Abstract
We give some characterizations of Fourier frames and tilings and obtain a more general form of characterizations of spectra and tilings.
1. Introduction
A countable family of elements in a separable Hilbert space is called a frame if there are positive constants such that for all . and are called frame bounds. The sequence is called a tight frame if . The sequence is called Bessel if the second inequality above holds. In this case, is called the Bessel bound. Frames were first introduced by Duffin and Schaeffer [1] in the context of nonharmonic Fourier series, and today they have applications in a wide range of areas. A frame can be considered as a generalized basis in the sense that every element in can be written as a linear combination of the frame elements.
In this paper, we consider Fourier frames for a special separable Hilbert space. Let have positive Lebesgue measure and let be a discrete subset of . The inner product and the norm on are We write If is a frame or an orthonormal basis for , then and are called a Fourier frame and a spectral pair, respectively. In the case of the spectral pair, the is then called a spectrum for and is called a spectral set. We follow the terminology of [2] and consider the packing and tiling in by compact set of the following kind.
A compact set in is a regular region if it has positive Lebesgue measure, is the closure of its interior , and has a boundary of measure zero. If is a regular region, then a discrete set is a packing set for if the sets have disjoint interiors or the intersections for in have measure zero. It is a tiling set if, further, the translates cover up to measure zero. In these cases, we say that is a packing or tiling of , respectively. Equivalently, we call a packing pair or a tiling pair, respectively.
It is well known that spectral sets and tilings are connected by the following conjecture of Fuglede [3].
Spectral Set Conjecture
A set in is a spectral set if and only if it tiles by translations.
Many people attempt to prove the spectral set conjecture for some special sets, although the conjecture is false in many cases (see [4–7]). For example, Jorgensen and Pedersen [8] conjectured that is a spectral pair if and only if is a tiling pair. They established the conjecture for dimension and for all when is a discrete periodic set. Iosevich and Pedersen [9] simultaneously and independently established the above-mentioned conjecture by a different approach based on a geometric argument. Kolountzakis [10] gave an alternative proof of this fact, which is based on a characterization of translational tiling by a Fourier analytic criterion. Lagarias et al. [2] related the spectra of sets to tiling in the Fourier space and obtained the following characterization of spectra and tilings.
Theorem 1. Let be a regular region in and let be such that the set of exponentials is orthogonal for . Suppose that is a regular region with such that is a packing of . Then, is a spectrum for if and only if is a tiling of .
Li [11] presented an elementary approach to obtain a more general form of Theorem 1. Enlightened by the ideas from [11, 12], we give some characterizations of Fourier frames and tilings and extend several results in [2] and [11].
2. Main Results and Their Proofs
Throughout this section, let and be two regular regions in . By the definition of frames, we may get that the following lemma.
Lemma 2. Let be a discrete set. If is a frame for with frame bounds , then where is the Fourier transform of the characteristic function .
Proof. By the frame inequality (1), for any , we have Similarly, we get .
Remark 3. In the case for , if is a tight frame for with the frame bound , then If is a Bessel sequence for with the Bessel bound , then Moreover, is an orthonormal basis for if and only if Since , for all , if we substitute “−” for “+” in (4), (7), (8), and (9), all the above results also hold.
In the remainder of this paper, we assume that is a discrete subset and and are two finite subsets of such that and are two direct sums.
Theorem 4. If is a frame for with frame bounds , and is a tiling pair, then where denotes the cardinality of some set.
Proof. Let and with and . Since is a frame for with frame bounds , it follows from Lemma 2 that Note that is a tiling pair, from the Plancherel’s formula on , we have the following: The bottom third equality holds by Lebesgue dominated convergence theorem and the last inequality follows from (11). Thus, . Similarly, we get . Hence, the proof is completed.
Since an orthonormal basis is also a tight frame with frame bounds , we get the following corollary.
Corollary 5. If is an orthonormal basis for , and is a tiling pair, then .
Lemma 6. Let be such that the set of exponentials is a Bessel sequence for with the Bessel bound . If is a tiling of with , then
Proof. Keep the assumptions on and in the above proof. Since is a Bessel sequence for with the Bessel bound , it follows from Remark 3 that Since is a tiling of and , for any , then we have which yields for almost every in . Since is arbitrary, (16) holds for almost every in . By the continuity of the function on the left side of (16), we see that (16) holds for every in .
Theorem 7. If is orthogonal in and is a tiling pair with , then is an orthonormal basis for .
Proof. The proof is straightforward by the above lemma.
Theorem 8. Suppose that is a packing of with . If is a frame for with the frame bounds , then is a tiling of .
Proof. Since is a frame for with the frame bounds , then (11) holds. If is a packing of , then it follows from (11) and that Thus, is a tiling of .
It is clear that the above theorem yields the following corollary.
Corollary 9. Suppose that is a packing of with . If is an orthonormal basis for , then is a tiling of .
Combining Theorem 7 with Corollary 9, we obtain a more general form of the theorem in [11] and Theorem 1.
Theorem 10. Suppose that is orthogonal in , and is a packing pair with . Then, is a spectral pair if and only if is a tiling pair.
Example 11. Let be two positive integers. Take the following: We see that , is a spectral pair and is a tiling pair.
Acknowledgment
This work was supported by the National Natural Science Foundation of China 11271148 and 11271114.