You are an expert at summarizing long articles. Proceed to summarize the following text: 
an operator @xmath0 is a projector if it is idempotent , i.e. , if it is endowed with property @xmath1 . the projection is along ( or parallel to ) its null space @xmath2 and onto its range @xmath3 . this entails that @xmath4 for @xmath5 and @xmath6 for @xmath7 . if the subspaces @xmath3 and @xmath2 are orthogonal the operator is called an orthogonal projector , which is the case if and only if @xmath0 is self - adjoint . otherwise it is called oblique projector . oblique projectors , though introduced early @xcite , have received less attention than orthogonal projectors . nevertheless , quite recently there has been a renewed interest in relation to their properties and applications @xcite . in particular , oblique projectors have been shown to be of significant relevance to signal processing techniques @xcite . the present effort is very much motivated by problems arising in the area of signal representation outside the traditional orthogonal basis setting @xcite . in such a context a signal @xmath8 , represented mathematically as an element of a vector space , is approximated as a linear expansion of the form [ eq1 ] f^k=_i=1^k c_i v_i . the vectors @xmath9 in are sometimes sequentially fed or chosen according to some optimality criterion . in such situations one needs to be in a position to effectively adapt the coefficients of the linear superposition so as to account for the possibility of changes in the model . this may entail a)increasing the order @xmath10 of the model by incorporating new terms in the expansion b)reducing the order by eliminating some terms in the expansion c)replacing some of the vectors in by different ones . assuming that the signal space is an inner product space , for @xmath11 given in to be the best approximation of a signal @xmath8 in a minimum distance sense , the coefficients in should be calculated in such a way that @xmath11 is the orthogonal projection of @xmath8 onto @xmath12 . this is the main reason for the popularity of orthogonal projectors in the context of approximation techniques . nevertheless , suppose that the observed signal is produced by the interference of two phenomena so that the model becomes [ eq2 ] _ i=1^k c_i v_i + _ i=1^n d_i w_i . if one were interested in discriminating the phenomena by splitting the signal , the component in @xmath12 could be obtained by an oblique projection operation mapping the other component to zero . there is a broad range of applications in which this procedure happens to be of assistance @xcite . thus , we felt motivated to find recursive equations for adapting oblique projectors . some of the equations to be proposed here are inspired by our previous work on recursive biorthogonalization for orthogonal projectors representation @xcite . we have recently been made aware that such a work is closely related to earlier one on recursive generalized inverses @xcite .    in spite of the fact that for most numerical implementations a projector is represented by a matrix , we prefer to think of projectors as operators acting by performing inner products . an important reason for this choice is the following : the equations can thereby be applied in general inner product spaces and comprise two very important cases in particular . namely , the euclidean inner product space , where a projector is indeed a matrix , and the space of functions of finite 2-norm . we like to see the proposed recursive equations as generalized gram - schmidt like procedures for generating sequences in inner product spaces . such sequences give rise to oblique projectors onto nested subspaces and , of course , to orthogonal projectors as special case . the paper is organized as follows : section  [ sec2 ] introduces the notation along with a discussion on the general construction of oblique projectors . section  [ sec3 ] provides the recursive equations for stepwise updating / downdating of such projectors . applications are illustrated in section  [ num ] by i)recovering a simulated x - ray diffraction peak from a background and ii)filtering impulsive noise from the register of the motion of a system consisting of the superposition of damped harmonic oscillators . the conclusions are drawn in section  [ con ] . as already mentioned we will work in a general inner product space @xmath13 , where the square norm @xmath14 is induced by the inner product that we represent as @xmath15 . given two closed subspaces , @xmath16 and @xmath17 , such that @xmath18 and @xmath19 , the _ oblique projector _ operator onto @xmath20 along @xmath21 will be represented as @xmath22 . then @xmath22 satisfies : ^2&= & + v&= & v , v + w&= & 0 , w . in the particular situation in which @xmath21 happens to be the orthogonal complement of @xmath20 in @xmath13 , i.e. if @xmath23 , the operator is self - adjoint and represents an orthogonal projection onto @xmath20 . we emphasize this special case by using the particular notation @xmath24 . in the sequel the orthogonal projector operator onto a subspace , say the subspace @xmath25 , will be indicated as @xmath26 . let us assume that in general @xmath27 and @xmath28 , with @xmath29 the orthogonal complement of @xmath21 . denoting as @xmath30 the standard orthonormal basis in @xmath31 , i.e. , the inner product @xmath32 with @xmath33 equal one if @xmath34 and zero otherwise , we define the operators @xmath35 and @xmath36 as @xmath37 thus the corresponding adjoint operators @xmath38 and @xmath39 are @xmath40 the operations @xmath41 and @xmath42 indicate that @xmath39 and @xmath38 act by performing inner products in @xmath13 . the inner product is defined in such a way that for @xmath43 and @xmath44 a complex constant the mapping @xmath45 produces a vector in @xmath31 of the form @xmath46 . the operation @xmath47 indicates the inner product in @xmath31 , thereby for @xmath48 the mapping @xmath49 yields a vector in @xmath20 of the form @xmath50 note that the matrix representation of @xmath51 has elements given by the inner products @xmath52 . the operator @xmath53 where @xmath54 denotes the moore - penrose pseudo - inverse , is known to be the oblique projector onto @xmath20 along @xmath21 @xcite . the particular choice [ udef ] u_i = v_i - _ v_i= _ v_i , i=1, ,k produces the expression for @xmath22 used in signal processing applications @xcite . certainly , setting @xmath55 one has the convenient equation [ fo2 ] = ( ^ _ ) ^^ _ that we adopt hereafter .    amongst the many properties of oblique projectors that have been studied we shall recall only the basic property needed for our purpose . it follows by applying @xmath56 on both sides of , i.e. , [ pro ] _ = _ ( ^ _ ) ^^_. since @xmath57 and @xmath58 , with the substitution @xmath59 the right hand side of takes the form @xmath60 . such an operator is the orthogonal projector onto @xmath61 . consequently , [ pro2 ] _ = _ , = ( _ ) = \ { _ v_i}_i=1^k . by denoting @xmath62 with @xmath63 the element @xmath64 of matrix @xmath65 , we can express @xmath22 as [ evw ] = _ i=1^k v_i u_i , . furthermore , from , , and [ euw ] _ = _ = _ i=1^k u_i u_i , . because @xmath56 is self - adjoint @xmath66 , and vice versa . on comparing and we see that the dual vectors @xmath67 are the same . this is of enormous assistance to derive the equations for adapting oblique projectors so as to account for the updating or downdating of the projecting subspace @xmath20 . this will allow us to give the proofs of the proposed recursive equations either by verification or by induction . it is appropriate to stress at this point that if we chose @xmath23 we would have @xmath68 and consequently @xmath69 . hence for such special situation @xmath70 and all the recursive equations of the subsequent sections would give rise to orthogonal projectors . in this section we provide the equations for updating and downdating oblique projectors in order to account for the following situations :    let us consider that the oblique projector @xmath71 onto the subspace @xmath72 along a given subspace @xmath21 is known . if the subspace @xmath73 is enlarged to @xmath74 by the inclusion of one element , i.e. , @xmath75 , we wish to construct @xmath76 from the availability of @xmath71 . on the other hand , if the subspace @xmath77 is reduced by the elimination of one element , say the @xmath78-th one , we wish to construct the corresponding oblique projector @xmath79 from the knowledge of @xmath71 . the subspace @xmath21 is assumed to be fixed . its orthogonal complement @xmath80 in @xmath81 changes with the index @xmath10 to satisfy @xmath82 , where @xmath83 denotes an orthogonal sum whilst the former is a direct sum , i.e. , @xmath84 . we assume that @xmath71 is known and write it in the explicit form [ evkw ] = _ i=1^k v_i u^k_i , . our aim is to find the vector @xmath85 , and to change the vectors @xmath86 to @xmath87 , so as to obtain [ evkwp ] = _ i=1^k+1 v_i u^k+1_i , . we will show that the duals @xmath88 can be constructed inductively from the dual of a single vector . [ lem1 ] for @xmath89 , with @xmath90 , operator @xmath91 is the oblique projector onto the span of the single vector @xmath92 along @xmath21 .    from the definition of @xmath93 ( c.f . it follows that the operator @xmath94 maps every vector in @xmath21 to the zero vector . suppose that @xmath8 is in the span of @xmath92 . then @xmath95 for some constant @xmath44 . since @xmath96 we have @xmath97 moreover @xmath98 , which concludes the proof that @xmath91 is the oblique projector onto the span of @xmath92 along @xmath21 .    in order to inductively construct from @xmath99 the duals @xmath100 we have to discriminate two possibilities    * @xmath101 , i.e. , @xmath102 * @xmath103 , i.e. @xmath104    let us consider first the case i ) . clearly if @xmath105 the corresponding @xmath106 belongs to @xmath107 , because @xmath108 yields @xmath109 . the proposition below prescribes how to modify the corresponding dual vectors in order to guarantee that @xmath110 . let @xmath105 and vectors @xmath111 in be given . for an arbitrary vector @xmath112 the dual vectors @xmath113 computed as [ utikp ] u_i^k+1= u_i^k- u_k+1 , u_i^ky_k+1 for @xmath114 and @xmath115 produce the identical oblique projector as the dual vectors @xmath116 . we use to explicitly express @xmath76 [ red ] _ i=1^k+1 v_i u_i^k+1 , & = & _ i=1^k v_i u_i^k , , - _ i=1^k v_i u_i^k , u_k+1 y_k+1 , + v_k+1 y_k+1 , + & = & - u_k+1 y_k+1 , + v_k+1 y_k+1 , + & = & - v_k+1 y_k+1 , + v_k+1 y_k+1 , , where the last equality holds because @xmath117 and @xmath118 for @xmath105 . hence , the left hand side of equals @xmath71 . the next proposition considers the case ii )    let vector @xmath119 and vectors @xmath120 in be given . thus the dual vectors @xmath113 computed as [ eq ] u_i^k+1=u_i^k- u_k+1^k+1 u_k+1 , u_i^k , where @xmath121 with @xmath122 , provide us with the oblique projector @xmath76 .    in order to organize the proof let us establish the following relations : [ a1 ] q_k+1,v_i&=&0 , i=1, ,k + [ a2 ] q_k+1,v_k+1&=&u_k+1 , v_k+1- v_k+1 , _ v_k+1 = || q_k+1||^2 . the first relation follows from the definition of @xmath123 and the fact that @xmath124 for @xmath114 @xmath125 on the other hand @xmath126 furthermore ||q_k+1||^2&= & q_k+1,u_k+1 - q_k+1 , _ u_k+1 = q_k+1 , u_k+1 + & = & q_k+1 , v_k+1 - q_k+1 , _ v_k+1= q_k+1 , v_k+1 + & = & u_k+1 , v_k+1 - v_k+1 , _ v_k+1 . we are now in a position to start the proof of the proposition by induction . from lemma [ lem1 ] we know that @xmath127 is the oblique projector onto @xmath128 along @xmath21 . assuming that @xmath71 is the oblique projector onto @xmath73 along @xmath21 we will prove that @xmath76 is the oblique projector onto @xmath74 along @xmath21 . for this we need to prove that the recursive equation yields the operator @xmath76 satisfying :    * @xmath129 * @xmath130 * @xmath131    we begin by using to express @xmath76 as [ obkp ] v_i u_i^k+1 , & = & v_i u_i^k , , - v_i u_i^k , u_k+1 u_k+1^k+1 , + v_k+1 u_k+1^k+1 , + & = & - u_k+1,+ v_k+1 , . for all @xmath132 in @xmath21 it holds that @xmath133 and @xmath134 . then from we conclude that condition iii ) is satisfied . every @xmath135 can be written as @xmath136 . thus , from and using relations and v & = & _ i=1^k c_i v_i + c_k+1v_k+1 - c_k+1 u_k+1 + c_k+1 v_k+1 + & = & _ i=1^k c_i v_i + c_k+1 v_k+1= v , which demonstrates condition ii ) . finally , since from and it follows that @xmath137 , we have ^2 & = & - u_k+1 , + v_k+1 , + & = & - u_k+1 , + v_k+1 , = .    [ re1 ] if vectors @xmath138 are linearly independent they are also biorthogonal to the dual vectors arising inductively from the recursive equation . the proof of this property is given in appendix a.    [ re2 ] if vectors @xmath138 are not linearly independent the oblique projector @xmath71 is not unique . indeed , if @xmath139 are dual vectors giving rise to @xmath71 then one can construct infinitely many duals as : [ nonu ] y_i= u_i^k+ y_i - _ j=1^k y_j v_j , u_i^ki=1, ,k , where @xmath140 are arbitrary vectors in @xmath13 . we use to write v_i y_i , & = & + v_i y_i , - v_i _ j=1^k u_i^k , v_j y_j , + & = & + v_i y_i , - _ j=1^kv_i u_i^k , v_j y_j , + & = & + v_i y_i , - _ j=1^k v_j y_j , + & = & . it follows from property [ re1 ] that if vectors @xmath138 are linearly independent equation yields the unique duals @xmath141 . suppose that by the elimination of the element @xmath78 the subspace @xmath73 is reduced to @xmath142 . in order to give the equations for adapting the corresponding dual vectors generating the oblique projector @xmath79 we need to consider two situations :    * @xmath143 i.e. , @xmath144 * @xmath145 , i.e. , @xmath146    the next proposition addresses i ) . let @xmath71 be given by and let us assume that removing vector @xmath147 from the spanning set of @xmath73 leaves the identical subspace , i.e. , @xmath148 . hence , if the remaining dual vectors are modified as follows : [ opeqb ] = u_i^k+ , the corresponding oblique projector does not change , i.e. @xmath149 . let us recall that @xmath148 implies @xmath150 . hence @xmath151 , as it is seen from the fact that @xmath152 , which implies @xmath153 if and only if for @xmath114 it holds that @xmath154 . this is not true if @xmath148 so that we can use to express @xmath79 as _ ^k v_i , & = & _ ^k v_i u_i^k , + _ ^k + & = & - v_j u_j^k,+ - v_j u_j^k , u_j + & = & - v_j u_j^k,+ - u_j^k , u_j + & = & . finally , proposition 4 addresses ii ) . let @xmath71 be given by and let us assume that the vector @xmath147 to be removed from the spanning set of @xmath73 is not in @xmath155 . in order to produce the oblique projector @xmath79 the appropriate modification of the dual vectors can be achieved by means of the following equation [ duba ] = u_i^k- . using we write : [ equp ] _ ^k v_i , & = & _ ^k v_i u_i^k , - _ ^k + & = & - v_j u_j^k , - + v_j u_j^k , + & = & - . we notice that @xmath156 is the orthogonal projector onto the span of the single vector @xmath157 and denote it as @xmath158 . thus the orthogonal projector onto @xmath159 can be expressed as @xmath160 . applying this operator on the right hand side of we obtain : @xmath161 and , since @xmath162 and @xmath163 , using the fact that @xmath164 it follows that ( _ - _ u^k_j ) ( -_u^k_j ) & = & _ - _ u^k_j - _ u^k_j + _ u^k_j + & = & _ - _ u^k_j = _ _ = _ _ . from the last equation and we gather that _ _ _ ^k v_i , - _ _ = 0 . for every vector @xmath165 we therefore have _ _ ( _ ^k v_i , - ) f = _ _ f = 0 , with @xmath166 . this implies that either @xmath167 is the zero operator or @xmath168 for every @xmath43 . the latter can not be true because from the definition of @xmath167 it is seen that @xmath169 and by hypothesis @xmath170 . hence @xmath167 should be the zero operator , which leads to the conclusion that = _ ^k v_i , . [ re3 ] the case of replacing a vector in @xmath171 , say @xmath147 by @xmath172 , is actually equivalent to augmenting the subspace @xmath173 to @xmath174 after the vector @xmath147 was deleted . some implementation issues arise , though . in order to modify the duals as prescribed in we need to compute a vector @xmath175 . for the sequential enlargement of the projecting subspace , discussed in section [ secup ] , the projector @xmath176 can be sequentially constructed by means of the orthonormal vectors @xmath177 . nevertheless , when replacing vectors sequentially we need to allow for the recalculation of the corresponding projectors . one possibility that could be considered is the recalculation of the orthogonal vectors @xmath178 @xcite . an alternative approach implies to use of the dual corresponding to the deleted vector for orthogonalization purposes . a discussion concerning the implementation of such a procedure is given in @xcite . the examples presented in this section aim at illustrating the application of our recursive construction of oblique projectors for signals in @xmath179 $ ] , the space of square integrable functions on @xmath180 $ ] . for @xmath8 and @xmath181 in @xmath179 $ ] , we define the inner product , according to the previously adopted convention , as @xmath182 where @xmath183 indicates the complex conjugate of @xmath8 . in the examples bellow all the integrals are numerically calculated . here the signal is simulated by emulating a crystallographic problem . it is assumed to be the x - ray diffraction intensity produced by a powder sample of a clay mineral consisting of very flat crystals . each such crystal is formed by the stacking of @xmath184 layers producing a diffraction intensity as given by @xcite i_n(x ) = , where the variable @xmath185 , given in radians , is related to the diffraction angle @xmath186 according to the equation x= 2 . [ xt ] the parameter @xmath187 in is the effective distance between two consecutive layers and characterizes the material . the parameter @xmath188 represents the wavelength of the incident radiation . we denote the diffraction intensity produced by the whole sample as @xmath189 . thus , [ smod ] f_1(x)= _ n=1^k c_n . as already mentioned @xmath184 indicates the possible number of layers forming a single crystal in the sample . the coefficients @xmath190 account for the proportions of crystals consisting of @xmath184 layers . here , for simulating the signal , the coefficients were considered to be @xmath191 the diffraction figure @xmath192 emerges from a background that is modelled as f_2(x)= 50 _ j=1 ^ 3 j e^-j(x- ) . the combined phenomenon gives rise to the signal @xmath193 plotted in the left graph of figure  1 on the interval relevant to the diffraction model , namely @xmath194.$ ]    we are interested in extracting the diffraction peak @xmath192 from the background . for this we will construct sequentially oblique projectors onto subspaces @xmath73 given as @xmath195.\ ] ] the final @xmath10-value is to be adjusted . the subspace @xmath21 is here @xmath196.\ ] ] since the order @xmath10 of the diffraction model is assumed unknown , it was adjusted as follows : firstly the order model was sequentially increased ( up to @xmath197 ) and then sequentially downdated . it was observed that the recovering of the signal was not very sensitive to the model order . in a range from @xmath198 to @xmath197 the approximations were totally equivalent . from @xmath199 to @xmath198 changes in the approximations were noticed but the approximations could still be considered ` practically ' equivalent . the recovered peak @xmath200 ( for @xmath198 ) is depicted in the right graph of figure  1 . it happens to coincide , in the scale of the figure , with the graph of the theoretical one ( c.f . ) . the convenience of the proposed adaptive technique in the determination of the order of the diffraction intensity model is clear : otherwise when changing the @xmath10 value as described above the whole projector would have to be recalculated for each different value of @xmath10 . however , the advantage of the proposed technique is even more significant when , as is the case in the next example , overestimation of the order in the signal model may result in the failure to discriminate the signals . [ f1 ]     measured in radians and related to the diffraction angle through . graph on the right : diffraction intensity extracted from the background.,title="fig:",width=283 ]   measured in radians and related to the diffraction angle through . graph on the right : diffraction intensity extracted from the background.,title="fig:",width=283 ]      in this case the signal is considered to be the register of the motion of a system consisting of uncoupled damped harmonic oscillators . the @xmath184-th oscillator is characterized by a frequency of @xmath201 hz . the corresponding equation for its motion @xmath202 as a function of time is given as x_n(t)= e^-t ( n t ) , n=1,  ,k . the distribution of frequencies is considered to be @xmath203 so that the motion of the system is registered by the signal [ smod2 ] f_1(t)=_n=1 ^ 100 , t . this signal ( shown in the right graphs of figure 2 ) is corrupted by impulsive noise , which represents a type of electrical noise appearing in some practical situations . the possible pulses are taking from the set of @xmath204 gaussian sparks @xmath205 . hence the corresponding subspace @xmath21 is @xmath206.\ ] ] first a random superposition of @xmath207 pulses is added to the signal @xmath192 to simulate the noisy one plotted in the top left graph of figure 2 . the result after the oblique projections along the subspace @xmath21 given above is shown in the top right graph ( it coincides with the theoretical signal @xmath192 given in ) . the left graph at the bottom of figure 2 corresponds to a different realization of the noise , in this case generated as a random superposition of @xmath208 pulses . the signal , after filtering by oblique projections along @xmath21 , is shown in the right graph ( it also coincides with the theoretical one ) . [ f2 ]      +    +    let us point out that an alternative way of splitting the signal would entail fitting both the signal and noise models , with the consequent increment in the dimension of the problem of determining the corresponding unknown parameters . in the example of this section , for instance , 400 more parameters ( coefficients of the noise model ) would be involved . recursive equations for updating / downdating oblique projectors have been proposed . the updating strategy can be regarded as a generalized gram - schmidt like procedure for generating a sequence giving rise to oblique projectors along a fixed subspace . the downdating strategy modifies such sequence to account for the removal of some elements . the equations are of the same nature as those for producing orthogonal projectors , but involve different vectors . orthogonal projectors arise within this framework as a particular case . the proposed technique has been applied to the problem of discriminating signals produced by different phenomena . the applications that have been considered assume that the signal model is determined by physical considerations and only the order of the model is to be adjusted . the task of setting the right order model is facilitated by the recursive nature of the proposed equations . in the two examples considered here the signal splitting is not very sensitive to the order of the signal model . however , an important difference between the two examples is the following : while in the first example an excessive overestimation of order model ( maximum possible number of layers present in a crystal ) does not prevent the extraction of the diffraction peak from the background , an excessive overestimation of the order of the model in the second example ( maximum possible frequency of an oscillator ) may produce the failure to separate the signal from the impulsive noise . the reason being that for very high frequencies the angle between the signal subspace and the noise subspace becomes very small , which generates an ill posed problem . the recursive feature of the proposed equations turns out to be even more important in those situations in which the signals splitting is achieved by stepwise selection of each component of the signal model . this is the subject of a recent work @xcite , where the present approach is shown to be of significant assistance . support from epsrc ( ep@xmath209d062632@xmath2091 ) is acknowledged . [ apen ] let us recall that if vectors @xmath210 are linearly independent , all the duals @xmath211 are generated by the recursive equation . we need to show that such vectors satisfy : @xmath212        * @xmath220 and @xmath221 + in this case @xmath222 ( c.f . ) . hence , from the recursive equation , we have @xmath223 * @xmath224 and @xmath225 + now v_l+1 , u^l+1_i & = & v_l+1 , u^l_i - v_l+1 , u^l_i + & = & v_l+1 , u^l_i so that , since @xmath226 ( c.f . ) , @xmath227 * @xmath224 and @xmath228 + this implies @xmath229 * @xmath220 and @xmath230 + in this case @xmath231                                                l.  rebollo - neira , recursive bi - orthogonalisation approach and orthogonal projectors , math - ph/0209026 ( 2002 ) ; l.  rebollo - neira , new topics in mathematical physics research , nova science publisher , new york , ( 2006 ) , ch . on non - orthogonal signal representation . m.  andrle , l.  rebollo - neira , experiments on orthogonalization by biorthogonal representations of orthogonal projectors , journal of computational and applied mathematics , in press , corrected proof available on line ( 2006 ) .
a generalization of the gram - schmidt procedure is achieved by providing equations for updating and downdating oblique projectors . 
 the work is motivated by the problem of adaptive signal representation outside the orthogonal basis setting . 
 the proposed techniques are shown to be relevant to the problem of discriminating signals produced by different phenomena when the order of the signal model needs to be adjusted . 
 = 1