A Paley–Wiener Type Theorem for Singular Measures on $(-1/2, 1/2)$

Abstract

For a fixed singular Borel probability measure $\mu$ on $(-1/2, 1/2)$, we give several characterizations of when an entire function is the Fourier transform of some $f \in L^2(\mu )$. The first characterization is given in terms of criteria for sampling functions of the form $\hat{f}$ when $f \in L^2(\mu )$. The second characterization is given in terms of criteria for interpolation of bounded sequences on ${\mathbb {N}}_{0}$ by $\hat{f}$. Both characterizations use the construction of Fourier series for $f \in L^2(\mu )$ demonstrated by Herr and Weber via the Kaczmarz algorithm and classical results concerning the Cauchy transform of $\mu$.

1 Introduction

The classical Paley–Wiener theorem states that an entire function F is the Fourier Transform for some $f \in L^{2}(-1/2,1/2)$ if and only if F is of exponential type at most $\pi$ and the restriction of F to ${\mathbb {R}}$ is square-integrable. An equivalent description of such entire functions is that F satisfy the following two conditions: (i) the values of F on the integer lattice are square-summable and (ii) for every $z \in {\mathbb {C}}$, F(z) can be recovered from the values of F on the integer lattice via cardinal interpolation, i.e. the Shannon–Whittaker–Kotelnikov Sampling Theorem [3]. While this latter description of the inhabitants of the Paley–Wiener space is less elegant than the former, we will demonstrate that it is amenable to generalization to singular measures on $(-1/2,1/2)$. Indeed, here is the question we shall answer: given a fixed singular Borel probability measure $\mu$ on $(-1/2, 1/2)$, when can an entire function F be written as

$$\begin{aligned} F(z) = \int _{-1/2}^{1/2} f(x) e^{- 2 \pi i x z} \ d \mu (x) \end{aligned}$$

(1)

for some $f \in L^{2}(\mu )$? Note that we are a priori fixing the measure $\mu$.

There are numerous results—in addition to the Paley–Wiener theorem—on when an entire function is the Fourier transform of a function, measure or distribution. The classical results in this regard include the Plancheral–Pólya theory [21], the “Bochner–Schoenberg–Eberlein conditions” contained collectively in [6, 12, 24], and the Beurling–Malliavin theory [4].

We consider the question from the viewpoint of the sampling and interpolating problems for bandlimited functions as presented in [14, 29]. In particular, in [29] Strichartz poses a more difficult question than the one we address in the present paper: for a compact set K, when is an entire function F the Fourier transform of a (complex) measure supported on K?

2 Main Results

For our characterization of those functions which admit the representation in Eq. (1), we require both the Fourier-Stieltjes transform $\widehat{\mu }$ and the Cauchy transform $\mu _{+}$ of $\mu$:

$$\begin{aligned} \widehat{\mu }(z) := \int _{-1/2}^{1/2} e^{- 2 \pi i z x} \ d \mu (x); \qquad \mu _{+}(z) := \int _{-1/2}^{1/2} \dfrac{ 1 }{ 1 - z e^{2 \pi i x} } \ d \mu (x). \end{aligned}$$

Note that $\mu _{+}$ is nonvanishing on ${\mathbb {D}}$ since $Re(1/1-z) > 1/2$ for $|z| < 1$; it is in fact the reciprocal of $\mu _{+}$ that we require. We also need a description of the Fourier series expansions for $f \in L^2(\mu )$; this is given in the following theorem, which uses the Kaczmarz algorithm [17], as well as the main result of [18]. See [15] for details; see also [22] for the original proof of the existence of Fourier series for $f \in L^2(\mu )$.

The Kaczmarz algorithm is an iterative method for reconstructing a vector x in a Hilbert space H given the data $\{ \langle x , \varphi _{n} \rangle \}_{n=0}^{\infty }$. It is defined as follows (assume $\Vert \varphi _{n} \Vert =~1$):

$$\begin{aligned} x_{0} = \langle x , \varphi _{0} \rangle \varphi _{0}; \qquad x_{k} = x_{k-1} + \langle x - x_{k-1} , \varphi _{k} \rangle \varphi _{k}. \end{aligned}$$

Only certain sequences $\{ \varphi _{n} \}_{n}$ have the property that $\Vert x_{k} - x\Vert \rightarrow 0$, those that do are called effective. Kaczmarz proves [18] that if $\{ \varphi _{n} \}_{n}$ is a periodic sequence that spans H, then $\Vert x_{k} - x \Vert \rightarrow 0$. A complete characterization of which sequences are effective is given in [13]. The main result of [18] says that if $\{ \varphi _{n} \}_{n}$ is a spanning, stationary sequence with singular spectral measure, then it is effective. Also in [18], if $\{ \varphi _{n} \}_{n}$ is an effective sequence, then there is a second sequence, called the auxiliary sequence and denoted by $\{ g_{n} \}_{n}$, such that for every $x \in H$,

$$\begin{aligned} x = \sum _{n=0}^{\infty } \langle x, g_{n} \rangle \varphi _{n}, \end{aligned}$$

with convergence in the norm. The auxiliary sequence is obtained by the following recursion:

$$\begin{aligned} g_{0} = \varphi _{0}; \qquad g_{k} = \varphi _{k} - \sum _{j=0}^{k-1} \langle \varphi _{k}, \varphi _{j} \rangle g_{j}. \end{aligned}$$

Here we denote the exponential functions by $e_{n}(x) := e^{2 \pi i n x}$; for a singular measure $\mu$, $\{ e_n \}_{n}$ is a stationary sequence which is effective.

Theorem A

Suppose $\mu$ is a singular Borel probability measure on $(-1/2, 1/2)$, and let $\{ \alpha _{n} \}$ be the sequence of Taylor coefficients of $\dfrac{1}{ \mu _{+}(z)}$. Define the sequence of functions $g_{n}(x) = \sum _{j=0}^{n} \overline{ \alpha _{n-j} } e^{2 \pi i j x}$. Then the sequence $\{ g_{n} \}_{n=0}^{\infty } \subset L^2(\mu )$ has the property that for all $f \in L^2(\mu )$,

$$\begin{aligned} f = \sum _{n=0}^{\infty } \langle f, g_{n} \rangle e_{n} = \sum _{n=0}^{\infty } \langle f , g_{n} \rangle g_{n} \end{aligned}$$

(2)

with the convergence of both series occuring in the norm. Moreover, Parseval’s identity holds: $\Vert f \Vert ^{2} = \sum _{n=0}^{\infty } | \langle f, g_{n} \rangle |^{2}$.

The $\{ g_{n} \}_{n}$ as defined in Theorem A is the auxiliary sequence of the exponential functions in $L^2(\mu )$ [15]. As a consequence of Parseval’s identity, the sequence $\{ g_{n} \}_{n=0}^{\infty }$ is a Bessel sequence and hence for any square-summable sequence $\{c_{n}\} \in \ell ^{2}({\mathbb {N}}_{0})$, the series $\sum _{n=0}^{\infty } c_{n} g_{n}$ also converges in norm.

2.1 Characterization Using Sampling Criteria

As noted previously, the Paley–Wiener theorem can be reformulated in terms of the Sampling Theorem. Our first characterization of which entire functions $F = \hat{f}$ for some $f \in L^2(\mu )$ is analogous.

Theorem 1

Suppose $\mu$ is a singular Borel probability measure on $(-1/2, 1/2)$, and let $\{ \alpha _{n} \}_{n=0}^{\infty }$ be the Taylor coefficients for $\dfrac{1}{\mu _{+}(z)}$. An entire function F admits the representation in Eq. (1) for some $f \in L^2(\mu )$ if and only if the following conditions hold:

• (i)
  $$\begin{aligned} \sum _{n=0}^{\infty } \left| \sum _{j=0}^{n} \alpha _{n-j} F(j) \right| ^{2} < \infty ; \end{aligned}$$

• (ii) for all $z \in {\mathbb {C}}$,

  $$\begin{aligned} F(z) = \sum _{n=0}^{\infty } \left( \sum _{j=0}^{n} \alpha _{n-j} F(j) \right) \left( \sum _{k=0}^{n} \overline{\alpha _{n-k}} \widehat{\mu }(z - k) \right) . \end{aligned}$$

  (3)

Proof

For $F = \hat{f}$, note that $\sum _{j=0}^{n} \alpha _{n-j} F(j) = \langle f, g_{n} \rangle$, so the necessity of (i) follows by the Parseval identity. The necessity of (ii) follows by the previous observation and applying the Fourier transform to the second series expansion of f in Eq. (2).

We turn now to the sufficiency. Combining (ii) with the fact that the sequence $\{ g_{n} \}_{n=0}^{\infty } \subset L^2(\mu )$ is a Bessel sequence, we define the function

$$\begin{aligned} f = \sum _{n = 0}^{\infty } \left( \sum _{j=0}^{n} \alpha _{n-j} F(j) \right) g_{n}. \end{aligned}$$

As this series converges in $L^2(\mu )$, we obtain

$$\begin{aligned} \hat{f}(z)&= \sum _{n = 0}^{\infty } \left( \sum _{j=0}^{n} \alpha _{n-j} F(j) \right) \widehat{g_{n}}(z) \\&= \sum _{n=0}^{\infty } \left( \sum _{j=0}^{n} \alpha _{n-j} F(j) \right) \left( \sum _{k=0}^{n} \overline{\alpha _{n-k}} \widehat{\mu }(z - k) \right) \\&= F(z) \end{aligned}$$

by Item (iii). $\square$

2.2 Characterization Using Interpolation Criteria

We consider now whether the sampling condition in Eq. (3) of Theorem 1 can be replaced by a different criteria. Our approach here is to view the characterization from an interpolation viewpoint rather than a sampling viewpoint. The question then becomes the following: when is $\{ F(n) \}_{n=0}^{\infty }$ the sequence of Fourier moments of some $f \in L^2(\mu )$? In other words: given an entire function F does there exist some $f \in L^2(\mu )$ such that for all $n \in {\mathbb {N}}_{0}$,

$$\begin{aligned} F(n) = \int _{-1/2}^{1/2} f(x) e^{-2 \pi i n x} \ d \mu (x) ? \end{aligned}$$

(4)

Certainly this is a necessary condition for $F = \hat{f}$, and in all (Theorem 2) but the extremal case (Theorem 3) concerning the support of $\mu$ this is sufficient.

The interpolation problem can be decided using the model subspaces of $H^2({\mathbb {D}})$, where ${\mathbb {D}}$ denotes the open unit disc in ${\mathbb {C}}$. For a singular measure $\mu$ on $(-1/2,1/2)$, there exists a unique inner function b on ${\mathbb {D}}$ given by the Herglotz Representation [20], i.e. there exists a unique inner function b such that:

$$\begin{aligned} Re \left( \dfrac{1 + b(z)}{1 - b(z)} \right) = \int _{-1/2}^{1/2} \dfrac{1 - |z|^2 }{| z - e^{2 \pi i x} |^2} d \mu (x). \end{aligned}$$

This inner function defines a backwards shift invariant space ${\mathcal {H}}(b) = H^{2} \ominus b H^{2}$ as a consequence of Beurling’s theorem [2]. Observe that $G \in {\mathcal {H}}(b)$ if and only if $T_{\overline{b}} G = 0$, where $T_{\varphi }$ is the Toeplitz operator on $H^{2}({\mathbb {D}})$ with symbol $\varphi$. Clark proves [7] that the Normalized Cauchy transform $V_{\mu }$ defined as

$$\begin{aligned} V_{\mu } : L^2(\mu ) \rightarrow {\mathcal {H}}(b) : f \mapsto \dfrac{ \displaystyle {\int _{-1/2}^{1/2} \dfrac{ f(x) }{ 1 - z e^{-2 \pi i x}} \ d \mu (x) }}{ \mu _{+}(z) } . \end{aligned}$$

(5)

is a unitary operator. If $G \in {\mathcal {H}}(b)$, then there exists a unique $g \in L^2(\mu )$ such that $G = V_{\mu }g$. We denote it by $g := G^{\star }$, and call $G^{\star }$ the $L^2(\mu )$-boundary of G because

$$\begin{aligned} \lim _{r \rightarrow 1^{-}} \Vert g( x ) - G(r e^{2 \pi i x} ) \Vert _{\mu } = 0. \end{aligned}$$

(6)

This limit was demonstrated by Poltoratskiĭ [22] (see also [1]).

The Normalized Cauchy Transform can be expressed in terms of the sequence $\{ g_{n} \}$ appearing in Theorem A [15]:

$$\begin{aligned} V_{\mu } f (z) = \sum _{n=0}^{\infty } \langle f, g_{n} \rangle z^{n}, \qquad f \in L^2(\mu ). \end{aligned}$$

(7)

Therefore, we have the following characterization of the interpolation problem posed in Eq. (4).

Lemma 1

Suppose $\mu$ is a singular Borel probability measure on $(-1/2, 1/2)$, b is the inner function on ${\mathbb {D}}$ associated to $\mu$ via the Herglotz representation, and suppose $\vec {a} := \{ a_{n} \}_{n = 0}^{\infty } \subset {\mathbb {C}}$. The following conditions are equivalent:

• (i) there exists a function $f \in L^2(\mu )$ with the property that

  $$\begin{aligned} a_{n} = \int _{-1/2}^{1/2} f(x) e^{- 2 \pi i n x} \ d \mu (x); \end{aligned}$$

  (8)

• (ii) the series $\sum _{n=0}^{\infty } a_{n} z^{n}$ has a radius of convergence of at least 1, and the following inclusion holds:

  $$\begin{aligned} G_{\vec {a}}(z) := \dfrac{ \sum _{n=0}^{\infty } a_{n} z^{n} }{ \mu _{+}(z) } \in {\mathcal {H}}(b). \end{aligned}$$

Proof

Observe that

$$\begin{aligned} G_{\vec {a}} (z) = \left( \sum _{m=0}^{\infty } \alpha _{m} z^{m} \right) \left( \sum _{n=0}^{\infty } a_{n} z^{n} \right) = \sum _{n=0}^{\infty } \left( \sum _{j=0}^{n} \alpha _{n-j} a_{j} \right) z^{n}. \end{aligned}$$

(9)

Suppose that the moment problem in Eq. (8) has a solution for some $f \in L^2(\mu )$. Combining Eqs. (9) and (7) demonstrates that $G_{\vec {a}} = V_{\mu } f$. Therefore, by (5) we obtain that $G_{\vec {a}} \in {\mathcal {H}}(b)$.

Conversely, if $G_{\vec {a}} \in {\mathcal {H}}(b)$, then reversing the previous argument yields the existence of a function $f \in L^2(\mu )$ such that $G_{\vec {a}}(z) = \sum _{n=0}^{\infty } \langle f , g_{n} \rangle z^{n}$. Since we have for every n

$$\begin{aligned} \sum _{j=0}^{n} \alpha _{n-j} a_{j} = \langle f , g_{n} \rangle = \sum _{j=0}^{n} \alpha _{n-j} \int _{-1/2}^{1/2} f(x) e^{ - 2 \pi i j x } \ d \mu (x), \end{aligned}$$

it now follows that Eq. (8) holds. $\square$

For an entire function F of exponential type, we use $h_{F}$ to denote the Phragmén-Lindelöf indicator function.

Theorem 2

Suppose $\mu$ is a singular Borel probability measure with support in $[\alpha , \beta ] \subset (-1/2,1/2)$. Let b be the inner function associated to $\mu$ via the Herglotz Representation. The entire function F admits the representation in Eq. (1) if and only if

• (i) F is of exponential type;

• (ii) the indicator function of F satisfies $h_{F} ( \dfrac{\pi }{2} ) \le 2 \pi \beta$ and $h_{F} ( - \dfrac{ \pi }{2} ) \le - 2 \pi \alpha$;

• (iii) the following inclusion holds:

  $$\begin{aligned} G_{F}(z) := \dfrac{ \sum _{n = 0}^{\infty } F(n) z^{n} }{ \mu _{+}(z) } \in {\mathcal {H}}(b) \end{aligned}$$

  i.e. the function $G_{F}$ is in the kernel of the Toeplitz operator $T_{\overline{b}}$.

Proof

If F admits the representation in Eq. (1), then F satisfies (i) and (ii) using standard estimates (see, e.g. [19]; see also Lemma 2 below). Additionally, (iii) follows from Lemma 1.

Conversely, if F satisfies (i), (ii), and (iii), then by Lemma 1, there exists a $f \in L^2(\mu )$ such that $\hat{f}(n) = F(n)$ for $n \in {\mathbb {N}}_{0}$. Moreover, $\hat{f}$ also satisfies conditions (i) and (ii), i.e. $\hat{f}$ is of exponential type and the indicator function of $\hat{f}$ satisfies $h_{\hat{f}} ( \dfrac{\pi }{2} ) \le 2 \pi \beta$ and $h_{\hat{f}} ( - \dfrac{ \pi }{2} ) \le - 2 \pi \alpha$. Therefore, we must have $F = \hat{f}$ by Carlson’s Theorem ([5, Theorem 9.2.1]). $\square$

Corollary 1

Suppose $\mu$ is a singular Borel probability measure with support in $[-1/2 + \epsilon , 1/2 - \epsilon ]$, where $0< \epsilon < 1/2$. Let b be the inner function associated to $\mu$ via the Herglotz Representation. The entire function F admits the representation in Eq. (1) if and only if

• (i) F is of exponential type at most $2\pi (1/2 - \epsilon )$;

• (ii) the following inclusion holds:

  $$\begin{aligned} G_{F}(z) := \dfrac{ \sum _{n = 0}^{\infty } F(n) z^{n} }{ \mu _{+}(z) } \in {\mathcal {H}}(b) \end{aligned}$$

  i.e. the function $G_{F}$ is in the kernel of the Toeplitz operator $T_{\overline{b}}$.

As noted previously, for $f \in L^2(\mu )$, standard arguments can be used to show that $| \hat{f}(z)| \le A e^{\pi |z|}$ (see also Theorem 7.23 in [23]). We require a stronger growth estimate on $\hat{f}$.

Lemma 2

If $\mu$ is a Borel measure on $(-1/2, 1/2)$, then for $f \in L^2(\mu )$, $\hat{f}$ satisfies the estimate

$$\begin{aligned} | \hat{f}(z) | \le \varepsilon (|z|) e^{ \pi | z | }, \qquad \varepsilon ( r ) = o(1). \end{aligned}$$

(10)

Proof

For $z = x + iy$ with $y > 0$, we estimate $\Vert e^{2 \pi i t z } \Vert _{\mu }^{2}$ as follows: let $\{ x_{n} \}$ be an increasing sequence with $x_0 = -1/2$, $x_{n} < 1/2$, $x_{n} \rightarrow 1/2$ and $\gamma _{n} = \mu ( \left( x_{n-1}, x_{n} \right] )$. We have

$$\begin{aligned} \int _{-1/2}^{1/2} | e^{2 \pi i t z} |^2 d\mu (t) = \sum _{n} \int _{(x_{n-1},x_{n}]} e^{4 \pi t y } d \mu (t) \le O(1) + \sum _{n} \gamma _{n} e^{ 4 \pi x_{n} y}. \end{aligned}$$

Since $\sum \gamma _{n} < \infty$, we obtain

$$\begin{aligned} \dfrac{ \int _{-1/2}^{1/2} | e^{2 \pi i t z} |^2 d\mu (t) }{ e^{2 \pi y } } \lesssim \sum _{n} \gamma _{n} e^{ 2 \pi (2x_n - 1) y } = o(1) \end{aligned}$$

by Lebesgue’s Convergence Theorem. The same estimate holds for $y < 0$. Equation (10) now follows from the Cauchy-Schwarz inequality. $\square$

Theorem 3

Suppose $\mu$ is a singular Borel probability measure on $(-1/2, 1/2)$, and let b be the inner function associated to $\mu$ by the Herglotz Representation. The entire function F admits the representation in Eq. (1) if and only if

• (i) $| F(z) | \le \varepsilon (|z|) e^{ \pi | z | }$ with $\varepsilon (r) = o(1)$;

• (ii) the following inclusions hold:

  $$\begin{aligned} G_{+}(z) := \dfrac{ \sum _{n = 0}^{\infty } F(n) z^{n} }{ \mu _{+}(z) } \in {\mathcal {H}}(b), \qquad G_{-}(z) := \dfrac{ \sum _{n = 0}^{\infty } \overline{F(-n)} z^{n} }{ \mu _{+}(z) } \in {\mathcal {H}}(b); \end{aligned}$$

• (iii) the $L^2(\mu )$-boundaries of $G_{+}$ and $G_{-}$ satisfy the relationship

  $$\begin{aligned} \overline{G_{+}^{\star }} = G_{-}^{\star }. \end{aligned}$$

Proof

The necessity of Item (i) follows from Lemma 2; the necessity of Item (ii) follows from Lemma 1; and Item (iii) follows from a routine calculation.

For the converse, the issue again is when can the sequence $\{ F(n) \}_{n=-\infty }^{\infty }$ be interpolated by a function of the form $\hat{f}$ for some $f \in L^2(\mu )$. If Item (ii) holds, then by Lemma 1, there exist $f_{+},f_{-} \in L^2(\mu )$ such that for all $n \in {\mathbb {N}}_{0}$,

$$\begin{aligned} F(n) = \int _{-1/2}^{1/2} f_{+}(x) e^{- 2\pi i n x} \ d \mu (x); \qquad \overline{F(-n)} = \int _{-1/2}^{1/2} f_{-}(x) e^{- 2 \pi i n x} \ d \mu (x). \end{aligned}$$

(11)

Since $V_{\mu } f_{+} = G_{+}$ and $V_{\mu } f_{-} = G_{-}$, we have $f_{+} = G_{+}^{\star }$ and $f_{-} = G_{-}^{\star }$. Therefore, if in addition Item (iii) holds, then we have for all $n \in {\mathbb {Z}}$

$$\begin{aligned} F(n) = \int _{-1/2}^{1/2} f_{+}(x) e^{-2 \pi i n x} \ d \mu (x). \end{aligned}$$

(12)

Consequently, if Item (i) holds, F and $\hat{f}_{+}$ are both entire functions satisfying the same estimate by Lemma 2 and agree on ${\mathbb {Z}}$. Therefore, Carlson’s Theorem in the form given in [5, Corollary 9.4.4] guarantees that $F = \hat{f}$. $\square$

3 The Paley–Wiener Theorem and Exponential Frames

The power and beauty of the Paley–Wiener theorem is that the Paley–Wiener space is the intersection of two simple collections: entire functions of exponential type at most $\pi$ and $L^2({\mathbb {R}})$. In general, this simplicity cannot be replicated. There can be no integrability characterization for entire functions F that admit a representation as in Eq. (1), at least in the sense we make precise presently. We note that in [26] (see also [25, 27]), Strichartz demonstrated that for measures $\mu$ which are “uniformly $\beta$-dimensional”, the function $\hat{f}$ satisfies the following integrability/Plancherel identity condition:

$$\begin{aligned} \limsup _{R \rightarrow \infty } \dfrac{1}{R^{1-\beta }} \int _{-R}^{R} | \hat{f}(t) |^2 dt \simeq \Vert f \Vert _{\mu }^{2}. \end{aligned}$$

(13)

However, this condition does not characterize the functions $\hat{f}$.

For our purposes here, let us denote $PW(\mu ) = \{ \hat{f} | f \in L^2(\mu ) \}$, and let us denote by ${\mathcal {C}}_{\tau }$ the collection of all entire functions of exponential type at most $\tau$ and bounded on ${\mathbb {R}}$ ($0< \tau \le \pi$). By a weight w on ${\mathbb {R}}$ we mean a nonnegative measurable function; we denote $L^2(w) := \{ f | \int _{{\mathbb {R}}} | f(x) |^2 w(x) dx < \infty \}$. The following is a folklore result, which we include here for completeness.

Theorem B

For a singular probability measure $\mu$ on $(-1/2, 1/2)$, there is no weight or measure w such that $PW(\mu ) = {\mathcal {C}}_{\tau } \cap L^2(w)$.

Proof

For convenience, assume that $L^2(\mu )$ is at least 3 dimensional. If so, there exists a function $F \in PW(\mu )$ that has at least two zeros, say $z_{1}$ and $z_{2}$. If $F \notin {\mathcal {C}}_{\tau } \cap L^2(w)$ we are done; so suppose $F \in {\mathcal {C}}_{\tau } \cap L^2(w)$. The function

$$\begin{aligned} G(z) := \dfrac{F(z)}{(z - z_{1})(z - z_{2})} \in {\mathcal {C}}_{\tau } \cap L^2(w). \end{aligned}$$

We now have $G(x) \in L^2({\mathbb {R}})$, so by the classical Paley–Wiener theorem, there exists $g \in L^2(-\tau , \tau )$ such that $G = \hat{g}$. Thus, $G \notin PW(\mu )$. $\square$

We can quantify this discrepancy further by considering the problem of the existence of exponential bases and frames in $L^2(\mu )$.

Definition 1

An exponential frame for $L^2(\mu )$ is a sequence $\{ \omega _{n} e^{2 \pi i \lambda _{n} x} \}_{n \in {\mathbb {Z}} } \subset L^2(\mu )$ such that there exist constants $0<A\le B< \infty$ such that for all $f \in L^2(\mu )$,

$$\begin{aligned} A \Vert f \Vert _{\mu }^{2} \le \sum _{n} | \langle f , \omega _{n} e^{2 \pi i \lambda _{n} \cdot } \rangle |^2 \le B \Vert f \Vert _{\mu }^{2}. \end{aligned}$$

An exponential Riesz basis is an exponential frame that ceases to be a frame if any of its elements are removed.

Jorgensen and Pedersen [16] prove that the uniform measure $\mu _{3}$ on the middle-third Cantor set is not spectral, meaning it does not have an orthonormal basis of exponentials. Strichartz posed the problem of whether $\mu _{3}$ has an exponential frame in [28]; this problem is still unresolved. We illustrate that a description of those functions F that admit the representation in Eq. (1) in terms similar to the classical Paley–Wiener theorem is related to the question of when the measure $\mu$ possesses an exponential frame or Riesz basis.

Theorem 4

Suppose $PW(\mu ) = {\mathcal {C}}_{\tau } \cap L^2(w)$ for some $\tau \in (0,\pi ]$ and some weight or measure w on ${\mathbb {R}}$ with

$$\begin{aligned} \Vert f \Vert _\mu \simeq \Vert \hat{f} \Vert _{w}. \end{aligned}$$

(14)

Then there exists a Riesz basis of the form

$$\begin{aligned} \{ \omega _{n} e^{2 \pi i \lambda _{n} x} \}_{n \in {\mathbb {Z}}} \subset L^2(\mu ) \end{aligned}$$

(15)

for some sequence $\{ \lambda _{n} \} \subset {\mathbb {R}}$ and $\omega _{n} > 0$. More generally, if $PW(\mu ) \subset {\mathcal {C}}_{\tau } \cap L^2(w)$ is a closed subspace satisfying Eq. (14), then there exists a frame of the form (15).

Proof

If $PW(\mu ) = {\mathcal {C}}_{\tau } \cap L^2(w)$, then $PW(\mu )$ endowed with the $\Vert \cdot \Vert _{w}$ norm is a de Branges space. Indeed, by [8, Theorem 23], we need to verify the conditions identified as (H1), (H2), and (H3). Clearly ${\mathcal {C}}_{\tau } \cap L^2(w)$ satisfies (H1) and (H3). For (H2), note that if $w \in {\mathbb {C}}$ and $F \in PW(\mu )$, then by our assumption that the norms are equivalent, we obtain

$$\begin{aligned} | F(w) | \le \Vert f \Vert _{\mu } \Vert e^{2 \pi i w \cdot } \Vert _{\mu } \lesssim \Vert F \Vert _{w}. \end{aligned}$$

(16)

It follows by Eq. (16) that $PW(\mu )$ is complete.

Consequently, by [8, Theorem 22] there exists a sequence $\{ \lambda _{n} \} \subset {\mathbb {R}}$ such that for all $f \in L^2(\mu )$,

$$\begin{aligned} \Vert \hat{f} \Vert ^{2}_{w} = \sum _{n \in {\mathbb {Z}}} \dfrac{ | \hat{f}(\lambda _{n}) |^2 }{ K(\lambda _{n}, \lambda _{n}) } \simeq \Vert f \Vert ^{2}_{\mu }. \end{aligned}$$

(17)

Here K is the reproducing kernel for the space; the sequence of kernels $\{ K(\lambda _{n}, \cdot ) \}$ form a complete orthogonal set in the space. This combined with Eq. (17) proves the claim, where $\omega _{n} = K(\lambda _{n}, \lambda _{n})^{-1/2}$. $\square$

Measures which possess a Riesz basis or frame of exponentials are rare (see e.g. [9, 11]). An alternate proof of the second statement in Theorem 4 is found in [10].

Remark 1

We point out that our techniques in the present paper are quite similar to the alternative proof of the Paley–Wiener theorem given in [5, p. 106].