THE “FULL MÜNTZ THEOREM” IN L_{p}[0, 1] FOR 0 < p < ∞

Tamás Erdélyi and William B. Johnson

Abstract. Denote by span{f_{1}, f_{2}, … } the collection of all finite linear combinations of the functions f_{1}, f_{2}, … over ℝ. The principal result of the paper is the following.

Theorem (Full Müntz Theorem in L_{p}(A) for p ∈ (0, ∞) and for compact sets A ⊂ [0, 1] with positive lower density at 0). Let A ⊂ [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) . Then span{x^{λ_{1}}, x^{λ_{2}}, … } is dense in L_{p}(A) if and only if

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} = ∞ .

Moreover, if

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} < ∞ ,

then every function from the L_{p}(A) closure of span{x^{λ_{1}}, x^{λ_{2}}, … } can be represented as an analytic function on {z ∈ ℂ \ (−∞, 0] : |z| < r_{A}} restricted to A ∩ (0, r_{A}), where

r_{A} := sup{y ∈ ℝ : m(A ∩ [y, ∞)) > 0}

(m(·) denotes the one-dimensional Lebesgue measure).

This improves and extends earlier results of Müntz, Szász, Clarkson, Erdős, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of span{x^{λ_{1}}, x^{λ_{2}}, … } are also considered.

1991 Mathematics Subject Classification. Primary: 41A17. Key words and phrases. Müntz Theorem, denseness in L_{p}[0, 1], strictly singular operator. Research of T. Erdélyi is supported, in part, by NSF under Grant No. DMS–9623156. Research of

W.B. Johnson is supported, in part, by NSF under Grants No. DMS–9623260, DMS–9900185, and by Texas Advanced Research Program under Grant No. 010366–163.

Typeset by AMS-TEX

1. Introduction and Notation

Müntz’s beautiful classical theorem characterizes sequences (λ_{j})_{j=0}^{∞} with

0 = λ_{0} < λ_{1} < λ_{2} < ⋯

for which the Müntz space span{x^{λ_{0}}, x^{λ_{1}}, … } is dense in C[0, 1]. Here, and in what follows, span{x^{λ_{0}}, x^{λ_{1}}, … } denotes the collection of finite linear combinations of the functions x^{λ_{0}}, x^{λ_{1}}, … with real coefficients, and C[a, b] is the space of all real-valued continuous functions on [a, b] ⊂ ℝ equipped with the uniform norm. Müntz’s Theorem [Bo-Er3, De-Lo, Go, Mü, Szá] states the following.

Theorem 1.1 (Müntz). Suppose (λ_{j})_{j=0}^{∞} is a sequence with

0 = λ_{0} < λ_{1} < λ_{2} < · · · .

Then span{x^{λ_{0}}, x^{λ_{1}}, … } is dense in C[0, 1] if and only if ∑_{j=1}^{∞} 1/λ_{j} = ∞.

The original Müntz Theorem proved by Müntz [Mü] in 1914, by Szász [Szá] in 1916, and anticipated by Bernstein [Be] was only for sequences of exponents tending to infinity. The point 0 is special in the study of Müntz spaces. Even replacing [0, 1] by an interval [a, b] ⊂ [0, ∞) in Müntz’s Theorem is a non-trivial issue. This is, in large measure, due to Clarkson and Erdős [Cl-Er] and Schwartz [Sch] whose works include the result that if ∑_{j=1}^{∞} 1/λ_{j} < ∞ then every function belonging to the uniform closure of

span{x^{λ_{0}}, x^{λ_{1}}, … }

on [a, b] can be extended analytically throughout the region {z ∈ ℂ \ (−∞, 0] : |z| < b}.

There are many variations and generalizations of Müntz’s Theorem [An, Be, Boa, Bo1, Bo2, Bo-Er1, Bo-Er2, Bo-Er3, Bo-Er4, Bo-Er5, Bo-Er6, Bo-Er7, B-E-Z, Ch, Cl-Er, De-Lo, Go, Lu-Ko, Op, Sch, So]. There are also still many open problems. In [Bo-Er6] it is shown that the interval [0, 1] in Müntz’s Theorem can be replaced by an arbitrary compact set A ⊂ [0, ∞) of positive Lebesgue measure. That is, if A ⊂ [0, ∞) is a compact set of positive Lebesgue measure, then span{x^{λ_{0}}, x^{λ_{1}}, … } is dense in C(A) if and only if ∑_{j=1}^{∞} 1/λ_{j} = ∞. Here C(A) denotes the space of all real-valued continous functions on A equipped with the uniform norm. If A contains an interval then this follows from the already mentioned results of Clarkson, Erdős, and Schwartz. However, their results and methods cannot handle the case when, for example, A ⊂ [0, 1] is a a Cantor type set of positive measure.

In the case that ∑_{j=1}^{∞} 1/λ_{j} < ∞, analyticity properties of the functions belonging to the uniform closure of span{x^{λ_{0}}, x^{λ_{1}}, … } on A are also established in [Bo-Er6].

From Theorem 1.1 we can easily obtain the following L_{p}[0, 1] version of the Müntz Theorem.

Theorem 1.2 (Müntz). Let p ∈ (0, ∞). Suppose (λ_{j})_{j=0}^{∞} is a sequence with

0 ≤ λ_{0} < λ_{1} < λ_{2} < · · · .

Then span{x^{λ_{0}}, x^{λ_{1}}, … } is dense in L_{p}[0, 1] if and only if ∑_{j=1}^{∞} 1/λ_{j} = ∞.

The main result of this paper is the following.

Theorem 1.3 (Full Müntz Theorem in L_{p}(A) for p ∈ (0, ∞) and for compact sets A ⊂ [0, 1] with positive lower density at 0). Let A ⊂ [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) . Then span{x^{λ_{1}}, x^{λ_{2}}, … } is dense in L_{p}(A) if and only if

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} = ∞ .

Moreover, if

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} < ∞ ,

then every function from the L_{p}(A) closure of span{x^{λ_{1}}, x^{λ_{2}}, … } can be represented as an analytic function on {z ∈ ℂ \ (−∞, 0] : |z| < r_{A}} restricted to A ∩ (0, r_{A}), where

r_{A} := sup{y ∈ ℝ : m(A ∩ [y, ∞)) > 0}

(m(·) denotes the one-dimensional Lebesgue measure).

This corrects, improves, and extends earlier results of Müntz [Mü], Szász [Szá], Clarkson and Erdős [Cl-Er], P. Borwein and Erdélyi [Bo-Er3, Bo-Er4], and Operstein [Op]. Related issues about the denseness of span{x^{λ_{1}}, x^{λ_{2}}, … } are also considered.

The notations

∥f∥_{A} := sup_{x∈A} |f(x)| ,

∥f∥_{L_{p,w}(A)} := (∫_{A} |f(x)|^{p}w(x) dx)^{1/p} ,
∥f∥_{L_{∞,w}(A)} := inf{α ∈ ℝ : |f(x)|w(x) ≤ α a.e. on A} ,
∥f∥_{L_{p}(A)} := (∫_{A} |f(x)|^{p} dx)^{1/p}
∥f∥_{L_{∞}(A)} := inf{α ∈ ℝ : |f(x)| ≤ α a.e. on A} ,

are used throughout this paper for real-valued measurable functions f defined on a measurable set A ⊂ ℝ with positive Lebesgue measure, for nonnegative measurable weight functions w defined on A, and for p ∈ (0, ∞). The space of all real-valued continuous functions on a set A ⊂ ℝ equipped with the uniform norm is denoted by C(A). For 0 < p ≤ ∞ the space L_{p,w}(A) is defined as the collection of equivalence classes of real-valued measurable functions for which ∥f∥_{L_{p,w}(A)} < ∞. The equivalence classes are defined by the equivalence relation f ∼ g if fw = gw almost everywhere on A. When A := [a, b] is a finite closed interval, we use the notation L_{p,w}[a, b] := L_{p,w}(A). When w = 1, we use the notation L_{p}[a, b] := L_{p,w}[a, b]. It is always our understanding that the space L_{p,w}(A) is equipped with the L_{p,w}(A) norm. Denote by span{f_{1}, f_{2}, … } the collection of all finite linear combinations of the functions f_{1}, f_{2}, … over ℝ.

The lower density of a measurable set A ⊂ [0, ∞) at 0 is defined by

d(A) := lim inf_{y→0+} \frac{m(A ∩ [0, y])}{y} .

2. Auxiliary Results

In [Bo-Er3, Section 4.2], [Op], and partially in [Bo-Er4] the following two theorems are proved.

Theorem 2.1 (Full Müntz Theorem in C[0, 1]). Suppose (λ_{j})_{j=1}^{∞} is a sequence of distinct positive real numbers. Then span{1, x^{λ_{1}}, x^{λ_{2}}, … } is dense in C[0, 1] if and only if

∑_{j=1}^{∞} \frac{λ_{j}}{λ_{j}^{2} + 1} = ∞ .

Moreover, if

∑_{j=1}^{∞} \frac{λ_{j}}{λ_{j}^{2} + 1} < ∞ ,

then every function from the C[0, 1] closure of span{1, x^{λ_{1}}, x^{λ_{2}}, … } is infinitely many times differentiable on (0, 1).

Theorem 2.2 (Full Müntz Theorem in L_{p}[0, 1] for p ∈ [1, ∞)). Suppose p ∈ [1, ∞). Suppose (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) . Then span{x^{λ_{1}}, x^{λ_{2}}, … } is dense in L_{p}[0, 1] if and only if

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} = ∞ .

Moreover, if

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} < ∞ ,

then every function from the L_{p}[0, 1] closure of span{x^{λ_{1}}, x^{λ_{2}}, … } is infinitely many times differentiable on (0, 1).

Unfortunately each of the works mentioned above has some shortcomings in proving the sufficiency part of Theorem 2.2. Hence in Section 4 we present the correct arguments to prove the sufficiency part of Theorem 2.2. This part is based on discussions with Peter Borwein.

Theorems 2.3 and 2.4 are restatements of some earlier results giving sufficient conditions for the non-denseness of span{x^{λ_{1}}, x^{λ_{2}}, … } in L_{p}[0, 1] when 0 < p < ∞ and (λ_{j})_{j=1}^{∞} is a sequence of distinct nonnegative numbers. See Theorems 6.1 and 5.6 in [Bo-Er6].

Theorem 2.3. Suppose (λ_{j})_{j=1}^{∞} is a sequence of distinct nonnegative numbers satisfying ∑_{j=1}^{∞} 1/λ_{j} < ∞ . Suppose that A ⊂ [0, ∞) is a set of positive Lebesgue measure, w is a nonnegative-valued, integrable weight function on A with ∫_{A} w > 0, and p ∈ (0, ∞) . Then span{x^{λ_{1}}, x^{λ_{2}}, … } is not dense in L_{p,w}(A) .

Moreover, if the gap condition

(2.1)  inf{λ_{j+1} − λ_{j} : j = 1, 2, … } > 0

holds, then every function f ∈ L_{p,w}(A) belonging to the L_{p,w}(A) closure of span{x^{λ_{1}}, x^{λ_{2}}, … } can be represented as

f(x) = ∑_{j=1}^{∞} a_{j}x^{λ_{j}} , x ∈ A ∩ [0, r_{w}) ,

where

r_{w} := sup {x ∈ [0, ∞) : ∫_{A∩(x,∞)} w(t) dt > 0} .

If the gap condition (2.1) does not hold, then every function f ∈ L_{p,w}(A) belonging to the L_{p,w}(A) closure of span{x^{λ_{1}}, x^{λ_{2}}, … } can still be represented as an analytic function on

{z ∈ ℂ \ (−∞, 0] : |z| < r_{w}}

restricted to A ∩ (0, r_{w}) .

Theorem 2.4. Suppose ∑_{j=1}^{∞} 1/λ_{j} < ∞ . Let s > 0 and p ∈ (0, ∞) . Then there exists a constant c depending only on Λ := (λ_{j})_{j=1}^{∞} , s , and p (and not on ϱ , A , or the “length” of f) so that

∥f∥_{[0,ϱ]} ≤ c ∥f∥_{L_{p}(A)}

for every f ∈ span{x^{λ_{1}}, x^{λ_{2}}, … } and for every set A ⊂ [ϱ, 1] of Lebesgue measure at least s .

Now we offer a sufficient condition for a sequence (λ_{j})_{j=1}^{∞} of distinct real numbers greater than −(1/p) converging to −(1/p) , to guarantee the nondenseness of span{x^{λ_{1}}, x^{λ_{2}}, … } in L_{p}[0, 1], where p ∈ (0, ∞) .

Theorem 2.5. Let p ∈ (0, ∞) . Suppose that (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) satisfying

∑_{j=1}^{∞} (λ_{j} + (1/p)) =: η < ∞ .

Then span{x^{λ_{1}}, x^{λ_{2}}, … } is not dense in L_{p}[0, 1] . Moreover, every function in the L_{p}[0, 1] closure of span{x^{λ_{1}}, x^{λ_{2}}, … } can be represented as an analytic function on ℂ \ (−∞, 0] restricted to (0, 1).

Proof. The theorem is a consequence of D. J. Newman’s Markov-type inequality [Bo-Er3, Theorem 6.1.1 on page 276] (see also [Ne]) and a Nikolskii-type inequality [Bo-Er3, page 281] (see also [Bo-Er5]). We state these as Theorems 2.6 and 2.7. Indeed, it follows from Theorem 2.7 that

(2.2)  ∥x^{1/p}Q(x)∥_{L_{∞}[0,1]} ≤ (18 · 2^{p}η)^{1/p} ∥Q∥_{L_{p}[0,1]}

for every Q ∈ span{x^{λ_{1}}, x^{λ_{2}}, … } . Now repeated applications of Theorem 2.6 with the substitution x = e^{−t} imply that

∥(e^{−t/p}Q(e^{−t}))^{(m)}∥_{L_{∞}[0,∞)} ≤ (9η)^{m}∥e^{−t/p}Q(e^{−t})∥_{L_{∞}[0,∞)} , m = 1, 2, …,

in particular

|(e^{−t/p}Q(e^{−t}))^{(m)}(0)| ≤ (9η)^{m}∥e^{−t/p}Q(e^{−t})∥_{L_{∞}[0,∞)} , m = 1, 2, …,

for every Q ∈ span{x^{λ_{1}}, x^{λ_{2}}, … } . By using the Taylor series expansion of e^{−t/p}Q(e^{−t}) around 0, we obtain that

(2.3)  |z^{1/p}Q(z)| ≤ c_{1}(K, η)∥x^{1/p}Q(x)∥_{L_{∞}[0,1]} ,  z ∈ K ,

for every Q ∈ span{x^{λ_{1}}, x^{λ_{2}}, … } and for every compact K ⊂ ℂ \ {0}, where

c_{1}(K, η) := ∑_{m=0}^{∞} \frac{(9η)^{m}( max_{z∈K} |log z|)^{m}}{m!} = exp (9η max_{z∈K} | log z|)

is a constant depending only on K and η. Now combining (2.2) and (2.3) gives

(2.4)  |Q(z)| ≤ c_{2}(K, p, η)∥x^{1/p}Q(x)∥_{L_{p}[0,1]} ,  z ∈ K ,

for every Q ∈ span{x^{λ_{1}}, x^{λ_{2}}, … } and for every compact K ⊂ ℂ \ {0}, where

c_{2}(K, p, η) := c_{1}(K, η) max_{z∈K} |log z|^{−(1/p)} = exp (9η max_{z∈K} | log z|) max_{z∈K} |log z|^{−(1/p)}

is a constant depending only on K, p, and η. Now (2.4) shows that if

Q_{n} ∈ span{x^{λ_{1}}, x^{λ_{2}}, … }

converges in L_{p}[0, 1], then it converges uniformly on every compact K ⊂ ℂ \ {0}, and the theorem is proved. □

Theorem 2.6 (Markov-Type Inequality for Müntz Polynomials). Suppose that γ_{1}, γ_{2}, …, γ_{n} are distinct nonnegative numbers. Then

∥xQ′(x)∥_{[0,1]} ≤ 9 (∑_{j=1}^{n} γ_{j}) ∥Q∥_{[0,1]}

for every Q ∈ span{x^{γ_{1}}, x^{γ_{2}}, …, x^{γ_{n}}} .

Theorem 2.7 (Nikolskii-Type Inequality for Müntz Polynomials). Let p ∈ (0, ∞) . Let λ_{1}, λ_{2}, …, λ_{n} be distinct real numbers greater than −(1/p) . Then

∥x^{1/p}Q(x)∥_{L_{∞}[0,1]} ≤ (18 · 2^{p} ∑_{j=1}^{n} (λ_{j} + (1/p)))^{1/p} ∥Q∥_{L_{p}[0,1]}

for every Q ∈ span{x^{λ_{1}}, x^{λ_{2}}, …, x^{λ_{n}}} .

Our next tool is an extension of the above Nikolskii-type inequality.

Lemma 2.8 (Another Nikolskii-Type Inequality for Müntz Polynomials). Let p ∈ (0, ∞) . Let B ⊂ [0, b] be a measurable set satisfying m(B ∩ [0, β]) ≥ δβ for every β ∈ [0, b]. Let λ_{1}, λ_{2}, …, λ_{n} be distinct real numbers greater than −(1/p) . Suppose that

∑_{j=1}^{n} (λ_{j} + (1/p)) = η ≤ δb/36 ,

where δ ∈ (0, 1]. Then

∥x^{1/p}Q(x)∥_{L_{∞}[0,b]} ≤ ((2/δ)b · 2^{p})^{1/p} ∥Q∥_{L_{p}(B)} ,

and hence

max_{z∈K} |Q(z)| ≤ c(K, p, η, b, δ)∥Q∥_{L_{p}(B)}

for every Q ∈ span{x^{λ_{1}}, x^{λ_{2}}, …, x^{λ_{n}}} and for every compact K ⊂ ℂ \ {0} , where the constant c(K, p, η, b, δ) depends only on K, p, η, b, and δ.

Proof. The proof of the lemma is easy. By using a linear scaling if necessary, without loss of generality we may assume that b = 1. Let Q ∈ span{x^{λ_{1}}, x^{λ_{2}}, …, x^{λ_{n}}} , and pick a point y for which

|y^{1/p}Q(y)| = max_{t∈[0,1]} |t^{1/p}Q(t)| .

Then using the Mean Value Theorem and applying Theorem 2.6 (Markov-Type Inequality for Müntz Polynomials) to

x^{1/p}Q(x) ∈ span{x^{λ_{1}+(1/p)}, x^{λ_{2}+(1/p)}, …, x^{λ_{n}+(1/p)}} ,

we obtain for x ∈ [(δ/2)y, y] that

( max_{t∈[0,1]} |t^{1/p}Q(t)|) − |x^{1/p}Q(x)| ≤ |y^{1/p}Q(y)| − |x^{1/p}Q(x)|

≤ |y^{1/p}Q(y) − x^{1/p}Q(x)| ≤ (y − x) max_{t∈[x,y]} |(t^{1/p}Q(t))′|

≤ y \frac{1}{x} max_{t∈[x,y]} |t(t^{1/p}Q(t))′| ≤ \frac{2}{δ} x \frac{9η}{x} max_{t∈[0,1]} |t^{1/p}Q(t)|

≤ \frac{18η}{δ} max_{t∈[0,1]} |t^{1/p}Q(t)| ≤ \frac{1}{2} max_{t∈[0,1]} |t^{1/p}Q(t)| .

Hence, for x ∈ [(δ/2)y, y] we have

|x^{1/p}Q(x)| ≥ \frac{1}{2} max_{t∈[0,1]} |t^{1/p}Q(t)| .

Using the assumption on the set B, we conclude that

m(B ∩ [(δ/2)y, y]) ≥ δy − (δ/2)y = (δ/2)y

and hence

∥Q∥_{L_{p}(B)}^{p} = ∫_{B} |Q(t)|^{p} dt ≥ ∫_{B∩[(δ/2)y,y]} |Q(t)|^{p} dt

≥ (δ/2)y2^{−p} (y^{−(1/p)})^{p} ( max_{t∈[0,1]} |t^{1/p}Q(t)|)^{p}

≥ (δ/2)2^{−p} ( max_{t∈[0,1]} |t^{1/p}Q(t)|)^{p}

This finishes the proof of the first inequality of the lemma when b = 1. As we have already remarked the case of an arbitrary b > 0 follows by a linear scaling. The second inequality of the lemma follows from the first one and from (2.3) applied with Q̃(x) = Q(bx), where Q̃ ∈ span{x^{λ_{1}}, x^{λ_{2}}, … }. □

Corollary 2.9. Let p ∈ (0, ∞) and δ ∈ (0, 1]. Let B ⊂ [0, b] be a measurable set satisfying m(B ∩ [0, β]) ≥ δβ for every β ∈ [0, b]. Let (λ_{j})_{j=1}^{∞} be a sequence of distinct real numbers greater than −(1/p) satisfying

∑_{j=1}^{∞} (λ_{j} + (1/p)) =: η ≤ δb/36 .

Then span{x^{λ_{1}}, x^{λ_{2}}, … } is not dense in L_{p}(B) . Moreover, every function from the L_{p}(B) closure of span{x^{λ_{1}}, x^{λ_{2}}, … } can be represented as an analytic function on ℂ \ (−∞, 0] restricted to B \ {0} .

Proof. The corollary is a consequence of D. J. Newman’s Markov-type inequality formulated in Theorem 2.6, and our Nikolskii-type inequality given by Lemma 2.8. Indeed, it follows from Lemma 2.8 and Theorem 2.6 by the substition z = e^{−t} and by the Taylor expansion of e^{−t/p}Q(e^{−t}) around 0 that

|z^{1/p}Q(z)| ≤ c(K, p, b, δ)∥Q∥_{L_{p}(B)}

whenever p ∈ (0, ∞), B ⊂ [0, b] is a measurable set satisfying m(B ∩ [0, β]) ≥ δβ for every β ∈ [0, b], (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) satisfying

∑_{j=1}^{n} (λ_{j} + (1/p)) = η ≤ δb/36 ,

δ ∈ (0, 1], Q ∈ span{x^{λ_{1}}, x^{λ_{2}}, … }, K ⊂ ℂ is bounded, and z ∈ K, where c(K, p, b, δ) is a constant depending only on K, p, b, and δ. □

Corollary 2.10. Let p ∈ (0, ∞) . Let A ⊂ [0, 1] be a measurable set with lower density δ > 0 at 0. Let (λ_{j})_{j=1}^{∞} be a sequence of distinct real numbers greater than −(1/p) satisfying

∑_{j=1}^{∞} (λ_{j} + (1/p)) < ∞ ,

Then span{x^{λ_{1}}, x^{λ_{2}}, … } is not dense in L_{p}(B) . Moreover, every function from the L_{p}(B) closure of span{x^{λ_{1}}, x^{λ_{2}}, … } can be represented as an analytic function on ℂ \ (−∞, 0] restricted to A \ {0} .

Proof. The corollary follows easily from Corollary 2.9. To see this, choose b ∈ (0, 1] such that with B := A ∩ [0, b] we have m(B ∩ [0, β]) ≥ δβ for every β ∈ [0, b]. Then choose N ∈ ℕ such that

∑_{j=N+1}^{∞} (λ_{j} + (1/p)) =: η ≤ δb/36 .

Let U be the L_{p}(A) closure of

span{x^{λ_{1}}, x^{λ_{2}}, … } .

Let V be the L_{p}(A) closure of

span{x^{λ_{N+1}}, x^{λ_{N+2}}, … } .

Since the space

W := span{x^{λ_{1}}, x^{λ_{2}}, …, x^{λ_{N}} }

is finite dimensional, we have U ⊂ V + W. Therefore, by Corollary 2.9 every function from the L_{p}(A) closure of span{x^{λ_{1}}, x^{λ_{2}}, … } can be represented as an analytic function on ℂ \ (−∞, 0] restricted to A \ {0} . □

Finally in this section we restate a Nikolskii-type inequality that is proved in [Bo-Er3, pages 216–217] for 1 ≤ p < ∞ .

Theorem 2.11. Let p ∈ [1, ∞) . Suppose that (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) satisfying

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} < ∞ .

Then for every ε > 0 there exists a constant c_{ε} > 0 depending only on ε so that

|Q(x)| ≤ c_{ε}x^{−(1/p)}∥Q∥_{L_{p}[0,1]}

for every Q ∈ span{x^{λ_{1}}, x^{λ_{2}}, … } and for every x ∈ [0, 1 − ε) .

We suspect that the above theorem may extend to all 0 < p < ∞ and would offer a natural approach to prove one half of the “Full Müntz Theorem in L_{p}[0, 1]” when 0 < p < 1 . However, we are unable to prove this extension. Nevertheless we can still prove the “Full Müntz Theorem in L_{p}[0, 1]” for all 0 < p < ∞ with the help of Theorems 2.3 – 2.8 and Theorem 3.5. This “Full Müntz Theorem in L_{p}[0, 1]” for all 0 < p < ∞ is formulated by Theorem 3.6.

3. New Results

The new results of the paper include the resolution of the conjecture that the “Full Müntz Theorem in L_{p}[0, 1]” remains valid when 0 < p < 1 . Theorems 3.1 and 3.2 offer the right sufficient conditions for the denseness of span{x^{λ_{1}}, x^{λ_{2}}, … } in L_{p}[0, 1] when 0 < p < 1 . The “easy case” when Λ := (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) tending to −(1/p) + α , where α > 0 , is handled by Theorem 3.1.

Theorem 3.1. Let p ∈ (0, ∞) . Suppose Λ := (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) tending to −(1/p)+α , where α > 0 . Then span{x^{λ_{1}}, x^{λ_{2}}, … } is dense in L_{p}[0, 1] .

In the much more interesting case, when Λ := (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) tending to −(1/p) , our next theorem offers a sufficient condition for the denseness of span{x^{λ_{1}}, x^{λ_{2}}, … } in L_{p}[0, 1] , p ∈ (0, 1] .

Theorem 3.2. Let p ∈ (0, ∞) . Let Λ := (λ_{j})_{j=1}^{∞} be a sequence of distinct real numbers greater than −(1/p) tending to −(1/p) . Suppose that

∑_{j=1}^{∞} (λ_{j} + (1/p)) = ∞ .

Then span{x^{λ_{1}}, x^{λ_{2}}, … } is dense in L_{p}[0, 1] .

Our next theorem establishes a sufficient condition for the non-denseness of span{x^{λ_{1}}, x^{λ_{2}}, … } in L_{p}(A) where 0 < p < ∞ and A ⊂ [0, 1] is a compact set with positive lower density at 0. It extends one direction of the “Full Müntz Theorem” in L_{p}[0, 1] proved earlier for p ∈ [1, ∞) , see Theorem 2.2. Moreover, the statement about the L_{p}(A) closure of span{x^{λ_{1}}, x^{λ_{2}}, … } in the non-dense case is new even for A = [0, 1] and 1 ≤ p < ∞.

Theorem 3.3. Let A ⊂ [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) . Suppose

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} < ∞ .

Then every function from the L_{p}(A) closure of span{x^{λ_{1}}, x^{λ_{2}}, … } can be represented as an analytic function on {z ∈ ℂ \ (−∞, 0] : |z| < r_{A}} restricted to A ∩ (0, r_{A}), where

r_{A} := sup{y ∈ ℝ : m(A ∩ [y, ∞)) > 0}

(m(·) denotes the one-dimensional Lebesgue measure).

The key to the proof of Theorem 3.3 is a combination of Theorems 2.3 – 2.7 with the following functional analytic theorem.

Theorem 3.4. Let p ∈ (0, ∞) . Assume that W and V are closed linear subspaces of L_{p}[0, 1] such that

∥f∥_{L_{∞}[0,1/2]} ≤ C_{1}∥f∥_{L_{p}[0,1]}

for every f ∈ W , and

∥f∥_{L_{∞}[1/2,1]} ≤ C_{2}∥f∥_{L_{p}[0,1]}

for every f ∈ V , where C_{1} and C_{2} are positive constants depending only on W and V , respectively. Then W + V is closed in L_{p}[0, 1] .

A straightforward modification of the proof of the above theorem yields

Theorem 3.5. Let p ∈ (0, ∞) . Let A_{1}, A_{2} ⊂ ℝ be sets of finite positive measure with A_{1} ∩ A_{2} = ∅. Assume that W and V are closed linear subspaces of L_{p}(A_{1} ∪ A_{2}) such that

∥f∥_{L_{∞}(A_{1})} ≤ C_{1}∥f∥_{L_{p}(A_{1}∪A_{2})}

for every f ∈ W , and

∥f∥_{L_{∞}(A_{2})} ≤ C_{2}∥f∥_{L_{p}(A_{1}∪A_{2})}

for every f ∈ V , where C_{1} and C_{2} are positive constants depending only on W and V , respectively. Then W + V is closed in L_{p}(A_{1} ∪ A_{2}) .

Theorem 3.6 (Full Müntz Theorem in L_{p}(A) for p ∈ (0, ∞)) and for compact sets A ⊂ [0, 1] with positive lower density at 0. Let A ⊂ [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) . Then span{x^{λ_{1}}, x^{λ_{2}}, … } is dense in L_{p}(A) if and only if

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} = ∞ .

Moreover, if

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} < ∞ ,

then every function from the L_{p}(A) closure of span{x^{λ_{1}}, x^{λ_{2}}, … } can be represented as an analytic function on {z ∈ ℂ \ (−∞, 0] : |z| < r_{A}]} restricted to A ∩ (0, r_{A}), where

r_{A} := sup{y ∈ ℝ : m(A ∩ [y, ∞)) > 0}

(m(·) denotes the one-dimensional Lebesgue measure).

It may be interesting to compare Theorem 3.6 with Theorems 3.A and 3.B below proved in [Bo-Er7]. Let

∥f∥_{L_{p,w}(A)} := (∫_{A} |f(x)|^{p}w(x) dx)^{1/p} .

The space L_{p,w}(A) is the collection of all real-valued measurable functions on A for which ∥f∥_{L_{p,w}(A)} < ∞ .

Theorem 3.A (Full Müntz Theorem in L_{p}(A) for p ∈ (0, ∞) when A ⊂ [0, 1] is compact and inf A > 0). Suppose (λ_{j})_{j=−∞}^{∞} is a sequence of distinct real numbers satisfying

∑_{{j=−∞ \atop λ_{j}≠0}}^{∞} \frac{1}{|λ_{j}|} < ∞

with λ_{j} < 0 for j < 0 and λ_{j} ≥ 0 for j ≥ 0. Suppose A ⊂ [0, ∞) is a set of positive Lebesgue measure with inf A > 0, w is a nonnegative-valued, integrable weight function on A with ∫_{A} w > 0, and p ∈ (0, ∞). Then

span{x^{λ_{j}} : j ∈ ℤ}

is not dense in L_{p,w}(A).

Suppose the gap condition

inf{λ_{j} − λ_{j−1} : j ∈ ℤ} > 0

holds. Then every function f ∈ L_{p,w}(A) belonging to the L_{p,w}(A) closure of

span{x^{λ_{j}} : j ∈ ℤ}

can be represented as

f(x) = ∑_{j=−∞}^{∞} a_{j}x^{λ_{j}}, x ∈ A ∩ (a_{w}, b_{w}),

where

a_{w} := inf {y ∈ [0, ∞) : ∫_{A∩(0,y)} w(x) dx > 0}

and

b_{w} := sup {y ∈ [0, ∞) : ∫_{A∩(y,∞)} w(x) dx > 0} .

If the above gap condition does not hold, then every function f ∈ L_{p,w}(A) belonging to the L_{p,w}(A) closure of

span{x^{λ_{j}} : j ∈ ℤ}

can still be represented as an analytic function on

{z ∈ ℂ \ (−∞, 0] : a_{w} < |z| < b_{w}}

restricted to A ∩ (a_{w}, b_{w}).

Theorem 3.B (Full Müntz Theorem in L_{p}(A) for p ∈ (0, ∞) when A ⊂ [0, 1] is compact and inf A > 0, Part 2). Suppose (λ_{j})_{j=−∞}^{∞} is a sequence of distinct real numbers. Suppose A ⊂ (0, ∞) is a bounded set of positive Lebesgue measure, inf A > 0, w is a nonnegative-valued integrable weight function on A with ∫_{A} w > 0, and p ∈ (0, ∞). Then

span{x^{λ_{j}} : j ∈ ℤ}

is dense in L_{p,w}(A) if and only if

∑_{{j=−∞ \atop λ_{j}≠0}}^{∞} \frac{1}{|λ_{j}|} < ∞ .

Finally our next theorem offers an upper bound for the L_{p}[0, 1] distance from x^{m} to

span{x^{λ_{1}}, x^{λ_{2}}, …, x^{λ_{n}}} ,

when Λ := (λ_{j})_{j=1}^{∞} is a sequence of real numbers tending to −(1/p) and m = −(1/p) + α for some α > 0 .

Theorem 3.7. Let p > 0 . Let Λ := (λ_{j})_{j=1}^{∞} be a sequence of real numbers tending to −(1/p) . Let m = −(1/p) + α for some α > 0 . Then there are

R_{n} ∈ span{x^{λ_{1}}, x^{λ_{2}}, …, x^{λ_{n}}} .

such that

∫_{0}^{1} |x^{m} − R_{n}(x)|^{p} dx

≤ \frac{c(Λ, α)^{p}}{p min_{1≤j≤n} (λ_{j} + (1/p))} exp (−p (\frac{1}{2α} − \frac{1}{2}) ∑_{j=1}^{n} (λ_{j} + (1/p)))

whenever min_{1≤j≤n} (λ_{j} + (1/p)) ≤ α , where c(Λ, α) is a constant depending only on Λ and α .

4. Proof of Theorems 3.1, 3.2, 3,7, and the sufficiency part of Theorem 2.2

To prove the sufficiency part of Theorem 2.2 we need the following; see [1, page 191].

Blaschke’s Theorem. Suppose (β_{j})_{j=1}^{∞} is a sequence in D := {z ∈ ℂ : |z| < 1} satisfying

∑_{j=1}^{∞} (1 − |β_{j}|) = ∞ .

Denote the multiplicity of β_{k} in (β_{j})_{j=1}^{∞} by m_{k} . Assume that f is a bounded analytic function on D having a zero at each β_{j} with multiplicity m_{j} . Then f = 0 on D .

The proof below is based on the Riesz Representation Theorem for continuous linear functionals on L_{p}[0, 1] , valid for p ∈ [1, ∞) , so the assumption p ∈ [1, ∞) in Theorem 2.2 is essential for our arguments.

Proof of the sufficiency part of Theorem 2.2. Choosing a subsequence if necessary, without loss of generality we may assume that one of the following three cases occurs. Case 1: λ_{j} ≥ 1 for each j = 1, 2, … with ∑_{j=1}^{∞} (1/λ_{j}) = ∞ . Case 2: (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers tending to −(1/p) + α , where α > 0 . Case 3: −(1/p) < λ_{j} ≤ 0 for each j = 1, 2, … with ∑_{j=1}^{∞} (λ_{j} + (1/p)) = ∞ and lim_{j→∞} λ_{j} = −(1/p) .

In Case 1, Theorem 2.1 (Full Müntz Theorem in C[0, 1]) yields that span{1, x^{λ_{1}}, x^{λ_{2}}, … } is dense in C[0, 1]. From this we can easily deduce that span{x^{λ_{1}}, x^{λ_{2}}, … } is dense in L_{p}[0, 1] .

In Case 2, Theorem 3.1 implies that span{x^{λ_{1}}, x^{λ_{2}}, … } is dense in L_{p}[0, 1] .

In Case 3, we argue as follows. By the Hahn-Banach Theorem and the Riesz Representation Theorem for continuous linear functionals on L_{p}[0, 1] we know that

span{x^{λ_{1}}, x^{λ_{2}}, … }

is not dense in L_{p}[0, 1] if and only if there exists a 0 ≠ h ∈ L_{q}[0, 1] satisfying

(4.1)  ∫_{0}^{1} t^{λ_{j}}h(t) dt = 0 ,  j = 1, 2, …,

where q is the conjugate exponent of p defined by p^{−1} + q^{−1} = 1 . Suppose there exists a 0 ≠ h ∈ L_{q}[0, 1] such that (4.1) holds. Let

f(z) := ∫_{0}^{1} t^{z}h(t) dt , Re(z) > −(1/p) .

We can easily show by using Hölder’s inequality that

g(z) := (z + 1)^{2}f(z + 1 − (1/p))

is a bounded analytic function on the open unit disk, that satisfies

g(λ_{j} + (1/p) − 1) = 0 .

Now

∑_{j=1}^{∞} (1 − |λ_{j} + (1/p) − 1|) = ∑_{j=1}^{∞} (1 − (1 − λ_{j} − (1/p)) = ∑_{j=1}^{∞} (λ_{j} + (1/p)) = ∞ .

Hence Blaschke’s Theorem with β_{j} := λ_{j} + (1/p) − 1 , j = 1, 2, …, yields that g = 0 on the open unit disk. Therefore f = 0 on the open disk with diameter [−(1/p), 2 − (1/p)] . Now observe that f is an analytic function on the half plane {z ∈ ℂ : Re(z) > −(1/p)} , hence f(z) = 0 whenever Re(z) > −(1/p) , so

f(n) = ∫_{0}^{1} t^{n}h(t) dt = 0 , n = 0, 1, 2, ….

Now the Weierstrass Approximation Theorem yields that

∫_{0}^{1} u(t)h(t) dt = 0

for every u ∈ C[0, 1]. This implies

∫_{0}^{x} h(t) dt = 0

for all x ∈ [0, 1] , so h(x) = 0 almost everywhere on [0, 1] , a contradiction. □

Proof of Theorem 3.1. Let

λ_{j}^{∗} = λ_{j} + (1/p) − (α/2) ,

where the assumptions on Λ insure that λ_{j}^{∗} > (α/4) for all sufficiently large j . Let m ≥ (α/2) . Then by Theorem 2.1 (Full Müntz Theorem in C[0, 1]), for every ε > 0 there is Q_{ε} ∈ span{x^{λ_{1}^{∗}}, x^{λ_{2}^{∗}}, … } such that

∥x^{m−(α/2)+(1/p)} − Q_{ε}∥_{[0,1]} < ε .

Let

R_{ε}(x) := x^{(α/2)−(1/p)}Q_{ε}(x) ∈ span{x^{λ_{1}}, x^{λ_{2}}, … } .

Then

∫_{0}^{1} |x^{m} − R_{ε}(x)|^{p} dx = ∫_{0}^{1} |x^{(α/2)−(1/p)} (x^{m−(α/2)+(1/p)} − Q_{ε}(x))|^{p} dx

≤ (∫_{0}^{1} x^{p(α/2)−1} dx) ∥x^{m−(α/2)+(1/p)} − Q_{ε}(x)∥_{L_{∞}[0,1]}^{p}

≤ \frac{ε^{p}}{p(α/2)} .

Hence the monomials x^{m} are in the L_{p}[0, 1] closure of span{x^{λ_{1}}, x^{λ_{2}}, … } for all sufficiently large m. Now Theorem 2.1 (Full Müntz Theorem in C[0, 1]) implies that the elements f of C[0, 1] with f(0) = 0 are contained in the L_{p}[0, 1] closure of span{x^{λ_{1}}, x^{λ_{2}}, … }, and since all such functions form a dense set in L_{p}[0, 1] , the theorem is proved. □

Proof of Theorem 3.2. The case p ∈ [1, ∞) is handled by Theorem 2.2 (the part of Theorem 2.2 needed here is proved in the beginning of this section). So in the rest of the proof we assume that p ∈ (0, 1).

Step 1. For t > 0 we define f_{t}(x) := x^{t}(1 − log x)^{b} , x ∈ (0, 1] , and f_{t}(0) := 0. Let b ∈ [1, ∞) . We show that

span{1 ∪ {f_{t} : t > 0}}

is dense in C[0, 1]. To see this, for a given ε > 0 we get a polynomial P so that

∥(1 − log x)^{−b} − P (x)∥_{L_{∞}[0,1]} < ε .

This can be done by the Weierstrass Theorem. For m ≥ 1 multiply through by the factor x^{m}(1 − log x)^{b} to see that

∥x^{m} − x^{m}(1 − log x)^{b}P (x)∥_{L_{∞}[0,1]} < ε∥x(1 − log x)^{b}∥_{L_{∞}[0,1]} .

Step 2. It is elementary calculus to show that for x ∈ (0, 1], a ∈ (0, 1), and b ∈ [1, ∞), we have

x^{a}(1 − log x)^{b} ≤ (b/a)^{b} .

Step 3. Suppose (γ_{n})_{n=1}^{∞} is a strictly decreasing sequence tending to 0. Suppose

∑_{j=1}^{∞} γ_{j} = ∞  and  b ∈ [1, ∞) .

Let

f_{0} := 1  and  f_{n}(x) := x^{γ_{n}}(1 − log x)^{b} , x ∈ (0, 1] , f_{n}(0) := 0 , n = 1, 2, ….

We show that span{f_{n} : n = 0, 1, 2, … } is dense in C[0, 1].

Suppose to the contrary that span{f_{n} : n = 0, 1, 2, … } is not dense in C[0, 1]. Then by the Hahn Banach Theorem and the Riesz Representation Theorem there is a nonzero finite signed measure μ on [0, 1] so that for each n = 0, 1, 2, … we have

∫_{0}^{1} f_{n}(x) dμ(x) = 0 .

For z ∈ ℂ with Re(z) > 0 we define

F (z) = ∫_{0}^{1} x^{z}(1 − log x)^{b} dμ(x) .

So F is analytic and bounded on

{z ∈ ℂ : Re(z) > a}

for all a > 0. Now for any z in the open unit disk, we define g(z) := (1 + z)^{2b}F (z + 1). Observe that g(γ_{n} − 1) = 0 for each n = 1, 2, . . .. Step 2 implies that g is bounded on the open unit disk, so by Blaschke’s Theorem and the hypothesis on γ_{n} we conclude that g = 0 on the open unit disk, hence F (z) = 0 for all z ∈ ℂ with Re(z) > 0 by the Unicity Theorem of analytic functions.

Step 4. Now assume that Λ := (λ_{j})_{j=1}^{∞} satisfies the assumptions of the theorem. Up to now b ∈ [1, ∞) was arbitrary. Now take b > 1/p, so that x^{−(1/p)}(1 − log x)^{−b} is in L_{p}[0, 1]. Let

γ_{j} := λ_{j} + (1/p) , j = 1, 2, ….

For m ≥ 1 and ε > 0 we use Step 3 to get an n ∈ ℕ and coefficients a_{1}, a_{2}, …, a_{n} ∈ ℝ so that

∥x^{m+(1/p)}(1 − log x)^{b} − ∑_{j=1}^{n} a_{j}x^{λ_{j}+(1/p)}(1 − log x)^{b}∥_{L_{p}[0,1]} < ε .

Then

∥x^{m} − ∑_{j=1}^{n} a_{j}x^{λ_{j}} ∥_{L_{p}[0,1]} ≤ ε∥x^{−(1/p)}(1 − log x)^{−b}∥_{L_{p}[0,1]} .

Hence x^{m} is in the L_{p}[0, 1] closure of span{x^{λ_{1}}, x^{λ_{2}}, … } for every integer m ≥ 1. Since the polynomials with constant term 0 form a dense set in L_{p}[0, 1], the theorem is proved. □

Proof of Theorem 3.7. Let m = −(1/p) + α with α > 0 . Let k = k(n) be such that

λ_{k} = min_{1≤j≤n} λ_{j} .

For j = 1, 2, …, n let

λ_{j}^{∗} := λ_{j} + (1/p) > 0 ,  μ_{j}^{∗} = λ_{j}^{∗} − (λ_{k}^{∗}/2) > 0 ,  μ̃_{j} := μ_{j}^{∗} − (1/2) > −(1/2) .

Note that

(4.2)  0 < λ_{j}^{∗}/2 ≤ μ_{j}^{∗} ≤ λ_{j}^{∗}

for every j = 1, 2, …. Assume that λ_{k} + (1/p) = λ_{k}^{∗} ≤ α . By [Bo-Er3, page 173], there is a

P_{n} ∈ span{xμ̃_{1}, xμ̃_{2}, …, xμ̃_{n}}

such that

∥x^{m−(λ_{k}^{∗}/2)+(1/p)−(1/2)} − P_{n}(x)∥_{L_{2}[0,1]}

≤ \frac{1}{\sqrt{2m − λ_{k}^{∗} + (2/p)}} |∏_{j=1}^{n} \frac{(m − (λ_{k}^{∗}/2) + (1/p) − (1/2)) − μ̃_{j}}{(m − (λ_{k}^{∗}/2) + (1/p) − (1/2)) + μ̃_{j} + 1}|

≤ \frac{1}{\sqrt{2α − λ_{k}^{∗}}} |∏_{j=1}^{n} (1 − \frac{2μ̃_{j} + 1}{(m − (λ_{k}^{∗}/2) + (1/p) − (1/2)) + μ̃_{j} + 1})|

≤ \frac{1}{\sqrt{α}} |∏_{j=1}^{n} (1 − \frac{2μ_{j}^{∗}}{α + μ_{j}^{∗} − (λ_{k}^{∗}/2)})| ≤ \frac{c_{1}(Λ, α)}{\sqrt{α}} | ∏_{{j=1 \atop μ^{∗}_{j}≤α/2}}^{n} (1 − \frac{2μ_{j}^{∗}}{2α} )|

≤ \frac{c_{1}(Λ, α)}{\sqrt{α}} | ∏_{{j=1 \atop μ^{∗}_{j}≤α/2}}^{n} (1 − \frac{λ_{j}^{∗}}{2α})| ≤ c_{2}(Λ, α) exp (−\frac{1}{2α} ∑_{{j=1 \atop μ^{∗}_{j}≤α/2}}^{n} λ_{j}^{∗})

≤ c_{3}(Λ, α) exp (−\frac{1}{2α} ∑_{j=1}^{n} λ_{j}^{∗}) ,

where c_{1}(Λ, α) , c_{2}(Λ, α) , and c_{3}(Λ, α) are constants depending only on Λ and α . Now let

Q_{n}(x) := x^{1/2}P_{n}(x) ∈ span{x^{μ_{1}^{∗}}, x^{μ_{2}^{∗}}, …, x^{μ_{n}^{∗}}} .

Then, combining the Nikolskii-type inequality of [Bo-Er3, page 281] (see Theorem 2.7 of this paper) and the above L_{2}[0, 1] estimate, we obtain

∥x^{m−(λ_{k}^{∗}/2)+(1/p)} − Q_{n}(x)∥_{L_{∞}[0,1]}

= ∥(x^{m−(λ_{k}^{∗}/2)+(1/p)−(1/2)} − P_{n}(x)) x^{1/2}∥_{L_{∞}[0,1]}

≤ 6\sqrt{2} (α + ∑_{j=1}^{n} (μ̃_{j} + (1/2)))^{1/2} ∥x^{m−(λ_{k}^{∗}/2)+(1/p)−(1/2)} − P_{n}(x)∥_{L_{2}[0,1]}

≤ 6\sqrt{2} (α + ∑_{j=1}^{n} (μ̃_{j} + (1/2)))^{1/2} c_{3}(Λ, α) exp (−\frac{1}{2α} ∑_{j=1}^{n} λ_{j}^{∗})

= 6\sqrt{2} (α + ∑_{j=1}^{n} μ_{j}^{∗})^{1/2} c_{3}(Λ, α) exp (−\frac{1}{2α} ∑_{j=1}^{n} λ_{j}^{∗})

≤ 6\sqrt{2} (α + ∑_{j=1}^{n} λ_{j}^{∗})^{1/2} c_{3}(Λ, α) exp (−\frac{1}{2α} ∑_{j=1}^{n} λ_{j}^{∗})

≤ c_{4}(Λ, α) exp (− (\frac{1}{2α} − \frac{1}{2}) ∑_{j=1}^{n} λ_{j}^{∗})

with a constant c_{4}(Λ, α) > 0 depending only on Λ and α . Now we define

R_{n}(x) = x^{(λ_{k}^{∗}/2)−(1/p)}Q_{n}(x) ∈ span{x^{λ_{1}}, x^{λ_{2}}, …, x^{λ_{n}}} .

Then

∫_{0}^{1} |x^{m} − R_{n}(x)|^{p} dx = ∫_{0}^{1} |x^{(λ_{k}^{∗}/2)−(1/p)} (x^{m−(λ_{k}^{∗}/2)+(1/p)} − Q_{n}(x))|^{p} dx

≤ (∫_{0}^{1} x^{p(λ_{k}^{∗}/2)−1} dx) ∥x^{m−(λ_{k}^{∗}/2)+(1/p)} − Q_{n}(x)∥_{L_{∞}[0,1]}^{p}

≤ \frac{c_{4}(Λ, α)^{p}}{p(λ_{k}^{∗}/2)} exp (−p (\frac{1}{2α} − 1) ∑_{j=1}^{n} λ_{j}^{∗}) ,

and the theorem is proved □

5. Proof of Theorems 3.3 and 3.6

Proof of Theorem 3.3. Let p ∈ (0, ∞) . Suppose (λ_{j})_{j=1}^{∞} is a sequence of distinct real numbers greater than −(1/p) . Suppose

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} < ∞ .

Then {λ_{j} : j = 1, 2, … } = {γ_{j} : j = 1, 2, … } ∪ {δ_{j} : j = 1, 2, … } , where (γ_{j})_{j=1}^{∞} is a strictly decreasing sequence of distinct real numbers greater than −(1/p) satisfying

∑_{j=1}^{∞} (γ_{j} + (1/p)) < ∞

and (δ_{j})_{j=1}^{∞} is a strictly increasing sequence of positive numbers satisfying

∑_{j=1}^{∞} \frac{1}{δ_{j}} < ∞ .

Let A ⊂ [0, 1] be a compact set with lower density δ > 0 at 0. Choose b ∈ (0, 1] such that m(A ∩ [0, β]) ≥ δβ for every β ∈ [0, b]. Then choose N ∈ ℕ such that

∑_{j=N+1}^{∞} (γ_{j} + (1/p)) =: η ≤ δb/36 .

Let U be the L_{p}(A) closure of

span{{x^{λ_{1}}, x^{λ_{2}}, … } \ {x^{γ_{1}}, x^{γ_{2}}, …, x^{γ_{N}} }} ,

Let V be the L_{p}(A) closure of

span{x^{γ_{N+1}}, x^{γ_{N+2}}, … } ,

and let W be the L_{p}(A) closure of span{x^{δ_{1}}, x^{δ_{2}}, … } . Then by Theorem 2.3 every f ∈ W can be represented as an analytic function on

D_{r_{A}} := {z ∈ ℂ \ (−∞, 0] : |z| < r_{A}}

restricted to A ∩ (0, r_{A}) . Further, by Corollary 2.10 every f ∈ V can be represented as an analytic function on ℂ \ (−∞, 0] restricted to A \ {0}. Finally, by Theorems 2.4 and 2.8, W and V satisfy the assumptions of Theorem 3.5. Hence W + V is closed in L_{p}[0, 1] , and every function from W + V can be represented as an analytic function on D_{r_{A}}. Since U ⊂ W + V, every function from U can be represented as an analytic function on D_{r_{A}}. Now let Y be the L_{p}(A) closure of

span{x^{λ_{1}}, x^{λ_{2}}, … } .

Since

Z := span{x^{λ_{1}}, x^{λ_{2}}, … x^{λ_{N}} }

is a finite-dimensional vector space, we have Y = U + Z, hence every function from Y can be represented as an analytic function on D_{r_{A}}. This finishes the proof. □

Proof of Theorem 3.6. If

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} < ∞ ,

then the theorem follows from Theorem 3.3. If

∑_{j=1}^{∞} \frac{λ_{j} + (1/p)}{(λ_{j} + (1/p))^{2} + 1} = ∞ ,

then the theorem follows from Tietze’s Extension Theorem and from Theorems 2.1 and 3.2. We omit the trivial details. □

6. Proof of Theorems 3.4 and 3.5

In this section we prove Theorem 3.4. Since the ideas that underly the proof led to some new results about quasi-Banach spaces that may be useful elsewhere, we present some general results that include more information than what is needed for the proof of Theorem 3.4. We thank Nigel Kalton for several very useful and illuminating e-discussions about the contents of this section and related matters.

A quasi-norm is a real valued function ∥ · ∥ on a (real or complex) vector space X which satisfies the axioms for a norm except that the triangle inequality is replaced by the condition

∥x + y∥ ≤ k(∥x∥ + ∥y∥)

for some constant k. The smallest such k is called the modulus of concavity of the quasinorm. For 0 < p ≤ 1, a quasi-norm ∥ · ∥ is p-subadditive provided

∥x + y∥^{p} ≤ ∥x∥^{p} + ∥y∥^{p}

for all vectors x and y. A p-subadditive quasi-norm is called a p-norm. A quasi-norm ∥ · ∥ with modulus of concavity k is equivalent to a p-norm with 2^{1/p} = 2k. A p-norm is obviously also a q-norm for all 0 < q < p.

These and many other basic facts about quasi-norms and p-norms are discussed in the first few sections of [K-P-R]. This book also contains much of the deeper theory of p-normed spaces.

In this section all spaces are p-normed spaces for some fixed 0 < p ≤ 1. B(X, Y) denotes the space of bounded (same as continuous for p-normed spaces) linear operators, p-normed by ∥T ∥ := sup{∥T x∥_{Y} : ∥x∥_{X} ≤ 1} .

We recall that a linearly independent sequence {x_{n}}_{n=1}^{∞} in a p-normed space is basic provided that the natural partial sum projections P_{n} from the linear span of {x_{n}}_{n=1}^{∞} onto the span of {x_{k}}_{k=1}^{n} are uniformly bounded. A sequence {y_{n}}_{n=1}^{∞} of nonzero vectors is called a block basis of {x_{n}}_{n=1}^{∞} provided that there is a strictly increasing sequence {n_{k}}_{k=1}^{∞} of natural numbers so that for each k, y_{k} is in span{x_{j} : n_{k} ≤ j < n_{k+1}}. A block basis of a basic sequence is again a basic sequence.

Just as for normed spaces, basic sequences play an important role in studying the structure theory of quasi-normed spaces (see [K-P-R, I.5ff]). However, in quasi-normed spaces it typically is difficult to construct basic sequences.

The main functional analytical concept we study in this section is that of strictly singular operator. An operator T in B(X, Y) is called strictly singular provided that for every infinite dimensional subspace X_{0} of X, the restriction T_{|X_{0}} of T to X_{0} is not an isomorphism. Here it is convenient to work with nonclosed subspaces but the definition is obviously equivalent if we add “closed” before “subspace”. The space of all strictly singular operators from X to Y is denoted by SS(X, Y).

Lemma 6.1. Assume that T, S are in SS(X, Y). Then T +S is strictly singular provided that either

(1) Every infinite dimensional closed subspace of X contains a basic sequence; or

(2) X is complete and ker T = {0}.

Proof. The proof of (1) is just like the proof when X is a normed space (of course, every normed space X satisfies the hypothesis of (1); see [Li-Tz 1.a.5]): Consider any closed subspace X_{0} of X which has a basis {x_{n}}_{n=1}^{∞}. Since for every N the restriction of T to span{x_{n} : n > N} is not an isomorphism, get a normalized block basis {y_{n}}_{n=1}^{∞} of {x_{n}}_{n=1}^{∞} so that ∥T y_{n}∥ → 0 arbitrarily quickly. Using then the strict singularity of S, get a normalized block basis {z_{n}}_{n=1}^{∞} of {y_{n}}_{n=1}^{∞} so that ∥Sz_{n}∥ → 0. If ∥T y_{n}∥ → 0 fast enough, then necessarilly ∥T z_{n}∥ → 0, so that T + S is not an isomorphism on X_{0}.

Part (2) is not needed in the sequel, so we present the proof at the end of this section. □

Remark 6.1. Something is needed to guarantee that the sum of strictly singular operators is strictly singular. Suppose that X contains a subspace E with dim E = 2 so that every closed infinite dimensional subspace of X contains E. Then for some Y there exist T, S in SS(X, Y) with T + S an isomorphic embedding. (Take Q_{X_{1}}: X → X/X_{1} , Q_{X_{2}}: X → X/X_{2} , Y = X/X_{1} ⊕ X/X_{2} , T = Q_{X_{1}} ⊕ {0} , S = {0} ⊕ Q_{X_{2}} . Here dim X_{1} = dim X_{2} = 1 with X_{1} ∩ X_{2} = {0} and X_{1} ∪ X_{2} ⊂ E. Q_{Z} is the quotient map from X to X/Z .) There exists such a strange space X: In Theorem 5.5 of [Ka] Kalton constructs for every n a p-Banach space X and an n dimensional subspace E so that every closed infinite dimensional subspace of X contains E.

Definition 6.1. We say that X has property (B) if every infinite dimensional subspace of X contains a basic sequence.

Remark 6.2. If the completion of a p-normed space X has a basic sequence, then so does X (the usual normed space perturbation argument [Li-Tz, 1.a.9] works). Thus if every infinite dimensional closed subspace of X contains a basic sequence, then X has property (B).

Definition 6.2. Given a sequence {x_{n}}_{n=1}^{∞} in X, say that {x_{n}}_{n=1}^{∞} has a lower ∞-estimate if there is δ > 0 which satisfies

∥∑ a_{k}x_{k}∥ ≥ δ max_{k} |a_{k}|

for all finitely nonzero sequences {a_{n}}_{n=1}^{∞} of scalars.

Obviously a normalized basic sequence has a lower ∞-estimate. This was used implicitly in the proof of Lemma 6.1. Remark 6.3. Obviously the following are equivalent.

(i) {x_{n}}_{n=1}^{∞} has a lower ∞-estimate. (ii) x_{n} ↦ e_{n} extends to a bounded linear operator from span{x_{n}}_{n=1}^{∞} into c_{0}. (iii) There is an equicontinuous sequence {x_{n}^{∗}}_{n=1}^{∞} ⊂ (span{x_{n}}_{n=1}^{∞})^{∗} so that {x_{n}, x_{n}^{∗}}_{n=1}^{∞}

is biorthogonal. (iv) There is a bounded linear operator T from span{x_{n}}_{n=1}^{∞} into some space Y so that

{T x_{n}}_{n=1}^{∞} has a lower ∞-estimate.

For Banach spaces, the next lemma is a standard exercise in text books. The extension to the p-normed setting is routine.

Lemma 6.2. Let X be a p-Banach space and W, V closed subspaces with W ∩ V = {0}. Then W + V is closed if and only if dist (S_{W} , V) > 0, where S_{W} := {w ∈ W: ∥w∥ = 1} .

Proof. Assume that W +V is not closed. Take w_{n} ∈ W, v_{n} ∈ V with w_{n}+v_{n} → z ∉ W +V . If sup ∥w_{n}∥ = ∞ , then without loss of generality ∥w_{n}∥ → ∞ , so ∥\frac{w_{n}}{∥w_{n}∥} + \frac{v_{n}}{∥w_{n}∥} ∥ → 0 and hence dist (S_{W} , V) = 0 . If sup ∥w_{n}∥ ≠ ∞ , then still {w_{n}}_{n=1}^{∞} cannot have a Cauchy subsequence (else z would be in W + V), so we can assume that there exists δ > 0 that δ < ∥w_{n} − w_{m}∥ < C for n ≠ m. Then ∥\frac{w_{n}−w_{n+1}}{∥w_{n}−w_{n+1}∥} + \frac{v_{n}−v_{n+1}}{∥w_{n}−w_{n+1}∥} ∥ → 0 , hence again dist (S_{W} , V) = 0 .

The other direction is even easier (and anyway is not needed in the sequel). □

Proposition 6.3. Let X be a p-Banach space and W, V closed subspaces. If W + V is not closed then there exist {w_{n}}_{n=1}^{∞} ⊂ W, {v_{n}}_{n=1}^{∞} ⊂ V so that

(1) ∥w_{n}∥ = 1

(2) ∥w_{n} + v_{n}∥ → 0

(3) {w_{n}}_{n=1}^{∞} has a lower ∞-estimate.

Proof. First assume that W ∩ V = {0} . Under the assumptions of the lemma, by Lemma 6.2 we can pick {w_{n}}_{n=1}^{∞} ⊂ W and {v_{n}}_{n=1}^{∞} ⊂ V with ∥w_{n}∥ = 1 and ∥w_{n}+v_{n}∥ → 0 . Define a p-norm on W by |w|^{p} = dist (w, V) = ∥Q_{V} w∥^{p}. This is a p-norm since W ∩ V = {0} is weaker than ∥ · ∥, so by [K-P-R, Theorem 4.7], {w_{n}}_{n=1}^{∞} has a subsequence which has a lower ∞-estimate.

In the general case pass to X/(W ∩ V) . Q_{W ∩V} W is closed there since it is isometric to W/(W ∩ V) and similarly for Q_{W ∩V} V . Also W + V = Q_{W ∩V}^{−1} (Q_{W ∩V} W + Q_{W ∩V} V) , so since W + V is not closed, neither is Q_{W ∩V} W + Q_{W ∩V} V . Thus we get {w_{n}}_{n=1}^{∞} ⊂ W and {v_{n}}_{n=1}^{∞} ⊂ V so that ∥Q_{W ∩V} w_{n} + Q_{W ∩V} v_{n}∥ → 0 , ∥Q_{W ∩V} w_{n}∥ = 1 , and {Q_{W ∩V} w_{n}}_{n=1}^{∞} has a lower ∞-estimate. By adding some z_{n} ∈ W ∩ V to w_{n} and subtracting z_{n} from v_{n} we can assume that ∥w_{n}∥ → 1 . Pick x_{n} ∈ W ∩ V so that ∥w_{n} + v_{n} + x_{n}∥ → 0 . Then v_{n} + x_{n} ∈ V and {w_{n}}_{n=1}^{∞} has a lower ∞-estimate since {Q_{W ∩V} w_{n}}_{n=1}^{∞} does. □

Proposition 6.4. Assume the p-Banach space X has property (B), W, V are closed subspaces, and there are T, S ∈ B(X, X) so that T_{|W} and S_{|V} are strictly singular and I = T + S , where I is the identity operator on X. Then W + V is closed.

Proof. Suppose W + V is not closed. Then get {w_{n}}_{n=1}^{∞} ⊂ W and {v_{n}}_{n=1}^{∞} ⊂ V by Proposition 6.3 and take δ > 0 so that for all finitely nonzero sequences of scalars {a_{n}}_{n=1}^{∞} ,

∥∑ a_{n}w_{n}∥ ≥ δ max |a_{n}| .

By passing to a subsequence, assume that

∑_{j=n}^{∞} ∥w_{j} + v_{j}∥^{p} < \frac{1}{n} .

Let V_{0} be an infinite dimensional subspace of span{v_{n}}_{n=1}^{∞} . Since T_{|W} is strictly singular, we can get x_{n} = ∑_{k_{n}+1}^{k_{n+1}} a_{j}w_{j} , ∥x_{n}∥ = 1 , with y_{n} := ∑_{k_{n}+1}^{k_{n+1}} a_{j}v_{j} ∈ V_{0} so that ∥T x_{n}∥ → 0 . Then

∥x_{n} + y_{n}∥^{p} ≤ ∑_{j=k_{n}+1}^{k_{n+1}} |a_{j}|^{p}∥w_{j} + v_{j}∥^{p} ≤ ( max_{k_{n}+1≤j≤k_{n+1}} |a_{j}|^{p}) ∑_{j=k_{n}+1}^{∞} ∥w_{j} + v_{j}∥^{p} ≤ δ^{−p}n^{−1}

so that 1 − δ^{−p}n^{−1} ≤ ∥y_{n}∥^{p} . But

∥T y_{n}∥^{p} ≤ ∥T x_{n}∥^{p} + ∥T ∥^{p}∥x_{n} + y_{n}∥^{p} → 0 .

So T_{|V_{0}} is not an isomorphism. This proves that the restriction of T to V_{1} := span{v_{n}}_{n=1}^{∞} is strictly singular, hence I_{|V_{1}} = T_{|V_{1}} + S_{|V_{1}} is strictly singular by Lemma 6.1, a contradiction. □

Theorem 3.4 is a corollary of Proposition 6.4.

Corollary 6.5. Suppose W, V are closed subspaces of L_{p} := L_{p}[0, 1], 0 < p < ∞, and

\begin{aligned}∥1_{(0,1/2)}f∥_{L_{∞}[0,1]} & ≤ C∥f∥_{L_{p}[0,1]} ,  f ∈ W \\ ∥1_{(1/2,1)}f∥_{L_{∞}[0,1]} & ≤ C∥f∥_{L_{p}[0,1]} ,  f ∈ V . \end{aligned}

Then W + V is closed in L_{p}.

Proof. The formal identity mapping I_{∞,p} : L_{∞}[0, 1] → L_{p}[0, 1] is strictly singular. When p = 2, this is contained in elementary text books (see e.g. [Ro, Chapter 10, # 41, # 55]). The case p < 2 follows formally from this, and the case p > 2 follows via a simple extrapolation argument (see e.g. [Jo-Li, Section 10]). Thus if T: L_{p}[0, 1] → L_{p}[0, 1] , S: L_{p}[0, 1] → L_{p}[0, 1] are defined by

T f = 1_{(0,1/2)}f , Sf = 1_{(1/2,1)}f ,

we have that T_{|W} and S_{|V} are strictly singular. Also, every closed infinite dimensional subspace of L_{p}[0, 1] contains an isomorphic copy of ℓ_{r} for some r < 2 by Bastero’s [Ba] extension of the Krivine-Maurey stable theory. Thus L_{p}[0, 1] has property (B) by Remark 6.2 and Proposition 6.4 applies. An easier proof of the fact that L_{p}[0, 1] has property (B) was given by Tam [Ta]. His proof uses Dvoretzky’s theorem rather than the theory of stable spaces. □

Remark 6.4. The case 1 < p < ∞ in Corollary 6.5 was proved jointly with G. Schechtman several years ago. Since this case is much simpler than the case 0 < p < 1, we present here the proof.

Proof. Recall [Wo, III.D] that a weakly compact operator from L_{∞} (or any other C(S)- space) is completely continuous; that is, sends weakly compact sets to norm compact sets. Thus the composition of two weakly compact operators with the middle space L_{∞}[0, 1] must be a compact operator. Therefore we get from the hypotheses that the operators T : W → L_{p}[0, 1] and S : V → L_{p}[0, 1] are compact, where T f := f1_{(0,1/2)} and Sf := f1_{(1/2,1)}. Thus if you fix 0 < δ < 1/2, there are closed, finite codimensional subspaces W_{0} ⊂ W and V_{0} ⊂ V so that ∥T_{W_{0}}∥ < δ and ∥S_{V_{0}}∥ < δ . This implies that W_{0} + V_{0} is closed in L_{p}[0, 1] (check that the unit spheres are a positive distance apart), and hence W + V is also closed. □

We turn now to the proof of part (2) of Lemma 6.1.

Proof. For x ∈ X define |x|_{T} := ∥T x∥_{Y} . This is a p-norm on X because T is one-to-one. | · |_{T} is strictly weaker than ∥ · ∥_{X} on every infinite dimensional subspace of X because T is strictly singular. Let X_{0} be an infinite dimensional closed subspace of X and let {x_{n}}_{n=1}^{∞} be a normalized sequence in X_{0} so that |x_{n}|_{T} → 0. By [K-P-R, Theorem 4.7], by passing to a subsequence we can assume that {x_{n}}_{n=1}^{∞} has a lower ∞-estimate and

that ∑_{n=1}^{∞} ∥T x_{n}∥_{Y}^{p} < ∞ . Since S is strictly singular, there exists for each n a unit vector

z_{n} ∈ span{x_{k}}_{k=n}^{∞} so that ∥Sz_{n}∥_{Y} → 0. Since {x_{n}}_{n=1}^{∞} has a lower ∞-estimate, the coefficients in the expansions of the y_{n}’s in terms of the x_{n}’s are uniformly bounded, and

hence ∥T y_{n}∥_{Y} → 0 because ∑_{n=1}^{∞} ∥T x_{n}∥_{Y}^{p} < ∞ and | · |_{T}^{p} satisfies the triangle inequality.

Thus T + S is not an isomorphism on X_{0}. □

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Department of Mathematics, Texas A&M University, College Station, Texas 77843

   E-mail address: terdelyi@math.tamu.edu (Tamás Erdélyi)

Department of Mathematics, Texas A&M University, College Station, Texas 77843

   E-mail address: johnson@math.tamu.edu (William Johnson)