Time and Band Limiting for Matrix Valued Functions, an Example
The main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of p...
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| Cite this: | Time and Band Limiting for Matrix Valued Functions, an Example / F.A. Grünbaum, I. Pacharoni, I.N. Zurrián // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. |
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| citation_txt | Time and Band Limiting for Matrix Valued Functions, an Example / F.A. Grünbaum, I. Pacharoni, I.N. Zurrián // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. |
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| description | The main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example showing in a non-commutative setup that a bispectral property implies that the corresponding global operator of ''time and band limiting'' admits a commuting local operator. This is a noncommutative analog of the famous prolate spheroidal wave operator.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 044, 14 pages
Time and Band Limiting for Matrix Valued Functions,
an Example?
F. Alberto GRÜNBAUM †, Inés PACHARONI ‡ and Ignacio Nahuel ZURRIÁN ‡
† Department of Mathematics, University of California, Berkeley 94705, USA
E-mail: grunbaum@math.berkeley.edu
‡ CIEM-FaMAF, Universidad Nacional de Córdoba, Córdoba 5000, Argentina
E-mail: pacharon@famaf.unc.edu.ar, zurrian@famaf.unc.edu.ar
Received February 11, 2015, in final form May 30, 2015; Published online June 12, 2015
http://dx.doi.org/10.3842/SIGMA.2015.044
Abstract. The main purpose of this paper is to extend to a situation involving matrix
valued orthogonal polynomials and spherical functions, a result that traces its origin and its
importance to work of Claude Shannon in laying the mathematical foundations of informa-
tion theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak.
To our knowledge, this is the first example showing in a non-commutative setup that a bis-
pectral property implies that the corresponding global operator of “time and band limiting”
admits a commuting local operator. This is a noncommutative analog of the famous prolate
spheroidal wave operator.
Key words: time-band limiting; double concentration; matrix valued orthogonal polynomials
2010 Mathematics Subject Classification: 33C45; 22E45; 33C47
1 Introduction
In a ground-breaking paper laying down the mathematical foundations of communication theory,
Claude Shannon [21, 22] considers a basic problem in harmonic analysis and signal processing:
how to best concentrate a function both in physical and frequency space. This issue was an
important part of the work of C. Shannon for several years after the publication of this paper.
The problem was appeared earlier in several versions and one should at least mention the role
of the Heisenberg inequality in this context: for a nice and simple proof it – due to W. Pauli –
see Hermann Weyl’s book [32].
What is really novel in Shannon and coworkers’s look at this problem is the following question:
suppose you consider an unknown signal f(t) of finite duration, i.e., the signal is non-zero only
in the interval [−T, T ]. The data you have are the values of the Fourier transform Ff(k) of f
for values of k in the interval [−W,W ]. What is the best use you can make of this data?
In practice, the values of Ff(k) will be corrupted by noise and one is dealing with a typical
situation in signal processing: recovering an image from partial and noisy data in the presence
of some apriori information.
This problem was treated originally by Shannon himself but a full solution had to wait for
the joint work, in different combinations, of three remarkable workers at Bell labs in the 1960’s:
David Slepian, Henry Landau and Henry Pollak, see [12, 13, 25, 27, 29].
A very good account of this development is a pair of papers by David Slepian [26, 28]. The
first one is essentially the second Shannon lecture given at the International Symposium on
Information Theory in 1975. The abstract starts with the sentence “It is easy to argue that real
?This paper is a contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of
Luc Vinet. The full collection is available at http://www.emis.de/journals/SIGMA/ESSA2014.html
mailto:grunbaum@math.berkeley.edu
mailto:pacharon@famaf.unc.edu.ar
mailto:zurrian@famaf.unc.edu.ar
http://dx.doi.org/10.3842/SIGMA.2015.044
http://www.emis.de/journals/SIGMA/ESSA2014.html
2 F.A. Grünbaum, I. Pacharoni and I.N. Zurrián
signals must be bandlimited. It is also easy to argue that they cannot be so”. The ideas in this
paper took their definite form in [14].
The second paper is on the occasion of the John von Neumann lecture given at the SIAM
30th anniversary meeting in 1982. Here is a quote from the second paper: “There was a lot of
serendipity here, clearly. And then our solution, too, seemed to hinge on a lucky accident-namely
that we found a second-order differential operator that commuted with an integral operator that
was at the heart of the problem”.
What these workers found was that instead of looking for the unknown f(t) itself one should
consider a certain integral operator with discrete spectrum in the open interval (0, 1) and a re-
markable “spectral gap”: about [4WT ] (integer part of 2W × 2T ) eigenvalues are positive, and
all the remaining ones are essentially zero. They argue that in the presence of noisy data one
should try to compute the projection of f(t) on the linear span of the eigenfunctions with “large”
eigenvalues. The effective computation of these eigenfunctions is made possible by replacing the
integral operator by the commuting differential one alluded to by D. Slepian (both have simple
spectrum). From a theoretical point of view these eigenfunctions are the same, but using the
differential operator instead of the integral one, we have a manageable numerical problem. For
a very recent account of several computational issues see [15, 11] and [1]. For new areas of
applications involving (sometimes) vector-valued quantities on the sphere, see [10, 20, 23, 24].
We still have to answer the question: How do you explain the existence of this local commuting
operator?
To this day nobody has a simple explanation for this miracle. Indeed there has been a system-
atic effort to see if the “bispectral property” first considered in [3], guarantees the commutativity
of these two operators, a global and a local one. A few papers where this question has been
taken up, include [4, 6, 5, 7, 8, 9, 18, 19].
The results in the present paper are a (first) natural extension of the work in [4, 8], where
the classical orthogonal polynomials played a central role, to a matrix-valued case involving
matrix orthogonal polynomials. In a case such as in [4, 8] or in the present paper where physical
and frequency space are of a different nature (one is continuous and the other one is discrete)
one can deal (as explained in [4, 8]) with either an integral operator or with a full matrix.
Each one of these global objects will depend on two parameters that play the role of (T,W )
in Shannon’s case, and one will be looking for a commuting local object, i.e., a second-order
differential operator or a tridiagonal matrix. In this paper we will be dealing with a full matrix
and a (block) tridiagonal one.
Finally a word about possible applications. The paper [8] was written with no particular
application in mind, but with the expectation that the analysis of functions defined on the
sphere could benefit from it. A few years later some applications did emerge, see [23, 24] and
its references. One can only hope that the present paper dealing with matrix valued functions
defined on spheres will find a natural application in the future.
2 Preliminaries
Let W = W (x) be a weight matrix of size R in the open interval (a, b). By this we mean
a complex R × R-matrix valued integrable function W on the interval (a, b) such that W (x)
is positive definitive almost everywhere and with finite moments of all orders. Let Qw(x) be
a sequence of real valued matrix orthonormal polynomials with respect to the weight W (x).
Consider the following two Hilbert spaces: The space L2((a, b),W (t)dt), or simply denoted
by L2(W ), of all matrix valued measurable matrix valued functions f(x), x ∈ (a, b), satisfying∫ b
a tr(f(x)W (x)f∗(x))dx <∞ and the space `2(MR,N0) of all real valued R×R matrix sequences
(Cw)w∈N0 such that
∞∑
w=0
tr(CwC
∗
w) <∞.
Time and Band Limiting for Matrix Valued Functions, an Example 3
The map F : `2(MR,N0) −→ L2(W ) given by
(Aw)∞w=0 7−→
∞∑
w=0
AwQw(x)
is an isometry. If the polynomials are dense in L2(W ), this map is unitary with the inverse
F−1 : L2(W ) −→ `2(MR,N0) given by
f 7−→ Aw =
∫ b
a
f(x)W (x)Q∗w(x)dx.
We call our map F to remind ourselves of the usual Fourier transform. Here N0 takes up the
role of “physical space” and the interval (a, b) the role of “frequency space”. This is, clearly,
a noncommutative extension of the problem raised by C. Shannon since he was concerned with
scalar valued functions and we are dealing with matrix valued ones.
The time limiting operator, at level N acts on `2(MR,N0) by simply setting equal to zero all
the components with index larger than N . We denote it by χN . The band limiting operator,
at level α, acts on L2(W ) by multiplication by the characteristic function of the interval (a, α),
α ≤ b. This operator will be denoted by χα.
One could consider restricting the band to an arbitrary subinterval (a1, b1). However, the
algebraic properties exhibited here, see Section 5 and beyond, hold only with this restriction.
A similar situation arises in the classical case going all the way back to Shannon.
Consider the problem of determining a function f , from the following data: f has support on
the finite set {0, . . . , N} and its Fourier transform Ff is known on a compact set [a, α]. This
can be formalized as follows
χαFf = g = known, χNf = f.
We can combine the two equations into
Ef = χαFχNf = g.
To analyze this problem we need to compute the singular vectors (and values) of the operator
E : `2(MR,N0) −→ L2(W ). These are given by the eigenvectors of the operators
E∗E = χNF−1χαFχN and S2 = EE∗ = χαFχNF−1χα.
The operator E∗E, acting in `2(MR,N0) is just a finite dimensional block-matrix M , and each
block is given by
(M)m,n = (E∗E)m,n =
∫ α
a
Qm(x)W (x)Q∗nw(x)dx, 0 ≤ m,n ≤ N.
The second operator S2 = EE∗ acts in L2((a, α),W (t)dt) by means of the integral kernel
k(x, y) =
N∑
w=0
Qw(x)Q∗w(y).
Consider now the problem of finding the eigenfunctions of E∗E and EE∗. For arbitrary N
and α there is no hope of doing this analytically, and one has to resort to numerical methods
and this is not an easy problem. Of all the strategies one can dream of solving this problem,
none sounds so appealing as that of finding an operator with simple spectrum which would have
the same eigenfunctions as the original operators. This is exactly what Slepian, Landau and
Pollak did in the scalar case, when dealing with the real line and the actual Fourier transform.
They discovered (the analog of) the following properties:
4 F.A. Grünbaum, I. Pacharoni and I.N. Zurrián
• For each N , α there exists a symmetric tridiagonal matrix L, with simple spectrum,
commuting with M .
• For each N , α there exists a selfadjoint differential operator D, with simple spectrum,
commuting with the integral operator S2 = EE∗.
3 From the real line to the sphere for matrix valued functions
In this paper the role of the real line will be taken up by the n-dimensional sphere. We will
consider 2 × 2 matrix valued functions defined on the sphere with the appropriate invariance
that makes them functions of the colatitude θ and we will use x = cos(θ) as the variable. The
role of the Fourier transform will be taken by the expansion of our functions in terms of a basis of
matrix valued orthogonal polynomials described below. This is similar to the situation discussed
in [8] except for the crucial fact that our functions are now matrix valued. This situation has,
to the best of our knowledge, not been considered before.
The matrix valued orthogonal polynomials considered here are those studied in [16], arising
from the spherical functions of fundamental representations associated to the n-dimensional
sphere Sn ' G/K, where (G,K) = (SO(n+ 1),SO(n)), studied in [31].
These spherical functions give rise to sequences {Pw}w≥0 of matrix orthogonal polynomials
depending on two parameters n and p ∈ R such that 0 < p < n
Pw(x) =
1
n+ 1
C
n+1
2
w (x) + 1
p+wC
n+3
2
w−2(x)
1
p+ w
C
n+3
2
w−1(x)
1
n− p+ w
C
n+3
2
w−1(x)
1
n+ 1
C
n+1
2
w (x) +
1
n− p+ w
C
n+3
2
w−2(x)
,
where Cλw(x) denotes the w-th Gegenbauer polynomial
Cλw(x) =
(2λ)w
w!
2F1
(
−w,w + 2λ
λ+ 1/2
;
1− x
2
)
, x ∈ [−1, 1].
We recall that Cλw is a polynomial of degree w whose leading coefficient is 2w(λ)w
w! , where (a)w =
a(a+ 1) · · · (a+ w − 1) denotes the Pochhammer’s symbol.
In particular we have
P0 =
1
n+ 1
I, P1 =
x 1
p+1
1
n− p+ 1
x
,
P2 =
(n+ 3)
2
x2 − p
2(p+ 2)
(n+ 3)
p+ 2
x
(n+ 3)
n− p+ 2
x
(n+ 3)
2
x2 − n− p
2(n− p+ 2)
.
Let us observe that the deg(Pw) = w and the leading coefficient of Pw is a non singular scalar
matrix
2w
(
n+1
2
)
w
(n+ 1)w!
Id.
The matrix polynomials {Pw}w≥0 are orthogonal with respect to the matrix valued inner product
〈P,Q〉 =
∫ 1
−1
P (x)W (x)Q(x)∗dx,
Time and Band Limiting for Matrix Valued Functions, an Example 5
where the weight matrix is given by
W (x) = Wp,n(x) =
(
1− x2
)n
2−1
(
px2 + n− p −nx
−nx (n− p)x2 + p
)
, x ∈ [−1, 1]. (1)
Let us observe that by changing p by n−p, the weight matrices are conjugated to each other.
In fact, by taking J = ( 0 1
1 0 ) we get
JWp,nJ
∗ = Wn−p,n. (2)
As a consequence of this fact (or directly from the explicit definition of Pw) we have
that (Pw)22, the entry (2, 2) of Pw, is the same that the entry (Pw)11 by replacing p by n − p.
Also the entry (2, 1) of Pw, (Pw)21 is the entry (Pw)12, if we replace p by n− p.
We have that 〈Pw, Pw〉 is always a diagonal matrix. Moreover one can verify that
〈Pw, Pw〉 = ‖Pw‖2 =
√
π2[w/2]Γ
(
n
2 + 1 + [w2 ]
) [(w−1)/2]∏
k=0
(n+ 2k + 1)
w!(n+ 1)(n+ 2w + 1)Γ
(
n
2 + 3
2
)
×
p(n− p+ w + 1)
p+ w
0
0
(n− p)(p+ w + 1)
n− p+ w
.
We consider the orthonormal sequence of matrix polynomials
Qw = SwPw, (3)
where Sw = ‖Pw‖−1 is the inverse of the matrix ‖Pw‖.
We display the first elements of the sequence {Qw}.
Q0 =
√
Γ
(
n
2 + 3
2
)
√
πΓ
(
n
2 + 1
)
1√
n− p+ 1
0
0
1√
p+ 1
,
Q1 =
√
2Γ
(
n
2 + 5
2
)
√
πΓ
(
n
2 + 1
)
√
(p+ 1)
p(n− p+ 2)
x
1√
p(n− p+ 2)(p+ 1)
1√
(n− p)(p+ 2)(n− p+ 1)
√
(n− p+ 1)
(n− p)(p+ 2)
x
. (4)
4 The matrix M
Given the sequence of matrix orthogonal polynomials {Pw}w≥0 we fix a natural number N and
α ∈ (−1, 1) and consider the matrix M of total size 2(N + 1)× 2(N + 1),
M =
M0,0 M0,1 . . . M0,N
M1,0 M1,1 . . . M1,N
. . . . . . . . . . . .
MN,0 MN,1 . . . MN,N
,
whose (i, j)-block is the 2× 2 matrix obtained by taking the inner product of the i-th and j-th
normalized matrix valued orthogonal polynomials in the interval [−1, α] with α ≤ 1
M i,j =
∫ α
−1
Qi(x)W (x)Qj(x)∗dx for 0 ≤ i, j ≤ N. (5)
6 F.A. Grünbaum, I. Pacharoni and I.N. Zurrián
It should be clear that the restriction to the interval [−1, α] implements “band-limiting” while
the restriction to the range 0, 1, . . . , N takes care of “time-limiting”. In the language of [8]
where we were dealing with scalar valued functions defined on spheres the first restriction gives
us a “spherical cap” while the second one amounts to truncating the expansion in spherical
harmonics.
We gather here a few important properties of the matrix M .
The entries of the matrix M = (Mrs)1≤r,s≤2(N+1) are related with the entries of the block
matrices M i,j =
(
M i,j
11 M i,j
12
M i,j
21 M i,j
22
)
by
M2i+1,2j+1 = M i,j
11 , M2i+1,2j = M i,j−1
12 , M2i,2j+1 = M i−1,j
21 , M2i,2j = M i−1,j−1
22 .
From the definition (5) it is clear that since M j,i = (M i,j)∗, M is a symmetric matrix
M = M∗.
The weight matrices Wp,n and Wn−p,p are conjugated to each other by (2). Let us denote
M i,j(p) the 2 × 2 matrix with parameter p, we get that the entry 1, 2 of M i,j is equal to the
entry 2, 1 by replacing p by n− p, i.e.,
M i,j
12 (p) = M i,j
21 (n− p), M i,j
22 (p) = M i,j
11 (n− p). (6)
In Section 7 we will give detailed information about the entries of M .
5 The commutant of M contains block tridiagonal matrices
The aim of this section is to find all block tridiagonal symmetric matrices L such that
LM = ML.
Notice that in principle there is no guarantee that we will find any such L except for a scalar
multiple of the identity. For the problem at hand we need to exhibit matrices L that have
a simple spectrum, so that its eigenvectors will automatically be eigenvectors of M .
Our finding is that the vector space of such matrices L is of dimension 4. It consist of the
linear span of the matrices L1, L2 and L3 given below. Of course we can always add a multiple
of the identity matrix. We do not have at the moment nice and clean proofs of these facts.
What we do have are careful symbolic computations, done (independently) in Maxima and in
Maple, showing that the space of solutions of the equation ML = LM is always of dimension
four and spanned by L1, L2, L3 and the identity matrix.
We are only indicating explicitly the non-zero entries of these symmetric matrices, for k =
1, . . . , N + 1 we have
(L1)2k,2k =
(n− p+ k − 1)(p+ k)
(p+ 1)(n− p)
,
(L1)2k−1,2k =
−(n− p+N + 1)(p+N + 1)
α(p+ 1)(n− 2p)
√
p(p+ k)(n− p+ k − 1)
(n− p+ k)(p+ k − 1)(n− p)
,
(L1)2k,2k+2 =
(N − k + 1)(N + n+ k + 1)
α(p+ 1)(n− p)
√
k(n− p+ k − 1)(p+ 1 + k)(n+ k)
(n+ 2k − 1)(n+ 2k + 1)(p+ k)(n− p+ k)
,
(L2)2k−1,2k−1 =
(n− p+ k)(p+ k − 1)
p(n− p+ 1)
,
Time and Band Limiting for Matrix Valued Functions, an Example 7
(L2)2k−1,2k =
(n− p+N + 1)(p+N + 1)
α(n− p+ 1)(n− 2p)
√
(n− p+ k)(p+ k − 1)(n− p)
p(p+ k)(n− p+ k − 1)
,
(L2)2k−1,2k+1 =
(N − k + 1)(N + n+ k + 1)
αp(n− p+ 1)
×
√
k(p+ k − 1)(n− p+ 1 + k)(n+ k)
(n+ 2k − 1)(n+ 2k + 1)(p+ k)(n− p+ k)
.
and finally
(L3)2k,2k =
(k − 1)(n+ k)p(n− p+ 1)
(n+ 2)(p+ 1)(n− p)
,
(L3)2k−1,2k−1 =
(k − 1)(n+ k)
n+ 2
,
(L3)2k−1,2k =
(k − 1)(n− p+N + 1)(p+N + 1)(n+ k)
√
p
α(p+ 1)(n+ 2)
√
(n− p)(p+ k − 1)(p+ k)(n− p+ k)(n− p+ k − 1)
,
(L3)2k−1,2k+1 =
(N − k + 1)(N + n+ k + 1)
α(n+ 2)
√
k(p+k−1)(n−p+1+k)(n+ k)√
(n+2k−1)(n+2k+1)(p+k)(n−p+k)
,
(L3)2k,2k+2 =
p(n− p+ 1)(N − k + 1)(N + n+ k + 1)
√
k(n−p+k−1)(p+k+1)(n+k)
α(n+ 2)(n− p)(p+ 1)
√
(n+2k−1)(n+2k+1)(p+k)(n−p+k)
.
In terms of the 2× 2 blocks we have that
L =
L0,0 L0,1 0 . . . 0
L1,0 L1,1 L1,2 . . . 0
0 L2,1 L2,2
...
. . .
...
0 . . . LN,N
,
where the blocks are of the form
(
L1
)j,j
=
(
0 (L1)2j+1,2j+2
(L1)2j+1,2j+2 (L1)2j+2,2j+2
)
, j = 0, . . . , N,
(
L1
)j−1,j
=
(
0 0
0 (L1)2j,2(j+1)
)
, j = 1, . . . , N,
(
L2
)j,j
=
(
(L2)2j+1,2j+1 (L2)2j+1,2j+2
(L2)2j+1,2j+2 0
)
, j = 0, . . . , N,
(
L2
)j−1,j
=
(
(L2)2j−1,2j+1 0
0 0
)
, j = 1, . . . , N,
(
L3
)j,j
=
(
(L3)2j+1,2j+1 (L3)2j+1,2j+2
(L3)2j+1,2j+2 (L3)2j+2,2j+2
)
, j = 0, . . . , N,
(
L3
)j−1,j
=
(
(L3)2j−1,2j+1 0
0 (L3)2j,2(j+1)
)
, j = 1, . . . , N.
These matrices L are pentadiagonal matrices, moreover the off-diagonal two-by-two blocks
are diagonal matrices.
In the scalar case the matrix L is unique up to shifts and scaling, in the 2× 2 case at hand
this is no longer true.
8 F.A. Grünbaum, I. Pacharoni and I.N. Zurrián
6 The spectrum of L1
For the purpose at hand it is enough to exhibit one matrix with simple spectrum satisfying
ML = LM . Here we give a “hands on” argument to prove that the matrix L1 has simple
spectrum. From this, and the commutativity established earlier, it follows that every eigenvector
of L1 is automatically an eigenvector of M , and we have achieved our goal: for each value of
the parameters (α,N) we have a numerically efficient way to compute the eigenvectors of M .
As a referee has suggested, a similar argument could be made to work in the case of L2, L3.
First observe the structure of the matrix L1. For N = 3 we have
L1 =
0 `12 0 0 0 0 0 0
`12 `22 0 `24 0 0 0 0
0 0 0 `34 0 0 0 0
0 `24 `34 `44 0 `46 0 0
0 0 0 0 0 `56 0 0
0 0 0 `46 `56 `66 0 `68
0 0 0 0 0 0 0 `78
0 0 0 0 0 `68 `78 `88
,
where all indicated entries `i,j = (L1)i,j are nonzero, see Section 5.
For general N , with the notation Ei,j for the standard basis of matrices, we have that L1 is
of the form
L1 =
N+1∑
j=1
`2j,2jE2j,2j + `2j−1,2j (E2j−1,2j + E2j,2j−1) + `2j,2j+2 (E2j,2j+2 + E2j+2,2j)
(note that `0,2 = `2N+2,2N+4 = 0). Its 2× 2 blocks are of the form
(L1)
j,j =
(
0 `2j+1,2j+2
`2j+1,2j+2 `2j+2,2j+2
)
, (L1)
j−1,j =
(
0 0
0 `2j,2j+2
)
,
where the coefficients `i,j = (L1)i,j were given in Section 5.
One can see by induction that
det(L1) = (−1)N+1
(
`1,2`3,4 · · · `2N+1,2N+2
)2
= (−1)N+1
N+1∏
j=1
(
`2j−1,2j
)2
.
Assume that λ is an eigenvalue of L1 and thus non-zero, and denote by X one of its eigen-
vectors. We will show that X is a scalar multiple of a certain vector that depends only on the
matrix L1 and the eigenvalue in question. This shows that the geometric multiplicity of λ is
one.
If (x1, x2, x3, . . . , x2N+2) are the components of X, the scalar equations resulting from
L1X = λX
break up into two different groups: the equations given by the odd entries, 1, 3, . . . , 2N + 1, of
the vector equation above, namely
`2j−1,2jx2j − λx2j−1 = 0, j = 1, . . . , N + 1, (7)
and the equations given by the even entries, 2, 4, . . . , 2N + 2, namely
`2j,2j+2x2j+2 + (`2j,2j − λ)x2j + `2j−1,2jx2j−1 + `2j−2,2jx2j−2 = 0,
for j = 1, . . . , N + 1, with the convention `0,2 = `2N+2,2N+4 = 0.
Time and Band Limiting for Matrix Valued Functions, an Example 9
The first set of equations allows us to write x1, x3, x5, . . . , x2N+1 in terms of x2, x4, x6, . . . ,
x2N+2. We get
x2j−1 =
`2j−1,2j
λ
x2j , j = 1, 2, . . . , N + 1. (8)
By replacing (8) in (7) we get, for j = 1, . . . , N + 1,
`2j,2j+2x2j+2 +
(
`2j,2j − λ+
(`2j−1,2j)
2
λ
)
x2j + `2j−2,2jx2j−2 = 0. (9)
This set of equations allows us to find, successively, the entries x2j+2 in terms of x2, for
j = 1, . . . , N . For j = 1 we obtain
x4 = − 1
`2,4
(
`2,2 − λ+
(`1,2)
2
λ
)
x2.
For j = 2 and using this expression for x4, we get
x6 =
1
`2,4`4,6
(
−(`2,4)
2 +
(
`4,4 − λ+
(`3,4)
2
λ
)(
`2,2 − λ+
(`1,2)
2
λ
))
x2.
Our strategy is now clear, by successively solving the equations (9) and using the relation
between x2j and x2j−1 we obtain expressions for x2, x3, x4, . . . etc. as quantities built out of L1
and λ all multiplied by the free parameter x1.
This goes on till we try to solve the last equation in (9), with j = N + 1, where we meet our
only restriction, namely(
`2N+2,2N+2 − λ+
(`2N+1,2N+2)
2
λ
)
x2N+1 + `2N,2N+2x2N = 0.
With the variables x2, x3, x4, . . . , x2N+1, x2N+2 all expressed in the desired form (i.e., as
certain multiples of x1) the last equation becomes a polynomial in elements of L1 and the
unknown quantity λ all multiplied by x1. Except for this factor this is just the characteristic
polynomial for L1 and by assumption λ is a root of it.
7 Explicit expression for the matrix M
Writing down nice and explicit expressions for the entries of M is not easy. We display here
some of the entries whose expressions are not too involved.
We start with computing explicitly the entries of the matrixM0,0 =
∫ α
−1
Q0(x)W (x)Q0(x)∗dx.
By the definition of the weight matrix given in (1) and the explicit expression of the Q0 given
in (4) we have
M0,0 =
Γ
(
n
2 + 3
2
)
√
πΓ
(
n
2 + 1
) ∫ α
−1
(
1− x2
)n
2
−1
px2 + n− p
n− p+ 1
− nx√
(p+ 1)(n− p+ 1)
− nx√
(p+ 1)(n− p+ 1)
(n− p)x2 + p
p+ 1
=
(
M0,0
11 M0,0
12
M0,0
21 M0,0
22
)
.
10 F.A. Grünbaum, I. Pacharoni and I.N. Zurrián
It is easy to verify that
M0,0
12 = M0,0
21 =
Γ
(
n
2 + 3
2
)(
1− α2
)n
2√
π(p+ 1)(n− p+ 1)Γ
(
n
2 + 1
) .
We also have
M0,0
11 =
Γ
(
n
2 + 3
2
)
(n− p+ 1)
√
πΓ
(
n
2 + 1
)
×
(
1
3px
3
2F1
(
−n
2
+1, 3
2
5
2
;x2
)
+ (n− p)x 2F1
(
−n
2
+1, 1
2
3
2
;x2
))∣∣∣∣α
−1
.
Computing the hypergeometric function at x = 1 we get
M0,0
11 =
Γ
(
n
2 + 3
2
)(
1
3pα
3
2F1
(
−n/2+1,3/2
5/2
;α2
)
+ (n− p)α 2F1
(
−n/2+1,1/2
3/2
;α2
))
(n− p+ 1)
√
πΓ
(
n
2 + 1
) +
1
2
. (10)
The expression of M0,0
22 can be obtained from M0,0
11 by changing p by n− p, see (6).
7.1 The matrix M̃
To simplify some calculations that follow and to avoid some square roots, we consider the
following matrix M̃ . This matrix is close to the matrix M and it is defined as a matrix of size
2(N + 1)× 2(N + 1), whose (i, j)-block is the 2× 2 matrix given by
M̃ i,j =
∫ α
−1
Pi(x)W (x)Pj(x)∗dx for 0 ≤ i, j ≤ N.
We observe that M̃ is defined in the same way as M by using the sequence (Pw)w≥0 of matrix
orthogonal polynomial instead of the orthonormal polynomials (Qw)w≥0.
The sequences of orthogonal polynomials (Pw)w≥0 and (Qw)w≥0 are related by Qw = SwPw,
where Sw = ‖Pw‖−1 (see (3)). Thus the blocks of the matrices M and M̃ are related by
M i,j = SiM̃
i,jS∗j for 0 ≤ i, j ≤ N.
Since the matrices Sj are diagonal matrices, we have that
M i,j
11 = (Si)11(Sj)11M̃
i,j
11 , M i,j
22 = (Si)22(Sj)22M̃
i,j
22 ,
M i,j
12 = (Si)11(Sj)22M̃
i,j
12 , M i,j
21 = (Si)22(Sj)11M̃
i,j
21 .
Explicitly,
Sj =
√√√√√√ j!(n+ 1)(n+ 2j + 1)Γ
(
n
2 + 3
2
)
√
π2[j/2]Γ
(
n
2 + 1 + [ j2 ]
) [(j−1)/2]∏
k=0
(n+ 2k + 1)
×
√
p+ j
p(n− p+ j + 1)
0
0
√
n− p+ j
(n− p)(p+ j + 1)
.
Time and Band Limiting for Matrix Valued Functions, an Example 11
7.2 Non diagonal elements for all the blocks of M
By direct computations we obtained the following expression for the non diagonal elements of
each 2× 2 block M̃ i,j .
For 0 ≤ i, j ≤ N we get
M̃ i,j
1,2 =
(
1− α2
)n/2
p(n− p)
(n+ 1)2(p+ i)(n− p+ j)
C
n+1
2
i (α)C
n+1
2
j (α),
M̃ i,j
2,1 =
(
1− α2
)n/2
p(n− p)
(n+ 1)2(p+ j)(n− p+ i)
C
n+1
2
i (α)C
n+1
2
j (α),
where C
n+1
2
n (α) denotes the n-th Gegenbauer polynomial in α.
We observe that these expressions determine the entries M2i+1,2j and M2i,2j+1 of the mat-
rix M , for 1 ≤ i, j ≤ N + 1, are given by
M2i+1,2j = M i,j−1
1,2 = (Si)11(Sj−1)22M̃
i,j
1,2, M2i,2j+1 = M i−1,j
2,1 = (Si−1)22(Sj)11M̃
i,j
2,1.
7.3 The diagonal entries of non diagonal blocks of M
Here we give the diagonal entries of the non diagonal blocks of M̃ , i.e., the elements M̃ j,k
11
and M̃ j,k
22 , with j 6= k.
For 1 ≤ j, k ≤ N , j 6= k we have
M̃ j−1,k−1
11 =
p
(
1− α2
)n/2
(k − j)(j + k + n− 1)(n+ 1)2
(
j(n− p+ j − 1)(n− p+ k)
n+ 2j − 1
Cj(α)Ck−1(α)
− k(n− p+ k − 1)(n− p+ j)
n+ 2k − 1
Cj−1(α)Ck(α)
+
(p+ j)(n+ j − 1)(n− p+ j)(n− p+ k)
(p+ j − 1)(n+ 2j − 1)
Cj−2(α)Ck−1(α)
−(p+ k)(n+ k − 1)(n− p+ j)(n− p+ k)
(p+ k − 1)(n+ 2k − 1)
Cj−1(α)Ck−2(α)
)
.
The element M̃ j−1,k−1
22 is obtained from M̃ j−1,k−1
11 by changing p by n− p.
These expressions determine the entries M2j−1,2k−1 and M2j,2k of the matrix M , for 1 ≤ j 6=
k ≤ N + 1.
M2j−1,2k−1 = M j−1,k−1
1,1 = (Sj−1)11(Sk−1)11M̃
j−1,k−1
1,1 ,
M2j,2k = M j−1,k−1
2,2 = (Sj−1)22(Sk−1)22M̃
j−1,k−1
2,2 .
7.4 The main diagonal of M
At this point we have explicit expressions of all non diagonal entries of M (and M̃). Now we
want to describe how to obtain the elements M i,i
11 = M2i+1,2i+1 and M i,i
22 = M2i+2,2i+2, in terms
of the known coefficients.
Starting with the equation (ML−LM)2j,2j+2 = 0, with L = L1 and j = 1, . . . , N , we obtain
M2j,2j −M2j−1,2j−1
= (L2j−1,2j)
−1(M2j−1,2jL2j,2j +M2j−1,2j−2L2j−2,2j +M2j−1,2j+2L2j,2j+2
)
.
12 F.A. Grünbaum, I. Pacharoni and I.N. Zurrián
The equation (ML− LM)2j,2j+2 = 0 is
M2j+2,2j+2 −M2j,2j = (L2j,2j+2)
−1(M2j,2j+2(L2j+2,2j+2 − L2j,2j) +M2j,2j+1L2j+1,2j+2
−M2j−1,2j+2L2j−1,2j +M2j,2j+4L2j+2,2j+4 −M2j−2,2j+2L2j−2,2j
)
.
By using (6) we have that the element M j−1,k−1
22 is obtained from M j−1,k−1
11 by changing p
into n− p.
Since we already know the explicit value of M1,1 (see (10)), these expressions allow us to
determine all the diagonal elements of M .
8 Some numerical results
In this last section we display the results of some numerical computations. This should make
clear the importance of having found, as above, a matrix such as L1 for a given M .
Our point becomes very clear even if we use a small value of N and a value of α pretty close
to 1. If we had chosen a larger value of N the phenomenon in question would be present for
an even larger range of values of α. For our illustration we choose n = 27, p = 15, and finally
N = 2 and α = 9/10.
If our task is to compute the eigenvectors of M we can use the QR algorithm as implemented
in LAPACK. The results are recorded below, where we denote by X the matrix of eigenvectors
(normalized and given as columns of X) and by D the diagonal matrix of eigenvalues. The
matrix X is
X =
0.046636 −.0318428 0.294888 −.953861 −0.0995136 −.0406174
−.0424748 −.0353152 −.241903 −.0756026 .852604 −.429742
.214609 −.17084 .649236 −.216917 .0981185 −.654981
−.19526 −.189172 −.589859 −.18561 .494728 0.569823
.706579 −.64988 −.215899 0.0105047 −.0312226 .174508
−.642327 −.714438 .19614 .0530856 −.0883495 −.171814
The question is: should we trust the result produced by this high quality numerical package?
One could be quite satisfied by observing that the difference
MX −XD
is indeed very small. On the other hand LAPACK reports for eigenvalues of M , with appropriate
rounding-off, the values 1.0, 1.0, 1.0, 1.0, 1.0, 1.0. This should be a red flag.
Recall that the eigenvectors of L1 should agree (up to order) with those of M . If we denote
the matrix made up of the normalized eigenvectors of L1 by Y we get
Y =
.641473 .227019 .0318478 −.674518 .280734 −.0466363
.688247 .247073 .0350924 .628845 −.258435 .0424743
−.229364 .613934 .170768 .280102 .645602 −.214607
−0.24584 .667491 .187976 −.260873 −.593721 .195257
0.028825 −.172614 .649883 −.0405441 −.214796 −.706573
.0308702 −.187518 0.71478 .0377295 .197372 .642335
For the eigenvalues of L1, LAPACK returns the values 6.46314, 6.55601, 6.63761, −5.61601,
−5.54541, −5.4863 a reasonably spread out spectrum.
If we compute the matrix of inner products given by
Y TX
Time and Band Limiting for Matrix Valued Functions, an Example 13
we expect to have the identity matrix up to some permutation and possibly some signs due to
the normalization of the eigenvectors which are the columns of X and Y . In our case we get for
the moduli of the entries of Y TX the matrix
4.65e−7 1.71e−4 .019 .667 .671 .021
2.83e−6 9.61e−4 .013 0.234 .2244 .9602
9.85e−6 .9999 7.801e−4 2.385e−4 4.395e−7 8.81e−4
7.49e−7 1.61e−4 8.569e−4 0.707 .707 .278
3.93e−6 7.89e−4 .9997 .0147 .007 .0115
1.0 9.84e−6 3.89e−6 9.61e−7 1.46e−6 2.55e−6
The reader will observe that some of the entries of this matrix are indeed very close to the
theoretically correct values, while others are terribly off. The reason is that for this choice of α
there are a few eigenvalues of M that are just too close together. This produces numerical
instability in the computation of the corresponding eigenvectors. On the other hand all the
eigenvalues of L1 are nicely separated, and the corresponding eigenvectors can be trusted.
In summary, a good way to obtain good numerical values for the eigenvectors of M is to
forget about M altogether and to compute numerically the eigenvectors of L1. Not only we will
then be dealing with a very sparse matrix for which the QR algorithm works very fast (most
of the work is avoided) but the problem is numerically very well conditioned. For a discussion
of the sensitivity of eigenvectors and their dependence on the separation of the corresponding
eigenvalues one can consult [2, 30] as well as [17, p. 15 and p. 222].
As a referee has pointed out, these numerical problems are not new in our situation involving
matrix valued functions, but already appear in the classical scalar case, and this phenomenon
is well documented. The important point is that even in the matrix valued case we can exhibit
commuting tridiagonal matrices that play the same role of the “prolate spheroidal differential
operator” in the scalar case.
Acknowledgements
This research was supported in part by the Applied Mathematical Sciences subprogram of the
Office of Energy Research, USDOE, under Contract DE-AC03-76SF00098, by AFOSR grant
FA95501210087 through a subcontract to Carnegie Mellon University, by CONICET grant PIP
112-200801-01533, by SeCyT-UNC and by the Oberwolfach Leibniz Fellows Program.
References
[1] Bonami A., Karoui A., Uniform approximation and explicit estimates for the prolate spheroidal wave func-
tions, arXiv:1405.3676.
[2] Demmel J.W., Applied numerical linear algebra, Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, PA, 1997.
[3] Duistermaat J.J., Grünbaum F.A., Differential equations in the spectral parameter, Comm. Math. Phys.
103 (1986), 177–240.
[4] Grünbaum F.A., A new property of reproducing kernels for classical orthogonal polynomials, J. Math. Anal.
Appl. 95 (1983), 491–500.
[5] Grünbaum F.A., Some new explorations into the mystery of time and band limiting, Adv. in Appl. Math.
13 (1992), 328–349.
[6] Grünbaum F.A., Band-time-band limiting integral operators and commuting differential operators, St. Pe-
tersburg Math. J. 8 (1997), 93–96.
[7] Grünbaum F.A., The bispectral problem: an overview, in Special Functions 2000: Current Perspective
and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, Kluwer Acad. Publ.,
Dordrecht, 2001, 129–140.
http://arxiv.org/abs/1405.3676
http://dx.doi.org/10.1137/1.9781611971446
http://dx.doi.org/10.1007/BF01206937
http://dx.doi.org/10.1016/0022-247X(83)90123-3
http://dx.doi.org/10.1016/0022-247X(83)90123-3
http://dx.doi.org/10.1016/0196-8858(92)90015-O
http://dx.doi.org/10.1007/978-94-010-0818-1_6
14 F.A. Grünbaum, I. Pacharoni and I.N. Zurrián
[8] Grünbaum F.A., Longhi L., Perlstadt M., Differential operators commuting with finite convolution integral
operators: some nonabelian examples, SIAM J. Appl. Math. 42 (1982), 941–955.
[9] Grünbaum F.A., Yakimov M., The prolate spheroidal phenomenon as a consequence of bispectrality, in
Superintegrability in Classical and Quantum Systems, CRM Proc. Lecture Notes, Vol. 37, Amer. Math.
Soc., Providence, RI, 2004, 301–312, math-ph/0303041.
[10] Jahn K., Bokor N., Revisiting the concentration problem of vector fields within a spherical cap: a commuting
differential operator solution, J. Fourier Anal. Appl. 20 (2014), 421–451, arXiv:1302.5261.
[11] Jamming P., Karoui A., Spektor S., The approximation of almost time and band limited functions by their
expansion in some orthogonal polynomial bases, arXiv:1501.03655.
[12] Landau H.J., Pollak H.O., Prolate spheroidal wave functions, Fourier analysis and uncertainty. II, Bell
System Tech. J. 40 (1961), 65–84.
[13] Landau H.J., Pollak H.O., Prolate spheroidal wave functions, Fourier analysis and uncertainty. III. The
dimension of the space of essentially time- and band-limited signals, Bell System Tech. J. 41 (1962), 1295–
1336.
[14] Melkman A.A., n-widths and optimal interpolation of time- and band-limited functions, in Optimal Es-
timation in Approximation Theory (Proc. Internat. Sympos., Freudenstadt, 1976), Editors C.A. Michelli,
T. Rivlin, Plenum, New York, 1977, 55–68.
[15] Osipov A., Rokhlin V., Xiao H., Prolate spheroidal wave functions of order zero. Mathematical tools for
bandlimited approximation, Applied Mathematical Sciences, Vol. 187, Springer, New York, 2013.
[16] Pacharoni I., Zurrian I., Matrix ultraspherical polynomials: the 2 × 2 fundamental cases, Constr. Approx.,
to appear, arXiv:1309.6902.
[17] Parlett B.N., The symmetric eigenvalue problem, Prentice-Hall Series in Computational Mathematics,
Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980.
[18] Perlstadt M., Chopped orthogonal polynomial expansions – some discrete cases, SIAM J. Algebraic Discrete
Methods 4 (1983), 94–100.
[19] Perlstadt M., A property of orthogonal polynomial families with polynomial duals, SIAM J. Math. Anal.
15 (1984), 1043–1054.
[20] Plattner A., Simons F.J., Spatiospectral concentration of vector fields on a sphere, Appl. Comput. Harmon.
Anal. 36 (2014), 1–22, arXiv:1306.3201.
[21] Shannon C.E., A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–423.
[22] Shannon C.E., A mathematical theory of communication, Bell System Tech. J. 27 (1948), 623–656.
[23] Simons F.J., Dahlen F.A., Spherical Slepian functions on the polar gap in geodesy, Geophys. J. Int. 166
(2006), 1039–1061, math.ST/0603271.
[24] Simons F.J., Dahlen F.A., Wieczorek M.A., Spatiospectral concentration on a sphere, SIAM Rev. 48 (2006),
504–536, math.CA/0408424.
[25] Slepian D., Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many
dimensions; generalized prolate spheroidal functions, Bell System Tech. J. 43 (1964), 3009–3057.
[26] Slepian D., On bandwidth, Proc. IEEE 64 (1976), 292–300.
[27] Slepian D., Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. The discrete case, Bell
System Tech. J. 57 (1978), 1371–1430.
[28] Slepian D., Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev. 25 (1983), 379–393.
[29] Slepian D., Pollak H.O., Prolate spheroidal wave functions, Fourier analysis and uncertainty. I, Bell System
Tech. J. 40 (1961), 43–63.
[30] Stewart G.W., Matrix algorithms. Vol. II. Eigensystems, Society for Industrial and Applied Mathematics
(SIAM), Philadelphia, PA, 2001.
[31] Tirao J.A., Zurrián I.N., Spherical functions of fundamental K-types associated with the n-dimensional
sphere, SIGMA 10 (2014), 071, 41 pages, arXiv:1312.0909.
[32] Weyl H., The theory of groups and quantum mechanics, Dutton, New York, 1931.
http://dx.doi.org/10.1137/0142067
http://arxiv.org/abs/math-ph/0303041
http://dx.doi.org/10.1007/s00041-014-9324-7
http://arxiv.org/abs/1302.5261
http://arxiv.org/abs/1501.03655
http://dx.doi.org/10.1002/j.1538-7305.1961.tb03977.x
http://dx.doi.org/10.1002/j.1538-7305.1961.tb03977.x
http://dx.doi.org/10.1002/j.1538-7305.1962.tb03279.x
http://dx.doi.org/10.1007/978-1-4614-8259-8
http://arxiv.org/abs/1309.6902
http://dx.doi.org/10.1137/0604012
http://dx.doi.org/10.1137/0604012
http://dx.doi.org/10.1137/0515081
http://dx.doi.org/10.1016/j.acha.2012.12.001
http://dx.doi.org/10.1016/j.acha.2012.12.001
http://arxiv.org/abs/1306.3201
http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x
http://dx.doi.org/10.1002/j.1538-7305.1948.tb00917.x
http://dx.doi.org/10.1111/j.1365-246X.2006.03065.x
http://arxiv.org/abs/math.ST/0603271
http://dx.doi.org/10.1137/S0036144504445765
http://arxiv.org/abs/math.CA/0408424
http://dx.doi.org/10.1002/j.1538-7305.1964.tb01037.x
http://dx.doi.org/10.1109/PROC.1976.10110
http://dx.doi.org/10.1002/j.1538-7305.1978.tb02104.x
http://dx.doi.org/10.1002/j.1538-7305.1978.tb02104.x
http://dx.doi.org/10.1137/1025078
http://dx.doi.org/10.1002/j.1538-7305.1961.tb03976.x
http://dx.doi.org/10.1002/j.1538-7305.1961.tb03976.x
http://dx.doi.org/10.1137/1.9780898718058
http://dx.doi.org/10.1137/1.9780898718058
http://dx.doi.org/10.3842/SIGMA.2014.071
http://arxiv.org/abs/1312.0909
1 Introduction
2 Preliminaries
3 From the real line to the sphere for matrix valued functions
4 The matrix M
5 The commutant of M contains block tridiagonal matrices
6 The spectrum of L1
7 Explicit expression for the matrix M
7.1 The matrix M"0365M
7.2 Non diagonal elements for all the blocks of M
7.3 The diagonal entries of non diagonal blocks of M
7.4 The main diagonal of M
8 Some numerical results
References
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| id | nasplib_isofts_kiev_ua-123456789-147111 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T12:02:52Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Grünbaum, F.A. Pacharoni, I. Zurrián, I.N. 2019-02-13T16:51:23Z 2019-02-13T16:51:23Z 2015 Time and Band Limiting for Matrix Valued Functions, an Example / F.A. Grünbaum, I. Pacharoni, I.N. Zurrián // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33C45; 22E45; 33C47 DOI:10.3842/SIGMA.2015.044 https://nasplib.isofts.kiev.ua/handle/123456789/147111 The main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example showing in a non-commutative setup that a bispectral property implies that the corresponding global operator of ''time and band limiting'' admits a commuting local operator. This is a noncommutative analog of the famous prolate spheroidal wave operator. This paper is a contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet. The full collection is available at http://www.emis.de/journals/SIGMA/ESSA2014.html. This research was supported in part by the Applied Mathematical Sciences subprogram of the Of fice of Energy Research, USDOE, under Contract DE-AC03-76SF00098, by AFOSR grant FA95501210087 through a subcontract to Carnegie Mellon University, by CONICET grant PIP 112-200801-01533, by SeCyT-UNC and by the Oberwolfach Leibniz Fellows Program. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Time and Band Limiting for Matrix Valued Functions, an Example Article published earlier |
| spellingShingle | Time and Band Limiting for Matrix Valued Functions, an Example Grünbaum, F.A. Pacharoni, I. Zurrián, I.N. |
| title | Time and Band Limiting for Matrix Valued Functions, an Example |
| title_full | Time and Band Limiting for Matrix Valued Functions, an Example |
| title_fullStr | Time and Band Limiting for Matrix Valued Functions, an Example |
| title_full_unstemmed | Time and Band Limiting for Matrix Valued Functions, an Example |
| title_short | Time and Band Limiting for Matrix Valued Functions, an Example |
| title_sort | time and band limiting for matrix valued functions, an example |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147111 |
| work_keys_str_mv | AT grunbaumfa timeandbandlimitingformatrixvaluedfunctionsanexample AT pacharonii timeandbandlimitingformatrixvaluedfunctionsanexample AT zurrianin timeandbandlimitingformatrixvaluedfunctionsanexample |