ETNA
Electronic Transactions on Numerical Analysis.
Volume 25, pp. 278-283, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
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A NOTE ON THE SHARPNESS OF THE REMEZ-TYPE INEQUALITY FOR
HOMOGENEOUS POLYNOMIALS ON THE SPHERE
M. YATTSELEV
Dedicated to Ed Saff on the occasion of his 60th birthday
Abstract. Remez-type inequalities provide upper bounds for the uniform norms of polynomials on given
compact sets
provided that
for every
where is a subset of of small measure. In this
note we obtain an asymptotically sharp Remez-type inequality for homogeneous polynomials on the unit sphere in
Key words. Remez-type inequalities, homogeneous polynomials
AMS subject classification. 41A17
1. Introduction. For any !#"$ define the space of homogeneous polynomials as
,
%')
& (+* - .0/
29 2 2 ":; 9 "<: &>= ? @
1 231 465 '87
7
&
where ABCA D stands for the EFD -norm of GH"I JLK
Denote by
M 'N &PORQTS U( *WVYX[Z]\_^`a^bCced4ihg4 j ( " % ' & lknmpo &q D sr &q D O k SutvQ &q DPw `
^f`L^ b cgd
&q D (U*yx 9 "z: & ( A 9 A *y{| is the unit sphere in : & (with respect to the usual E} where o
(+*€ƒ‚„3…†  A ~ O 9 S A for any continuous function& ~ on an arbitrary compact
norm, ABCA ),
f
^
8
~
f
^

&
q
O
S
set ‡ˆ and r
D B stands for the Lebesgue surface measure in : .
The classical inequality of Remez [4] (see also [2]) was generalized in numerous ways
during the past decades. In particular, in the recent paper by A. Kroó, E. B. Saff, and the
author [3] a result for homogeneous polynomials on star-like domains was obtained. Roughly
& is a star-like ‰ -smooth (Šp‹Œ‰ tŽ )
speaking, a simply connected compact set ‡ in :
&q D which is Lipschitz continuous
domain if its boundary is given by an even mapping of o
% ' & such
of order ‰ K Then, by the result mentioned above, for any Šƒ‹ Q ‹ {  and any "
`
that
we have
where
r & q D O x 9 "<‘’‡ ( A ` O 9 S 3A “ {| SutvQ &q D
{
tv˜TO ‡ Sš™8›aORQTS !<”
•P– ^f`L^—
,
- . Q › D Š]‹v‰z‹ {
Q ž
™ › RO QTS U( *
‰ *Ÿ{
Q ”œ•–
{ ‹v‰ tv K
Received June 9, 2005. Accepted for publication October 20, 2005. Recommended by D. Lubinsky.
Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Ten-
nessee, 37240 (maxym.l.yattselev@vanderbilt.edu)
278
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279
SHARPNESS OF THE REMEZ-TYPE INEQUALITY ON THE SPHERE
For instance, in the case of the unit sphere, it follows that
{ M 'N &PORQTSutv˜TO o & q D S Q K
! ”œ•–
<
&q
D S in
The goal of this note is to obtain asymptotically sharp expression for the constant ˜TO o
the previous inequality.
T HEOREM 1.1. Let x Q ' |'¡ 5 D be a sequence of positive numbers tending to zero such that
O BS
and ¥
* ¤
'”
£ ¢ € ¡ ! Q ' 0
stand for the Gamma function. Then for any integer ƒ¦
M ' N &PORQ ' S
€
* §&
¨
”
œ
•
–
'”œ£ ¢ ¡
! Q'
(1.1)
where
(1.2)

we have
&q
­¯ { ¥ « °
­ { DY´—µ D ¶ K
§ & (+* © { ª¬« ®
 ±³±
²
In particular, in the case of the unit circle we obtain
C OROLLARY 1.2. Let x Q ' |'¡ 5 D be as above. Then
M '3N } ORQ ' S {
€
* ¯ K
”
P
•
–
'”œ£ ¢ ¡
! Q'
2. Proofs. The proof of Theorem
} % } D 1.1 explores a connection between the restriction of
to the unit sphere in : , ' O o S , and ·L} ' OR¸sS the space of complex polynomials of
% ' } O o D S there
degree at most  ! restricted to the unit circle. Namely, for any Oº¹ » S "
`
exists ¼ O¾½CS "¿·L} ' O¾¸sS such that
% '}
A ` RO ¹ Y » S A * A ¼ ¾O ½CS UA for any
½ * ¹ÁÀz »Ã" ¸ K
It will allow us to use the known Remez inequality for polynomials in ·8} ' OR¸ÄS . The following
result that we shall apply later is due to V. Andrievskii and can be found in [1].
T HEOREM 2.1. Let !#"<$; Q ¦pŠ[ and ¼Å"· ' OR¸ÄS be such that
r D x ½ " ¸ ( A ¼ OR½S A¦ {T| tÆQ K
Then
O¾Q  ¯ S ' K
 ¯ Sʱ
This estimate is sharp in the asymptotic sense. Namely, let x ¼ ' |
Fekete polynomials for the set
À
¼^ ^—Ç t « { É VV ¢
ORÈ Q
•
Ë  (+*ÍÌ ½ *WΐÏÑÐ " ¸
where normalization means that ¼ '
^ Ø^ ×Ù
'”œ£ ¢ € ¡ A ¼ ' O {
be a sequence of normalized
( Ò "#Ó
­ ª f­ Q  ÔPÕ Ó Q   ª 6Ô Ö ®
*Ÿ{ K Then
¯
S A D´ ' * { À É VV ¢œORÈ Q O¾ Q ¯  S S K
•
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280
M. YATTSELEV
Next we shall need an auxiliary lemma which will reduce the problem to the twodimensional case.
&q D (U*žx 9 * Oº¹ D KfKfK ¹’&FS ": & ( A 9 A *ž{ ¹’& ¦yŠ | denote the upper halfLet o J
sphere. Any two-dimensional plane containing the line x ¹ D * BfBfB * ¹’&q D * Š | can be
described as follows:
Ú
* xÛ fB Ü ÀÞÝ fB ß & à( Û Ý "<: | Ð 0
&q } Ó Š[ ª gÔ â Ó ­ ª   ª  Ô & qäã Ø ß & +( * O Š KfKK gŠ[ { S ": & and Ü * ORå D KKfK å &q D S "
where Ò "<á
&o q } which can beU( * represented
&q D as O { Ò S or O ­ { Ò S K
in the spherical coordinates of :
&q D be such that ß & "¿k
L EMMA 2.2. Let 泓pŠ and ç"¿$ be fixed. Further, let kèmWo J
&
q
D
and r &q D O k S * æ Ú . Then
&q }Pî tv & ´—µ &q D¶ § & æ Àðï3O æ S as æ;ñòŠ[
Õ
(2.1)
¢œÈ3é>ê rDìë Ð k®í ( Ò ")á
where § & is defined by (1.2).
&
&q D by the rule
Proof. Define a projection · & ( : ñó:
· &COº¹ D KfKfK ¹ä&q D ¹ä&S (U* Oº¹ D KfKK ¹’&q D S K
õàö
For any ôÁ“pŠ denote by
ö
(+* · & q D O¾÷ &q D SøÕ o J &q D
ö
ö
&q D
a spherical cap around
unit sphere which is the preimage of the ball ÷
õ ö & point ß & ÷ on&q the
&
q
D
D
9
9
(U*Ÿx "ù:
( A A t ô | . Let ô O æ S be chosen in such
under the projection · , where
a way that r &q D O
µœú¾¶ S * æ &q D K Denote by
k Ð *ûÌü "ÞÓý­ { { Ô ( O ü Ò S " · &CO k SÖ &q } are spherical coordinates in : &q D K
where O ü Ò S "<: â á
First we are going to showÚ that
Õ k í ( Ò "á &q }Pî tW ‚Tþ É V O ô O æ SS K
r
(2.2)
ì
D
ë
Ð
¢œÈ3é ê
¢
È
&
q
}
we have that
Suppose O 2.2 S is false, i.e., for any Ú Ò "<á
r Dìë Ð Õ k í “  ‚þ É V ¢œÈ O ô O æ SS K
ö
The last claim can be restated as
ÿ
ÿ ö µœúR¶
ü
ü
j { ü } “ q µœúR¶ { ü } for all Ò "á & q } ­
­
which can be written in the following form
(2.3)
Since
ÿ
ö
ö
j Fh q µÑúR¶ N µœú¾¶
ÿ ö ö
ü
“
q µÑúR¶ N µœú¾¶ hgj
{ ­ ü}
ü { ­ ü}
for all Ò
ö ö
ö ö
ü D +( * j h q € œ¢µ Èú¾¶ N µÑúR¶ A ü A &q } ¦ q µœú¾ƒ
€ ¶ N ‚Tµœ„úR¶ hgj A ü A &q } *Å(Tü }T
"<á & q } K
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SHARPNESS OF THE REMEZ-TYPE INEQUALITY ON THE SPHERE
281
inequality O 2.3 S implies that
ö ö
ü &D q } ü
A ü A & q } ü ¦ ÿ
j h q µÑúR¶ N µœú¾¶ { ü }
j h q µœú¾¶ N µÑúR¶ { ü }
­
­
&ü q }
ÿ ö ö
ÿ ö ö
&q }
“ q N hgj } } ü ¦ q N hgj A ü A } ü
µœúR¶ µœú¾¶
{ ­ ü
µœú¾¶ µœúR¶
{ ­ ü
ö
and consequently
ÿ
ÿ ö µœú¾¶ A ü A &q }
A ü A &q }
j { ü } ü “ q µœú¾¶ { ü } ü
­
­
&q } K Then
for all Ò "á
q DY´ }
&q
ÿ
ÿ
ü A &q } ü Ò
&æ q D * r &q D O k S * ÿ j { ­ / D ¹ }
A
9 *
O
S
Ò
j
ö µ ¶
5 D
{ ­ ü}
g
c
d
c
q
DY´ }
&/ q D
ÿ
ÿ ö µœú¾¶ A ü A &q }
ÿ
}
OÒ S
“
9
} ü Ò<* 4 { ­ 5 D ¹
õ
ö
q
{
ü
œ
µ
¾
ú
¶
­
cgd
cgd
* r &q D µœú¾¶ * æ &q D &q } O Ò S is the Jacobian of the spherical transformation in : &q D K Thus, we have
where A ü A
ÿ
ö
ö
obtained a contradiction.
Now, to prove O 2.1 S we need to getõ an
ö upper estimate for ô
ö
we have
&q Dìë ÷ & œµ q ú¾¶ D í t r & q D
ö
&q D
&q D ÷ D&q D ô &q D O æ S * { « ô § O æ &)S ±
&q D O B S stands for the usual Lebesgue measure in : &q D . From the above we obtain
æ &q D Àzï3O æ &q D S *
where that
O æ S K Since
ö
}
œµ ú¾¶ t O { À ô O æ S  S &q Dë ÷ &µœq ú¾¶ D í &q Dë ÷ & µœq úR¶ D í *
ô O æ S *  D´fµ &q šD ¶ § & æ Àðï3O æ S which completes the proof.
Proof of Theorem 1.1. We start by showing the upper estimate for the limit in O 1.1S K Let
% &
"
o &q D with r &q D O k S®t Q '&q D K Without loss of generality we may assume
j mû
`that ' and4 hgkŽ
* { and ` attains maximum of its modulus at ß & "vk K Then the auxiliary
^f`L^fbcgd
D
lemma ensures that there exists a one-dimensional sphere o which goes through the ß & with
the property
rD k Õ o D t ¯ § &Q ' Àzï3O¾Q ' S where ï3ORQ
'S
is understood in the following sense
q D
'P”œ£ ¢ € ¡ ï3ORQ ' S B Q ' * Š K
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282
M. YATTSELEV
D
Since restricted to o is a homogeneous polynomial of two variables, problem can be
`
reduced to the two-dimensional case.
}
The unit sphere in : can be viewed as the unit circle ¸ in the complex plane ! , which
D
allows us to establish a relationship between homogeneous polynomials on o and polynomials with complex coefficients on ¸ K
} À { " ½ } ­ { ' q " &¼ % OR½ } S
/'
/'
" ' q "
½
"
"
* $5 # ` «  ½²± « T Â6½ó±
* ½' ` RO ¹ Y» S * " $5 # ` ¹ »
"
where ½
* ¹ÅÀz »
' RO ¸sS K Moreover
A ` Oº¹ » S A * A ¼ % OR½ } S AU ½ * ¹ÁÀz »Ã" ¸ K
and ¼'%]"¿·
Which, in particular, means
A ` O É • V Ò V ¢
È
ª
for any Ò "ÞÓ Š[ Ô K Since
r D x ½ * ¹ÁÀÞ »ç" ¸ (
Ò S A * A ` O É • V O ª À Ò S V œ¢ È O ª À Ò S S A *)(( ¼
%*
Î } ÏÑÐ
(
(
ª
A ` Oº¹ » S A[“ {|³*  D xÒ "ÞÓ Š[ Ô ( A ` O É • V Ò V œ¢ È Ò S A3“ {T|
t ¯ § &FQ ' z
À ï3O¾Q ' S we obtain
ª
D Ì Ò "ÞÓ Š[  Ô (+(( ¼'%* ÎFÏÑÐ (( “ { Ö
*  D ÌCÒ "#Ó Š[ ª Ô +
( (( ¼ % Î } ÏÑÐ (( “ { Ö]t ¯ § & Q ' Àzï3O¾Q ' S K
rD x ½ " ¸ ( A ¼&% OR½S A3“ {|³*
Thus we can apply Theorem 2.1, which yields
'
4 * ¼'% t « { À V ¢œÈ O § & Q ' Àðï3ORQ ' SS K
4 *
É V O § F& Q ' Àzï3O¾Q ' SYS ±
^f`L^fb cgd
^`a^ b
^ ^Ç
•
The last inequality implies
{ M 'N &PORQ ' S;t
É O &Q ' Àðï3ORQ ' SS * § &Q ' Àðï3ORQ ' S O À O &FQ ' Àzï3O¾Q ' SYSS ­
! ”œ•–
<
”œ•– { V œ¢ È §
”
•P– • V §
which gives us the desired upper bound for the limit in O 1.1S K
Now we turn our attention to the lower estimate. For Š0‹ æz‹ { consider the ! -th
Chebyshev polynomials for the interval Ó
­ { À æF { ­Hæ Ô i.e.
¹
á ' ú Oº¹’S (U* á ' « { ­#æ ± © ¹ } S ' ÀÍOR¹ © ¹ } S ' 
where á ' Oº¹’S * x OR¹À
­ {
­
­ { |T is the classical ! -th Chebyshev
polynomial. It satisfies
(i) A á ' ú Oº¹’S A t { for ¹ "HÓý­ { À æF { ­Hæ Ô-,
„ . † q D N D A á ' ú OR¹äS A * A á ' ú O { S A * (( á ' ë D q D ú í (( K
(ii) €ƒ‚
(
(
Due to the symmetry of Óý­ { À æF { ­Hæ Ô we can write á ' ú Oº¹’S in the next form:
, 2' 14"3 5 OR¹ } "} S
. /
0
­ 5 ! * 8 76,
D 6
á ' ú Oº¹’S *
} ­ 5 "} S ! * 8 7èÀ { K
/ ' ¹ 1 " 3 5 D Oº¹ 9
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SHARPNESS OF THE REMEZ-TYPE INEQUALITY ON THE SPHERE
This leads to the following homogeneous polynomials of degree ! :
ú 9
`'O S *
,
5 D O{ 9
­ 5 "} S6¹ }& ­65 "} OR¹ }D À fB BB ÀÞ¹ }&q
' ä¹ & 14"3 5 D O { ­95 "} S6¹ }&ì­65 "} OR¹ }D À BfBB Àz¹
'
-.
/
/
1
"3
which enjoys the property
D S ! * 8 76,
}&q D S ! * 8 7nÀ {
,
ú 9
ú
4
` ' O S A b cgd * á ' ºO ¹’&S and consequently
4
ú
ú K
^f` ' ^fbcgd * A á ' O { S A
Then the exceptional set k ú (i.e. k³ú (+*¬x 9 "¿o &q D ( A ` 'ú O 9 S A3¦ T{ | ) can be described as
k³ú * Ì 9 "¿o &q D ( A ¹ä& A3¦ { ­#æ Ö K
ö
q
D
Thus, k ú * · &
ë ÷ &µœq ú¾¶ D í where · & is the orthogonal projection from Lemma 2.2 and
ô O æ S * æ O¾ìÀ æ S K We choose æ in such a way that r &q D O k ú S * Q ' &q D K As was shown before
 æ ORQ ' SøÀ æ } O¾Q ' S * § &Q ' Àðï3ORQ ' S where § & is defined by O 1.2S K So, we get
{ M '3N &PO¾Q ' S ¦ {
{
(
(
ú6µ  ; ¶ * {
'
(á ' «
(
'
(
(
¾
O
Q
S
±
{
!”
•P–
!<”
•P*
– :: `
:
<
!
”
P
•
–
H
­
æ
(
(
:
:
:
}
'
'

R
O
Q
S
À
R
O
Q
S
{
{
{
¦ ”
•P– { ­Hæ ORQ ' S À=< æ O { ­#æ O¾Q æ ' SS } À !<
”œ•– 
{
{ À  æ O¾Q ' S À æ } ORQ ' SøÀzï ë æ ORQ ' S í
* !<
”
•P– 
{
{ À &Q ' Àzï3O¾Q ' S K
* !<
”
•P–  §
We complete the proof by dividing the both sides of the inequality above by Q ' and taking
the limit when !ùñ ¤ K
REFERENCES
[1] V. A NDRIEVSKII , A note on a Remez-type inequality for trigonometric polynomials, J. Approx. Theory, 116
(2002), pp. 416–424.
[2] P. B ORWEIN AND T. E RD ÉLYI , Polynomials ans Polynomial Inequalities, Springer-Verlag, New-York, 1995.
[3] A. K RO Ó , E. B. S AFF , AND M. YATTSELEV , A Remez-type theorem for homogeneous polynomials, J. London
Math. Soc., 73 (2006), pp. 783–796.
[4] E. J. R EMEZ , Sur une properiété des polynômes de Tchebycheff, Comm. Inst. Sci. Kharkov, 13 (1936), pp. 93–
95.