Research Statement (pdf) - Department of Mathematics

RESEARCH STATEMENT
BENJAMIN AARON BAILEY
1. Introduction
1.1. Post-doctoral research. My post-doctoral research has focused on the behavior of convergent
cardinal series. A convergent cardinal series is a particular expansion that under certain
circumstances produces an entire function of exponential type π with prescribed values at the integers
(see equation (1) for a precise definition). The general condition on the prescribed function
values which ensures convergence of the corresponding convergent cardinal series is given in Section
2.1 along with related rates of convergence of the resulting series. We also show that the growth
rate along the real axis of a convergent cardinal series tracks with the growth rate of its values on
Z. In Section 2.2, we present conditions on a function f in terms of its growth along R, its degree
of smoothness, properties of its Fourier transform, or values on Z, which allow us to conclude that
f is a convergent cardinal series. We conclude our investigation of cardinal series in Section 2.3
with a connection to cardinal spline interpolation.
1.2. Doctoral research. My doctoral studies investigated interpolatory sampling theory regarding
(primarily) multivariate bandlimited functions. The focus was recovery of bandlimited functions
from unequally spaced sampled data where the sampling nodes arise from either an exponential
frame or an exponential Riesz basis condition. In Section 3.1 we consider approximation of multivariate
bandlimited functions from sampled data by Lagrangian polynomial interpolants in lieu of a
biorthogonal series expansion. Proceeding to Sections 3.2 and 3.3, recovery formulae are presented
that demonstrate a certain stability with respect to l ∞ errors in otherwise ideal l 2 data through,
respectively, oversampling of data and use of Gaussian radial-basis functions.
1.3. Masters research. My master’s research (recently revisited and generalized) considered the
hyperbolic area of hyperbolic polygons in the disc model for hyperbolic geometry. In Section 4.1
we present a simple computational formula for the hyperbolic area of a hyperbolic polygon in
terms of the coordinates of its vertices, along with a geometric reformulation. We then apply this
computational formula in Section 4.2 to describe (and show existence of) all solutions to a wellknown
isoperimetric problem: for a given number of sides and hyperbolic perimeter, determine the
hyperbolic polygons with the greatest hyperbolic area.
1.4. Future research. In the next phase of my career I intend to collaborate with others to answer
questions which have been raised by my previous research, and enter the fields of signal processing
and compressed sensing.
2. Post-doctoral research
2.1. Basic convergence properties of convergent cardinal series. We define a convergent
cardinal series (CCS) to a be a function f : C → C which can be written in the form
(1) f(z) = lim
N∑
N→∞
n=−N
a n sinc(z − n) := lim f N(z), where sinc(z) = sin(πz) ,
N→∞ πz
and convergence in (1) is uniform on compact subsets of C. Note that the CCS above (provided
it converges) interpolates {(n, c n )} n∈Z . CCSes have been extensively studied in the literature; see
[6, 17, 21, 22, 24, 26, 39] for further reference. The article [21] is an important survey that includes
historical information. If f is a function on C or Z, then the even and odd parts of f are defined by
f e (z) = [f(z)+f(−z)]/2 and f o (z) = [f(z)−f(−z)]/2, and f = f e +f o . Note that f is a CCS if and
1

2
only if f o and f e are both CCSes. The set of entire functions f which satisfy |f(z)| ≤ M ɛ e (σ+ɛ)|z|
is denoted by E σ , the functions of exponential type σ. For convenience we consider σ = π.
Theorem 2.1 below (a consolidation of Theorems from [6]) presents necessary and sufficient conditions
on a sequence {c n } n∈Z so that some CCS f interpolates {(n, c n )} n∈Z , a rate of convergence
of f N to f, and bounds on the growth of the resulting function f.
Theorem 2.1 (B, Madych). Suppose {c n } n∈Z ⊂ C is a sequence with even and odd parts {a n } n∈Z
and {b n } n∈Z .
1) {f N } N∈N converges uniformly on compact subsets of C to an entire function f if and only if both
∞∑
(−1) n a n
∞
n 2 , and ∑
(−1) n b n
n
converge.
n=1
2) Let f be a CCS. If K ⊂ C is compact, there exists M > 0 such that
∣ ∞∑ ∣∣∣∣∣
sup |f e (z)−f e,N (z)| ≤ M sup
(−1) j a j
z∈K
n≥N ∣ j 2 and sup |f o (z)−f o,N (z)| ≤ M sup
z∈K
n≥N ∣
3) If f is a CCS, then
In particular, f ∈ E π .
j=n+1
n=1
|f e (z)|e −π|Im(z)| = o(|z| 2 ) and |f o (z)|e −π|Im(z)| = o(|z|) as |z| → ∞.
∞∑
j=n+1
(−1) j b j
j
2.2. Uniqueness of convergent cardinal series. Theorem 2.2 (see [9]), Corollary 2.3, and Theorem
2.7, (a generalization of Theorem 5 in [6]), present conditions on f ∈ E π which allow us to
conclude that f is a CCS.
Theorem 2.2 below, (a partial converse of item 3) in Theorem 2.1) provides a wide class of
unbounded functions in E π which are CCSes.
Theorem 2.2 (B, Madych). The following are true.
1) Let f ∈ E π be odd. If lim x→∞ f (n) (x) = 0 for some n ≥ 0 and f(k) = o(k) as k → ∞ over the
integers, then f is a CCS.
f
2) Let f ∈ E π be even. If lim (n) (x)
x→∞ x
= 0 for some n ≥ 0 and f(k) = o(k 2 ) as k → ∞ over the
integers, then f is a CCS.
Theorem 2.2 has the following corollary.
Corollary 2.3 (B, Madych). Let f ∈ E π .
1) If f e (x) = o(|x|) and f o (x) = o(1) as |x| → ∞, then f is a CCS.
2) If lim |x|→∞ f(x) = 0, then f is a CCS.
3) Let f (n)∣ ∣
R
∈ L p (R) for some 1 < p < ∞ and n ≥ 0. If f o (k) = o(k) and f(k) = o(k 2 ) as k → ∞,
then f is a CCS.
Item 2) in Corollary 2.3 is of particular interest, being a partial extension of a theorem of
Plancherel and Polya in [26, page 152]. Counterexamples may be found in [9] which show item 2)’s
limitations as a generalization.
Theorem 2.4 (a version of which appears in [8]) follows somewhat quickly from Corollary 2.3.
Theorem 2.4 (B, Madych). Let h ∈ L 1 [−π, π], and define
f(z) = z
∫ π
−π
h(ξ)e iξz dξ.
The function f is a CCS if and only if the Fourier series for h(ξ) + h(−ξ) converges at ξ = π.
∣ .

Theorem 2.4 has the following corollaries. Corollary 2.5 appears in [8], and Corollary 2.6 originates
in [6] as a stand-alone result with a different proof.
Corollary 2.5 (B, Madych). Let σ < π. If f ∈ E σ satisfies f(x)/x ∈ L 2 (R), then f is a CCS.
Corollary 2.6 (B, Madych). Let µ be a bounded Borel measure on [−π, π], and define
∫
f(z) = e izξ dµ(ξ) for all z ∈ C.
[−π,π]
The function f can be expressed as follows:
f(z) = i[µ{π} − µ{−π}] sin(πz) + lim
N∑
N→∞
n=−N
f(n)sinc(z − n).
Corollary 2.6 may be generalized to Theorem 2.7, which is notable for its multidimensional
nature.
Theorem 2.7 (B). For z = (z 1 , · · · , z d ) ∈ C d , define SINC(z) = sinc(z 1 ) · . . . · sinc(z d ). Let µ be a
bounded Borel measure on [−π, π] d , and define
∫
f(z) = e i(z 1ξ 1 +...+z d ξ d ) dµ(ξ) for all z = (z 1 , · · · , z d ) ∈ C d .
[−π,π] d
For each z ∈ C d , the limit
exists, and
S f (z) := lim
∑
N→∞
‖n‖ ∞≤N
f(n)SINC(z − n)
|f(z) − S f (z)| ≤ 2e π(|Im(z 1)|+···+|Im(z d )|) |µ|(∂([−π, π] d )).
Consequently, if µ is supported on (−π, π) d , then f(z) = S f (z) for all z ∈ C d .
2.3. Splines and convergent cardinal series. Let L k (x) be the fundamental spline of order 2k
that interpolates {(n, δ n0 )} n∈Z . L k satisfies
(2) lim k(x) = sinc(x)
k→∞
when x ∈ R.
Define the spline
N∑
(3) S k (x) = lim c n L k (x − n).
N→∞
n=−N
For certain classes of functions f ∈ E π , (see [33, 34, 36]), equation (3) converges when c n = f(n),
and
lim
k→∞ S k(x) = f(x).
While the cited classes of functions are not necessarily CCSes, they do not contain all CCSes.
However, in light of equation (2) and the form of equation (3), it is reasonable to expect that (3)
holds when f is a CCS. This is exactly the content of Theorem 2.8 in [7].
Theorem 2.8 (B, Madych). If f is CCS and S k is a spline of order 2k which both interpolate the
data {(n, f(n))} n∈Z , then f(x) = lim k→∞ S k (x) uniformly on compact subsets of R.
See [28, 31, 37, 35, 34] for other results regarding spline summability and its extensions.
3. Doctoral research
Let E ⊂ R d be Lebesgue measurable. P W E denotes the set of functions in L 2 (R d ) whose Fourier
transform F(f) is zero a.e. in R d \ E.
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4
3.1. { Multivariate bandlimited functions and Lagrangian polynomial interpolation. Let
e
i〈(·),t } n〉
be a Riesz basis for L
n∈Z d 2 ([−π, π] d ). Any f ∈ P W [π,π] d can be recovered from the
sample values {f(t n )} n∈Z d by expanding against an associated Riesz basis, each element of which is
an infinite product which is not computational in nature ( see [27, Chapter 4], [30], and [32]). Theorem
3.2 investigates when any f ∈ P W [−π,π] d can be concretely approximated using multivariate
polynomial interpolants. For this we require the notion of a uniformly invertible Riesz basis (see
[3]).
Definition 3.1. For each l ≥ 1, let C l,d = {−l, · · · , l} d . Let {t n } n∈Z d ⊂ R d . A sequence
{
e
i〈(·),t } n〉
is a uniformly invertible Riesz basis (UIRB) for L
n∈Z d 2 ([−π, π] d ) if the following two
conditions are satisfied.
1) For each l ≥ 0, the sequence {f l,n } n∈Z d = { e i〈(·),tn〉} ∪ { e i〈(·),n〉} is a Riesz basis
n∈C l,d n∈Z d \C l,d
for L 2 ([−π, π] d ). That is, the map L l
(
e
i〈(·),n〉 ) = f l,n is an onto isomorphism.
2) sup l≥0 ‖L −1
l
‖ < ∞.
The following are examples of UIRBs:
1) When d = 1 and lim sup |n|→∞ |t n − n| < 1/4, the sequence { e itn(·)} n∈Z is a UIRB for L 2[−π, π].
It is a classic result that { e itn(·)} Riesz bases (see [25]).
n∈Z
2) When d > 1 and sup n∈Z d ‖t n − n‖ ∞ < ln(2)
πd , the sequence { e i〈(·),tn〉} is a UIRB for
n∈Z d
L 2 ([−π, π] d ). The fact that { e i〈(·),tn〉} is Riesz basis follows from results in [1].
n∈Z d
Before stating Theorem 3.2 (proven in [3]), we need a definition. For x ∈ R d , define
Q d,l (x) =
l∏
k 1 =1
( )
1 − x2 1
k1
2
· . . . ·
l∏
k d =1
( )
1 − x2 d
kd
2 .
Theorem 3.2 (B). Let {t n } n∈Z d ⊂ R d be a sequence such that { e i〈(·),tn〉} n∈Z d is a UIRB for
L 2 ([−π, π] d ). The following are true.
1) For all f ∈ P W [−π,π] d, there exists a unique sequence of polynomials {Ψ l } l∈N
such that Ψ l :
R d → R has coordinate degree at most 2l, and Ψ l (t n ) = f(t n ) for all n ∈ C l,d .
2) If f ∈ P W [−π,π] d where {Ψ l } is the sequence of interpolating polynomials from 1), then
f(t) = lim
l→∞
Ψ l (t) SINC(t)
Q d,l (t)
where the limit exists in both L 2 (R d ) and L ∞ (R d ) metrics.
uniformly on compact subsets of R d .
In particular, f(t) = lim l→∞ Ψ l (t)
3.2. Oversampling and recovery of bandlimited functions from sampled data. The following
theorem from [5] (motivated by [13] and [18] and generalized from results in [4]) presents a
recovery formula from unequally spaced data nodes for a broad class of bandlimited functions. Under
certain conditions on the sampling nodes, the recovery formula is stable under l ∞ perturbations
in the sampled values.
Theorem 3.3 (B). Let 0 ∈ E ⊂ R d be compact, symmetric about 0, and satisfy E ⊂ Int(βE) for
all β > 1. Choose {t n } n∈N
⊂ R d such that { e i〈(·),tn〉} n , is a frame for L 2(E) with frame operator
S, and let λ 0 > 1.

1) There exists a rapidly decaying real-valued function g ∈ C ∞ (R d ) such that if λ ≥ λ 0 and
f ∈ P W E ,
(
(4) f(t) = 1 ∑ ∑
( ) ) (
tn
λ d B kn f g t − t )
k
, t ∈ R d ,
λ
λ
k∈N n∈N
where B kn = 〈S −1 f n , S −1 f k 〉 E . The sum converges with respect to the L 2 (R d ) and uniform metrics.
2) Let {ɛ n } n∈N be a sequence of real numbers such that sup n |ɛ n | = ɛ < ∞, and B be the N × N
matrix with the entries B kn in item 1). If B : l ∞ → l ∞ (interpreted as matrix multiplication) is
well-defined, then
(
˜f(t) = 1 ∑ ∑
( ) ]
tn
λ d B kn
[f
) (
+ ɛ n g t − t )
k
, t ∈ R d
λ λ
k∈N n∈N
converges, and there exists a real number M independent of f and ɛ such that ||f − ˜f|| L∞(R d ) ≤ Mɛ.
In Theorem 3.3 if E = [−π, π] d , where { e i〈(·),tn〉} n∈N is a Riesz basis for L 2([−π, π] d ), we can
replace B by a sequence of finite matrices and obtain a computationally manageable version of
equation (4). We omit this formula for the sake of brevity.
3.3. Nonuniform sampling and Gaussian radial-basis functions. Theorem 3.4 below, (see
[10]) gives a asymptotic recovery formula for a class of bandlimited functions involving Gaussian
radial-basis functions.
Theorem 3.4 (B, Schlumprecht, Sivakumar). Let B 2 be the closed unit ball in L 2 (R d ). Let Z ⊂ R d
be convex and symmetric about the origin such that δB 2 ⊂ Z ⊂ B 2 for some δ ∈ ( √ 2/3, 1]. Let
{t n } n∈N
⊂ R d be a sequence such that { e 〈(·),tn〉} n∈N is a Riesz basis for L 2(Z). Given f ∈ P W B2 ,
let f S = {f(t k )} k∈N
. For λ > 0, define the N × N matrix A by A nm = e −λ‖tn−tm‖2 2 . The following
statements hold:
1) The map x ↦→ Ax from l 2 (N) to itself is linear, bounded, invertible, with bounded inverse.
2) The map
f ↦→ I λ (f) := ∑ n∈N(A −1 f S ) n e −λ‖(·)−tn‖2 2
is bounded linear from P W Z to L 2 (R d ). Additionally, I λ (f) ∈ C 0 (R d ).
3) Let 0 < β < √ 3δ 2 − 2. For all f ∈ P W βB2 , we have f = lim λ→0 + I λ (f) in L 2 (R d ) and uniformly
on R d .
Consider when d = 1 and Z = [−π, π]. In this case Theorem 3.4 enjoys a strong stability property.
If we reindex all terms in Theorem 3.4 by Z, then the entries of A −1 satisfy |A −1
n,m| ≤ Me −α|n−m|
for some M, α > 0. This implies that A −1 is l ∞ → l ∞ continuous, so we have stability of I λ (f)
given l ∞ perturbations of the sampled data. It is not known to the author whether Riesz bases
for L 2 (B 2 ) exist. It is known however that Riesz bases consisting of exponentials exist for L 2 (P)
where P is a filled convex symmetric polygon with an even number of sides (see [29]). There are
certainly polygons Z of this type that satisfy the somewhat delicate conditions of Theorem 3.4.
4. Master’s research
The closed unit disc D ⊂ C endowed with the Poincaré metric density ds = 2
1−|z| 2 |dz| is the disc
model for hyperbolic geometry. The terms geodesic, hyperbolic arclength (h-length), and hyperbolic
area (h-area) may be defined naturally in terms of the resulting metric, and the orientationpreserving
isometries are the Möbius transformations from D onto itself, see [12]. A hyperbolic
n-gon (h-n-gon) is a Jordan curve in D whose boundary consists of n geodesics.
5

6
4.1. The hyperbolic area formula. Theorem 4.1 below appears in [2].
Theorem 4.1 (B). If (x n+1 , y n+1 ) := (x 1 , y 1 ), . . . , (x n , y n ) are the distinct positively oriented vertices
of an h-n-gon, then the h-area of the h-n-gon is
n∑
( )
(5) A = 2tan −1 xk y k+1 − y k x k+1
.
1 − x k x k+1 − y k y k+1
k=1
If the h-n-gon has l concave sides and m convex sides, with angle measures α k and β k respectively,
then equation (5) takes the form
(6) A =
l∑
α k −
k=1
m∑
β k .
Equation (6) compares favorably to the classical Gauss-Bonnet formula, and each can be used
to prove the other.
4.2. The isoperimetric theorem. Theorem 4.1 can be applied to prove the isoperimetric result
below.
Theorem 4.2 (B, unpublished). For each n ≥ 0, there exists an h-n-gon with h-perimeter P with
maximal h-area. If (w 1 , . . . , w n ) are the vertices of any such h-n-gon, there is an isometry M, and
R ∈ (0, 1) (depending only on P and n) such that
( ) 2πik
M(w k ) = R exp .
n
Versions of Theorem 4.2 (not all equivalent) have appeared elsewhere, see [11, 15, 16].
k=1
5. Questions for future research
5.1. Questions relating to convergent cardinal series. We present the following reworded
Theorem from [38].
Theorem 5.1 (Seip). Let {t n } n∈Z be a real sequence such that
(7) sup |t n − n| < 1/4.
n∈Z
Define the entire functions
G(z) = (z − t 0 )
∞∏
n=1
Let 0 < σ < π. For all f ∈ E σ ∩ L ∞ (R),
(
1 − z
t n
) (
1 − z
t −n
)
(8) f(t) = lim
uniformly on compact subsets of R.
N∑
N→∞
n=−N
and H n (z) =
f(t n )H n (t)
G(z)
G ′ (t n )(z − t n ) .
Theorem 5.1 has the following statement as a corollary (which may also be deduced for Corollary
2.5). Let 0 < σ < π. If f ∈ E σ ∩ L ∞ (R), then f is a CCS. This is certainly not true if σ = π;
simply consider f(z) = sin(πz). If f ∈ E π ∩ L ∞ (R), it is unknown whether or not the CCS which
interpolates {(n, f(n))} n∈Z always converges. A partial result in this direction is Theorem 5.2 from
[9].

Theorem 5.2 (B, Madych). Let f ∈ E π .
If f is bounded on R, there exists an increasing sequence of positive integers {N j } j∈N and a constant
C such that
f(z) = C sin(πz) + lim
j→∞
N j
∑
n=−N j
f(n) sinc(z − n).
In light of the corollary mentioned above, Theorem 5.2, and the function z ↦→ sin(πz), we ask
the following question.
1) If f ∈ E π ∩ L ∞ (R), when does there exist a constant C such that the equality below holds?
f(z) = C sin(πz) + lim
N∑
N→∞
n=−N
f(n) sinc(z − n).
It is well known that if {t n } is any real sequence such that {e i(·)tn } is a Riesz basis for L 2 [−π, π],
then B = {sinc((·) − t n )} is a Riesz basis for P W [−π,π] . By [30] (which also develops the corresponding
L p theory) and [32], B has a biorthogonal Riesz basis {H n (·)} with an infinite product
expansion analogous to the expression given in Theorem 5.1. By Kadec’s ”1/4” Theorem in [25],
the expansion (8) is the biorthogonal expansion for f ∈ P W [−π,π] . In this setting, Seip’s Theorem
shows that a nontrivial expansion which lives most naturally in a Hilbert space is also valid in an
L ∞ space! Taking these statements together, we ask the following question.
2) Let 1 < p < ∞, and {t n } be a complete interpolating sequence for E π ∩ L p (R). In what spaces
of entire functions of exponential type σ < π does the corresponding expansion analogous to (8)
(as developed in [30]) hold?
Multidimensional generalizations of Theorems 2.1 and 2.2 are also of interest.
5.2. Questions relating to Theorem 3.2. Examples of UIRBs are given in Section 2, but no
comprehensive characterization has been given. In [5], a more general concept of uniform invertibility
is developed, so that given any Riesz basis {f n } n for separable Hilbert space, there exists
an orthogonal basis (playing a role similar to that of { e i〈(·),n〉} presented here), such that
n∈Z d \C l,d
{f n } n is a UIRB with respect to that basis. The proof is currently non-constructive, and given an
exponential Riesz basis, it would be useful to have a construction of such a concrete orthogonal
basis which is amenable to computation through the Fourier transform.
3) If { e i〈(·),t k〉 } n∈Z d is a Riesz basis for L 2 ([−π, π] d ), when is it a UIRB?
Multidimensional generalizations of Theorem 3.2 are also of interest.
5.3. Questions relating to Theorem 3.3. There are unresolved issues regarding the matrix
from Theorem 3.3.
4) What are criteria for the infinite matrix B from Theorem 3.3 to be defined (and in this case
necessarily bounded) as a map from l ∞ to itself?
As a special case, what is the rate of decay of the entries of B off the main diagonal, and how
does this relate to the rate of decay of the entries of (B −1 ) nm = sincπ(t n −t m ) when B is invertible?
Even when t n = n + δ n and δ n is small, the off-diagonal decay of B −1 is somewhat pathological,
and the following theorem doesn’t apply.
7

8
Theorem 5.3 (Jaffard, [23]). If A is boundedly invertible on l 2 (Z d ) and |a kl | = O(|k−l| −s ) for some
s > d, then its inverse B = A −1 has the same polynomial-type off-diagonal decay |b kl | = O(|k−l| −s ).
Unfortunately, Corollary 11 in [20] (a generalization of Theorem 5.3) doesn’t appear to apply
either, so the problem appears genuinely difficult.
5.4. Questions relating to Theorem 3.4. The following variant of Theorem 3.4 was proven by
Schlumprecht and Sivakumar, (see [37]).
Theorem 5.4. Let {t n } n∈N ⊂ R be a sequence such that { e 〈(·),tn〉} n∈N is a Riesz basis for L 2([−π, π]).
Given f ∈ P W [−π,π] , let f S = {f(t k )} k∈N . For A as defined in Theorem 3.4, and λ > 0, define
The following statements hold:
I λ (f)(·) = ∑ n∈N(A −1 f S ) n e −λ((·)−tn)2 .
1) The map f ↦→ I λ (f) is bounded linear from P W [−π,π] to L 2 (R). Additionally, I λ (f) ∈ C 0 (R).
2) For all f ∈ P W [−π,π] , we have f = lim λ→0 + I λ (f) in L 2 (R) and uniformly on R.
Schlumprecht and Sivakumar have proven the d-dimensional version of Theorem 5.4 (unpublished)
with [−π, π] replaced by B 2 , but in order to be non-vacuous, their generalization requires
the existence of Riesz bases of exponentials for L 2 (B 2 ). For this reason we ask whether or not
Theorem 5.4 holds in d-dimensions when {t n } n∈N ⊂ R d is chosen such that { e 〈(·),tn〉} is a frame
n∈N
for L 2 (B 2 ) instead of a Riesz basis. In this setting, Beurling has shown in [14] that exponential
frames exist in abundance.
5.5. A Question relating to Theorem 4.1. The proof of Theorem 4.1 makes extensive use of
the geometry of Möbius transformations and standard integral theorems. The transformations in
the analogous three dimensional model of hyperbolic geometry are quite similar in spirit (see [12])
to their two dimensional cousins, though we no longer have a field structure to exploit. Suitably
modified, the proof of Theorem 4.1 should generalize to give a hyperbolic volume formula for
hyperbolic polyhedra.
References
[1] B. A. Bailey, An asymptotic equivalence between two frame perturbation theorems, in: M. Neamtu, L. Schumaker
(Eds.), Proceedings of Approximation Theory XIII: San Antonio 2010, Springer pp. 1-7.
[2] B. A. Bailey, Area of polygons in hyperbolic geometry, Masters thesis, Texas Tech University Department of
Mathematics and Statistics, (2004).
[3] B. A. Bailey, Multivariate polynomial interpolation and sampling in Paley-Wiener spaces, J. Approx. Theory 164
(2012), no. 4, 460-487.
[4] B. A. Bailey, Sampling and recovery of multidimensional bandlimited functions via frames, J. Math. Anal. App.
370 (2010), no. 2, 374-388.
[5] B. A. Bailey, Studies in interpolation and approximation of multivariate bandlimited functions, Doctoral dissertation,
Texas A&M University Department of mathematics, (2011).
[6] B. A. Bailey, W. R. Madych, Convergence of Classical Cardinal Series and Band Limited Special Functions, J.
Fourier. Anal. Appl. DOI 10.1007/s00041-013-9291-4.
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