NORMS OF POLYNOMIALS AND CAPACITIES ON BANACH SPACES

NORMS OF POLYNOMIALS AND CAPACITIES ON BANACH SPACES

MIGUEL LACRUZ

Added in proof. Although I never submitted this paper for publication anywhere else, by a mistake I sent it to Integral Equations Operator Theory who printed it, also by a mistake, in Vol. 34 (4) (1999), 494^499.

1. Introduction

Recall that a functionP defined on a complex Banach space X is a homo- geneous polynomial of degree 0 ifPis constant, a homogeneous polynomial of degreed 1 if there is a bounded, symmetricd-linear formAon X such thatP…x† ˆA…x;. . .;x†for allx2X, and a polynomial of degreed if Pcan be written as PˆP₀‡P₁‡ ‡P_d, where each P_k is a homogeneous polynomial of degree k. There is a natural norm on the vector space of polynomials; it is given by the expression

kPk ˆsupfjP…x†j:x2B_Xg:

This paper concerns a general procedure for estimating norms of poly- nomials. Such procedure can be described as follows. Let S be a subset of BX, put

jPj_S ˆsupfjP…x†j:x2Sg;

c_d…S† ˆsupfkPk: degPˆd;jPj_S 1g;

and observe that the inequality jPj_S kPk c_d…S†jPj_S holds for every polynomial P of degree d on X. Thus, giving upper bounds on c_d…S†pro- vides a way for estimatingkPkin terms of jPj_S.

Received February 5, 1997. The paper has appeared in Integral Equations Operator Theory 34 (4) (1999), 494^499 because of the following course of events. The author submitted adif- ferentpaper to IEOT February 16, 1998. This was accepted March 3, 1998. By a mistake the author sent the manuscript of the wrong paper to IEOT October 22, 1998, believing it to be the proof of the paper he submitted to IEOT. IEOT did not discover that they had received the wrong paper, and the wrong paper was printed and published July 30, 1999. Math. Scand. was informed about the wrong publication November 22, 1999, too late to make large changes in the page proof of vol 85.2.

Durand [2] introduced this scheme for polynomials in one complex vari- able and computed the precise values of c_d…S† for some specific subsets of the unit disc, obtaining the results that are shown in the table below.

S cd…S)

fz2C:jzj rg r^ÿd

fe^it2C:jtÿj ^"₂g ¹₂h…tan^"₈†^d‡ …tan^"₈†^ÿdi

‰0;1Š ¹₂ …3‡ 

p2

†^d‡ …3ÿ2 

p2

†^d

h i

fz2C:z^dkˆ1g;k2 hsin^kÿ1†_2k iÿ1

Aron, Beauzamy, and Enflo [1] analyzed the problem of comparing real and complex norms of polynomials in many variables. They looked at poly- nomialsPof degreed in Nvariables

P…x₁;. . .;x_N† ˆ X

jjd

ax₁¹ x^ÿN_N

and considered the norms

kPk_RˆsupfjP…x₁;. . .;x_N†j:ÿ1x₁;. . .;x_N1g;

kPk_CˆsupfjP…e^it1;. . .;e^itN†j:0t1;. . .;tN 2g;

for which they obtained the inequality kPk_C…3 

p2

‡4†^d‡ …3 

p2 ÿ4†^d

2 kPk_R:

This estimate represents an upper bound on c_d…S† when X ˆ`^N†1 and Sˆ ‰ÿ1;1Š^N. Notice that the bound is independent of the number of vari- ables.

Siciak [6] improved their inequality later on. Using the notions of extremal functions and capacities inC^N allowed him to obtain the sharp estimate

cd‰ÿ1;1Š^N

ˆ …1‡ 

p2

†^d:

The aim of this paper is on the one hand to give upper bounds onc_d…S†for certain subsets of general Banach spaces, and on the other hand to extend Durand's inequalities to polynomials in many variables.

2. Results on general Banach spaces

This section is devoted to give an upper estimate oncd…S†whenSis an"^net of B_X and to compute c_d…S† when S is a ball of radius r>0 centered at a2X.

Theorem 2.1. Let X be a complex Banach space, let d1, and let 0< " <log2=…ed†. If S_"is an"^net of B_X then

c_d…S_"† 1

2ÿe^ed":

The proof of Theorem 2.1 relies on an infinite^dimensional version of Bernstein's inequality due to Harris [3], who stated it only for homogeneous polynomials, although it also works for arbitrary polynomials, as Tonge and Lacruz [4] pointed out.

Lemma2.2 (Harris^Bernstein inequality). Let P be a polynomial of degree d on X and let D^kP denote its kth Fre¨chet derivative, where 1kd. If x;y2B_X then

jD^kP…x†…y^k†j d^d

…dÿk†^dÿkkPk:

Proof of Theorem2.1. LetPbe a polynomial of degreed onX, let >0, and pick anx2BX with jP…x†j …1ÿ†kPk. Now take a point y2S" such thatkxÿyk< "and expandPin Taylor series aroundyto obtain

P…x† ÿP…y† ˆX^d

kˆ1

kxÿyk^k

k! D^kP…y† xÿy kxÿyk

_k

: It follows from Harris^Bernstein inequality that

jP…x† ÿP…y†j kPkX^d

kˆ1

d^d …dÿk†^dÿk

"^k k!;† but‰d=…dÿk†Š^dÿke^k so the last expression is

kPkX^d

kˆ1

…ed"†^k

k! kPkÿe^ed"ÿ1 :

Hence, …1ÿ†kPk jP…x† ÿP…y†j ‡ jP…y†j kPk…e^ed"ÿ1† ‡ jP…y†j, which leads to the inequality jP…y†j kPk…2ÿe^ed"ÿ†. Since >0 is arbitrary, it follows thatjPj_S" kPk…2ÿe^ed"†.

Remarks. (i) Notice thatc_d…S_"† !1 as "!0‡, so that the error when approximatingkPkin terms ofjPj_S" can be made as small as desired.

(ii) LetT be a subset ofB_X such that kPk ˆ jPj_T for everyP. The proof of Theorem 2.1 shows that the same result holds whenS"is just an"^net of T. This is the case when X ˆ`^N†1 , T is the distinguished boundary of the polydisc, and S" consists of the points in C^N all of whose coordinates are dkth roots of unity. This can be regarded as an extension of Durand's fourth result.

Theorem2.3. Let X be a complex Banach space, let a2X, and let r>0. If d1then

cd…a‡rBX† 1‡ kak r

_d

; and the estimate is best possible.

Proof of Theorem 2.3. First of all consider the case aˆ0. Let >0, pick anx2BX such thatjP…x†j …1ÿ†kPk, and definef…z† ˆP…zx†. Then f…z†is a polynomial of degree d in one complex variable. Now apply Dur- and's first result to get

1

r^dsupfjf…z†j:jzj rg kfk jf…1†j ˆ jP…x†j …1ÿ†kPk

so that there exists a z2C with jzj r and jP…zx†j …1ÿ†r^dkPk. This gives the desired inequality, sincekzxk rand >0 is arbitrary.

Next, no longer assume aˆ0, consider the polynomial Q…x† ˆP…xÿa†

and observe that 1

r^djPj_a‡rBX ˆ 1

r^djQj_rBX kQk ˆsup jQ……1‡ kak†x†j:kxk 1 1‡ kak

1

…1‡ kak†supfjQ……1‡ kak†x†j:kxk 1g

ˆ 1

…1‡ kak†supfjQ…y†j:kyk 1‡ kakg

1

…1‡ kak†supfjQ…x‡a†j:kxk 1g ˆ 1

…1‡ kak†dkPk;

so the inequality follows. Finally, consider the polynomial f…z† ˆ

‰…zÿa†=rŠ^d, wherea2Candr>0, and conclude that the inequality is best possible.

3. Extremal functions and capacities

According to Siciak [5], the extremal function associated with a subset SC^N is defined by

_S…x† ˆsup

d1



_d…x†

pd

;

where d…x† ˆsupfjP…x†j: degPˆd;jPj_S ˆ1g: The ^capacity of S is de- fined by

1

…S†ˆsupf_S…x†:kxk 1g:

Notice that these definitions also make sense whenSis a subset of a complex Banach spaceX. The following result is a well^known fact about sequences of real numbers.

Lemma3.1. Letfa_dg¹_dˆ1 be a sequence of positive numbers that satisfies the condition a_d‡d⁰ a_da_d⁰ for all d;d⁰1. Then

d!1lim



ad

pd

ˆsup

d1ad:

Since the product of a polynomial of degreed and a polynomial of degreed⁰ is a polynomial of degreed‡d⁰, it follows from Lemma 3.1 that

S…x† ˆ lim

d!1



d…x†

pd

:

There is a strong connection between the ^capacity of a set S and the quantitiesc_d…S†; it is given by the following

Lemma3.2. If S is a subset of a complex Banach space then 1

…S†ˆsup

d1



c_d…S†

pd

ˆ lim

d!1



c_d…S†

pd

Proof of Lemma3.2. First of all, notice that 1

…S†ˆ sup

x2BX

S…x† ˆsup

x2BX

supd1



d…x†

pd

ˆsup

d1 sup

x2BX



d…x†

pd

ˆsup

d1



cd…S†

pd

:

The second equality follows from Lemma 3.1, taking into account the sub- multiplicativity of the norm of polynomials.

Notice that, as an application of Lemma 3.2, the result obtained in Theo- rem 2.3 admits of an equivalent formulation in terms of capacities, namely,

…a‡rBX† ˆ r

1‡ kak:

4. Estimates for polynomials in many variables

The following result is due to Siciak [5]; it concerns the computation of the maximal function for a product of subsets of the complex plane.

Lemma 4.1. Let S₁;. . .;S_NC and let SˆS₁ S_N C^N. If zˆ …z₁;. . .;z_N† 2C^N then

S…z† ˆmaxfS1…z1†;. . .; SN…zN†g:

A straightforward consequence of Lemma 4.1 is the following

Lemma 4.2. Suppose S1;. . .;SN C and let SˆS1 SN`^N†1 . Then

…S† ˆminf…S₁†;. . .; …S_N†g:

Proof of Lemma4.2. Just notice that 1

…S†ˆ sup

kzk11_S…z† ˆ sup

jzkj1maxf_S1…z₁†;. . .; _SN…z_N†g

ˆmax sup

jz1j1S1…z1†;. . .; sup

jzNj1SN…zN†

( )

ˆmax 1

…S1†;. . .; 1 …SN†

;

and the desired identity follows at once.

Now everything is ready to extend Durand's second result to polynomials in many variables.

Theorem 4.3. Let Xˆ`^N†1 and ÿ ˆ f…e^it1;. . .;e^itN† 2C^N :jtkÿkj

"k=2g, where1;. . .; N are fixed real numbers and0< "1;. . .; "N <2. Then cd…ÿ† cot "

8

d;

where"ˆminf"1;. . .; "Ng.

Proof of Theorem 4.3. For each 1kN, put ÿ_kˆ f…e^itk 2C: jtkÿkj "k=2g, so that ÿ ˆÿ1 ÿN. Durand's second result in combination with Lemma 3.2 leads to

1

…ÿk†ˆ lim

d!1



cd…ÿk† pd

ˆ lim

d!1

cd‡1…ÿk†

c_d…ÿ_k† ˆcot "k

8 : Now Lemma 4.2 gives

…ÿ† ˆminf…ÿ₁†;. . .; …ÿ_N†g

ˆmin tan "1

8 ;. . .;tan "N

8

n o

ˆtan "

8 : Thus,c_d…ÿ† 1=…ÿ†^d ˆcot…₈^"†^d.

REFERENCES

1. R. Aron, B. Beauzamy, and P. Enflo,Polynomials in many variables: real vs. complex norms, J. Approx. Theory 74, no. 2, 1993.

2. A. Durand,Quelques aspects de la the¨orie analytique des polynoªmes, Lecture Notes in Math.

1415, 1990.

3. L. A. Harris, Bounds on the derivatives of holomorphic functions of vectors, Colloque d'Analyse, ed. L. Nachbin, Rio de Janeiro, 1972.

4. M. Lacruz and A. M. Tonge,Polynomials on Banach spaces:zeros and maximal points, J.

Math. Anal. Appl. 192, no. 2, 1995.

5. J. Siciak,On some extremal functions and their applications in the theory of several complex variables, Trans. Amer. Math. Soc. 105, no. 2, 1962.

6. J. Siciak,Wiener's type sufficient conditions inC^N, preprint.

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