ETNA
Electronic Transactions on Numerical Analysis.
Volume 26, pp. 178-189, 2007.
Copyright  2007, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
PROBABILITY AGAINST CONDITION NUMBER AND SAMPLING OF
MULTIVARIATE TRIGONOMETRIC RANDOM POLYNOMIALS
ALBRECHT BÖTTCHER AND DANIEL POTTS
Abstract. The difficult factor in the condition number of a large linear system is the
spectral norm of . To eliminate this factor, we here replace worst case analysis by a probabilistic argument. To
be more precise, we randomly take from a ball with the uniform distribution and show that then, with a certain
probability close to one, the relative errors and satisfy with a constant that involves
only the Frobenius and spectral norms of . The success of this argument is demonstrated for Toeplitz systems and
for the problem of sampling multivariate trigonometric polynomials on nonuniform knots. The limitations of the
argument are also shown.
Key words. condition number, probability argument, linear system, Toeplitz matrix, nonuniform sampling,
multivariate trigonometric polynomial
AMS subject classifications. 65F35, 15A12, 47B35, 60H25, 94A20
1. The general setting. Let "!# be an invertible linear operator. We equip
the space with the $&% norm ')(*' and denote the corresponding operator norm ( + spectral
norm) also by ',(-' . Let .0/ +2143657 8'39';:<.*= , pick > in .0?@/ BA 10C= and put
D
D
D
D
+EF> . If B
G> 5HG.I/ is a perturbation to > , then ,JK>MLN
G> OP+
LQG with G +RSG> . We have
D
'4,
G> 'T:U'WVXY''@G ' and hence
'4,
G> '
'>'
D
'@G '
D
V-X '
:Z'
'
D
'
'
'
'[>'\
which is usually written in the form
(1.1)
'&]^>'^:"_ X J`
D
>
O@'&]
\
'
where
'
_aXbJc
>aO is the
\
VX 'F'>'e9'>' (see
condition number with respect to a vector defined by _-XYJc >
OdK+
\
e.g. [1, p. 605]).
On the other hand, let D G 5fG.g/g be a perturbation to D . Then '9D h 'T:i'@j'F'>Ih ' and thus
D
'@G
'
'
D
'
:Z'j'
'
'4I
G> '
D
'@
'
V-X
D
'
'[>'lk
This can be written as
(1.2)
'&]
where now
D
_%mJc
D \
'@eo' '
'8'F'WVX
D
Ot:"_gJ`^O
\
D
D
'^:_ % Jc
D
\
O@'@]^>'
the condition number with respect to a vector given by _a%mJc D OnK+
(see e.g. [3]). Note that always _aXYJc >
Op:q_gJ`^OpK+r'j''sV-\ Xb' and
O
\
.
Now take an arbitrary linear (not necessarily invertible) operator u !vw and
suppose > and G> are independently and randomly drawn from .?@/ and G.g/ with the uniform
distribution, respectively. Assume we know that
_%Jc
(1.3)
„
xsyz
'[>'{:Z'>'^:Q|'[>'}~i€)f‚
xSy&z
\
'0,
G> 'T:i'@Sj
G> '^:Q|'4,
G> '@}ƒ~6€)f‚
\
Received March 10, 2006. Accepted for publication November 28, 2006. Recommended by M. Gutknecht.
Fakultät für Mathematik, Technische Universität Chemnitz, D - 09107 Chemnitz, Germany (aboettch,
potts@mathematik.tu-chemnitz.de).
178
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179
PROBABILITY, CONDITION NUMBER, AND SAMPLING
x
z
O . Note that
is the probability of the event … , ‚i5‡† C €4O , and |ˆ5‡JcC
x
\
\
\Š‰
J‹'[>'s+ŒCmO^+EC and J‹'>'s+ŒCmO{+RC , so that these two cases can be ignored. It follows
that
where
x
J`…,O
'0,
G> '
'[>'
:
z
€
'S,
G> '
|
:
'>'
'@Sj
G> '
z
|
+
'F>'
z
'
D
'@G '
D
'
or, equivalently,
(1.4)
|
zQ'@]
'@]^>'^:
with a probability of at least
'
. One similarly obtains that
J€)‚mO %
(1.5)
D
'@]
with a probability of at least
D
|
z
'P:
'&]{>'
.
J€)‚mOŽ%
In many cases the estimation of ',VX' or 'WVX D ' is difficult, so that the basic quantities
D
_
XYJc
>
O in (1.1) or _%J`
O in (1.2) are not available. In contrast to this, it is frequently easy
\
\
to establish inequalities of the form (1.3). Consequently, we can invoke (1.4) or (1.5) to prove
that the equation F>+ D is well-posed at least with a certain probability.
then
In [3] it was shown that if
'@{39'@%
‘
is randomly drawn from the uniform distribution,
3n5QT/ '@j'@“%
+
'@j' % '39' %9’
'@j' %•”
%
\—–
'@{39'%

:
'@j' % '39' %’
L
”
˜
˜
\
where … and % denote the expected value and the variance and 'o(' “ stands for the Frobenius
–
norm). Consequently, Chebyshev’s inequality yields
norm ( + Hilbert-Schmidt
x
'^39'%
™
™
'@39' %
™
t
'8'“%
'@j'“%
”
fšo'8' %
™ :Hšo'j' %
™
”
™
™
whence
x

’
~›€)
’
™
'@39' %
:i'@{39' %
'@j'@“%
:ˆ
”
Notice that this estimate is independent of  . Thus, if
give
(1.6)
'@]^>' %
:
'@j'“% e ”
'@j' “% e ”
LBšo'8' %
Ldšo'8'%
'&]
ƒšo'8' %
 €)
J ”
L
%
˜
˜
O[š %’
k
D
˜
L
˜
'@39' %
’
'@j'4“% e ”œ
with a probability of at least
(1.7)
J ”
' %
O[š % \
’
šo'@j'%
~i€)
J ”
L
˜
˜
O[š % k
, then (1.3) and (1.4)
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180
A. BÖTTCHER AND D. POTTS
2. Toeplitz systems. Let žŸ+¡ ž9eY¢{ž and pick £d5¥¤¦pJ`ž0O . Put
The Toeplitz operator § J`£O is the linear operator on $&%J`…,ž O defined by
+1m€
\ k&k@k \
” =
.
J¨§
­ > ­
Jc£O¨>
Oª©T+¬«
£Y©
V
­4®¯I°
±
J³²´5p
ž O
\
where 14£ = ®Yµ ° is the sequence of the Fourier coefficients of £ . The function £ is usuw
w
ally referred to as the symbol of § Jc£O . The question of whether the condition numbers
_IJ¨§
Jc£OŽOP+'•§
Jc£O@''•§sV-X Jc£O@' remain bounded has been studied for a long time (see, e.g.,
[5], [6], [7]). Criteria are known for all dimensions ¶ , but these are effectively verifiable for
¶·+¸€ only. Moreover, for smooth functions £ƒ5¥¹ŸJcO ( ¶·+¸€ ) it turns out that generically
_IJ¨§
Jc£OŽO either remains bounded or grows exponentially fast to infinity (see [4] and [12]).
The following result tells a completely different story. It shows that from the probabilistic
point of view the systems § J`£ºO¨>+ D are asymptotically always well-conditioned. We put
'£'
+»4¼½¼)¼½¾9¿ £
JcÃoO
Â\
À ®YÁ °RÂ
¦
È4ÉËÊ
where  ¶mà  +  ¶
JcÈ0ÉËʽÌ
\ k@k&k \
T HEOREM 2.1. If šo'£'@%
°
O
¦¸Í
x
-'@]^>' %
:‘˜
Â
+¶m‚
'£'
'@£a'@%
X k@k&k
%
¶m‚
, then
%
ž
£aJ¨ÃoO %
Â
Á °gÂ
+ĐÅ
J
. Note that always
J‹'£'% B
L šo'£'%
%
¦
'£' % fš9'@£' %
¦
%
O
'&]
D
' %
Â
¶Ã
˜YÆ
'@£a'
~ˆ€)
’
X½Ç‹%
Â
O ž’
\
:Z'£'
%
¦
.
%
˜
” ž š %9’
for all sufficiently large ” . In particular,
ÎÐϳÑ
Ò
x
¦
 '@]^>' %
:7˜
J‹'£'% B
L šo'£'%
%
¦
'@£' % f9
š '@£' %
¦
%
O
D
'&]
' %
’
+Z€
k
Proof. From (1.6) and (1.7) we deduce that
x
-'&]^>' %
:
Jc£O@'@“% e ” žL¥š9'§
J`£ºO&'%
' §
•
Jc£O@' “% e ”gž š9'§
`J £ºO&' %
'•§
'&]
D
' %
~‡g€t
’
J ”gž L
%
˜
˜
O[š %9’
k
It is well known that '§ J`£ºO&'%´!Ó'£'@% , and the multivariate version of the Avram-Parter
¦
theorem (for which see [13]) says that '§ Jc£O@'@“% e ” ž{!#'@£a'@% . Ô
%
3. Limitations. The argument employed here has of course its limitations. One of them
is that passage to (1.6) requires that '@j'@“% e ”‡œ šo'j'@% and that š œ C cannot be made
arbitrarily small because (1.7) must be a number less than one. In several cases of interest,
is the principal ”Õ” truncation of a bounded linear operator Ö on $0%mJc× O . In other
Z+i
words, 7+R
is the compression of Ö to $ % J[1*€
' converges
” =0O . It follows that '@
k&k@k \
to '•Ö)' . The critical term is ' '@“% e ” . If this term\ goes
to zero sufficiently fast, we cannot
guarantee the inequality ' '@“% e ”ƒœ š9'@ '@% with an š such that ” š%{!
.
‰
Suppose, for example, Ø is a compact operator on $4%mJc× O . Then Ù >;+ D is expected
to be an asymptotically ill-posed equation. In [3, Theorem 5.1] it was shown that indeed
'Ù
'@“% e ” !uC , which confirms that expectation.
The Toeplitz matrices § Jc£O make sense for every £H5N¤)XbJ`O . However, they are the
principal ”NÕ¥” truncations of a bounded operator if and only if £R5¡¤)¦pJ`O . Theorem
2.1 for ¶H+ڀ says that '§ J`£ºO&'“% e9J ” '§ J`£ºO&'%0OM!l'£'% e9'£'@% if £5U¤¦pJ`O . In [3] it
%
¦
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PROBABILITY, CONDITION NUMBER, AND SAMPLING
was proved that '•§ Jc£O@'@“% eJ ” '•§ Jc£O@'@%&OM!ÛC if £"56¤XbJ`O A ¤¦pJcO . Thus, the systems
D
§
Jc£OÜ>+
are not tractable by our approach if £´5p¤)XbJ`O A ¤¦pJcO . The same must be said
of the principle ”ƒÕ·” truncations of the Hankel matrix ÖN+EJ`£ ©½Ý ­
O½¦ ­•ß
generated by the
VX ©½Þ
X
Fourier coefficients of a function £5¤tXbJcO . In this case always '@ '“% eJ ” '@ '@%&OF!uC (see
[3, Theorem 5.2]). Notice that this also happens for £5H¤ ¦ JcO , in which case the Hankel
matrix Ö induces a bounded operator on $4%mJc× O .
4. Sampling of trigonometric polynomials. Let àSáâ+¡1m{ã
ãŒ= . We now em\ k&k@k \
bark on the problem of finding the coefficients > ­ ( ä5à žá ) of a trigonometric polynomial
(4.1)
from the values >IJcå9©0O at given points å
> ­ È %‹è É
«
­ ®Yæ °ç
0
>IJ¨åaOF+
åëS5B† C
X \ k&k@k \
we have to solve the linear equation î>+
îU+¸ï`È %‹è É
\
D
€@ìܞ
. Thus, letting
ë
+RJK>J¨å © OŽO © ß
D
>+EJí> ­ O ­4®Yæ °ç
\
­0é ê
X \
with
­0é ê@ð&ñ
© ®*ò
°ç
Þôóôóôó Þ ë•õÞ ­4®Yæ
X
k
. Together with the ö Ք system î>+ D we also consider the ”^Ք system
î
î>H+
and the ö Õ ö system îWî >H+ D . In these three systems, > and D need not be
the same vectors, but we always assume that > is randomly drawn from a ball of appropriate
dimension with the uniform distribution.
In what follows we are concerned with asymptotic results. We therefore assume that ¶
remains fixed but that 1&å
and that ã and ö go to infinity.
å ë = is a set depending on ã
X \ k&k@k \
As for random sampling, we remark that Bass and Gröchenig [2] work with a randomly
å
ëb= of sampling points and prove that there exist constants 5
chosen set 14å
X \ k@k@k \
\½÷
JcC
O depending only on ã
and ¶ such that
Put ”
+EJ
ã7LM€0OŽž
˜D
\Š‰
xsy
JŽ€)ƒšO[ö9'@39' %
for every š65‡JcC
implies that
\
€0O
:Z'&îS39' %
:UJ€LBšO[ö9'39' % }f~›€)d{È V
ø
and all trigonometric polynomials
xQý
'&]^>'^:6þ
€Ldš
€)ƒš
'@]
D
3
ë½ùŽú
Ç@ûËX
ÝùŽü
of the form (4.1). Clearly, this
'•ÿŒ~6€)d{È V
ø
ë½ùŽú
Ç@ûËX
ÝùŽü
x
and hence that if we fix a threshold ¹ œ € , then J½'&]{>'8:E¹M'@] D '@O converges to € exponentially fast as ã remains fixed and ö goes to infinity (oversampling). In contrast to this,
z
å
ëb= deterministically and show that there are constants
|¥5BJ`C
O
we here pick 14å
X \ k@k@k \
\
\Š‰
and ‚Ÿ5J`C €0O such that
\
x
y z
z
'@39'^:U'4îS39'^:H|'@39' }
~i€)‚
if 3 is randomly taken and that e0|Q!—€ and ‚·!—€ if ö and ã are appropriately adjusted
x
and go to infinity. Our conclusion is that J‹'@]^>'W:6¹M'@] D '@Ot! € as the two appropriately
adjusted parameters ã and ö go to infinity. The techniques employed here do not yield
exponentially fast convergence.
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182
A. BÖTTCHER AND D. POTTS
5. The spectral norm.
For our purposes we need upper bounds for the spectral norms
of î , §Zô+Zî î , and K+ZîWî . Clearly, '4îM'@%^+7'•§8'P+7' ' . The matrix § is a Toeplitz
matrix,
ë
§+EJ¨§ ­ O ­ Þ ‹®Yæ °ç
while
§ ­ Þ +
\
È V
%‹è É û
«
© ß
­
ü é ê&ð
V
\
X
is of the form
+EJ
©4O
ë
Þ© ß
©^+NÙ;J¨å
X \
få9©0O
«
Ù;J¨å
OF+
\
®Yæ °ç
w
éê
È %½è É w
k
å
ëb= , it is also useful to
In order to compensate for clusters in the sampling set 1&å
X \ k@k&k \
incorporate weights © œ C into our problem, i.e., to consider the weighted approximation
problem
ë
©
«
© ß
Ï
D
Â
Ñ8Ï
% !
©t·>IJcå©4O Â
k
X
J
)ë4O we are therefore also interested in bounds for the spectral
Letting lK+
X \ k&k@k \
norm of the Toeplitz matrix § given by
ë
§
Eîi+RJ¨§ ­ O ­ Þ ‹®Yæ °ç
ô+6î
§ ­ Þ +
\
© ß
§ ­ ? È %½è É
«
­ ®Yæ °ç
4
ë
­4é ê
+
«
­ ®Yæ °ç
4
© ß
norm of the symbol of the matrix.
­4é ê ê ð ü
û V
+
È %½è É
«
k
X
The norm '•§j' can be estimated from above by the ¤)¦
In the present case, this is the function
£aJ¨å
O+
ü é ê&ð
­
©&È V
%‹è É û V
«
X
ë
«
Ù;J¨åMå©0O
© ß
X
k
Consequently,
(5.1)
ë
Ñ
'•§j'{:
êY® ? Þ
ÙHJ¨åMå©bO ™
° ™ «
™ © ß
™k
™
™
X
X
On the other hand, to get an upper bound for the
spectral norm of , we can invoke Gersh
gorin’s theorem. Since the diagonal entries of are all ” , we obtain that
(5.2)
'
'P:
”
L
Ñ
©
« ß ©
Â
©
Â
+
”
L
Ñ
©
« ß ©
Â
ÙHJ¨å
å©4O
Âk
Of course, it would suffice to work with only one of the estimates (5.1) and (5.2). We rather
want to emphasise that two frequently used tools, estimates for Toeplitz matrices through the
symbol on the one hand and estimates for diagonal dominant matrices via Gershgorin on the
other, lead to essentially the same problem in the case at hand.
åëY= depending on
We consider two ways of characterising a sequence of knots 1&å
X \ k&k@k \
D
ã . First, for å
5p ž and ,+RJ
O5 ž we define
\
' '
¦
+
Ñ!
X \ k@k&k \
J O
 X  \ k@k&k \  ž  \
ž
Ï
¼#"IJ¨å
\
D
O+
џÏ
­4®Yµ ° 'åM
D
L;ä-'
¦
k
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PROBABILITY, CONDITION NUMBER, AND SAMPLING
The separation distance $ and mesh norm % of the set
defined as
џÏ
Ï
ß ¼#"IJcå ©
© (
$n+
å O
\
å
ëb='&r† C
X \ k@k@k \
€@ìܞ&
\
ž
are
Ñ
Ñ Ï)
Ÿ
Ï
ê®YÁ ° © ß ÞôóôóôóôÞ ë ¼*"J¨å å © O
˜
\
\
X
%S+
\
1&å
respectively. Clearly, $:HöºVX½ÇŠžW:'% . In what follows,
always denotes a positive constant
÷ ž
depending only on ¶ , but not necessarily the same at each occurrence.
Ñ8Ï
L EMMA 5.1. If
© '•å´å © '
, then
~+$
¦
ë
«
© ß
Â
Ù;J¨å´å © O
ý
:
Â
÷
X
 ã
ž
Î
,
€
L
ž
€
$
ž ÿ
dã
$ ’
k
Proof. This can be shown by using standard arguments (see, for example, [9]) and the
estimates
Â
Â
ÙHJcå
O
ÙHJcå
O
+
Â
+
Â
and so on.
Â
Â
ÙHJÙHJ-
-
X \ k@k&k \
-
X \ k@k&k \
O
ž
O
ž
J
:
Â
J
:
Â
ã
LN€4OŽž&V-X
˜
if
-/.
Â
˜ Â
LN€4OŽž&Va%
ã
˜ 0
- .
Â
Â@Â
-21
$:
if
$:
Â
- .
Â
-3.
Â
€
:
Â
\
˜
Â
:
€
$f:
\
˜
Â
- 1
€
:
Â
˜
\
Ô
A second way of generating a sequence of nonuniform sampling points is as follows. Let
4
4
4
and
J`ãZO
+
JªãZO be natural numbers that go to infinity as ã
!
X
X
ž
ž
\ k&k@k \
‰
4
4
divide the cube † C €ì ž into
congruent boxes
4
+
\
576
(5.3)
X k@k&k
X
4
ž
6
\
X
X
4
576
LN€
Õ
Õ
k@k&k
X98
6
ž
4
\
ž
ž
4
LN€
ž8
k
åëY= where ¤ is indeSuppose each of these boxes contains at most ¤ points of 1&å
X \ k&k@k \
pendent of ã . Notice that such a distribution of the sampling points allows the separation
distance $ to be arbitrarily small.
L EMMA 5.2. Take one of the boxes (5.3) containing å and denote by
box and the :*ž)Q€ neighboring boxes. Then
Î),; 4
«
® ¯
© m
Ç
Â
ÙHJ¨åMå©bO
:
Â
÷
¤NPã
ž
L
4
Î),; 4
X
k@k&k
’
X
Proof. Proceed as in the proof of Lemma 5.1.
oã
L
4
ž
Bã
’
ž
the union of this
Ô
T HEOREM 5.3. We have
(5.4)
(5.5)
'•§j't+R'
'•§j't+R'
'{:
'{:
 ã
ž
÷
÷
ž
¤N9ã
€
L
L
$
Î),;
$ ’
Î
, 4
4
ž
€
X
X
’
\
k&k@k
Î),; 4
ã
L
4
ž
ž
’
k
ž
’
k
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184
A. BÖTTCHER AND D. POTTS
First proof. We use (5.1) to estimate
$ . Thus, by Lemma 5.1,
'•åMå©'
¦
. There are at most :ºž indices
'§j'
²
such that
Í
'•§j'^:
< ê
«
ÙHJ¨åfå © O L
Â
ê&ð <*=>@? Â
V
:: ž J
˜
ý
L€0O ž L
ã

ã
ž
÷
< ê
«
Ù;J¨å´å © O
Â
ê@ð <*=A(? Â
V
Î),;
€
L
$
ž
€
ž ÿ
dã
$ ’
\
which implies (5.4). Combining the same kind of argument with Lemma 5.2, we obtain (5.5).
Ô
Second proof. This time we employ (5.2) to tackle ' ' . Fix ²n5;1m€
ö*= . There are
\ k@k&k \
at most : ž indices B such that 'å då © '
$ , and hence Lemma 5.1 yields (5.4) as in the
¦
Í
previous proof. Also as in the preceding proof one can deduce (5.5) from Lemma 5.2. Ô
6. The system C!D+EGF . We first consider a sequence of sampling knots governed by a
constraint for the separation distance $ .
X
%
¹tá
and
xSy
γÏÐÑ
€
œ
D
'&]{>'^:Q¹tád'@]
á
\
¹áˆ+Z€
Pá
\
¦
Ò
M
ÝJ
T HEOREM 6.1. If $Z~IH ö*V-XŽÇŠž and H ãEXŽÇ‹%
X
%
constants H , H , HNK , O , then there are two sequences
:‘öXŽÇŠž:9HK» ¿'L ã
with positive
1b¹ á =0¦á ß
and 1/P á =bá¦ ß such that
X
X
γÏÐÑ
€
Páˆ+Z€
á
Í
\
Ò ¦
.
'}n~+Pá
Proof. In the case at hand, the constant in (1.6) is
'&îM'@“% e ”
'&îM' “% e ”
(6.1)
Obviously, '&îM'@“% +6ö ” . The quantity
the constant (6.1) is not greater than
€L
(6.2)
€)
÷
ž
÷
ž
LBš9'4îM'%
fš9'4îM' % k
'&îM'@%S+¡'•§j'{+¡'
šbömV-XYJªã
L›J€beQ$gO
šbö V-X Jªã
L›J€beQ$gO
Î
,
Î
,
'
can be estimated by (5.4). Thus,
J€beQ$gOŽO½ž
J€beQ$gOŽO ž k
By assumption,
Î
,
€
$
ÝJ
Choose šs+Nã¡Važ•Ç‹%
€
$
+
+
y
öYO+SRŸJ¨ö X½Ç‹ž ã
XŽÇ
K
O
k
. Then
ž•Ç‹%
šbö VX J`ã2L›J€0e$gO
y
Î
,
+SRŸJ¨ö XŽÇŠž
ã
VX½Ç‹%
ã
XŽÇ‹%
ÝJ
since we may assume that
ÝJ
Î
,
JŽ€0eQ$gO½O ž
ǽ% ö VX½Ç‹ž ã
ÝJ
VX½ÇT
Nj% ö V-XŽÇŠž ã
s€beWPLXO9e
VX½Ç½%
L;ã
˜
Í
C
ÝJ
Nj% RŸJ€0O}
Nj% ö V-XŽÇŠž RŸJ¨ö X½ÇŠž ã
ž
+VUJŽ€4O
\
. Consequently, (6.2) is
€LXUJŽ€4O
€)YUJŽ€4O
+jm¹ á %
k
XŽÇ
K
O }
ž
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185
PROBABILITY, CONDITION NUMBER, AND SAMPLING
From (1.6) and (1.7) we now obtain that
xsy
'@]^>' %
:¹ á %
D
'@]
' % }ƒ~ˆ€)
J ”
%
˜
L
˜
+jPá
O[š %9’
k
Since
J ”
L
˜
we finally see that P
O š %
˜
áˆ!
as ã
€
€
+ZRŒ
ã
!
ž
.
‰
J
ž&V
ã
ž
+ZR
’
V
ï ã
J
ž
ñ
+SUJ€0O
\
Ô
In the following theorem, we consider a sequence of knots as before Lemma 5.2.
T HEOREM 6.2. If
=0¦á ß
and 1/P á =b¦á
10¹ á
X
¹ á
y
x
and
€
œ
X
y
D
~+Pá
' }
'@]^>'T:"¹tá'&]
Važ•Ç
ã
K
P á
}
\
P á
€
Í
then there are two sequences
C
œ
such that
+E€L\R
¹ á
\
with a constant H
öZ~[HãEž
ß
\
y
+Z€)]R
Važ•Ç
ã
K
}
k
Proof. By (5.5), the constant (6.1) is at most
€Ldš
€)ƒš
ž
y
÷
ž
¤ö*VX
÷
ã
y
¤ö VX
L›J
ã
L›J
Î
, 4
Î
, 4
4
O‹e
X
X
O‹e
y
}
X
4
X
k@k&k
}
ã
y
LNJ
ã
k@k&k
LNJ
Î),; 4
Î),; 4
4
O‹e
ž
ž
4
O‹e
}
ž
k
}
ž
K
Taking into account that öºV-XP+SRŸJªãŒVaž0O and choosing šs+6ã¡Važ•Ç , we see that this is
€LRŸJ`ã
Vž•Ç
€)]RŸJ`ã
Vž•Ç
K
K
O
xsy
From (1.6) and (1.7) we infer that
Páâô+␀)
†ÐJ
˜
ã
with
+j¹ á %
O
'@]^>'T:Q¹ á
D
'@]
'}n~+P á
%
˜
L€0O ž L
¹táˆ+Z€LRŸJ`ã
˜
ìbã
V
%Šž•Ç
K
K
O
k
with
+E€)^RŸJªã
’
Vž•Ç
Važ•Ç
K
O
Ô
k
ÝJ
Theorems 6.1 and 6.2 are based on the hypothesis that öj~H ” X½Ç‹%
(with some O œ C )
%
and ö8~\H ” . Thus, ö has to go to infinity with a rate that does not lag too much in comparison
with ” . This restriction is caused by the reason outlined in Section 3.
7. The system _ `aC!DbEcF . We now consider a sequence of sampling knots controlled by a constraint for the mesh norm % . Let © be the area (volume) of the Voronoi
region of the point å © .
T HEOREM 7.1. If %p:dH-ãŒV-X with a positive constant H then there are two sequences
ß
and 1/Pá=b¦á ß such that
10¹tán=0¦á
X
¹tá
and
x
y
œ
X
€
\
'@]^>'T:"¹ á
¹tá‡+E€L\R
'&]
D
' }
~+P á
y
Važ•Ç‹% }
ã
k
\
Pá
Í
€
\
Páˆ+Z€)]R
y
ã
Važ•Ç‹% }
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186
A. BÖTTCHER AND D. POTTS
Proof. From [8, Theorem 5] we obtain that
ï
Eî´' “%
«
­0®Yæ °ç
+
©T+
«
© ß
LHÈ %‹èž
˜
ë
M
'
'T:
'•§
«
­0®Yæ °ç
X
€^+
”
áfe
ñ
%
and since
\
the constant in (1.6) is at most
€Ldš9J
L¥È0%‹èž
€)ƒš9J
˜
L¥È %‹èž
˜
áfe
O%
áfe
k
O %
Now take šW+›ã¡Važ•Ç‹% and proceed as in the proof of Theorem 6.1 Ô .
8. The system Chg;C!DiE[F . We now consider the ”BÕ” system î î>f+ D in which
> and the perturbation G> are taken from .I/ and G./ with the uniform distribution. We
establish the analogues of Theorems 6.1 and 6.2.
M
K
ÝJ
:‘öXŽÇŠž:9HK» ¿'l ã
T HEOREM 8.1. If $Z~IH ö*V-XŽÇŠž and H ã Çkj
with positive
X
%
constants H , H , HNK , O , then there are two sequences 1b¹ á =0¦á ß and 1/P á =bá¦ ß such that
X
%
¹tá
and
xSy
'&]{>'^:Q¹ á
γÏÐÑ
€
œ
á
D
\
\
€
Í
á
\
X
X
Páˆ+Z€
¦
Ò
.
'}n~+P á
'@]
Pá
¹áˆ+Z€
¦
Ò
γÏÐÑ
Proof. The constant in (1.6) is
îM' “% e ”
îM' “% e ”
'&î
(8.1)
'&î
Ldšo'4î
îM' %
ƒšo'4î
îM' %
'§j' “% e ”
'§j' “% e ”
+
LBšo'•§j' %
fšo'•§j' % k
The entries on the main diagonal of the ”´ÕW” matrix § are all equal to ö . Hence
and thus (8.1) is at most
ö %
ÝJ
Choosing š+qãŒVž•Ç½%
assertion. Ô
T HEOREM 8.2. If
10¹ á =0¦á ß
and 1/P á =0¦á
” ö0%
¹tá
œ
€
öZ~[Hã
X
ž
such that
y
¹tá‡+E€L\R
\
D
'&]{>'^:Q¹tád'@]
ƒšo'•§8' % k
and estimating as in the proof of Theorem 6.1, we arrive at the
ž
ß
X
xSy
~
ö % Ldšo'•§8' %
(8.2)
and
'§j'4“%
'}n~+Pá
Proof. Start with (8.2), take
with a constant H
Vž•Ç
ã
K
}
Pá
k
\
K
š·+‘ãŒVž•Ç
€
Í
then there are two sequences
C
œ
Pá‡+U€)]R
\
y
ã
Važ•Ç
K
}
, and proceed as in the proof of Theorem 6.2.
Ô
9. The system CmChg/DnEoF . In this section, we turn our attention to the ö Õ ö sysë
ë
tem îWî >Z+ D where > and G> are drawn from the balls ./ and G.I/ with the uniform
distribution.
K J
T HEOREM 9.1. If $"~ZH ö*V-XŽÇŠž and öX½ÇŠž:ZH ãpj•Ç V with positive constants H , H ,
X
%
X
%
O , then there are two sequences 1b¹ á =0¦á ß
and 1/P á =bá¦ ß such that
¹tá
œ
€
X
\
γÏÐÑ
á
Ò
¦
¹áˆ+Z€
\
X
Pá
Í
€
\
γÏÐÑ
á
Ò
Páˆ+Z€
¦
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187
PROBABILITY, CONDITION NUMBER, AND SAMPLING
and
y
x
D
.
~+Pá
' }
'@]^>'T:"¹tá'&]
Proof. Now the constant in (1.6) equals
(9.1)
'4îWî
'“% ebö)LBšo'&îWî
'%
'4îWî
' “% ebö^fšo'&îWî
' %
All entries of the main diagonal of the ö
Consequently, (9.1) does not exceed
+
” %Ldšo'
(9.2)
' “% e0öTfš9'
'%
' % k
are all equal to ” , whence
'
'0“%
~Hö ” %
.
'%
”-% ƒšo'
K
'@“% e0öPLBš9'
'
matrix
ö
Õ
'
' % k
ü¨Ý
Our assumption implies that ãÛ~H K ö Ç@ûqj•ž %sr with some (arbitrarily small) . œ C . Letting
Ý
šs+ömV-XŽÇ‹% %sr½ž and estimating as in the proof of Theorem 6.1 we arrive at the assertion.
Ô
The sequence of sampling knots in the following theorem is again as before Lemma 5.2.
Notice that although we make no assumption about the connection between ö and ã in this
4
theorem, the nature of the knot sequence involves that necessarily ö8:"¤ 4
, so that ö
X k@k&k
ž
is in fact majorized by ã .
T HEOREM 9.2. There are two sequences 10¹Páƒ=0¦á ß and 1/Páƒ=0¦á ß such that
¹tá
and
xSy
œ
€
\
'@]^>'T:"¹ á
y
¹tá‡+E€L\R
'&]
D
'@}n~+P á
Važ•Ç
K
ã
X
}
Pá
\
Í
€
X
y
Páˆ+Z€)]R
\
k
ã
Važ•Ç
K
}
K
Proof. This can be obtained from (9.2) with the choice šp+ãŒVž•Ç as in the proof of
Theorem 6.2. Ô
Thus, while for the matrices î and î î the number ö of sampling points has to be
greater than a certain power of ã to guarantee a good conditioning with high probability, the
situation is reverse for the matrix îWî : here ö must not exceed a certain quantity increasing
with ã in order to ensure a good conditioning with a probability close to one.
10. Example. Nonuniform sampling problems are currently emerging in more and more
applications. The matrix-vector multiplication with the Vandermonde-like matrix î can realized in an efficient way with the Fourier transform for nonequispaced data [11]. The irregular
sampling problem of band-limited functions can be solved with an iterative method such as
CGNR or CGNE. In practice, the numerical results are usually much better than the deterministic predictions [10]. As an example we consider the Fourier transform on a polar grid (and
are thus in particular in the case ¶M+
). For the rather complicated deterministic analysis of
˜
the example see [2], [8], and [9], for example. The points of the polar grid lie on concentric circles around the origin, restricted to the unit square (see Figure 10.1, left). Instead of
concentric circles, the nodes of the linogram or pseudo-polar grid lie on concentric squares
around the origin. Thus, they are typically given by a slope and an intercept. Depending on
the slope, we distinguish two sets of nodes,
(10.1)
åut1 Þ ©
ô+

0v

²
0v
\ §
²

v
’
åxw1 Þ ©
\
ô+
0
 
²
§

0
²
\ 
’
We
›€ and +‡)§)e
§)e
N€ (see Figure 10.1, right). The
where ²n+‡PWe
0
˜ \ k@k&k \
˜
\ k&k@k \
öp+¡§{ . We choose +
ã
number of points
for
this
grid
is
about
and §7+9y*ã . This
0
gives %s+U€beJ ãZO . In the proof of Theorem 7.1 we established that
(10.2)
'@]^>' %
:
€LBš9J
€)fš9J
LHÈ4èOŽ%
˜
˜
LHÈ è O %
'&]
D
' %
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188
A. BÖTTCHER AND D. POTTS
F IG . 10.1. Polar grid (left) and linogram grid (right).
F IG . 10.2. Estimate for the “probabilistic” condition number #z4 with a probability of at least 0.9.
with a probability of at least
PŒô+Ѐ)
J
0
ã
% L
0˜
ã
%
L\:mO[š %9’
k
Figure 10.2 shows the constant in (10.2) and thus an estimate for the “probabilistic” condition number '@]^>'@e9'&] D ' for the probability P‡+¡C . Consequently, the reconstruction of
k{
a trigonometric polynomial from the values on the linogram grid can be realized very effi-
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PROBABILITY, CONDITION NUMBER, AND SAMPLING
189
ciently. Similar results can be obtained for the polar grid (Figure 10.1, left).
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