INTERPOLATION SEQUENCE for the SPACES (Φ)(Q 1)

Interpolation sequence for the spaces (φ)(q≥1)

Maher M.H. Marzuq

84 Raymond Road, Plymouth, MA 02360 USA

April 25, 2009

Abstract

Let φ be a subadditive increasing real valued function defined on [0, ∞) and which satisfies φ(x) = 0 if and only if x = 0. For q ≥1 we define (φ) to be the set of all functions f which are analytic in the open unit disc and satisfy

And (φ) to be the subspace of (φ) of functions which satisfy

=

In this paper we prove some interpolation theorems for (φ)

AMS subject Classification: 30, 46

Key Words and Phrases: The Class, as an F-space in the sense of Banach, Interpolation sequence, Universal interpolation sequence

1.  Introduction.

Let us recall some definitions. We call a real-valued function φ defined on [0, ∞) a modulus function if φ is an increasing continuous subadditive function that satisfies the condition that φ(x) = 0 if and only if x = 0.

Let q ≥1. By the class (φ) we mean the collection of all analytic functions f defined in the open unit disc Δ which satisfy

For q = 1 the spaces were studied in details in W. Deeb and M. Marzuq [2] , W. Deeb, R. Khalil and Marzuq [3]. If φ(x) = x, then (φ) becomes the usual Banach space . For φ(x) = xp, 0 < p ≤ 1 and q = 1, then

(φ) Becomes the usual F-spaces . If q = 1 and φ(x) = log (1+xp), then (φ) becomes Np. If q=1 and φ(x) = log (1+x), then (φ) becomes the class N of functions of bounded characteristic. For Np = N+ for 0 < p ≤ 1 Z. Jianzhong [8] where

Np = {f H(D) H (φp ):

And

N={fN: = .

In general the spaces (φ) are not F-Spaces; see N. Yanagihara [13, p.453] for an example.

We shall assume that φ satisfies the additional condition that φ(et) is

a convex function of t, and consequently (φ)N.

A function f є (φ) is said to belong to the class (φ) if

. =

For the class (φ), which is a vector space , we define a distance function

by

(1)

The spaces (p≥1), (0<p≤1) P.L. Duren [5], N+ N. Yanagihara [13] (p>0) and (log+H)α(a>1)

M. Stoll [12] becomes special cases of (φ).

Let α =( αn)=( α1,…., αn,….), be a sequence of real numbers such that αn à 0.

Let

L(φ,α)={(cn):cn є C and d((cn),0) =

Let X is a class of analytic functions in Δ and {zn} is a given sequence in Δ.

When a complex sequence {cn} is given, the interpolation problem asks if a function f є X exists such that f{zn} = cn for all n.

Let Y be a class of complex sequences. If for every sequence (cn) є Y there is an f є X such that f{zn} = cn, then the sequence {zn} is called a universal interpolation sequence for the pair (X,Y), simply written zn is u.i.s for (X,Y). We are interested in the pair (φ), L(φ,α) where α = (1 – |zn|2).

A sequence {zn} in Δ is called uniformly separated (u.s.s) if

and

L. Carleson [1] showed that {zn} is u.i.s for (H∞, l∞) if and only if {zn} is u.s.s. For the pair ((φ),l(φ,α)) we have the following results: for q ≥1, φ(x) = x, Shapiro and Shields [11] proved that {zn} is a u.i.s for (H(φ),l(φ,α)) if and only if {zn} is u.s.s. The same result was proved for q = 1 and φ(x) = xp, 0 < p ≤ 1 by V. Kabaila [9]. For q = 1, φ(x) = log (1+x), N. Yanagihara [14] showed that {zn} is u.s.s, then {zn} is u.i.s. and if {zn} is u.i.s, then

where

In this paper we obtain results which generalize the above mentioned results.

In section 2 we will prove that (φ) and l(φ,α) are F-spaces in the sense of Banach, N. Dunford and J.T. Shwartz [4].

In section 3 we prove that if {zn} is u.s.s then {zn} is u.i.s. for ((φ),l(φ,α), we also proved that if φ satisfies lim φ-1(ax) φ-1(1/x) < ∞ for all real a & Tφ,q ((H(φ)) = l(φ,α) then {zn} is u.s.s.

where Tφ,q (f(z)) = ((φ-1(1-|zn|2))(1/q)-1f(zn))

2.  The spaces (φ), l (φ,α).

In this section we will show that the spaces (φ) and l(φ,α) are F-spaces.

Theorem 1. (φ) is an F-space in the sense of Banach, N. Dunford and J.T. Shwartz [4]. That is

(i)  Let fn be functions in (φ) such that ρ(fn,0)à0 as nà∞. Then for any α є C, ρ(αfn,0)à0 as n à∞.

(ii)  Let αn є C be such that αnà 0. Then for each f є (φ), ρ(αn f,0)à0 as nà∞.

(iii)  (φ) is complete with respect to the metric (1.1).

Proof.

Suppose {fn} is a sequence in (φ), ρ(fn,0)à0 and β є C. Now

Where [|β|] is the greatest integer in |β|.

(i)  Suppose βn є C, βnà0 and f є (φ), without loss of generality we may assume |βn| ≤1, so

.

a.e. Hence by Lebesgue convergence theorem we get

Thus ρ(βnf,0)à 0 as nà∞

(ii)  Suppose {fn} is a Cauchy sequence in (φ). By Lemma 3 in W. Deeb and M. Marzuq [2] applied to [φ(|f|)which is subharmonic for q ≥1 M. Hasumi & S. Kataoka [6, Lemma 5.1], we get

Therefore

n,m>N(ε) and for all z є {w:|w| ≤r<1}. Hence {fn(z)} is a Cauchy sequence in C. Since {fn(z)} converges uniformly on compact subsets of Δ and fn(z) is analytic, then {fn} converges to an analytic function f. Clearly, {φ(|fn|)} converges uniformly on compact subsets to φ(|f|). Therefore

Hence, f є (φ). But (φ), so a.e. and fn(θ)àf(θ) in measure. Now choose a subsequence such that a.e., then

Thus if nj and n are sufficiently large, we have

Ρ(f, fn)à0, as nà∞.

It remains to show that f є (φ). Since a.e., then there exists a compact set E such that m(E)> 2π – ε and uniformly on E, hence

. for large j.

Also,

so

letting jà ∞,

for any ε>0. Hence

which proves that f ε (φ).

Theorem 2. lis an F-space.

Proof. Parts (i),(ii) are exactly the same as in Theorem 2.1. For completeness, suppose {xn} is a Cauchy sequence, let ε > 0 be given, then there exists N such that

d(xk, xm) < ε for all k,m > N(ε).

hence

Thus () is a Cauchy sequence in C for each n. So it must converge to some xn є C.

Let x = (xn), then Minkowski’s inequality gives

but {xn} is a Cauchy sequence, hence the second term of the right side is bounded by M which does not depend on k, also

by (2). Thus xnà x and x є lq(φ,α).

3.  Interpolation Theorems

We will assume an additional condition on f, for each g є H1 the function is integrable on the unit circle.

Now we prove a theorem which generalizes theorem 1 in N. Yanagihara [14].

Theorem 3. If {zn} is u.s.s then {zn}is u.i.s. for ((φ),l(φ,α)),

where α = (1- |zn|2)

Proof. Let c = (cn) є l(φ,α) and let

Clearly g is analytic in Δ since Also g є, because

Since is integrable, hence by K. Hoffman [7,p.53] there exists f1 є such that . Since for all k Hence, |f1(zn)| = |cn|+1 and f1(zn)= (|cn|+1), where cn = |cn|and f3(zn)= , then (f1f2 – f3)(zn)= cn. Since , we have f1 є (φ) and so is f1f2 – f3. By Carleson’s Theorem there exists functions f2, f3, in H ∞ such that f2(zn) =

The following theorem generalizes the one given by Shapiro and Shields in [11].

Theorem 4. Suppose φ satisfies

For all real a and Tφ,q((φ))= l(φ,α), then {zn} is u.s.s.

We need the following lemma:

Lemma. The sequence E = {en} is bounded in l(φ,α) in the sense of topological vector space W. Rudin [10], where en = (0,0,….,0,1,0,….) and 1 appears in the nth place.

Proof. Let s> 0 be given and let Bs = {x є l(φ,α)||x||<s}. We need to show that there exists ro such that E r Bs for all r > ro. Let ro be such that (φq)-1(s)=, then for r > ro

which implies є Bs. Therefore E r Bs .

Proof of Theorem 4. Let

for all n}.

The quotient space is an F-space and Tφ,q induces a one-to-one bounded linear functional Δφ,q(since Tφ,q is bounded) from onto l(φ,α). Hence, the inverse is bounded which implies that is bounded on , i.e. there exists M > 0 such that BM, which means that for all n, there exists

fn є such that

||fn||

and

so by W. Deeb and M. Marzuq [2]

but

So

and by using (3) and (4) we get

for xk large, where xk = and Δ is a constant. Thus {zk} is u.s.s.

References

[1] L. Carleson, An interpolation problem for bounded analytic functions, Amer. J. of Math., 80 (1958), 921-930.

[2] W. Deeb and M. Marzuq, H(φ) spaces, Cand. Math. Bull., Vol. 29(3), (1986), 295-301.

[3] W. Deeb, R. Khalil and Marzuq, Isometric multiplication of Hardy-Orlicz spaces, Bull. Austral. Math. Soc. 34 (1986), 177-189.

[4] N. Dunford and J.T. Shwartz Linear operators, 1: General theory, Pure and Appl. Math., Vol.7, Interscience, N.Y. 1958.

[5] P.L. Duren, Theory of spaces, Academic Press, N.Y.1970.

[6] M. Hasumi & S. Kataoka, Remarks on Hardy-Orlicz classes, Arch. Math. 51(1988), 455-463.

[7] K. Hoffman, Banach spaces of analytic functions, Prentice Hall Inc., N.J., (1962).

[8] Z. Jianzhong, A note on Hardy-Orlicz spaces, Cand. Math. Bull. Vol.33(1),(1990), 29-33.

[9] V. Kabaila, Interpolation sequences for classes in the case p < 1, Litov. Mat. Sb., 3 (1963), #1, 141-147 (Russian).

[10] W. Rudin, Functional Analysis, McGraw-Hill, N.Y. (1973).

[11] H.S. Shapiro & A.L. Shields, On some interpolation problems for analytic functions, Amer. J. Math., 83(1961), 513-532.

[12] M. Stoll, Mean growth and Taylor coefficients of some topological algebras of analytic functions, Ann. Polon. Math. 35(1977), 139-158.

[13] N. Yanagihara, Multipliers and linear functional for the class , Trans. Amer. Math. Soc. 180(1973), 449-461.

[14] N. Yanagihara, Interpolation theorems for the class , III. J. Math. 18(1974), 427-435.

1