Accuracy of reconstruction of spike-trains with two near-colliding nodes

0.32 MB·

by  Andrey Akinshin

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1
Accuracy of reconstruction of spike-trains with two
0
2 near-colliding nodes

n
a Andrey Akinshin, Gil Goldman, Vladimir Golubyatnikov,
J
and Yosef Yomdin
5

] Abstract. We consider a signal reconstruction problem for signals F of the
A
form F(x)= Pdj=1ajδ(x−xj), from their moments mk(F) = R xkF(x)dx.
C Weassumemk(F)tobeknownfork=0,1,...,N,withanabsoluteerrornot
. exceeding ǫ>0.
h
We study the “geometry of error amplification” in reconstruction of F
t
a from mk(F), in situations where two neighboring nodes xi and xi+1 near-
m collide, i.e xi+1 −xi = h ≪ 1. We show that the error amplification is
governedbycertainalgebraiccurvesSF,i,intheparameterspaceofsignalsF,
[
alongwhichthefirstthreemomentsm0,m1,m2 remainconstant.

1
v
2
8
1. Introduction
4
1
Theproblemofreconstructionofspike-trains,andofsimilarsignals,fromnoisy
0
moment measurements, and a closely related problem of robust solving the clas-
.
1
sical Prony system, is a well-known problem in Mathematics and Engineering. It
0
is of major practical importance, and, in case when the nodes nearly collide, it
7
1 presentsmajormathematicaldifficulties. Itiscloselyrelatedtoaspike-train“super-
: resolutionproblem”,(see[3, 4, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 23]
v
as a small sample).
i
X The aim of the present paper is to investigate the possible amplification of the
r measurements error ǫ in the reconstruction process, caused by the fact that some
a
of the nodes of F near-collide. Recently this problem attracted attention of many
researchers. In particular, in [1, 4, 10, 13, 14] it was shown (in different settings
of the problem) that if s spikes of F are near-colliding in an interval of size h 1,
≪
then a strong “noise amplification” occurs: up to a factor of (1)2s 1. Specifically,
h −
in [1] a parametric setting (the same as in the present paper) was considered (see,
asasmallsample,[6, 7, 8, 23, 24, 25, 27]andreferencestherein). Inthissetting,
signals are assumed to be members of a parametric family with a finite number of
parameters. The parameters of the signal are then considered as unknowns, while
the measurements provide a system of algebraic equations in these unknowns.
2010 Mathematics Subject Classification. Primary42A38,94A12.
Key words and phrases. Signal reconstruction, spike-trains, Fourier transform, Prony
systems.
TheworkwassupportedbytheRFBRgrant15-01-00745A;ISF,GrantNo. 779/13.
1
2 A.AKINSHIN,G.GOLDMAN,V.GOLUBYATNIKOV,ANDY.YOMDIN
It was announced in [1] that the strongest “noise amplification” occurs along
the algebraic curves S (“Prony curves”), defined in the signal parameter space by
the 2s 1 initial equations of the classical “Prony system” (system 2.2 below).
−
However, [1] provided neither detailed proofs, nor explicit constants in the error
bound, nor the explicit description of the curves S.
In the present paper we consider reconstruction of spike-train signals of an
d
a priori known form F(x) = a δ(x x ), from their moments m (F),...,
j=1 j − j 0
m (F), N 2d, in the case where two nodes x ,x near-collide. That is,
N 1 i i+1
− ≥ P
x x =h 1.
i+1 i+1
− ≪
In Section2 we introduce the ǫ-errorset E (F), consistingof allsignals F , for
ǫ ′
whichthemomentsofF differfromthemomentsofF byatmostǫ. ThesetE (F)
′ ǫ
presents the distribution of all the possible reconstructed signals F , caused by
′
independent errors, not exceeding ǫ, in each of the moment measurements m (F).
k
ThusthegeometryofE (F)reflectsthepatternsofthepossibleerroramplification
ǫ
in the reconstruction process.
Inthispaperwearemostlyinterestedinthelowerboundsfortheerrorinnodes
reconstruction. We thus consider the projection Ex(F) of the error set E (F) to
ǫ ǫ
the nodes space. This set represents the error amplification in nodes reconstruc-
tion. In particular, the “radius” ρx(F) of Ex(F) provides a lower bound on the
ǫ ǫ
nodesreconstructionaccuracyofanyreconstructionalgorithm(seeamoredetailed
description of this fact in Section 2).
Oneofourtwomainresults-Theorem4.2inSection4-isthatfor F with two
nodes in a distance h, and for any ǫ of order h3 (or larger) we have ρx(F) Ch.
ǫ ≥
Consequently, the presence of near-colliding nodes implies a massive amplification
of the measurements error in the process of nodes reconstruction - up to h 2 times.
−
In order to prove Theorem 4.2 we start in this paper the investigation of the
geometry of the error sets E (F) and Ex(F) (which, as we believe, is important
ǫ ǫ
by itself). First we provide in Section 3 numerical simulations and visualizations,
which suggest that for ǫ h3 the ǫ-error set E (F) is an “elongated curvilinear
ǫ
∼
parallelepiped” of the width h, stretched up to the size 1 along a certain
∼ ∼
curve S (while its projection onto the nodes space, Ex(F), is stretched along the
F ǫ
projectionSx ofS ). Theseexperimentssuggestalsothatash 0,thesetsE (F)
F F → ǫ
and Ex(F) concentrate closer and closer around the curves S (respectively, Sx).
ǫ F F
Next,we giveinSection4anindependentdefinitionofthe“Pronycurves”S ,
F
“discovered” in Section 3: for each F the Prony curve S , passing through F in
F
the signal parameter space, is defined by the requirement that along it the first
three moments m ,m ,m do not change. An explicit parametric description of
0 1 2
the curves S is given in Section 5.1.
F
Oursecondmainresult-Theorem4.1inSection4-isthatindeed, as suggested
by visualizations in Section 3, the set E (F) contains a “sufficiently long” part of
ǫ
the Prony curve S around F. As a consequence, we obtain Theorem 4.2. Let us
F
stress that all the constants in Theorems 4.1 and 4.2 are explicit (and reasonably
realistic).
The proofs are given in Section 5.
Finally, in Section 6, we compare two approaches to the reconstruction prob-
lemforrealspike-trainsignals: fromtheirmoments,andfromtheirFouriersamples
(whichcanbeinterpretedasthemomentsofanappropriatesignalF˜ withcomplex
ACCURACY OF RECONSTRUCTION OF SPIKE-TRAINS 3
nodes). Recentlyin[2]atrigonometricreconstructionmethodforF˜ wassuggested,
which uses, as an input, only three complex moments m (F˜),m (F˜),m (F˜). Ac-
0 1 2
cordingtotheapproachofthepresentpaper,wewouldexpectforthetrigonometric
method (for F˜ with two nodes in a distance h 1, and for ǫ h3,) the worstcase
≪ ∼
reconstruction error of order √ǫ, while for the Prony inversion we show it to be of
order ǫ31. We pose some open questions related to this apparent contradiction.
The authors would like to thank the referee for a constructive criticism, as
well as for remarks and suggestions, which allowed us to significantly improve the
presentation.
2. Setting of the problem
Assume that our signal F(x) is a spike-train,that is, a linear combinationof d
shifted δ-functions:
d
(2.1) F(x)= a δ(x x ),
i i
−
i=1
X
wherea=(a ,...,a ) Rd, x=(x ,...,x ) Rd.Weassumethattheform(2.1)
1 d 1 d
∈ ∈
is a priori known, but the specific parameters (a,x) are unknown. Our goal is to
reconstruct(a,x)fromN 2dmomentsmk(F)= ∞ xkF(x)dx, k =0,...,N 1,
≥ −
which are known with a possible absolute error of n−o∞more than ǫ>0.
R
The moments m (F) are expressed through the unknown parameters (a,x) as
k
m (F)= d a xk. Henceourreconstructionproblemisequivalenttosolvingthe
k i=1 i i
(possiblyover-determined)Pronysystemofalgebraicequations,withtheunknowns
P
a ,x :
i i
d
(2.2) a xk =m (F), k=0,1,...,N 1.
i i k −
i=1
X
Thissystemappearsinmanytheoreticalandappliedproblems. Thereexistsavast
literature on Prony and similar systems, in particular, on their robust solution in
the presence of noise - see, as a small sample, [3, 7, 19], [22]-[26], and references
therein.
We present the spike train reconstruction problem in a geometric language of
spaces and mappings. Let us denote by = the parameter space of signals F,
d
P P
= (a,x)=(a ,...,a ,x ,...,x ) R2d, x <x <...<x ,
d 1 d 1 d 1 2 d
P { ∈ }
andby M=MN ∼=RN the moment space,consisting of the N-tuples of moments
(m ,m ,...,m ). We will identify signals F with their parameters (a,x) .
0 1 N 1
− ∈P
The Prony mapping PM =PM : is given by
d,N d N
P →M
PM(F)=µ=(µ ,...,µ ) , µ =m (F), k =0,...,N 1.
0 N 1 k k
− ∈M −
InversionofthePronymappingisequivalenttoreconstructionofaspike-trainsignal
F from its moments (or to solving Prony system (2.2)).
The aim of this paper is to investigate the amplification of the measurements
error ǫ in the reconstruction process, in case of two near-colliding nodes. We are
interestedineffects, causedbythe geometricnatureofsystem(2.2), independently
of the specific method of its inversion.
4 A.AKINSHIN,G.GOLDMAN,V.GOLUBYATNIKOV,ANDY.YOMDIN
The error amplification is reflected by the geometry of the ǫ-error set E (F),
ǫ
which is defined as follows:
Definition 2.1. The ǫ-error set E (F) consists of all signals F , for
ǫ ′ d
∈ P
which the moments of F differ from the moments of F by at most ǫ:
′
Eǫ(F)= F′ d, mk(F′) mk(F) ǫ, k =0,...,N 1 .
{ ∈P | − |≤ − }
Equivalently, Eǫ(F)=PMd−1(QNǫ (F)), where QNǫ (F)⊂MN is the N-dimensional
ǫ-cube centered at PM(F) .
N
∈M
Theǫ-errorsetE (F)presentsthedistributionofpossiblereconstructedsignals
ǫ
F , causedby the independent errors,notexceedingǫ, in eachofthe momentmea-
′
surements m (F). Its yet another convenient description is as the set of solutions
k
of the Prony system
d
(2.3) a xk =m (F)+ǫ , k=0,1,...,N 1,
i i k k −
i=1
X
with all the possible errors ǫ satisfying ǫ ǫ, k =0,1,...,N 1.
k k
| |≤ −
Notice that the ǫ-error set E (F) depends on N, the number of the moments
ǫ
whichweuse in reconstruction. Since N is assumedto be fixed, wedo notindicate
it in the notations.
Inthis paperwe aremainly interestedinthe accuracyofthe nodes reconstruc-
tionwhichis determinedby thegeometryofthe projectionEx(F)ofthesetE (F)
ǫ ǫ
onto the nodes space. Accordingly, we define the worst case error ρx(F) in the
ǫ
Prony reconstruction of the nodes of F as follows:
Definition 2.2. For F = (a,x) , the worst case error ρx(F) in the
∈ Pd ǫ
reconstruction of the nodes of F is defined by
(2.4) ρx(F)= sup x x ,
ǫ F′=(a′,x′) Eǫ(F)|| ′− ||
∈
where denotes the Euclidean norm in the space of the nodes.
||·||
Infact, 1ρx(F)boundsfrombelowtheworstcaseerrorinnodesreconstruction
2 ǫ
with any reconstruction algorithm A. Indeed, we can informally argue as follows:
let F = (a ,x ) E (F) be a signal for which the supremum in (2.4) is nearly
′′ ′′ ′′ ǫ
∈
achieved. Assume that we apply A to both signals F and F , and the (adversary)
′′
noise is zero for F and is equal to the difference of the moments of F and F in
′′
the second case. Thus A obtains as the input in both cases the moments of F.
Whatever result F˜ =(a˜,x˜) the algorithm A produces as an output, at least one of
the distances x x˜ or x x˜ will be not smaller than 1ρx(F).
|| − || || ′′− || 2 ǫ
3. Visualization of the error sets E (F)
ǫ
Fora givenh, 0<h<1,considerasignalF(x)= 1δ(x+h)+1δ(x h) .
2 2 − ∈P2
We put N =2d=4. The moments of F are
m (F)=1, m (F)=0, m (F)=h2, m (F)=0.
0 1 2 3
Foragivenǫ>0weconsidertheǫ-cubeQ4(F) centeredat(1,0,h2,0) ,
ǫ ⊂M4 ∈M4
and the ǫ-error set Eǫ(F) = PM2−,41(Q4ǫ(F)). Equivalently, Eǫ(F) is defined in P2
by the inequalities
m0(F′) 1 ǫ, m1(F′) ǫ, m2(F′) h2 ǫ, m3(F′) ǫ.
| − |≤ | |≤ | − |≤ | |≤
ACCURACY OF RECONSTRUCTION OF SPIKE-TRAINS 5
Figure 1. The error set E (F) and its projection Ex(F) for
ǫ ǫ
h=0.1, ǫ=2h3 =0.002 and F(x)= 1δ(x 0.1)+ 1δ(x+0.1).
2 − 2
Figure 2. The error set E (F) and its projection Ex(F) for
ǫ ǫ
h=0.05,ǫ=2h3 =0.00025andF(x)= 1δ(x 0.05)+1δ(x+0.05).
2 − 2
6 A.AKINSHIN,G.GOLDMAN,V.GOLUBYATNIKOV,ANDY.YOMDIN
The ǫ-error set Eǫ(F) is a four-dimensional subset of P2 ∼= R4, and its direct
visualization is problematic. Instead the following Figures 1 and 2 show the pro-
jectionofE (F)ontothe three-dimensionalcoordinatesubspaceof ,spannedby
ǫ 2
P
the two nodes coordinates x ,x and the first amplitude a , as well as its further
1 2 1
projection Ex(F) onto the nodes plane x ,x .
ǫ 1 2
Notice, that by the first of the Prony equations a +a =m (F)+ǫ in (2.3)
1 2 0 0
we have a =m (F) a +ǫ , with ǫ ǫ. Thus the projections of E (F) shown
2 0 1 0 0 ǫ
− | |≤
in Figures 1 and 2, give a rather accurate (up to ǫ) representationof the true error
set.
Let us stressa naturalscalingin our problem,reflectedin Figures 1 and2: the
scale in nodes is of order h, while the scale in the amplitudes is of order 1.
Figures1and2suggestthatE (F)isan“elongatedcurvilinearparallelepiped”,
ǫ
with the sizes of its two largest edges of orders 1 and h, respectively. (The third
and the fourth edges, of orders h2 and h3, respectively, are not visible). E (F)
ǫ
is stretched up to the size 1 along a certain curve S, depicted in the pictures.
∼
Respectively, Ex(F) is stretched up to the size h along the projection curve
ǫ ∼
Sx. A comparison between Figures 1 and 2 also suggests that as h (and ǫ h3)
∼
decrease, the error set concentrates closer and closer along the curve S. (Compare
a conjectured general description of E (F) at the end of Section 4).
ǫ
Belowweanalysethe structureofthe errorsetE (F)insomedetail, andshow
ǫ
that S is an algebraic curve, which we call the “Prony curve”. We show that
for F as above, the projection Sx of S onto the node subspace is the hyperbola
x x = h2, while a is expressed on this curve through x ,x as a = x2 . In
Se1ct2ion−5.1 we study1such “Prony curves” in detail. 1 2 1 x2−x1
Numerically, the figures above were constructed via the following procedure:
we construct a four-dimensional regular net Z Q4(F) , with a sufficiently
⊂ ǫ ⊂ M2
small step. For each point z Z, its Prony preimage w = PM 1(z) is
− 2
∈ ∈ P
calculated, and the projection of w onto the space (a ,x ,x ) is plotted.
1 1 2
Some other visualisation results can be found in [28].
In what follows we assume that inversion of the Prony map, or solving of
(2.3) (when possible) is accurate, and the reconstruction error is caused only by the
measurements errors ǫ .
k
4. Prony curves and error amplification: main results
We will consider signals F(x) of the form (2.1), with two near colliding nodes
x and x , 1 i d 1. In the present paper we study the geometry of the
i i+1
≤ ≤ −
reconstructionerror, allowing perturbations only of the cluster nodes x ,x , and
i i+1
of their amplitudes a ,a . Therefore the positions and the amplitudes of the
i i+1
other nodes are not relevant for our results. However, in order to avoid possible
collisions of the cluster nodes with their neighbors in the process of deformation,
we will alwaysassume that for x x =h>0, the distances to the neighboring
i+1 i
−
nodes from the left and from the right satisfy x x 3h, x x 3h.
i i 1 i+1 i+2
− − ≥ − ≥
We do not assume formally that h 1, but this is the case where the geometric
≪
patterns we describe become apparent.
ACCURACY OF RECONSTRUCTION OF SPIKE-TRAINS 7
For each signal F and index i the Prony curve S = S passing through F is
F,i
obtainedby varying only the nodes andamplitudes (a ,x ),(a ,x ), while pre-
i i i+1 i+1
serving the first three moments. More accurately, we have the following definition:
Definition 4.1. Let F(x)= d a δ(x x ) and let i, 1 i d 1,
j=1 j − j ∈Pd ≤ ≤ −
be fixed. The Prony curve S =S consists of all the signals
F,i d
P⊂P
d
F (x)= a δ(x x )
′ ′j − ′j ∈Pd
j=1
X
for which a =a ,x =x , j =i,i+1, and m (F )=m (F) for k=0,1,2.
′j j ′j j 6 k ′ k
By definition, we always have F S . In this paper we concentrate on an
F,i
∈
“h-local” part S (h) around F of the Prony curve S , consisting of all
F,i F,i
d
F (x)= a δ(x x ) S ,
′ ′j − ′j ∈ F,i
j=1
X
for which the nodes (x ,x ) belong to the disk D R2 of radius 1h centered at
′i ′i+1 ⊂ 2
(x ,x ). In particular, the node collision cannot happen on S (h) (see Lemma
i i+1 F,i
5.1 below). Compare also to Figure 3.
Notice,however,thatthePronycurvesS areglobal algebraic curves (possibly
F,i
singular). Their explicit global parametrization is described in Section 5.1 below,
and one can show that in some cases they can pass through the node collision
points(with amplitudes tendingto infinity). We believethatthe Pronycurvesand
their multi-dimensional generalizationsplay an important role in understanding of
multi-nodes collision singularities.
The following definition specifies the type of signals we will work with:
Definition 4.2. A signal F = d a δ(x x ) is said to form an
j=1 j − j ∈ Pd
(i,h,M)-cluster, with given h, 0 < h < 1, M > 0, and i, 1 i d 1, if
P ≤ ≤ −
x x =h, and a , a M.
i+1 i i i+1
− | | | |≤
ForF formingan(i,h,M)-cluster,letκ= xi+1+xi be the centerofthe interval
2
[x ,x ]. We put C(F)=18M(1+ κ)N (Notice thatC(F) depends onκ,M,but
i i+1
| |
not on h). One of our main results is the following:
Theorem 4.1. Let F form an (i,h,M)-cluster. Then for each ǫ
d
∈ P ≥
C(F)h3 the ǫ-error set E (F) contains the local Prony curve S (h).
ǫ F,i
8 A.AKINSHIN,G.GOLDMAN,V.GOLUBYATNIKOV,ANDY.YOMDIN
Figure 3. The error set E (F) and its projection Ex(F) for
ǫ ǫ
h = 0.1, ǫ = 2h3 = 0.002 and F(x) = 1δ(x 0.1)+ 1δ(x+0.1).
2 − 2
The circle depicts the boundary ofthe disk D. In bold is the local
prony curve S (h) and its projection.
F,i
Figure 3 suggests that in fact the ǫ-errorset E (F) “concentrates” around the
ǫ
localPronycurveS (h).AlreadythefactthatthiscurveisinsideE (F)(provided
F,i ǫ
by Theorem 4.1) implies important conclusion on the worst case reconstruction
error. Indeed, as we show below, the projection Sx (h) of the local Prony curve
F,i
S (h)ontothenodesspacehasalengthoforderh. Consequently,inthepresence
F,i
of an h-cluster in F, and for each ǫ C(F)h3, there are signals F E (F),
′ ǫ
≥ ∈
with the nodes x ,x at a distance h from x ,x . That is, a massive error
′i ′i+1 ∼ i i+1
amplification from h3 to h occurs in this case.
Our second main result presents this fact formally, in terms of the worst case
error:
Theorem 4.2. Let F form an (i,h,M)-cluster. Then for each ǫ
d
∈ P ≥
C(F)h3 the worst case reconstruction error ρx(F) in nodes of F is at least 1h.
ǫ 2
We see that a measurements error ǫ h3 can be amplified up to the factor
∼
h 2 in reconstruction of the nodes of F. In particular, for d = 2, i.e, in the
−
∼
case of exactly two nodes in F, and for N =4, we get (assuming that M =1 and
x ,x [ 1,1]) that C(F) 288. So the minimal accuracy ǫ required to keep the
1 2
∈ − ≤
error in the nodes reconstruction less than 1h is ǫ 288h3. For h = 0.01 we get
2 ≤
ǫ 0.0003.
≤
ACCURACY OF RECONSTRUCTION OF SPIKE-TRAINS 9
In[1],inordertoshowthatthelengthofthecurveSx (h)isoforderh,weuse
F,i
the inverse function theorem, combined with estimations from [5] of the Jacobian
of the Prony mapping. As a result, the constants become much less explicit.
We expect that the results of Theorems 4.1 and 4.2 can be extended to an
accurate description of the ǫ-error set in the case of clusters with more than two
nodes,usinganappropriateversionofthe “quantitativeinversefunctiontheorem”.
Informally, we expect the following general result to be true:
Let the nodes x ,...,x of F form a cluster of size h 1. Then for ǫ O(h2d 1)
1 d −
≪ ≤
theǫ-errorsetE (F)isa“non-linearcoordinateparallelepiped”Π (F)withrespect
ǫ h,ǫ
to the moment coordinates m (F ), centered at F. Its width in the direction of the
k ′
moment coordinate m , k = 0,...,2d 1, is of order ǫh k. In particular, the
k −
−
maximal stretching of Π (F), of order ǫh (2d 1), occurs along the Prony curve
h,ǫ − −
S (F).
2d 2
−
However,anapplicationoftheapproachbasedontheinversefunctiontheorem
will significantly reduce the domain of applicability of the results, and will make
the constants less explicit.
On the other hand, we believe that the explicit parametric description of the
Pronycurves,giveninSection5.1below,canbeextendedtoclusterswithmorethan
two nodes. It becomes significantly more complicated, but promises potentially
better understanding of the geometry of error amplification.
5. Proofs
The proofofTheorems 4.1and4.2, givenbelow,is basedona detailedexplicit
description of the Prony curves, on one hand, and of a behavior of the moments
m , k 3 on these curves, on the other.
k
≥
5.1. Parametrization of the Prony curves. WedenotebySx theprojec-
F,i
tion of the Prony curve S onto the node plane spanned by the node coordinates
F,i
x ,x .
′i ′i+1
Theorem 5.1. The curve Sx is a hyperbola in the plane x ,x defined by
F,i ′i ′i+1
the equation
m (F)x x m (F)(x +x )+m (F)=0.
0 ′i ′i+1− 1 ′i ′i+1 2
The original curve S is parametrized through x ,x in Sx as
F,i ′i ′i+1 F,i
m (F)x m (F) m (F)x +m (F)
a′i = 0 x ′i+1−x 1 , a′i+1 = − 0x ′i x 1 , (x′i,x′i+1)∈SFx,i.
′i+1− ′i ′i+1− ′i
Proof: Since all the nodes and amplitudes are fixed on the Prony curve S ,
F,i
but (a ,x ),(a ,x ), we can work only with the partial signals a δ(x x )+
i i i+1 i+1 i i
−
a δ(x x ). In otherwords,we canconsiderthe case ofexactlytwo nodes, i.e.
i+1 i+1
−
signals of the form F(x) = a δ(x x )+a δ(x x ) . In this case there is
1 1 2 2 2
− − ∈ P
only one choice i=1 for the index i in the definition of the Pronycurves S , and
F,i
we denote them by S .
F
Alternatively, we can consider algebraic curves S(m ,m ,m ) in , defined
0 1 2 2
P
by the equations
10 A.AKINSHIN,G.GOLDMAN,V.GOLUBYATNIKOV,ANDY.YOMDIN
a +a =m ,
1 2 0
(5.1) a x +a x =m ,
1 1 2 2 1
a x2+a x2 =m ,
1 1 2 2 2
for any moments m ,m ,m . If we put m = m (F), k = 0,1,2, we get S(m ,
0 1 2 k k 0
m ,m )=S .Sinceweareinterestedinthe behaviorofthenodesx ,x alongthe
1 2 F 1 2
curve S, we will consider also the node parameter space x = (x ,x ) , and the
P2 { 1 2 }
projections Sx(m ,m ,m ) x of the Prony curves S(m ,m ,m ) to the node
0 1 2 ⊂ P2 0 1 2
space x. The following proposition is an extended version of Theorem 5.1.
P2
Proposition 5.1. The curves Sx(m ,m ,m ) x are hyperbolas in the
0 1 2 ⊂ P2
plane x ,x defined by the equation
1 2
(5.2) m x x m (x +x )+m =0.
0 1 2 1 1 2 2
−
They form a two-parametric family, depending only on the ratio of the moments
(m : m : m ). The corresponding curves S(m ,m ,m ) are parametrized
0 1 2 0 1 2 2
⊂ P
as
m x m m x +m
(5.3) a = 0 2− 1, a = − 0 1 1, (x ,x ) Sx(m ,m ,m ).
1 2 1 2 0 1 2
x x x x ∈
2 1 2 1
− −
Proof: We get from the first two equations of (5.1) the following expressions for
a ,a through x ,x :
1 2 1 2
(5.4) a =m a , a x +(m a )x =m ,
2 0 1 1 1 0 1 2 1
− −
and hence
m x m m x +m
0 2 1 0 1 1
(5.5) a = − , a = − .
1 2
x x x x
2 1 2 1
− −
ThecurveS(m ,m ,m )isdefinedbyallthethreeequationsof(5.1). Substituting
0 1 2
(5.5)intothelastequationof(5.1),weseethattheprojectionSx =Sx(m ,m ,m )
0 1 2
of S(m ,m ,m ) onto the (x ,x )-subspace x is obtained in x as the
0 1 2 1 2 P2 ⊂ P2 P2
solution of the third degree equation
m x m m x +m
(5.6) 0 2− 1x2+ − 0 1 1x2 =m .
x x 1 x x 2 2
2 1 2 1
− −
An explicit description of the curve Sx can be obtained as follows: we can rewrite
the left hand side of equation (5.6) in the form
1
(5.7) [m (x2x x x2)+m (x2 x2)]= m x x +m (x +x ),
x x 0 1 2− 1 2 1 2− 1 − 0 1 2 1 1 2
2 1
−
which leads to the equation
(5.8) m x x m (x +x )+m =0
0 1 2 1 1 2 2
−
for the curve Sx(m ,m ,m ). So this curve is a hyperbola with the center at the
0 1 2
point (m1,m1), and with the asymptotes x = m1, x = m1. Equation 5.8 is
m0 m0 1 m0 2 m0