13, Corollary 1]: ( iv ) rank G ( w ) = r for all w ∈ R . The goal in this pap er is to obtain necessary and su fficien t conditions assurin g that w e can fin d r econstruction functions S j , j = 1 , 2 , . . . , s , in form u la (5) h a ving compact supp ort. T o this end, assume from now on that the generator ϕ and L ϕ are compactly supp orted. W e introdu ce the s × r matrix G ( z ) :=      g 1 ( z ) g 1 ( W z ) · · · g 1 ( W r − 1 z ) g 2 ( z ) g 2 ( W z ) · · · g 2 ( W r − 1 z ) . . . . . . . . . g s ( z ) g s ( W z ) · · · g s ( W r − 1 z )      (6) 5 where W := e − 2 π i/r and g j ( z ) := P n ∈ Z ( L ϕ )  n + ( j − 1) r /s  z n , j = 1 , 2 . . . , s . Note that the matrix G ( z ) has Laur en t p olynomials en tries, and G ( w ) = G ( e − 2 π iw ). On the other hand, if th e fu n ctions a j ( z ), j = 1 , 2 . . . , s , are Laurent p olynomials satisfying [ a 1 ( z ) , . . . , a s ( z )] G ( z ) = [1 , 0 , . . . , 0] , (7) then, th e trigonometric p olynomials a j ( w ) = a j ( e − 2 π iw ), j = 1 , 2 , . . . , s , satisfy (4). In this case, the corresp onding reconstruction functions S j , j = 1 , 2 , . . . , s , h a ve compact supp ort. In deed, in terms of the coefficients c j,n of a j ( z ), that is, a j ( z ) = P n ∈ Z c j,n z n , j = 1 , 2 , . . . , s , the reconstru ction function S j , j = 1 , 2 , . . . , s , can b e written as (see (2)): S j ( t ) = r X n ∈ Z c j,n ϕ ( t − n ) , t ∈ R . (8) The existence of p olynomial solutions of (7) is equiv alent to the existence of a left in verse of the matrix G ( z ) whose en tries are p olynomials. This problem has b een studied in [7] b y Cvetk o vi ´ c and V etterli in the filter banks setting. By u sing the S mith canonical form S ( z ) of the matrix G ( z ) (see App endix 6.1), a c h aracterizat ion for the existence of p olynomial solutions of (7) has b een found in [14]. Namely , assuming that the generator ϕ and L ϕ ha v e compact supp ort, there exists a p olynomial ve ctor [ a 1 ( z ) , a 2 ( z ) , · · · , a s ( z )] satisfying (7) if and only if the p olynomials i j ( z ), j = 1 , 2 , . . . , r , on the diagonal of th e Smith canonical form S ( z ) of the matrix G ( z ) are monomials. Assu me that the s × r matrix S ( z ) =             i 1 ( z ) 0 · · · 0 0 i 2 ( z ) · · · 0 . . . . . . . . . 0 0 · · · i r ( z ) 0 0 · · · 0 . . . . . . . . . 0 0 · · · 0             (9) is the Smith canonical form of th e m atrix G ( z ) (note that it is the case whenever α G > 0) and consider the unimo dular matrices V ( z ) and W ( z ), of dimension s × s and r × r resp ectiv ely , su c h that G ( z ) = V ( z ) S ( z ) W ( z ). Observe that if S ( z ) is the Smith form of the m atrix G ( z ) then, taking into account that V ( z ) and W ( z ) are unimo dular matrices, w e ha v e rank S ( z ) = rank G ( z ) for all z ∈ C . Therefore, it is straigh tforw ard to d educe that, for eac h j = 1 , 2 , . . . , r , the p olynomial i j ( z ) is a monomial if and only if rank S ( z ) = r for all z ∈ C \ { 0 } . This condition, under the ab o ve hyp otheses on ϕ and L ϕ , is equiv alen t to sa ying th at rank G ( z ) = r for all z ∈ C \ { 0 } . (10) (See [14] f or th e d etails). Th e main aim in this work is to search for an equiv alen t condition to (10 ) useful from a practical p oint of view. 6 3 Searc hing for an u seful equiv alen t con d ition The first step is to redu ce our p olynomial matrix G ( z ) to a matrix p encil in order to use the w ell-established theory . In so d oing w e need some preliminaries. Let f ( z ) = a m z m + a m − 1 z m − 1 + · · · + a 1 z 1 + a 0 b e an algebraic p olynomial of order m , and let n b e a p ositiv e in teger. F or eac h j = 0 , 1 , . . . , n − 1 let b f j ( z ) den ote th e su m of the monomials a r z r where r ≡ j ( mo d n ). Obvio usly , f ( z ) = P n − 1 j =0 b f j ( z ). Th e p olynomial b f j , 0 ≤ j ≤ n − 1, is the so-called n -harmonic of order j of the p olynomial f ; it satisfies: b f j ( e 2 π i/n z ) = e 2 π ij /n b f j ( z ) for all z ∈ C . Assume that su pp L ϕ is con tained in an inte rv al [0 , N ], where N ∈ N . Th us, the functions g j ( z ) are Lauren t p olynomials in the v ariable z . Consider p := min { q ∈ N : q r s > 1 } . It is easy to c h ec k that p = c + 1 where c denotes the quotien t in the euclidean division s | r . Hence, w e can w rite the Laur ent p olynomials g i ( z ), j = 1 , 2 . . . , s , as: g 1 ( z ) = L ϕ (1) z + L ϕ (2) z 2 + · · · + L ϕ ( N − 1) z N − 1 g 2 ( z ) = L ϕ  r s  + L ϕ  1 + r s  z + · · · + L ϕ  N − 1 + r s  z N − 1 . . . g p ( z ) = L ϕ  ( p − 1) r s  + L ϕ  1 + ( p − 1) r s  z + · · · + L ϕ  N − 1 + ( p − 1) r s  z N − 1 (11) g p +1 ( z ) = L ϕ  p r s − 1  z − 1 + · · · + L ϕ  N − 2 + p r s  z N − 2 . . . g s ( z ) = L ϕ  ( s − 1) r s − r + 1  z − ( r − 1) + · · · + L ϕ  N − r + 2 + ( s − 1) r s  z N − r +2 . The p olynomial g 1 ( z ) has at most N − 1 nonzero terms; the r est of p olynomials g j ( z ), 2 ≤ j ≤ s , ha ve at most N nonzero terms. In w hat follo ws, w e use the new matrix G ( z ) = G ( z ) U ( z ), w here U ( z ) = d iag  z ( r − 1) , ( W z ) ( r − 1) , ( W 2 z ) ( r − 1) , . . . , ( W r − 1 z ) ( r − 1)  . Th us, all th e en tries of the p olynomial matrix G ( z ) are algebraic p olynomials in z and, moreo ve r w e ha ve rank G ( z ) = rank G ( z ) for all z ∈ C \ { 0 } . W e denote b y e g j ( z ) the algebraic p olynomial z r − 1 g j ( z ), 1 ≤ j ≤ s . The strategy is to reduce the p olynomial matrix G ( z ) into another simpler one having the same rank for all z ∈ C \ { 0 } . Lemma 1 Consider the matrix b G ( z ) = [ b G 0 ( z ) b G 2 ( z ) . . . b G ( r − 1) ( z )] , wher e b G j ( z ) , 0 ≤ j ≤ ( r − 1) , denotes the c olumn ve ctor c onsisting of the r - harmonics of or der j of the p olynomials e g i ( z ) wher e 1 ≤ i ≤ s . Then G ( z ) = b G ( z )Ω r wher e Ω r denotes the F ourier matrix of or der r . 7 Pro of: F or eac h i = 1 , 2 , . . . , s we hav e that e g i ( z ) = P r − 1 j =0 b e g ij ( z ) w here b ˜ g ij ( z ) d enotes th e r -harmonic of order j of e g i . W e can write the matrix G ( z ) as G ( z ) =  b G 0 ( z ) + b G 1 ( z ) + · · · + b G r − 1 ( z ) b G 0 ( z ) + W b G 1 ( z ) + · · · + W r − 1 b G r − 1 ( z ) · · · · · · b G 0 ( z ) + W r − 1 b G 1 ( z ) + · · · + W ( r − 1) 2 b G r − 1 ( z )  Hence, in matrix form w e hav e G ( z ) =  b G 0 ( z ) b G 1 ( z ) . . . b G r − 1 ( z )  Ω r = b G ( z )Ω r where Ω r =        1 1 1 · · · 1 1 W W 2 · · · W r − 1 1 W 2 W 4 · · · W 2( r − 1) . . . . . . 1 W r − 1 W 2( r − 1) · · · W ( r − 1) 2        is the F ourier matrix of ord er r .  Observe that rank G ( z ) = rank b G ( z ) for all z ∈ C \ { 0 } . In what follo ws, w e assume that supp L ϕ ⊆ [0 , N ] and, in addition, w e also assume that N ≤ r . In this case, ha ving in mind the num b er of nonzero consecutiv e terms of the p olynomial e g j ( z ), we conclude th at the r -harm onic of order q , q = 0 , 1 . . . , r − 1, of the p olynomial e g i ( z ), 1 ≤ i ≤ s , is a monomial having the form c ip z k r + q where c iq ∈ C and k ∈ { 0 , 1 } . T his c hoice of r and, consequent ly , of th e sampling p erio ds T = r /s , r , s ∈ N and s > r , simplifies the str u cture of the matrix b G ( z ). First, let u s to give an illustrativ e example: Con s ider N = 3, r = 4 and s = 5; here T = 4 / 5, p = 2 and the p olynomials e g j ( z ) read ˜ g 1 ( z ) = ⋆z 4 + ⋆z 5 ˜ g 2 ( z ) = ⋆z 3 + ⋆z 4 + ⋆z 5 ˜ g 3 ( z ) = ⋆z 2 + ⋆z 3 + ⋆z 4 ˜ g 4 ( z ) = ⋆z + ⋆z 2 + ⋆z 3 ˜ g 5 ( z ) = ⋆ + ⋆z + ⋆z 2 Hence, the matrix b G ( z ) reads b G ( z ) =       ⋆z 4 ⋆z 5 0 0 ⋆z 4 ⋆z 5 0 ⋆ z 3 ⋆z 4 0 ⋆z 2 ⋆z 3 0 ⋆z ⋆z 2 ⋆z 3 ⋆ ⋆z ⋆z 2 0       (12) This example shows that the 3rd an d 4th columns ha ve the form z 2 C and z 3 C ′ where C, C ′ ∈ C s × 1 . Th e fir st and second columns do not share this prop erty . If w e righ t 8 m u ltiply the matrix b G ( z ) by diag[1 , z − 1 , z − 2 , z − 3 ], w e get the n ew matrix e G ( z ) :=       ⋆z 4 ⋆z 5 0 0 ⋆z 4 ⋆z 5 0 ⋆ z 3 ⋆z 4 0 ⋆z 2 ⋆z 3 0 ⋆z ⋆z 2 ⋆z 3 ⋆ ⋆z ⋆z 2 0           1 z − 1 z − 2 z − 3     =       ⋆z 4 ⋆z 4 0 0 ⋆z 4 ⋆z 4 0 ⋆ ⋆z 4 0 ⋆ ⋆ 0 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ 0       No w w e can go into th e general case for th e matrix b G ( z ). Ha vin g in mind Eqs.(11) and that e g i ( z ) = z r − 1 g i ( z ) w e obtain: max  grad e g j : 1 ≤ j ≤ s  = ( N − 1) + ( r − 1) = N + r − 2 < 2 r . Hence, the matrix b G ( z ) has the form b G ( z ) =    c 11 z k 11 r c 12 z k 12 r +1 · · · c 1 r z k 1 r r +( r − 1) . . . . . . . . . . . . c s 1 z k s 1 r c s 2 z k s 2 r +1 · · · c sr z k sr r +( r − 1)    where the coefficien ts k ij ∈ { 0 , 1 } . W e can easily obtain th e follo w ing result: Lemma 2 A ssume that N > 1 . Then, for e ach 1 ≤ j ≤ N − 1 ther e exist indic es i ′ 6 = i , 1 ≤ i, i ′ ≤ s , such that k ij 6 = k i ′ j . Otherwise, for e ach N ≤ j ≤ r it holds that k ij = k i ′ j for al l 1 ≤ i, i ′ ≤ s . Assume that N > 1 and recall that N ≤ r . T he en tries of the j th column of the m atrix b G ( z ), where N ≤ j ≤ r , h a ve the form ⋆z j − 1 ( ⋆ ∈ C ); they could h a ve the form ⋆z j − 1 or ⋆z r +( j − 1) whenev er 1 ≤ j ≤ N − 1. Dividing the j th col umn by z j − 1 , ob viously w e obtain a matrix with the same r ank than b G ( z ) for an y z ∈ C \ { 0 } . Thus, w e int ro duce the new p olynomial matrix e G ( z ): e G ( z ) := b G ( z ) Q ( z ) =  M ( z ) G  , where G ∈ C s × ( r − N +1) denotes an scalar matrix and Q ( z ) := diag[1 , z − 1 , . . . , z 1 − r ]. Wh en- ev er rank G < r − N + 1, we ha ve that rank e G ( z ) = rank b G ( z ) < r for all z ∈ C \ { 0 } and, hence, there is no a p olynomial left inv erse for b G ( z ). In the case rank G ( z ) = r − N + 1, there exists an inv ertible matrix R ∈ C s × s suc h that R G =  G ′ 0  , where G ′ ∈ C ( r − N +1) × ( r − N +1) is in v er tib le. T h u s, R e G ( z ) =  R M ( z ) R G  =  M 1 ( z ) G ′ M 2 ( z ) 0  The entries of the p olynomial matrix M ( z ) ∈ C s × ( N − 1) are of the form ⋆z r or constan ts; denoting λ = z r , the matrices M i ( z ), i = 1 , 2, can b e expressed as M i ( λ ) = M i 1 − λM i 2 donde M 1 i ∈ C ( r − N +1) × ( N − 1) y M 2 i ∈ C ( s − r + N − 1) × ( N − 1) . As a consequence, w e ha ve the follo wing resu lt: 9 Lemma 3 A ssume that rank G = r − N + 1 . Then, rank G ( z ) = r for al l z ∈ C \ { 0 } if and only if rank M 2 ( λ ) = N − 1 for al l λ ∈ C \ { 0 } . Next step is to c haracterize w hen the rank of th e matrix M 21 − λM 22 equals N − 1 for an y λ ∈ C \ { 0 } . T o this end, w e use the Kronec k er canonical form (K C F h erafter) of the matrix p encil M 2 ( λ ) (see the Ap p endix 6.2 for the details). By using the block structure notation A ⊕ B := diag ( A, B ), consider the K CF of the matrix p encil M 2 ( λ ), i.e., K ( λ ) := S r ig ht M 2 ( λ ) ⊕ J M 2 ( λ ) ⊕ N M 2 ( λ ) ⊕ S lef t M 2 ( λ ) where S r ig ht M 2 ( λ ) denotes the righ t singular p art of M 2 ( λ ), S lef t M 2 ( λ ) denotes the left singular part, J M 2 ( λ ) is the blo ck asso ciated with the fi nite eigen v alues of the p encil and, fin ally , N M 2 ( λ ) is the blo ck asso ciated w ith the in finite eigen v alue. Ha ving in mind the stru ctur e of the d ifferen t b lo cks app earing in the KCF of the m atrix p encil M 2 ( λ ), we can deriv e that the rank of K ( λ ), and consequen tly of M 2 ( λ ), is N − 1 for all λ ∈ C \ { 0 } if and only if K ( λ ) h as n ot righ t singular part and the only p ossibly fin ite eig en v alue is the zero one. In fact, we h a ve the follo wing resu lt: Lemma 4 The r ank of matrix M 2 ( λ ) is N − 1 for e ach λ ∈ C \ { 0 } if and only if the fol lowing c onditions hold: 1. The KCF of the matrix p encil M 2 ( λ ) has not right singular p art and, 2. If µ is a finite eigenvalue of the matrix p encil M 2 ( λ ) , then µ = 0 . No w, Lemma 4 allo ws us to decide when the rank of our initial p olynomial matrix G ( z ) is r for all z ∈ C \ { 0 } . Let us to remind all the giv en steps in reducing the initial p olynomial matrix G ( z ): G ( z ) G ( z ) b G ( z ) e G ( z )  M 1 ( z ) G ′ M 2 ( z ) 0  , where 1. G ( z ) = G ( z ) U ( z ), 2. b G ( z )Ω r = G ( z ), 3. e G ( z ) = b G ( z ) Q ( z ) = [ M ( z ) |G ], where G ∈ C s × ( r − N +1) and Q ( z ) = diag [1 , z − 1 , . . . , z 1 − r ], 4. If rank G = r − N + 1, there exists R ∈ C s × s in vertible s u c h that R e G ( z ) =  M 1 ( z ) G ′ M 2 ( z ) 0  where the matrix G ′ ∈ C ( r − N +1) × ( r − N +1) is in vertible, 5. The matrices M i ( z ), i = 1 , 2, can b e expressed as M i ( λ ) = M i 1 − λM i 2 with λ = z r . As a consequence, we hav e pro ved th e follo wing r esu lt: Theorem 2 A ssume th at supp L ϕ ⊆ [0 , N ] , wher e N ∈ N with N > 1 , and take N ≤ r < s . L et G ( z ) b e the c orr esp onding s × r p olynomial matrix give n in (6) . Then, rank G ( z ) = r for any z ∈ C \ { 0 } if and only if the fol lowing statements hold: 10 1. rank G = r − N + 1 and, 2. the KCF of the matrix p encil M 2 ( λ ) has not right singular p art, and the only p ossible finite eig e nvalue is µ = 0 . F or practical pu rp oses it is not necessary to compu te th e K CF of th e matrix p encil M 2 ( λ ) (if p ossible). T he n eeded information ab out M 2 ( λ ) is obtained from its GUPTRI form (Generalized UPer TRIangular form). See the App endix 6.3 for the details. As the matrix e G ( z ) dep ends on z r , in what follo ws w e iden tify the matrix e G ( z ) with e G ( λ ) w here λ = z r . 3.1 A t o y mo del in volving the quadratic B -spline The follo wing example illustrates the result giv en in Th eorem 2. Consider as generator ϕ the quadratic B-spline N 3 ( x ), i.e., N 3 ( x ) = x 2 2 χ [0 , 1) + ( − 3 2 + 3 x − x 2 ) χ [1 , 2) + 1 3 (3 − x ) 2 χ [2 , 3) , where χ [ a,b ) denotes the c h aracteristic fun ction of the in terv al [ a, b ). In this case, for th e iden tity s ystem L f = f for all f ∈ V ϕ w e h a ve supp L ϕ ⊆ [0 , 3], i.e., N = 3. T aking the sampling p erio d T = 4 / 5, i.e., r = 4 and s = 5, th e Laurent p olynomials g i ( z ) giv en by (11) read: g 1 ( z ) = 1 2 z + 1 2 z 2 g 2 ( z ) = 8 25 + 33 50 z + 1 50 z 2 g 3 ( z ) = 9 50 z − 1 + 37 50 + 2 25 z g 4 ( z ) = 2 25 z − 2 + 37 50 z − 1 + 9 50 g 5 ( z ) = 1 50 z − 3 + 33 50 z − 2 + 8 25 z − 1 F ollo wing the ab o ve steps we obtain b G ( z ) =         1 2 z 4 1 2 z 5 0 0 33 50 z 4 1 50 z 5 0 8 25 z 3 2 25 z 4 0 9 50 z 2 37 50 z 3 0 2 25 z 37 50 z 2 9 50 z 3 1 50 33 50 z 8 25 z 2 0         Righ t multiplic ation by the matrix diag[1 , z − 1 , z − 2 , z − 3 ] giv es: e G ( λ ) = [ M ( λ ) | G ] =         1 2 λ 1 2 λ 0 0 33 50 λ 1 50 λ 0 8 25 2 25 λ 0 9 50 37 50 0 2 25 37 50 9 50 1 50 33 50 8 25 0         11 where λ = z 4 . Th e matrix G ∈ C 5 × 2 has rank 2; p erformin g some elemen tary op erations on the r o ws of G we obta in G ′ =         9 50 37 50 0 8 25 0 0 0 0 0 0         =         0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 37 9 − 161 18 0 1 0 16 9 − 37 9 0 0 1                 0 0 0 8 25 9 50 37 50 37 50 9 50 8 25 0         = R G Therefore, R e G ( λ ) = [ R M ( λ ) R G ] =  M 1 ( λ ) G ′ M 2 ( λ ) 0  where M 2 ( λ ) =    1 2 λ 1 2 λ 5017 900 λ 2 25 + 161 900 λ 1 50 + 1157 450 λ 33 50 + 37 450 λ    In this case, a direct compu tation give s K M 2 ( λ ) = L ⊤ 2 ( λ ). As a consequence, Theorem 2 ensures that the corresp on d ing p olynomial matrix G ( z ) p ossess a p olynomial left in v erse. Next w e deal with the pr oblem of computing a p olynomial left inv erse of G ( z ) wheneve r it do es exist. 4 Computing a p olynomial left in v erse of the matrix G ( z ) First note that if we compute a p olynomial left inv erse of the matrix e G ( λ ) then we obtain a p olynomial left inv erse of the matrix G ( z ). Indeed, remind that e G ( z ) = G ( z ) U ( z )Ω − 1 r Q ( z ) where U ( z ) = diag[ z r − 1 , ( W z ) r − 1 , . . . , ( W r − 1 z ) r − 1 ], Ω r is the F ourier matrix of order r , and Q ( z ) = diag[1 , z − 1 , . . . , z 1 − r ]. Th u s, if L ( z ) is a p olynomial left in v erse of the matrix e G ( z ), then the matrix L G ( z ) = diag [1 , z , . . . , z r − 1 ]Ω r diag[ z 1 − r , ( W z ) 1 − r , . . . , ( W r − 1 z ) 1 − r ] L ( z ) will b e a p olynomial left inv erse of the matrix G ( z ). Hence, we concentrate ourselves in computing a p olynomial left inv erse of the matrix e G ( z ). T o this end, consider e G ( λ ) = A ⊤ − λ B ⊤ ( λ = z r ); b eing L ( λ ) a p olynomial left inv erse of the matrix e G ( λ ), we h a ve ( A − λ B ) L ⊤ ( λ ) = I r . Let us denote L ( λ ) := L ⊤ ( λ ). As w e are searching for s × r matrices L ( λ ), wh ose en tries are p olynomials, suc h that ( A − λ B ) L ( λ ) = I r w e can use the f ollo wing notation: L ( λ ) =  L 1 ( λ ) L 2 ( λ ) . . . L r ( λ )  , i.e., L i ( λ ) denotes th e i th column of L ( λ ) , L i ( λ ) = ℓ 0 i + ℓ 1 i λ + · · · + ℓ ν i λ ν , i = 1 , 2 , . . . , r , where ℓ k i ∈ C s , k = 0 , 1 , . . . , ν . As a consequence, ( A − λ B ) L ( λ ) = I r is equiv alen t to A ℓ 0 i + ( A ℓ 1 i − B ℓ 0 i ) λ + · · · + ( A ℓ ν i − B ℓ ν − 1 i ) λ ν − B ℓ ν i λ ν + 1 = I i r , i = 1 , 2 , . . . , r , (1 3) 12 where I i r denotes th e i th column of the iden tity matrix I r . Equ ating co efficien ts, for eac h i = 1 , 2 , . . . , r , we obtai n the set of linear equations A ℓ 0 i = I i r , A ℓ 1 i − B ℓ 0 i = 0 , . . . , A ℓ ν i − B ℓ ν − 1 i = 0 , −B ℓ ν i = 0 , or in matrix form          −B A −B A −B . . . A −B A               ℓ ν i ℓ ν − 1 i . . . ℓ 0 i      =           0 . . . 0 . . . 0 I i r           , i = 1 , 2 , . . . , r , (14) where the resu lting blo c k matrix has order ( ν + 2) r × ( ν + 1) s . Th e goal is to find ν ∈ N su c h that th e ab o ve r linear systems b ecome consistent. Next, we come b ac k to the example in Section 3.1. The e xample revisited: Consider again the example inv olving the quadr atic B-spline giv en in Section 3.1. In th is case, e G ( z ) = G ( z ) U ( z )Ω − 1 4 diag[1 , z − 1 , z − 2 , z − 3 ] and, taking λ = z 4 w e hav e e G ( λ ) =         1 2 λ 1 2 λ 0 0 33 50 λ 1 50 λ 0 8 25 2 25 λ 0 9 50 37 50 0 2 25 37 50 9 50 1 50 33 50 8 25 0         = A ⊤ − λ B ⊤ , where A =      0 0 0 0 1 50 0 0 0 2 25 33 50 0 0 9 50 37 50 8 50 0 8 25 37 50 9 50 0      y B =      − 1 2 − 33 50 − 2 25 0 0 − 1 2 − 1 50 0 0 0 0 0 0 0 0 0 0 0 0 0      . Here, the matrix S =  −B A −B A  of size 12 × 10 has rank 10. C h o osing the columns of L ( λ ) as L i ( λ ) = ℓ 0 i + ℓ 1 i λ ∈ C 5 × 1 , th e linear s ystems   −B A − B A    ℓ 1 i ℓ 0 i  =   0 0 I i 4   , i = 1 , 2 , 3 , 4 (15) ha ve a un ique solution. Observe that deleting the trivial equations 3 and 4, w e hav e 13 consisten t squ are sy s tems. By usin g Matlab TM w e obtain the left inv erse L ( λ ) = 10 3       4 . 4812 − 0 . 143 8 0 . 016 6 − 0 . 0043 − 3 . 4840 0 . 1118 − 0 . 012 8 0 . 003 1 1 . 6069 − 0 . 051 4 0 . 005 6 0 . 000 0 − 0 . 4125 0 . 0125 0 . 0000 − 0 . 0000 0 . 0500 0 . 0000 − 0 . 0000 0 . 0000       + + 10 3       − 0 . 0021 0 . 0001 − 0 . 000 0 0 . 000 0 0 . 0517 − 0 . 001 7 0 . 000 2 − 0 . 0000 − 0 . 4133 0 . 0133 − 0 . 001 5 0 . 000 4 1 . 6071 − 0 . 051 6 0 . 005 9 − 0 . 0015 − 3 . 4841 0 . 1118 − 0 . 012 9 0 . 003 3       λ A t this p oin t, the c hallenge problem is to giv e conditions on the matrix p encil A ⊤ − λ B ⊤ in ord er to obtain a left inv erse w ith p olynomial en tries (ha ving nonegativ e p o wers) by solving th e corresp ond ing linear systems (15). The answ er to this question is based on the KCF of the matrix p encil A ⊤ − λ B ⊤ . In our example the corresp onding KCF is N 1 ( λ ) ⊕ N 1 ( λ ) ⊕ L ⊤ 2 ( λ ), i.e., the p encil has not fi nite eigen v alues, all the blo cks asso ciated with the infin ite eigen v alue ha v e order 1, and the left singular part h as a u nique blo c k. In what follo ws, we pr o ve that these conditions for the KCF of the matrix p encil e G ( λ ) are sufficien t to giv e a p ositiv e answer to the raised problem in a very imp ortant p articular case: 4.1 The case where the ov ersampling rate is minim um for a fixed r ≥ N It corresp onds to the case wh ere N ≤ r and s = r + 1, i.e., the sampling p erio d is T = r / ( r + 1). Here, the matrix p encil e G ( λ ) = A ⊤ − λ B ⊤ has the f orm               0 · · · 0 0 · · · 0 0 0 0 · · · 0 0 · · · 0 0 ∗ 0 · · · 0 0 · · · 0 ∗ ∗ . . . . . . . . . . . . . . . . . . . . . 0 · · · 0 0 ∗ · · · ∗ ∗ 0 · · · 0 ∗ ∗ · · · ∗ ∗ 0 · · · ∗ ∗ ∗ · · · ∗ 0 . . . . . . . . . . . . . . . . . . . . . ∗ · · · ∗ ∗ ∗ · · · 0 0               − λ                ∗ · · · ∗ ∗ 0 · · · 0 ∗ · · · ∗ ∗ 0 · · · 0 ∗ . . . ∗ 0 0 · · · 0 . . . . . . . . . . . . . . . . . . ∗ . . . 0 0 0 · · · 0 0 · · · 0 0 0 · · · 0 0 · · · 0 0 0 · · · 0 . . . . . . . . . . . . . . . 0 · · · 0 0 0 · · · 0                , (16) i.e., denoting the ent ries of A ⊤ and B ⊤ b y A ⊤ ij and B ⊤ ij resp ectiv ely , we hav e A ⊤ ij = 0 if 2 + r < i + j < r + N + 1, B ⊤ 1 N = 0 and B ⊤ ij = 0 if i + j > N + 1. Ha ving in mind the structure of the m atrices A ⊤ and B ⊤ w e ha ve rank( A ⊤ ) ≤ r , rank( B ⊤ ) ≤ N − 1 and rank(  −B A −B  ) ≤ r + N − 1. Whenev er these matrices hav e m axim um rank, the f ollo wing result holds: Theorem 3 A ssume that the singular matrix p enci l A ⊤ − λ B ⊤ of size ( r + 1) × r satisfies the fol lowing c onditions: 14 1. The p encil has not finite eigenvalues, 2. rank( A ⊤ ) = r , 3. rank( B ⊤ ) = N − 1 , with N ≤ r , and 4. rank(  −B A −B  ) = r + N − 1 . Then, the N r × ( N − 1)( r + 1) matrix G r :=          −B A −B A −B . . . A −B A          has r ank ( N − 1)( r + 1) . First note that rank( A ⊤ ) = r imp lies th at the KCF of the matrix p encil A ⊤ − λ B ⊤ has not right sin gular p art (and also that 0 is not an eigen v alue). Thus, by u sing Theorem 2, the p encil A ⊤ − λ B ⊤ has a p olynomial left in verse. Before to pro ve Theorem 3, and in order to ease its p r o of, w e fi r st obtai n, un der the theorem hyp otheses, the K CF of the matrix p encil A ⊤ − λ B ⊤ : Lemma 5 The KCF of the matrix p encil A ⊤ − λ B ⊤ is r − N +1 M i =1 N 1 ( λ ) ! ⊕ L ⊤ N − 1 ( λ ) . Pro of of Lemma 5: Since the matrix p encil has neither fin ite eigen v alues nor righ t singular part, w e conclude that its KCF has the form N ( λ ) ⊕ L lef t ( λ ), where N ( λ ) denotes the blo c ks asso ciated with the infi nite eigen v alue and L lef t ( λ ) denotes the left singular p art. Since r + 1 is the n u m b er of ro ws of the matrix p en cil, r the n umb er of col umns, and th e rank of B is N − 1 it cannot app ear blo c ks of the form L ⊤ i ( λ ) for i ≥ N . Each left sin gular blo c k increases in one the num b er of rows with resp ect to the num b er of columns; hence, as the size of A ⊤ − λ B ⊤ is ( r + 1) × r , it can app ear only one left singular b lo c k in its K C F. F urtherm ore, we prov e that this only left singular blo ck corresp ond s to L ⊤ N − 1 ( λ ). Indeed, let K ⊤ A − λ K ⊤ B b e the KCF of the matrix p en cil A ⊤ − λ B ⊤ . Ob vious ly , w e hav e that rank( A ⊤ ) = rank( K ⊤ A ) = r , rank( B ⊤ ) = rank( K ⊤ B ) = N − 1 and rank  −B A −B  = rank  −K B K A −K B  = r + N − 1 . The rank of the matrix h −K B K A −K B i coincides with its num b er of nonzero ro ws b ecause the n u m b er of null rows of K B is r − N + 1, i.e., the n um b er of b lo c ks in N ( λ ); th e m atrix K A has n ot null r o ws so th at, the num b er of n onzero ro ws of h −K B K A −K B i is 2 r − ( r − N + 1) = r + N − 1. Assume that in the KCF of the matrix p encil A ⊤ − λ B ⊤ app ears a s in gular blo c k L ⊤ i ( λ ) with i < N − 1. Since the ran k of B ⊤ is N − 1, the regular p art in th e K CF has a 15 blo c k of the f orm N l ( λ ) w ith l ≥ 2. By rearranging the blo c ks, we obtain that the KCF of A ⊤ − λ B ⊤ is N l ( λ ) ⊕ · · · ⊕ L ⊤ i ( λ ); therefore  −K B K A −K B  =                    0 0 · · · 0 0 0 0 · · · 0 0 − 1 0 · · · 0 0 0 0 · · · 0 0 ∗ ∗ · · · ∗ ∗ 0 0 · · · 0 0 . . . . . . . . . . . . . . . . . . . . . . . . ∗ ∗ · · · ∗ ∗ 0 0 · · · 0 0 1 0 · · · 0 0 0 0 · · · 0 0 0 1 · · · 0 0 − 1 0 · · · 0 0 ∗ ∗ · · · ∗ ∗ ∗ ∗ · · · ∗ ∗ . . . . . . . . . . . . . . . . . . . . . . . . ∗ ∗ · · · ∗ ∗ ∗ ∗ · · · ∗ ∗                    In this case , the rank of h −K B K A −K B i is strictly sm aller than r + N − 1 b ecause the second ro w and the ( r + 1)th row are linearly dep end en t. This cont radicts the hyp otheses and , hence, the only left singular blo ck is L ⊤ N − 1 ( λ ). Ha ving in mind that rank( B ⊤ ) = N − 1, w e conclude that the KCF of the matrix p encil A ⊤ − λ B ⊤ is r − N +1 M i =1 N 1 ( λ ) ! ⊕ L ⊤ N − 1 ( λ ).  Pro of of Theorem 3: Once we ha v e determined the KCF of the matrix p en cil A ⊤ − λ B ⊤ w e compute the rank of the matrix G r . If K A − λ K B is the KCF of the matrix p encil A − λ B , it is obvious th at rank( G r ) = rank          −K B K A −K B K A −K B . . . K A −K B K A          As K ⊤ A − λ K ⊤ B is the KCF of the matrix p encil A ⊤ − λ B ⊤ , Lemma 5 giv es K ⊤ A =  I 0 0 L ⊤ A  , K ⊤ B =  0 0 0 L ⊤ B  where I = I ( r − N +1) denotes the iden tit y matrix of ord er r − N + 1, and L ⊤ A =        0 0 · · · 0 1 0 · · · 0 . . . . . . . . . . . . 0 0 · · · 0 0 0 · · · 1        ∈ C N × ( N − 1) ; L ⊤ B =        1 0 · · · 0 0 1 · · · 0 . . . . . . . . . . . . 0 0 · · · 1 0 0 · · · 0        ∈ C N × ( N − 1) 16 As a consequence, rank( G r ) = rank                     0 0 0 − L B I 0 0 0 0 L A 0 − L B I 0 0 0 0 L A 0 − L B . . . . . . I 0 0 0 0 L A 0 − L B I 0 0 L A                     A suitable in terc h ange of r ows and columns giv es rank( G r ) = rank                 0 · · · 0 0 0 I . . . I − L B 0 L A − L B . . . L A − L B 0 L A                 where the fi rst r − N + 1 = r − r ank( K B ) are null ro ws ; hence, the rank of G r equals ( N − 1)( r + 1) if and only if the remaining ( N − 1)( r + 1) ro ws are linearly indep endent. This is equiv alen t to the matrix L A , B =        − L B 0 L A − L B . . . L A − L B 0 L A        ∈ C N ( N − 1) × N ( N − 1) has fu ll rank. T o pro v e it, we u se the follo wing result in [11, p. 32]: Let x ( λ ) b e a nonzero v ector having the form x ( λ ) = x 0 + λ x 1 + λ 2 x 2 + · · · + λ ε x ε , x i ∈ C N × 1 suc h th at ( L A − λL B ) x ( λ ) = 0 . Then, necessarily , ε ≥ N − 1. No w, let us con tinue by con tradiction, and assume that the matrix L A , B has n ot fu ll rank. Then, there exists a nonzero v ector z ∈ C N ( N − 1) × 1 suc h th at L A , B z = 0 . Denoting z ⊤ = [ z ⊤ N − 2 . . . z ⊤ 1 z ⊤ 0 ] where z i ∈ C N × 1 , we obtai n that ( L A − λL B )( z 0 + λ z 1 + λ z 2 + · · · + λ N − 2 z N − 2 ) = 0 , 17 whic h con tradicts the minimal pr op ert y for N − 1. Th erefore, th e matrix L A , B has full rank and, fi nally , rank G r = ( N − 1)( r + 1).  Remark: Note that Th eorem 3 remains v alid for any singular matrix p encil A ⊤ − λ B ⊤ of size ( r + 1) × r su bstituting N − 1 b y p ∈ N which satisfies 0 < p < r . Consider the matrix p encil e G ( λ ) = A ⊤ − λ B ⊤ of size ( r + 1) × r with N ≤ r . Assu ming that the e G ( λ ) has p olynomial left in verses, the follo w ing resu lt giv es sufficient cond itions for computing one of such p olynomial left inv erses. Once w e ha ve got one solution, it is straigh tforward to derive the remaining solutions. Corollary 1 (Computing a p olynomial lef t inv erse of e G ( λ ) ) L et e G ( λ ) = A ⊤ − λ B ⊤ b e a singular matrix p encil of size ( r + 1) × r with N ≤ r . Assume that e G ( λ ) admits p olynomial lef t inverses, and that the fol lowing c onditions hold: 1. rank( A ⊤ ) = r , 2. rank( B ⊤ ) = N − 1 , and 3. rank(  −B A −B  ) = r + N − 1 Then, the line ar systems          −B A −B A −B . . . A −B A               ℓ N − 2 i ℓ N − 3 i . . . ℓ 0 i      =           0 . . . 0 . . . 0 I i r           , i = 1 , 2 , . . . , r , (17) wher e I i r denotes the i th c olumn of the identity matr ix I r , admit a unique solution. M or e- over, let [ ℓ N − 2 i ℓ N − 3 i . . . ℓ 0 i ] ⊤ ∈ C ( N − 1)( r +1) b e this solution for i = 1 , 2 , . . . , r , and c onsider the p olynomial ve ctor L i ( λ ) = ℓ 0 i + ℓ 1 i λ + · · · + ℓ N − 2 i λ N − 2 , i = 1 , 2 , . . . , r . Then, the ( r + 1) × r p olynomial matrix L ( λ ) := [ L 1 ( λ ) L 2 ( λ ) . . . L r ( λ )] satisfies L ⊤ ( λ ) e G ( λ ) = I r Pro of: Theorem 3 implies that the rank of the co efficien t matrix G r ∈ C N r × ( N − 1)( r +1) is ( N − 1)( r + 1) in (17). Having in mind (16), the last r − N + 1 rows of B are n u ll. Deleting these ro w s in the first ro w blo c k (which b ecome trivial equations in (17)), we obtain an s quare inv er tib le m atrix, and consequen tly (17) has a u nique solution f or eac h i = 1 , 2 , . . . , r . Recalling (14), w e finally obtain that L ⊤ ( λ ) is a p olynomial left in v er s e of e G ( λ ).  Observe that an y other p olynomial left in v erse A ( λ ) of the matrix e G ( λ ) is give n by A ( λ ) = L ⊤ ( λ ) + B ( λ )  I r +1 − e G ( λ ) L ⊤ ( λ )  , 18 where B ( λ ) is an arb itrary r × ( r + 1) p olynomial matrix. F or the matrix p encil e G ( λ ) = A ⊤ − λ B ⊤ of size ( r + 1) × r with N ≤ r , it is easy to giv e suffi cien t conditions in order to satisfy the cond itions 1-3 in Corollary 1. Namely: Corollary 2 Consider the singular matrix p encil e G ( λ ) = A ⊤ − λ B ⊤ of size ( r + 1) × r with N ≤ r . Denoting A ⊤ = [ A ⊤ ij ] and B ⊤ = [ B ⊤ ij ] , assume that the fol lowing c onditions hold: A ⊤ ij 6 = 0 if i + j = r + 2 or i + j = r + N + 1 (18) B ⊤ ij 6 = 0 if i + j = N + 1 and i ≥ 2 . (19) Then the c onditions 1-3 in Cor ol lary 1 ar e satisfie d. Pro of: Conditions (18) and (19) sa y that the en tr ies mark ed as • in the matrices b elo w are nonzero A =                     0 . . . . . . . . . . 0 0 · · · · · · 0 · · · 0 • 0 . . . . . . . . . . 0 0 · · · · · · 0 · · · • ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 . . . . . . . . . . 0 0 • · · · ∗ · · · ∗ ∗ 0 . . . . . . . . . . 0 • ∗ · · · ∗ · · · ∗ ∗ 0 . . . . . . 0 • ∗ ∗ · · · ∗ · · · ∗ • 0 . . . . . . • ∗ ∗ ∗ · · · ∗ · · · • 0 . . . . . . . . . . . . . . . . . . 0 • · · · ∗ ∗ ∗ · · · · · · • · · · 0 0                     =  A 11 A 12 A 21 A 22  ∈ C r × ( r +1) B =            ∗ ∗ · · · ∗ • 0 · · · 0 ∗ ∗ · · · • 0 0 · · · 0 . . . . . . . . . . . . . . . . . . . . . ∗ • · · · 0 0 0 · · · 0 0 0 · · · 0 0 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 0 · · · 0 0 0 · · · 0            =  B 11 0 0 0  ∈ C r × ( r +1) where A 22 ∈ C ( r − N +1) × ( r − N +1) and B 11 ∈ C ( N − 1) × N . T rivially , r an k ( A ⊤ ) = r and rank( B ⊤ ) = N − 1. Cond ition 3 comes by observing the form of th e m atrix  −B A −B  . In terc hanging r o ws and columns w e obtain that the m atrix  −B A −B  has the same rank than the m atrix     0 0 0 0 B 11 0 0 0 A 11 B 11 A 12 0 A 21 0 A 22 0     19 Since the matrix A 22 ∈ C ( r − N +1) × ( r − N +1) is in vertible, elemen tary ro w op erations give the new matrix     0 0 0 0 B 11 0 0 0 e A 11 B 11 0 0 A 21 0 A 22 0     . Finally , the ab o ve m atrix has rank 2( N − 1) + r − N + 1 = r + N − 1.  Remark that condition 1 in Theorem 3 can b e c hec k ed by using the algorithm guptri . In case that conditions 1-3 in Corollary 1 are satisfied, we could chec k directly the consistency of the linear systems (17) ; if they are not consisten t, w e deriv e that the p encil e G ( λ ) has not p olynomial left inv erses. 5 Conclusion Consider the problem of the reco very of any fun ction f in a shift-inv arian t space V ϕ from the sequence of samples {  L f  ( r n/s ) } n ∈ Z of its filtered version L f , where the p ositive in tegers r and s satisfy s > r , i.e., the o v ersamp ling setting. The existence of compactly supp orted reconstruction fu nctions for this s amp ling problem is intimatel y r elated to th e existence of a p olynomial left in v erse for a p olynomial m atrix G ( z ) asso ciated with th e sampling problem. This is equiv alen t to that the matrix G ( z ) h as fu ll rank for any z ∈ C \ { 0 } . Other c h aracterizat ions can b e f ound in the m athematical literature inv olving the S m ith canonical f orm of the p olynomial matrix G ( z ) or the Euclides algorithm f or the minors of ord er r in G ( z ). Un fortunately , w henev er the parameter r is large, the aforesaid metho ds are u seless from a practical p oint of view. In this w ork , b y assumin g that N ≤ r < s , where L ϕ ⊆ [0 , N ], we d eriv e a new c haracterizatio n for the existence of p olynomial left inv erses whic h in vo lv es the K ronec ker canonical form of a singular matrix p encil. T he adv anta ge, f rom a practical p oin t of view, of this n ew metho d is th at we can retrieve the needed inform ation fr om the KCF by using the so-called guptri algorithm. F urtherm ore, in the imp ortan t case w here s = r + 1, i.e., the ov ersampling rate is m inim u m (for a fixed r ), we pr op ose a metho d for the computation of a p olynomial left inv erse (and hence, the whole set of p olynomial left inv erses) of G ( z ). 6 App endix F or the sak e of completeness w e include here a br ief remin d er on the canonical form s alluded throughout th e pap er. 6.1 Smith c anonical form of a p olynomial matrix Recall that any m × n ( m ≥ n ) p olynomial matrix H ( z ) with rank H ( z ) = r (recall that the rank of a p olynomial matrix is the order of its largest minor that is not equal to the zero p olynomial) can b e written as the pro du ct H ( z ) = V ( z ) S ( z ) W ( z ) w here V ( z ) an W ( z ) are u nimo dular matrices (i.e., the determinants of V ( z ) and W ( z ) are nonzero constan ts) of size m × m and n × n resp ectiv ely , and S ( z ) is a diagonal m × n p olynomial 20 matrix S ( z ) := diag[ i 1 ( z ) , . . . , i r ( z ) , 0 , . . . , 0]. Moreo v er, the diagonal entries (the so-called in v ariant p olynomials of H ( z )) are giv en by i j ( z ) = d j ( z ) /d j − 1 ( z ), j = 1 , 2 , . . . , r , wh ere d j ( z ) is the greatest common d ivisor of all minors of H ( z ) of order j , j = 1 , 2 , . . . , r , and d 0 ( z ) ≡ 1. The matrix S ( z ) is called the Smith canonical form of the matrix H ( z ). S ee [15] for the details. 6.2 Kronec ker canonical form of a matrix p encil The K ronec ker canonical form (KCF) for matrix p encils H ( λ ) = A − λB , A, B ∈ C m × n , is a generalization of the Jordan canonical form to matrix p encils (see [11]): T h ere exist t wo nonsingular matrices U ∈ C m × m and V ∈ C n × n suc h that (in block stru cture notat ion): U ( A − λB ) V − 1 = S r ig ht H ( λ ) ⊕ J H ( λ ) ⊕ N H ( λ ) ⊕ S lef t H ( λ ) , where J H ( λ ) ⊕ N H ( λ ) is the r egular part of the matrix p encil, S r ig ht H ( λ ) is its righ t singular part, and S lef t H ( λ ) its left singular part. The b lo c k J H ( λ ) is asso ciated with the finite eigen v alues of the m atrix p encil, and it reads: J H ( λ ) = J l 1 ( µ 1 ) ⊕ · · · ⊕ J l g q ( µ q ) where J lj ( µ i ) is a l j × l j Jordan block asso ciate with the finite eigen v alue µ i , i.e., J l i ( µ i ) =       µ i 1 . . . . . . . . . 1 µ i       − λ       1 0 . . . . . . . . . 0 1       Recall that µ is a fin ite eigen v alue of the matrix p encil H ( λ ) if rank H ( µ ) < rank H ( λ ), b eing rank H ( λ ) the order of the largest minor that is n ot equal to the zero p olynomial. The b lo c k N H ( λ ) is asso ciated with the infi nite eigenv alue (if do es exist) and it reads: N H ( λ ) = N p 1 ( λ ) ⊕ · · · ⊕ N p g ∞ ( λ ) where N p i ∈ C p i × p i is the matrix N p i ( λ ) =       1 0 . . . . . . . . . 0 1       − λ       0 1 . . . . . . . . . 1 0       and g ∞ denotes th e geometric m ultiplicit y of the infinite eigenv alue whic h corresp onds to the num b er of Jordan b lo c ks f or the in finite eigenv alue. Recall that the p encil H ( λ ) has the infinite eig en v alue if its dual p encil H ♯ ( λ ) := λ H (1 /λ ) has the zero eigen v alue. If m 6 = n or det( A − λB ) = 0 for all λ ∈ C , then the matrix p encil also in cludes a singular part, S r ig ht H ( λ ) and/or S lef t H ( λ ), and we say th at th e matrix p encil is singular. F or the righ t singular part, we ha v e S r ig ht H ( λ ) = L ε 1 ( λ ) ⊕ · · · ⊕ L ε r 0 ( λ ) 21 where L ε i ( λ ) is a blo c k of size ε i × ( ε i + 1) defined b y L ε i ( λ ) =    0 1 . . . . . . 0 1    − λ    1 0 . . . . . . 1 0    . (20) L 0 is a blo c k of s ize 0 × 1 whic h con tributes to a column of zeros. Analogously , the left singular part has the form S lef t H ( λ ) = L T η 1 ( λ ) ⊕ · · · ⊕ L T η l 0 ( λ ) , where L ⊤ η i ( λ ) is a blo ck of size ( η i + 1) × η i , and L ⊤ 0 is a b lo c k of size 1 × 0 wh ic h con tribu tes to a row of zeros. 6.3 GUPTRI form The GUPTRI form (Generalized UP er TRIangular form) for singular matrix p encils w as done b y V an Do oren [19, 20] b y usin g unitary equiv alence transformations. It is a ge ner- alizati on of the Sc hur-staircase form f or m atrices. Giv en a singular matrix p encil H ( λ ) = A − λB with A, B ∈ C m × n , there exist t wo unitary matrices U and V of size m × m and n × n resp ectiv ely such that U ( A − λB ) V H =   A r − λB r ⋆ ⋆ 0 A r eg − λB r eg ⋆ 0 0 A l − λB l   , where the r ectangular blo ck up p er triangular A r − λB r and A l − λB l giv e the right and left singular structures of the matrix p encil, resp ectiv ely . The remaining square up p er triangular A r eg − λB r eg con tains all the fin ite and infi nite eigen v alues of H ( λ ). F urthermore, the regular p art A r eg − λB r eg is in stai rcase form: A r eg =   A 0 ⋆ ⋆ 0 A f ⋆ 0 0 A ∞   , B r eg =   B 0 ⋆ ⋆ 0 B f ⋆ 0 0 B ∞   where A 0 − λB 0 and A ∞ − λB ∞ rev eal the Jord an s tr uctures of the zero and infi nite eigen v alues, and A f − λB f , in generalize d Sc hur f orm , includ es the finite but nonzero eigen v alues. The eigen v alues µ i are computed as pairs of v alues, denoted by ( α i , β i ), α i in the diagonal of A r eg and β i in the d iagonal of B r eg as follo ws: If α i 6 = 0 and β i 6 = 0 then µ i is the fi nite nonzero eigen v alue µ i = α i /β i ; if α i = 0 and β i 6 = 0, µ i is the zero eigen v alue and; if α i 6 = 0 and β i = 0 then µ i is the in finite eigen v alue. T he case α i = 0 and β i = 0 do es n ot corresp on d to an eigen v alue, instead it b elongs to the singular part of the matrix p en cil. In [8, 9 ] is describ ed an efficien t algorithm f or computing the GUPTRI form of a matrix p encil. The implemen tation of this algorithm can b e found in http://w ww.cs.umu .se/research/nla/singular_pairs/guptri . Ac kno wledgmen t s: The authors gratefully ac knowle dge F. M. Dopico (Univ ersid ad Carlos I I I d e Madrid) for the fruitful discuss ions on the pro of of Theorem 3. 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