Embeddings of M\"{u}ntz spaces: the Hilbertian case

Given a strictly increasing sequence $\Lambda=(\lambda_n)$ of nonegative real numbers, with $\sum_{n=1}^\infty \frac{1}{\lambda_n}<\infty$, the M\"untz spaces $M_\Lambda^p$ are defined as the closure in $L^p([0,1])$ of the monomials $x^{\lambda_n}$. We discuss properties of the embedding $M_\Lambda^p\subset L^p(\mu)$, where $\mu$ is a finite positive Borel measure on the interval $[0,1]$. Most of the results are obtained for the Hilbertian case $p=2$, in which we give conditions for the embedding to be bounded, compact, or to belong to the Schatten--von Neumann ideals.


Introduction
The Müntz-Szasz Theorem states that, if 0 = λ 0 < λ 1 < · · · < λ n < . . . is an increasing sequence of nonnegative real numbers, then the linear span of x λn is dense in C([0, 1]) if and only if λn < ∞, the closed linear span of the monomials x λn in different Banach spaces that contain them is usually not equal to the whole space. In particular, if 1 ≤ p < +∞, the closed linear span of the monomials x λn , n ≥ 0, in L p ([0, 1]) is a proper subspace of L p ([0, 1]). These spaces, called Müntz spaces and denoted M p Λ , exhibit interesting properties that have not been very much investigated. We refer principally to the monographies [4,9]; recent results appear in [1,2,11,6].
Our starting point is the paper [6], which tries to find conditions under which the space M 1 Λ is continuously embedded in the Lebesgue space L 1 (µ). In full generality the problem is rather difficult; more precise results are obtained for special classes of sequences Λ and/or measures µ. One can see therein that, as a general rule, the embedding properties are related to the behavior of µ near the point 1, which for these problems acts as a kind of "distinguished boundary" of the unit interval.
The purpose of this paper is to investigate embedding results for other Müntz spaces; we will actually focus on the Hilbert space M 2 Λ , although occasionally other values of p also enter the picture. In the case of M 2 Λ we will also treat more refined properties of the embedding, namely its possible belonging to Schatten-von Neumann classes. As expected, the behavior of µ near the point 1 is again decisive. Examples are given to illustrate the difficulties.
The plan of the paper is the following. The next section contains necessary preliminaries. In Section 3 we associate to a sequence Λ a certain real valued function ψ; this function allows to formulate in a unitary manner some general embedding results. Sections 4 and 6 focus on particular classes of measures and sequences, while in Section 5 an interpolation result of Riesz-Thorin type is proved for embedding operators. Finally, Section 7 presents two interesting examples.
then we obtain a quasinorm for 0 < q < 1 and a norm for q ≥ 1, with respect to which S q (H 1 , H 2 ) is complete. It is immediate that T q ≥ T q ′ for q ≤ q ′ , hence S q ⊂ S q ′ . Again we gather in a lemma some properties that we will use; more information can be found, for instance, in [10,8]. For the sake of this lemma, we will denote by T ∞ the usual operator norm.
where the infimum is taken over all orthonormal bases (φ n ) n of H 1 .
(iii) If 2 ≤ q < ∞, then where the supremum is taken over all orthonormal bases (φ n ) n of H 1 .
Note that the right hand side in (i) and (ii) may be infinite, meaning that T ∈ S q . We will use the following corollary of Lemma 2.1 (iii) and Lemma 2.2 (ii).
In particular, T ∈ S q whenever the right hand side is finite for some Riesz basis (x n ).
The elements of S 2 (H 1 , H 2 ) are called Hilbert-Schmidt operators and by Lemma 2.2 the Hilbert-Schmidt norm of T ∈ S 2 (H 1 , H 2 ) is given by where (φ n ) n is any orthonormal basis of H 1 .

Müntz spaces and embeddings.
We denote by m the Lebesgue measure on [0, 1] and by · p the norm in L p (m) for 1 ≤ p ≤ ∞.
Suppose Λ = {λ n } n≥1 an increasing sequence of positive real numbers with ∞ n=1 1 λn < ∞. As discussed in the introduction, the Müntz space M p Λ is defined to be the closure of the monomials x λn , n ≥ 1 in L p (m); which is a proper subspace of L p (m) by the Müntz-Szasz Theorem. It is proven in [4,9] that the functions in M p Λ are actually real analytic on the interval (0, 1) and continuous on [0, 1). We will use the following two results concerning Müntz spaces that appear in [9].
For a fixed p ≥ 1, a positive measure µ on [0, 1] is called Λ p -embedding if there is a constant C > 0 such that g L p (µ) ≤ C g p for all polynomials g ∈ M p Λ . Whenever p is clear from the context, we will remove the subscript p and use the notation Λ-embedding.
It follows easily from the definition (see [6]) that a Λ p -embedding measure µ has to satisfy µ(1) = 0. Therefore, as in Remark 2.5 of [6], we may extend the embedding to all f ∈ M p Λ : if µ is Λ p -embedding, then M p Λ ⊂ L p (µ) and f L p (µ) ≤ C f p for all f ∈ M p Λ . For a Λ p -embedding µ we denote by i p µ the embedding operator i p µ : M p Λ ֒→ L p (µ), which is bounded. If 0 < ε < 1, then the interval [1 − ε, 1] will be denoted by J ε .
If T : E → F is a bounded operator on Banach spaces, we define by T e = inf K T + K the essential norm of an operator, where the infimum is taken over all compact operators K : E → F . This norm measures how far an operator is from being compact. In particular, T is compact if and only if T e = 0. If µ is a positive measure on [0, 1], we will denote by µ m the measure equal to µ on [0, 1− 1 m ] and 0 elsewhere, and µ ′ m = µ − µ m . The next proposition gathers some facts that are analogues of the corresponding results obtained in [6] for p = 1 and can be proved by adapting the methods therein; we state them without further comment.
Then i µ is bounded and i µ e = a 1/p .
We will have the occasion to use the following elementary lemma.
Then for any continuous, positive, increasing function g we have A sequence Λ is lacunary if for some γ > 1 we have λ n+1 /λ n ≥ γ, n ≥ 1. The main feature of lacunarity is that the monomials λ 1/p n x λn form a Riesz basis in each of the spaces M p Λ . In particular, the sequence (λ A more general class of sequences is defined as follows. The sequence Λ is called quasilacunary if for some increasing sequence {n k } of integers with N := sup k (n k+1 − n k ) < ∞ and some γ > 1 we have λ n k+1 /λ n k ≥ γ. It is easy to show that any quasilacunary sequence may be enlarged to one that is still quasilacunary and satisfies λ n+1 /λ n ≤ γ 2 . The main property of quasilacunary sequences is contained in the following lemma.

Embeddings in M 2 Λ
Most of our results pertain to the Hilbert space M 2 Λ . In particular, in this section we will consider only p = 2, and therefore we will drop the index p and write "Λ-embedding" and "i µ ". On the other hand, we will complicate things slightly by introducing M 2 Λ,a as the closure of the same monomials x λn in L 2 ([0, a]); so pp. 177-178) that the condition n 1/λ n < ∞ ensures that the system x λn is minimal in M 2 Λ , and that, if d n is the distance from x λn to the linear space spanned by x λm with m = n, then d n = e −γnλn , with γ n → 0. Let us denote The remarks above show that the sum is convergent for any x < 1, and defines an increasing function of x.
We have thus the estimate, for functions in M 2 Λ,a , In particular, if a = 1, we recapture (3.2) for k = 1.
Although the function ψ is a rather rough indicator of the properties of the sequences Λ, it is useful in obtaining sufficient conditions for embedding results. A first example is an analogue for M 2 Λ of [6, Theorem 2.6].
Proof. The proof consists in integrating with respect to µ the relation |f (x)| ≤ ψ(x) f 2 , which, as noted above, is the case k = 1 of (3.2).
We obtain then the analogue for M 2 Λ of [6, Corollary 2.7].
Proof. Similar to the L 1 case, the proof follows from Theorem 3.1 by applying Lemma 2.7 to g = ψ 2 .
More interesting, we may improve Proposition 2.6 (ii): if the support of µ is compact in [0, 1), then the embedding is not only compact, but inside any Schattenvon Neumann class.
Let us fix a positive integer k and a number b On the other hand, integration is a Hilbert-Schmidt operator on Consider then the embedding i ′ µ from M 2 Λ,b ′ into L 2 (µ). According to (3.3) and (3.4), i ′ µ is bounded and Finally, i µ = i ′ µ R k , and thus Choosing k such that 2/k ≤ q, inequality (3.5) proves the theorem for any q > 0.
If µ is a general measure, Theorem 3.3 can still be used in order to obtain sufficient conditions for the embedding to be in S 2 . Namely, we take a sequence b n ր 1 and define µ j = µ|[b j , b j+1 ); then i 2 µ = j i 2 µj , and thus we have (for q ≥ 1) i 2 µ q ≤ j i 2 µj q . We may then apply Theorem 3.3 to each of the measures µ j . The statements obtained depend on the arbitrary sequence (b j ), and are thus not very natural. We prefer to state a more elegant result, valid for Hilbert-Schmidt embeddings.
The function ψ is hard to compute precisely, but one can give estimates that can be used in the above embedding results. Here are some examples; the estimates for the power series are classical and can be obtained, for instance, by the techniques in [5].
(1) Suppose λ n = 2 n (a typical case of a lacunary sequence). Then (2 n/2 x 2 n ) forms a Riesz basis of M 2 Λ , so in (3.1) we have d n ∼ 2 −n/2 , and This estimate shows the fact that the function ψ reflects only partially the properties of the sequence Λ. It does not use the precise fact that (2 n/2 x 2 n ) is a Riesz basis, but only that it is a uniformly minimal sequence. For instance, one cannot use it to recapture the results obtained in Section 4 below. (2) λ n = n 2 . This is a typical case of what is called a "standard" sequence (which is defined by the fact that λ n+1 /λ n → 1). A computation essentially done in [6,Section 7] shows that for some constant C > 0.

Sublinear measures
As in [6], one can obtain much more precise results if one considers special classes of measures. In this section we will again consider different values of p, so we return to the notations i p µ and M p Λ .
Definition 4.1. A measure µ is called sublinear if there is a constant C > 0 such that for any 0 < ε < 1 we have µ(J ε ) ≤ Cε. The smallest such C will be denoted by µ S . The measure µ is called vanishing sublinear if lim ε→0 µ(Jε) ε = 0.
As one can see, sublinear measures satisfy the condition of Corollary 3.2 for ρ(ǫ) = ǫ. The next lemma gathers some results that are either contained or analogues of [6, Section 4]. 1] gdm.
Since µ is sublinear and the function x λn+λm is continuous, positive, and increasing, it follows from Lemma 2.7 that 1] x λn+λm dm, and thus g n , g m L 2 (µ) ≤ µ S g n , g m 2 . Thus, if we define the matrices A = ( g n , g m L 2 (µ) ) and B = ( µ S g n , g m 2 ), then the entries of A are nonegative and majorized by those of B. But B is bounded since (g n ) is a Riesz basis in M 2 Λ (by Lemma 2.1 (i)); it follows that A is also bounded. Therefore, by 2.1 (ii), the embedding i 2 µ is bounded, of norm of order µ as m → ∞. Since i µm is compact by Proposition 2.6 (ii), it follows that i µ is compact.

Interpolation
If Λ is lacunary and µ is sublinear, then µ is Λ 2 -embedding by Theorem 4.3, while it is Λ 1 -embedding by Theorem 5.5 from [6]. It is interesting that, although the Müntz spaces do not form an interpolation scale of spaces, we may still apply the proof of the Riesz-Thorin theorem in order to extend the result to values 1 < p < 2. We will actually obtain below a more general result concerning interpolation of embeddings.
If a positive measure µ on [0, 1] is Λ p0 -embedding and Λ p1 -embedding, then it is also Λ pt -embedding with where i µ ps is the operator norm of i µ : M ps Λ → L ps (µ) for 0 ≤ s ≤ 1. Proof. The proof follows the Riesz-Thorin Theorem (see, for instance, [3]), so we will just sketch the main steps. For any 1 ≤ p ≤ ∞, denote, as customary, by p ′ the conjugate exponent (satisfying 1/p + 1/p ′ = 1). Let P Λ be the space of all polynomials in span{x λ : λ ∈ Λ}. Then P Λ is dense in M p Λ and the theorem will be proved once we show that Fix then f ∈ P Λ with f pt = 1. If p ′ t is the exponent conjugate to p t , then Take then g continuous with g L p ′ t (µ) = 1. Define, for 0 ≤ ℜz ≤ 1, 1 The Three Lines Lemma yields then whence the theorem follows by (5.1).
(ii) for any vanishing sublinear measure µ the embedding i p µ : M p Λ → L p (µ) is compact for 1 ≤ p ≤ 2.

Schatten-von Neumann embeddings
We return now to the case p = 2 and the simplified notations of Section 3. A slight strengthening of the sublinearity condition implies that for quasilacunary sequences the embedding belongs already to all Schatten-von Neumann classes. Theorem 6.1. If Λ is quasilacunary and the positive measure µ satisfies µ(J ε ) ≤ Cε α for some α > 1, then i µ ∈ S q for all q > 0.
Proof. Again the letter C will be used for possibly different universal constants. Suppose then that {n k } is a sequence of integers with N := sup k (n k+1 − n k ) < ∞, such that for some γ > 1 we have γ ≤ λ n k+1 /λ n k +1 ≤ γ 2(N −1) . With the notations of Lemma 2.8, each subspace F k has dimension n k+1 − n k . If we choose an orthonormal basis in each of the spaces F k , it follows easily from Lemma 2.8 that the union of these bases is a Riesz basis in M 2 Λ . Let us denote by (φ i ) this basis; we may also assume it consists of real functions.
Suppose φ i ∈ F k . Applying Lemma 2.4 to φ i with β j = β = 1/(n k+1 − n k ) and Lemma 2.7 for g(x) = x 2βλn k +1 and ρ(x) = Cx α , we obtain The Euler beta function satisfies the following asymptotic formula: if t is large and s is fixed, then .
It is elementary to see (and can be found in [6], Lemma 5.4) that if f : [0, 1] → R is a nonconstant differentiable function, then We apply this inequality to f = φ 2 i . If the minimum is given by the first term above, then φ i 2 ∞ ≤ 4 φ i 2 2 = 4, whence from (6.1) it follows that If the minimum is given by the second term, that is, Therefore φ i ∞ ≤ Cλ 1/2 n k+1 , whence, again by (6.1), The factor (λ n k+1 /λ n k +1 ) 1/2 is bounded by γ N −1 , and therefore Note that the inequalities (6.2) and (6.3) have been obtained for i = n k + 1, . . . , n k+1 . They imply, if q ≤ 2, that By Corollary 2.3 (ii), it follows that i µ ∈ S q (M 2 Λ , L 2 (µ)) for all q ≤ 2, and therefore for all q > 0.
In particular, for Λ lacunary the condition µ(J ε ) ≤ Cε α for some α > 1 implies that the embedding is in all Schatten-von Neumann classes.
7. Examples 7.1. In the first example we intend to construct a measure µ and a sequence Λ such that µ is Λ p -embedding for p = 2 but not for p = 1. As above, we will use the same letter C for possibly different universal constants.
Take µ = k c k δ a k , with 0 < a k < 1. We will define recurrently λ n → ∞, a n → 1, and c n → 0 such as to have: (A) sup n λ n c n a λn n = ∞; (B) k λ n c k a 2λn k ≤ C ln n n 2 . First, one can start with λ 1 = 1, a 1 = 1/2 and c 1 = 1. Suppose then that λ k , a k , c k have been obtained for k ≤ n − 1. Choose first λ n sufficiently large such that (i) λ n k≤n−1 a λn k ≤ 1 n 2 ; (ii) λ n+1 ≥ n 4 λ n . Put a n = 1 − 2 ln n λn and c n = 2n 2 ln n λn . Then The first sum is smaller than 1 n 2 by (i). The second term is of order ln n n 2 by (7.1) and (7.2). For the third term, we have k≥n+1 λ n c k a 2λn k ≤ λ n k≥n+1 c k .
From (ii) it follows in particular that c n decreases faster than a geometric progression (which also proves the convergence of the sum defining µ), and thus for some constant C we have k≥n+1 c k ≤ Cc n+1 = C2(n + 1) 2 ln(n + 1) λ n+1 .
Applying again (ii), So we have estimated all three terms of (3.3) by ln n n 2 , whence (B) is satisfied. Now (ii) implies that Λ is lacunary and the functions g k (x) = λ On the other hand, According to (B), we have and thus k g k 2 L 2 (µ) < ∞. So it follows from (7.3) and (7.4) that Thus µ is Λ 2 -embedding. On the other side, for p = 1 1] λ n x λn dµ = k c k λ n a λn k ≥ c n λ n a λn n .

So by (A)
sup n λ n x λn L 1 (µ) ≥ sup n c n λ n a λn n = ∞, whence λ n x λn 1 ≤ 1 for all n = 1, 2, . . .. Hence µ is not Λ 1 -embedding. 7.2. In this example we consider the Hilbert space M 2 Λ . We will show that for any 0 < r < q we can construct a lacunary sequence Λ and a measure µ such that i 2 µ / ∈ S r but i 2 µ ∈ S q . Fix q > r > 0. Choose a sequence {α n } ∈ ℓ q with |α n | < 1 but {α n } / ∈ ℓ r , and a double sequence {β nm } ∞ n,m=1 with n m β nm < 1 4 . The measure will again be of the form µ = j c j δ aj , with 0 < a j < 1; we will construct a n , c n as well as λ n recurrently. Suppose a j , c j , λ j have been obtained for j ≤ n − 1. Choose first λ n sufficiently large such that Λ is lacunary, and for j = 1, . . . , n − 1, (7.6) Take then a n = e −1/2λn and c n = Since clearly i 2 µ g n 2 L 2 (µ) ≥ c n λ n a 2λn n we obtain If we define f n = i 2 µ g n / i 2 µ g n L 2 (µ) , then f n are nonnegative functions, and (7.9) implies that f n , f m L 2 (µ) ≤ eα −1 n α −1 m λ 1] x λn+λm dµ(x). The last quantity can be estimated using (7.6), (7.7), and (7.8); we have, for n > m, f n , f m L 2 (µ) 2 < e 4 by (7.10). Therefore Γ 0 ≤ Γ 0 S2 < √ e/2, whence Γ is invertible. This implies by Lemma 2.1 (i) that (f n ) is a Riesz sequence in L 2 (µ). By (7.9) and the choice of α n , the sequence ( i 2 µ g n ) is in ℓ q but not in ℓ r . Corollary 2.3 (i) implies then that i 2 µ is in S q but not in S r , as desired.

Final remarks
It is often the case when dealing with Müntz spaces that results that are valid for lacunary sequences can be extended, albeit sometimes after significant work, to the quasilacunary case. We have already seen such an situation in Section 6, where the proof of Theorem 6.1 would be actually simpler if one assumes Λ lacunary. In particular, the continuity of the embedding i 1 µ for sublinear measures is shown in [6] for quasilacunary sequences. However, a similar result is not yet proved for i 2 µ ; at least the proof of our Theorem 4.3 does not seem to extend easily to quasilacunary sequences. It is an open problem to provide such an extension.
Another interesting open question is the possible extension of Corollary 5.2 to the range 2 < p < ∞; in particular, is it true in that case that, if Λ is lacunary, then any sublinear measure is Λ p -embedding?
Finally, let us note that an important application of embedding theorems is the study of boundedness properties of composition and multiplication operators with domain Müntz spaces (see [1,2,6]). We will give such applications in a forthcoming paper.