Table Of ContentGEOMETRY AND SINGULARITIES OF THE PRONY MAPPING
DMITRYBATENKOVANDYOSEFYOMDIN
3
1
0 Abstract. Pronymappingprovides theglobalsolutionofthePronysystem
2 ofequations
n Σni=1Aixki =mk, k=0,1,...,2n−1.
a This system appears innumerous theoretical and appliedproblems arisingin
J SignalReconstruction. Thesimplestexampleistheproblemofreconstruction
7 of linear combination of δ-functions of the form g(x) = ni=1aiδ(x−xi),
with the unknown parameters ai, xi, i = 1,...,n, from the “moment mea-
] surements” mk = xkg(x)dx. P
A Global solution´ofthe Prony system, i.e. inversion ofthe Prony mapping,
encounters several typesofsingularities. Oneofthemostimportantonesisa
N
collisionofsomeofthepointsxi.Theinvestigationofthistypeofsingularities
. hasbeenstartedin[21]wheretheroleoffinitedifferences wasdemonstrated.
h
In the present paper we study this and other types of singularities of the
t
a Pronymapping,anddescribeitsglobalgeometry. Weshow,inparticular,close
m connections of the Prony mapping with the “Vieta mapping” expressing the
coefficientsofapolynomialthroughitsroots,andwithhyperbolicpolynomials
[
and“Vandermondemapping” studiedbyV.Arnold.
1
v
6
3
3
1. Introduction
1
.
1
Pronysystemappearsaswe try to solvea verysimple “algebraicsignalreconstruc-
0
tion” problem of the following form: assume that the signal F(x) is known to be a
3
linear combination of shifted δ-functions:
1
:
v d
i F(x)= ajδ(x xj). (1.1)
X −
j=1
X
r
a We shall use as measurements the polynomial moments:
m =m (F)= xkF (x)dx. (1.2)
k k ˆ
After substituting F into the integral defining m we get
k
d d
m (F)= xk a δ(x x )dx= a xk.
k ˆ j − j j j
j=1 j=1
X X
2000 Mathematics Subject Classification. 94A12 62J02, 14P10, 42C99.
Key words and phrases. Singularities, Signal acquisition, Non-linear models, Moments
inversion.
ThisresearchissupportedbytheAdamsFellowshipProgramoftheIsraelAcademyofSciences
andHumanities,ISFgrant264/09 andtheMinervaFoundation.
1
GEOMETRY AND SINGULARITIES OF THE PRONY MAPPING 2
Considering a and x as unknowns, we obtain equations
j j
d
m (F)= a xk, k =0,1,.... (1.3)
k j j
j=1
X
This infinite set of equations (or its part, for k = 0,1,...,2d 1), is called Prony
−
system. It can be traced at least to R. de Prony (1795, [19]) and it is used in a
wide variety of theoretical and applied fields. See [2] for an extensive bibligoraphy
on the Prony method.
In writing Prony system (1.3) we have assumed that all the nodes x ,...,x are
1 d
pairwise different. However, as a right-hand side µ = (m ,...,m ) of (1.3) is
0 2d 1
−
providedbytheactualmeasurementsofthesignalF,wecannotguaranteeapriori,
that this condition is satisfied for the solution. Moreover, we shall see below that
multiplenodesmaynaturallyappearinthesolutionprocess. Inordertoincorporate
possible collisions of the nodes, we consider “confluent Prony systems”.
AssumethatthesignalF(x)isalinearcombinationofshiftedδ-functionsandtheir
derivatives:
s dj−1
F(x)= a δ(ℓ)(x x ). (1.4)
j,ℓ j
−
j=1 ℓ=0
X X
def
Definition 1.1. For F(x) as above, the vector D(F) = (d ,...,d ) is the multi-
1 s
def
plicity vector of F, s = s(F) is the size of its support, T (F) = (x ,...,x ), and
1 s
rank(F) d=ef s d is its rank. For avoiding ambiguity in these definitions, it is
j=1 j
alwaysunderstoodthata =0forallj =1,...,s(i.e. d isthemaximalindex
P j,dj−1 6 j
for which a =0).
j,dj−1 6
For the moments m =m (F)= xkF(x)dx we now get
k k
´
s dj−1 k!
mk = aj,ℓ(k ℓ)!xjk−ℓ.
j=1 ℓ=0 −
X X
Considering x and a as unknowns, we obtain a system of equations
i j,ℓ
s dj−1 k!
(k ℓ)!aj,ℓxjk−ℓ =mk, k =0,1,...,2d−1, (1.5)
j=1 ℓ=0 −
X X
which is called a confluent Prony system of order d with the multiplicity vector
D =(d ,...,d ). The originalProny system (1.3) is a special case of the confluent
1 s
one, with D being the vector (1,...,1) of length d.
The system (1.5) arises also in the problem of reconstructing a planar polygon P
(or even an arbitrary semi-analytic quadrature domain) from its moments
m (χ )= zkχ dxdy, z =x+ıy,
k P ¨ P
R2
where χ is the characteristic function of the domain P R2. This problem is
P
⊂
important in many areas of science and engineering [11]. The above yields the
GEOMETRY AND SINGULARITIES OF THE PRONY MAPPING 3
confluent Prony system
s dj−1
mk = ci,jk(k−1)···(k−i+1)zjk−i, ci,j ∈C, zj ∈C\{0}.
j=1 i=0
X X
Definition 1.2. For a given multiplicity vector D = (d ,...,d ), its order is
1 s
s d .
j=1 j
APsweshallseebelow,ifwestartwiththemeasurementsµ(F)=µ=(m ,...,m ),
0 2d 1
−
then a natural setting of the problem of solving the Prony system is the following:
Problem 1.3 (Prony problem of order d). Given the measurements
µ=(m ,...,m ) C2d
0 2d 1
− ∈
in the right hand side of (1.5), findthe multiplicity vector D =(d ,...,d )of order
1 s
r= s d d, and find the unknowns x and a , which solve the corresponding
j=1 j ≤ j j,ℓ
confluent Prony system (1.5) with the multiplicity vector D (hence, with solution
P
of rank r).
It is extremely important in practice to have a stable method of inversion. Many
researcheffortsaredevotedtothistask(seee.g. [3,7,10,17,18,20]andreferences
therein). A basic question here is the following.
Problem 1.4 (Noisy Prony problem). Given the noisy measurements
µ˜ =(m˜ ,...,m˜ ) C2d
0 2d 1
− ∈
and an estimate of the error m˜ m ε , solve Problem 1.3 so as to minimize
k k k
| − |≤
the reconstruction error.
In this paper we study the global setting of the Prony problem, stressing its alge-
braic structure. In Section 2 the space where the solution is to be found (Prony
space) is described. It turns out to be a vector bundle over the space of the nodes
x ,...,x . We define also three mappings: “Prony”, “Taylor”, and “Stieltjes” ones,
1 d
which capture the essential features of the Prony problem and of its solution pro-
cess.
InSection3weinvestigatesolvabilityconditionsforthePronyproblem. Theanswer
leads naturally to a stratification of the space of the right-handsides, according to
the rank of the associatedHankel-type matrix and its minors. The behavior of the
solutionsnearvariousstrataturnsouttobehighlynontrivial,andwepresentsome
initial results in the description of the corresponding singularities.
InSection4,westudythemultiplicity-restrictedPronyproblem,fixingthecollision
pattern of the solution, and derive simple bounds for the stability of the solution
via factorization of the Jacobian determinant of the corresponding Prony map.
In Section 5 we consider the rank-restricted Prony problem, effectively reducing
the dimension to 2r instead of 2d, where r is precisely the rank of the associated
Hankel-type matrix. In this formulation, the Prony problem is solvable in a small
neighborhood of the exact measurement vector.
InSection6westudyoneofthemostimportantsingularitiesinthePronyproblem:
collisionofsome of the points x . The investigationof this type ofsingularities has
i
been started in [21] where the role of finite differences was demonstrated. In the
presentpaperweintroduceglobalbasesoffinite differences,study theirproperties,
GEOMETRY AND SINGULARITIES OF THE PRONY MAPPING 4
and prove that using such bases we can resolve in a robust way at least the linear
part of the Prony problem at and near colliding configurations of the nodes.
In Section 7 we discuss close connections of the Prony problem with hyperbolic
polynomials and “Vandermonde mapping” studied by V.I.Arnold in [1] and by
V.P.Kostov in [13, 14, 15], and with “Vieta mapping” expressing the coefficients
of a polynomial through its roots. We believe that questions arising in theoretical
studyofPronyproblemandinitspracticalapplicationsjustifyfurtherinvestigation
of these connections, as well as further applications of Singularity Theory.
Finally, in Appendix A we describe a solution method for the Prony system based
on Padé approximation.
2. Prony, Stieltjes and Taylor Mappings
Inthis sectionwe define “Prony”,“Taylor”,and“Stieltjes” mappings,whichcapture
someessentialfeatures ofthe Prony problemandof its solutionprocess. The main
idea behind the spaces and mappings introduced in this section is the following:
associate to the signal F(x) = d a δ(x x ) the rational function R(z) =
i=1 i − i
d ai . (In fact, R is the Stieltjes integral transform of F). The functions
Rio=b1taz−inxeid in this way can be wPritten as R(z) = P(z) with degP degQ 1,
P Q(z) ≤ −
and they satisfy R( ) = 0. Write R as R(z) = d zai . Developing the
summands into geom∞etric progressions we conclude tPhait=1R1(−zx)i=/z ∞k=0mk(z1)k+1,
with m = d a xk, so the moment measurements m in the right hand side
k i=1 i i k P
of the Prony system (1.3) are exactly the Taylor coefficients of R(z). We shall
P
see below that this correspondence reduces solution of the Prony system to an
appropriate Padé approximationproblem.
Definition 2.1. For each w = (x ,...,x ) Cd, let s = s(w) be the number
1 d
∈
of distinct coordinates τ , j = 1,...,s, and denote T (w) = (τ ,...,τ ). The
j 1 s
multiplicity vector is D = D(w) = (d ,...,d ), where d is the number of times
1 s j
the value τ appears in x ,...,x . The order ofthe values in T (w) is defined by
j 1 d
{ }
their order of appearance in w.
Example 2.2. Forw =(3,1,2,1,0,3,2)wehaves(w)=4,T (w)=(3,1,2,0)and
D(w)=(2,2,2,1).
Remark 2.3. Note the slight abuse of notations between Definition 1.1 and Defini-
tion 2.1. Note also that the order of D(w) equals to d for all w Cd.
∈
Definition 2.4. For each w Cd, let s=s(w), T (w)=(τ ,...,τ ) and D(w)=
1 s
∈
(d ,...,d ) be as in Definition 2.1.
1 s
(1) V is the vector space of dimension d containing the linear combinations
w
s dj−1
g = γ δ(ℓ)(x τ ) (2.1)
j,ℓ j
−
j=1 ℓ=0
X X
of δ-functions and their derivatives at the points of T(w). The “standard
basis” of V is given by the distributions
w
δ =δ(ℓ)(x τ ), j =1,...,s(w); ℓ=0,...,d 1. (2.2)
j,ℓ j j
− −
GEOMETRY AND SINGULARITIES OF THE PRONY MAPPING 5
(2) W is the vector space of dimension d of all the rational functions with
w
poles T(w) and multiplicities D(w), vanishing at :
∞
s
P (z)
R(z)= , Q(z)= (z τ )dj, degP (z)<degQ6d.
j
Q(z) −
j=1
Y
The “standard basis” of W is given by the elementary fractions
w
1
R = , j =1,...,s; ℓ=1,...,d .
j,ℓ (z τ )ℓ j
j
−
Nowwe arereadyto formallydefine thePronyspace andthe Stieltjes space .
d d
P S
Definition 2.5. The Prony space is the vector bundle over Cd, consisting of
d
P
all the pairs
(w,g): w Cd, g V .
w
∈ ∈
The topologyon is induced by the naturalembedding Cd , where is
d d
the space of distrPibutions on C with its standard topologyP. ⊂ ×D D
Definition 2.6. The Stieltjes space is the vector bundle over Cd, consisting of
d
S
all the pairs
(w,γ): w Cd, γ W .
w
∈ ∈
The topology on is induced by the naturalembedding Cd , where is
d d
S S ⊂ ×R R
the space of complex rational functions with its standard topology.
Definition 2.7. The Stieltjes mapping : is defined by the Stieltjes
d d
SM P → S
integral transform: for (w,g)
d
∈P
∞ g(x)dx
((w,g))=(w,γ), γ(z)= .
SM ˆ z x
−∞ −
Sometimes we abuse notation and write for short (g) = γ, with the under-
standing that is also a map :V W foSrMeach w Cd.
w w
SM SM → ∈
The following fact is immediate consequence of the above definitions.
Proposition2.8. is a linear isomorphism of the bundles and (for each
d d
w Cd, is aSliMnear isomorphism of the vector spaces VPand WS ). In the
w w
∈ SM
standard bases of V and W , the map is diagonal, satisfying
w w
SM
(δ )=( 1)ℓℓ!R (z).
j,ℓ j,ℓ
SM −
Furthermore, for any (w,g)
d
∈P
P (z)
(g)= , degP <degQ=rank(g)6d. (2.3)
SM Q(z)
irreducible
Definition 2.9. The Tay|lo{rzs}pace d is the space of complex Taylor polynomials
atinfinity of degree 2d−1 of the forTm 2kd=−01mk(1z)k+1. We shall identify Td with
the complex space C2d with the coordinates m ,...,m .
0 2d 1
P −
GEOMETRY AND SINGULARITIES OF THE PRONY MAPPING 6
Definition 2.10. The Taylor mapping : is defined by the truncated
d d
TM S →T
Taylor development at infinity:
2d−1 1 k+1 ∞ 1 k+1
((w,γ))= α , where γ(z)= α .
k k
TM z z
k=0 k=0
X X
We identify ((w,γ))as abÅoveãwith (α ,...,α ) C2d. SomÅetiãmes we write
0 2d 1
TM − ∈
for short (γ)=(α ,...,α ).
0 2d 1
TM −
Finally, we define the Prony mapping which encodes the Prony problem.
PM
Definition 2.11. The Prony mapping : C2d for (w,g) is defined
d d
PM P → ∈ P
as follows:
((w,g))=(m ,...,m ) C2d, m =m (g)= xkg(x)dx.
PM 0 2d−1 ∈ k k ˆ
By the above definitions, we have
= . (2.4)
PM TM◦SM
Solvingthe Pronyproblemfor a givenright-handside (m ,...,m ) is therefore
0 2d 1
−
equivalent to inverting the Prony mapping . As we shall elaborate in the sub-
PM
sequent section, the identity (2.4) allows us to split this problem into two parts:
inversion of , which is, essentially, the Padé approximation problem, and in-
TM
versionof , whichis,essentially,the decompositionofagivenrationalfunction
SM
into the sum of elementary fractions.
3. Solvability of the Prony problem
3.1. General condition for solvability. In this section we provde a necessary
and sufficient condition for the Prony problem to have a solution (which is unique,
as it turns out by Proposition 3.2). As mentioned in the end of the previous
section, our method is based on inverting (2.4) and thus relies on the solution of
the corresponding (diagonal) Padé approximation problem [4].
Problem3.1(DiagonalPadéapproximationproblem). Givenµ=(m ,...,m )
0 2d 1
C2d, find a rational function R (z)= P(z) with degP <degQ6d, such th−at ∈
d Q(z) ∈Sd
the first 2d Taylor coefficients at infinity of Rd(z) are {mk}2kd=−01.
Proposition 3.2. A solution to Problem 3.1, if exists, is unique.
Proof. WritingR(z)= P(z), R (z)= P1(z),withdegP <degQ6danddegP <
Q(z) 1 Q1(z) 1
degQ 6d, we get
1
PQ P Q
1 1
R R = − ,
1
− QQ
1
andthisfunction,ifnonzero,canhaveazerooforderatmost2d 1atinfinity. (cid:3)
−
Let us summarize the above discussion with the following statement.
GEOMETRY AND SINGULARITIES OF THE PRONY MAPPING 7
Proposition 3.3. The tuple
s
s, D =(d ,...,d ), r = d d, X = x s , A= a
( 1 s j ≤ { j}j=1 { j,ℓ}j=1,...,s;ℓ=0,...,dj−1)
j=1
X
is a (unique, up to a permutation of the nodes x ) solution to Problem 1.3 with
j
{ }
right-hand side
µ=(m ,...,m ) C2d
0 2d 1
− ∈
if and only if the rational function
RD,X,A(z)= s dj (−1)ℓ−1(ℓ−1)!(zaj,ℓx−1)ℓ =2d−1zmk+k1 +O z−2d−1
Xj=1Xℓ=1 − j Xk=0 (cid:0) (cid:1)
is a (unique) solution to Problem 3.1 with input µ. In that case,
R (z)= ∞ g(x)dx where g(x)= s dj−1a δ(ℓ)(x x ),
D,X,A ˆ z x j,ℓ − j
−∞ − Xj=1 Xℓ=0
i.e. R (z) is the Stieltjes transform of g(x).
D,X,A
Proof. This follows from the definitions of Section 2, (2.4), Proposition 3.2 and
the fact that the problem of representing a given rational function as a sum of
elementary fractions of the specified form (i.e. inverting ) is always uniquely
SM
solvable up to a permutation of the poles. (cid:3)
The next result provides necessary and sufficient conditions for the solvability of
Problem 3.1. It summarizes some well-known facts in the theory of Padé approx-
imation, related to “normal indices” (see, for instance, [4]). However, these facts
are not usually formulated in the literature on Padé approximationin the form we
need in relation to the Prony problem. Consequently, we give a detailed proof of
this result in Appendix A. This proof contains, in particular, some facts which are
important for understanding the solvability issues of the Prony problem.
Definition3.4. Givenavectorµ=(m ,...,m ),letM˜ denotethed (d+1)
0 2d 1 d
− ×
Hankel matrix
m m m ... m
0 1 2 d
m m m ... m
1 2 3 d+1
M˜d = ... ... ... ... ... . (3.1)
m m m ... m
d−1 d d+1 2d−1
For each e6d, denote by M˜ the e (e+1) submatrix of M˜ formed by the first
e d
×
e rows and e+1 columns, and let M denote the corresponding square matrix.
e
Theorem 3.5. Let µ=(m ,...,m ) be given, and let r6d be the rank of the
0 2d 1
Hankel matrix M˜ as in (3.1). Then P−roblem 3.1 is solvable for the input µ if and
d
only if the upper left minor M of M˜ is non-zero.
r d
| |
As an immediate consequence of Theorem 3.5 and Proposition 3.3, we obtain the
following result.
GEOMETRY AND SINGULARITIES OF THE PRONY MAPPING 8
Theorem 3.6. Let µ = (m ,...,m ) be given, and let r 6 d be the rank of
0 2d 1
the Hankel matrix M˜ as in (3.1). Th−en Problem 1.3 with input µ is solvable if
d
and only if the upper left minor M of M˜ is non-zero. The solution, if exists,
r d
| |
is unique, up to a permutation of the nodes x . The multiplicity vector D =
j
(d ,...,d ), of order s d =r, of the resul{ting}confluent Prony system of rank
1 s j=1 j
r is the multiplicity vector of the poles of the rational function R (z), solving
D,X,A
P
the corresponding Padé problem.
As a corollary we get a complete description of the right-hand side data µ C2d
∈
for which the Prony problem is solvable (unsolvable). Define for r = 1,...,d sets
Σ C2d (respectively, Σ C2d) consisting of µ C2d for which the rank
r ⊂ ′r ⊂ ∈
of M˜ = r and M = 0 (respectively, M = 0). The set Σ is a difference
d r r r
| | 6 | |
Σ =Σ1 Σ2 oftwoalgebraicsets: Σ1 isdefinedbyvanishingofallthes sminors
ofrM˜ , rr\<sr d, while Σ2 is definedrby vanishingof M . Inturn, Σ =×Σ′1 Σ′2,
withdΣ′1 =Σ1≤ Σ2 andΣ′r2 definedbyvanishing ofal|lthre| r r minor′rsofM˜r \.Thre
union Σr Σr∩conrsists ofrall µ for which the rank of M˜ =r×, which is Σ1 Σd′2.
r∪ ′r d r \ r
Corollary3.7. ThesetΣ(respectively,Σ)ofµ C2d forwhichthePronyproblem
′
∈
is solvable (respectively, unsolvable) is the union Σ = d Σ (respectively, Σ =
d Σ ). In particular, Σ µ C2d,detM =0 . ∪r=1 r ′
∪r=1 ′r ′ ⊂{ ∈ d }
So for a generic right hand side µ we have M = 0, and the Prony problem
d
| | 6
is solvable. On the algebraic hypersurface of µ for which M = 0, the Prony
d
| |
problem is solvable if M =0, etc.
d 1
− 6
Let us now consider some examples.
Example 3.8. Let us fix d = 1,2,.... Consider µ = (m ,...,m ) C2d,
0 2d 1
− ∈
the right hand sides of the Prony problem, to be of the form µ = µ = (δ ) =
ℓ kℓ
(0,...,0, 1 ,0,...,0),withallthem =0besidesm =1, ℓ=0,...,2d 1,
k ℓ
−
positionℓ+1
and let M˜ℓ be the corresponding matrix.
d |{z}
Proposition 3.9. The rank of M˜ℓ is equal to ℓ+1 for ℓ d 1, and it is equal
d ≤ −
to 2d ℓ for ℓ d. The corresponding Prony problem is solvable for ℓ d 1, and
− ≥ ≤ −
it is unsolvable for ℓ d.
≥
Proof. For d=5 and ℓ=2,4,5,9, the corresponding matrices M˜d are as follows.
ℓ
0 0 1 0 0 0 0 0 0 0 1 0
0 1 0 0 0 0 0 0 0 1 0 0
M˜2 = 1 0 0 0 0 0, M˜4 =0 0 1 0 0 0, (solvable)
5 5
0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0
M˜5 = 0 0 0 1 0 0, M˜9 =0 0 0 0 0 0. (unsolvable)
5 5
0 0 1 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 1
GEOMETRY AND SINGULARITIES OF THE PRONY MAPPING 9
Ingeneral,the matricesM˜ℓ havethe same patternasin the specialcasesabove,so
d
their rank is ℓ+1 for ℓ6d 1, and 2d ℓ for ℓ>d, as stated above. Application
− −
of Theorem 3.6 completes the proof. (cid:3)
In fact, µ is a moment sequence of
ℓ
1
F (x)= δ(ℓ)(x),
ℓ!
and this signal belongs to if and only if ℓ6d 1. In notations of Corollary3.7
d
P −
we have
µ Σ , ℓ6d 1,
ℓ ℓ+1
∈ −
µℓ ∈ Σ′2d−ℓ, ℓ>d.
It is easy to provide various modifications of the above example. In particular, for
µ=µ˜ =(0,...,0,1,1,...,1), the result of Proposition 3.9 remains verbally true.
ℓ
Example3.10. Anotherexampleisprovidedbyµ ,withallthem =0besides
ℓ1,ℓ2 k
m = 1, m = 1, 0 ℓ < d ℓ 2d 1. For ℓ < ℓ d+1 the rank of
ℓ1 ℓ2 ≤ 1 ≤ 2 ≤ − 1 2 −
the correspondent matrix M˜ is r =2d+ℓ ℓ +1 while M =0, so the Prony
d 1 2 r
− | |
problem for such µ is unsolvable. For d = 5 and ℓ = 2, ℓ = 8 the matrix is
ℓ1,ℓ2 1 2
as follows:
0 0 1 0 0 0
0 1 0 0 0 0
M˜(2,8) =1 0 0 0 0 0.
5
0 0 0 0 0 1
0 0 0 0 1 0
3.2. Near-singular inversion. The behavior of the inversion of the Prony map-
ping near the unsolvability stratum Σ and near the strata where the rank of M˜
′ d
drops, turns out to be pretty complicated. In particular, in the first case at least
one of the nodes tends to infinity. In the second case, depending on the way the
right-handsideµapproachesthelowerrankstrata,thenodesmayremainbounded,
or some of them may tend to infinity. In this section we provide one initial result
in this direction, as well as some examples. A comprehensive description of the in-
version of the Prony mapping near Σ and near the lower rank strata is important
′
bothin theoreticalstudy andin applicationsofProny-likesystems,andwe plan to
provide further results in this direction separately.
Theorem 3.11. As the right-hand side µ C2d Σ approaches a finite point
′
∈ \
µ Σ, at least one of the nodes x ,...,x in the solution tends to infinity.
0 ′ 1 d
∈
Proof. By assumptions, the components m ,...,m of the right-hand side µ=
0 2d 1
(m ,...,m ) C2d remain bounded as µ µ .−By Theorem 6.14, the finite
0 2d 1 0
differences co−ordi∈nates of the solution −1(→µ) remain bounded as well. Now, if
PM
allthe nodesarealsobounded,bycompactnessweconcludethat −1(µ) ω
PM → ∈
. By continuity in the distribution space (Lemma 6.6) we have (ω) = µ .
d 0
P PM
Hence the Prony problem with the right-hand side µ has a solution ω , in
0 d
∈ P
contradiction with the assumption that µ Σ. (cid:3)
0 ′
∈
GEOMETRY AND SINGULARITIES OF THE PRONY MAPPING 10
Example 3.12. Let us consider an example: d=2 and µ =(0,0,1,0). Here the
0
rankℓofM˜ is2,and M =0,sobyTheorem3.6wehaveµ Σ Σ. Consider
2 | 2| 0 ∈ ′2 ⊂ ′
now a perturbation µ(ǫ)=(0,ǫ,1,0) of µ . For ǫ=0 we have µ(ǫ) Σ Σ, and
0 2
6 ∈ ⊂
thePronysystemissolvableforµ . Letuswriteanexplicitsolution: thecoefficients
ǫ
c ,c of the polynomial Q(z)=c +c z+z2 we find from the system (A.⋆⋆):
0 1 0 1
0 ǫ c 1
0 = − ,
ǫ 1 c 0
1
whose solution is c1 = −1ǫ, cï0 = òǫ12ï. Hòencïe thòe denominator Q(z) of R(z) is
Q(z) = 1 1z+z2, and its roots are x = 1+ı√3, x = 1 ı√3. The coefficients
ǫ2 − ǫ 1 2ǫ 2 −2ǫ
b ,b of the numerator P(z)=b +b z we find from (A.⋆):
0 1 0 1
0 0 1 b
−ǫ = 1 ,
0 ǫ 1 b
0
i.e. b =0, b =ǫ. Thus the solution of the associated Padé problem is
1 0 ï òï ò ï ò
P(z) ǫ ǫ2 1 ǫ2 1
R(z)= = = .
Q(z) (z x1)(z x2) ı√3(z x1) − ı√3(z x2)
− − − −
Finally, the (unique up to a permutation) solution of the Prony problem for µ is
ǫ
ǫ2 ǫ2 1+ı√3 1 ı√3
a = , a = , x = , x = − .
1 2 1 2
ı√3 −ı√3 2ǫ 2ǫ
Asǫtendstozero,thenodesx ,x tendtoinfinitywhilethecoefficientsa ,a tend
1 2 1 2
to zero.
As it was shown above, for a given µ Σ (say, with pairwise different nodes) the
∈
rankofthematrixM˜ isequaltothenumberofthenodesinthesolutionforwhich
d
the corresponding δ-function enters with a non-zero coefficients. So µ approaches
a certain µ belonging to a stratum of a lower rank of M˜ if and only if some of
0 d
the coefficients a in the solution tend to zero. We do not analyze all the possible
j
scenarios of such a degeneration, noticing just that if µ Σ, i.e., the Prony
0 ′
∈
problem is unsolvable for µ , then Theorem 3.11 remains true, with essentially the
0
same proof. So at least one of the nodes, say, x , escapes to infinity. Moreover,
j
one can show that ajxj2d−1 cannot tend to zero - otherwise the remaining linear
combination of δ-functions would provide a solution for µ .
0
If µ Σ, i.e., the Prony problem is solvable for µ , all the nodes may remain
0 0
bound∈ed, or some xj may escape to infinity, but in such a way that ajx2jd−1 tends
to zero.
4. Multiplicity-restricted Prony problem
Consider Problem 1.4 at some point µ Σ. By definition, µ Σ for some
0 ∈ 0 ∈ r0
r d. Letµ = ((w ,g ))forsome(w ,g ) . Assumeforamomentthat
0 0 0 0 0 0 d
the≤multiplicity vPecMtor D =D(g )=(d ,...d )∈, Ps0 d =r , has a non-trivial
0 0 1 s0 j=1 j 0
collision pattern, i.e. d > 1 for at least one j = 1,...,s . It means, in turn, that
j 0
P
the function R (z) has a pole of multiplicity d . Evidently, there exists an
D0,X,A j
arbitrarilysmallperturbationµ˜ofµ forwhichthismultiplepolebecomesacluster
0
of single poles, thereby changing the multiplicity vector to some D = D . While
′ 0
6
we address this problem in Section 6 via the bases of divided differences, in this
section we consider a “multiplicity-restricted” Prony problem.
