Spectral self-affine measures on the spatial Sierpinski gasket

2017

Trends in Mathematics

Abstract. This paper gives a review of the recent progress in the study of Fourier bases and Fourier frames on self-affine measures. In particular, we emphasize the new matrix analysis approach for checking the completeness of a mutually orthogonal set. This method helps us settle down a long-standing conjecture that Hadamard triples generates self-affine spectral measures. It also gives us non-trivial examples of fractal measures with Fourier frames. Furthermore, a new avenue is open to investigate whether the Middle Third Cantor measure admits Fourier frames.

Section: Introductionmentioning

confidence: 83%

Fourier Bases and Fourier Frames on Self-Affine Measures

Dutkay

Lai

2017

Trends in Mathematics

“…More recently, the following additional results are obtained for the pair

(M, D)

given by (1.4) (see , ).…”

Section: Introductionmentioning

confidence: 92%

“…The proof of Theorem A(i) has been simplified by [, p. 308–309], and the number “256” in Theorem A(ii) has been improved to the best number “4” (see ), and the following more general result holds (see ).…”

Section: Introductionmentioning

confidence: 99%

“…The generalized spatial Sierpinski gasket corresponding to

M = [\begin{matrix} right p_{1} & right p_{4} & right p_{6} \\ right 0 & right p_{2} & right p_{5} \\ right 0 & right 0 & right p_{3} \end{matrix}] and D = \{((), \begin{matrix} 0 \\ 0 \\ 0 \end{matrix}), ((), \begin{matrix} 1 \\ 0 \\ 0 \end{matrix}), ((), \begin{matrix} 0 \\ 1 \\ 0 \end{matrix}), ((), \begin{matrix} 0 \\ 0 \\ 1 \end{matrix})\},

where,

p_{1}, p_{2}, p_{3} \in Z ∖ {0, \pm 1}

and

p_{4}, p_{5}, p_{6} \in Z

, there are little results on the spectrality of

μ_{M, D}

. For example, we know, from [, Proposition 3.2] that if any two of the three numbers

p_{1}, p_{2}, p_{3}

are in the set

2 Z

, then there are infinite families of orthogonal exponentials

E (Λ)

L^{2} (μ_{M, D})

with

Λ \subseteq {boldZ}^{3}

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Spectrality of certain self‐affine measures on the generalized spatial Sierpinski gasket

Mathematische Nachrichten

2015

Self Cite

Key words Iterated function system (IFS), self-affine measure, orthogonal exponentials, spectrality MSC (2010) 28A80, 42C05, 46C05The self-affine measure μ M,D corresponding to a upper or lower triangle expanding matrix M and the digit set D = {0, e 1 , e 2 , e 3 } in the space R 3 is supported on the generalized spatial Sierpinski gasket, where e 1 , e 2 , e 3 are the standard basis of unit column vectors in R 3 . We consider in this paper the existence of orthogonal exponentials on the Hilbert space L 2 (μ M,D ), i.e., the spectrality of μ M,D . Such a property is directly connected with the entries of M and is not completely determined. For this generalized spatial Sierpinski gasket, we present a method to deal with the spectrality or non-spectrality of μ M,D . As an application, the spectral property of a class of such self-affine measures are clarified. The results here generalize the corresponding results in a simple manner.

“…We know that the finiteness or infiniteness of orthogonal exponentials in the corresponding Hilbert space

L^{2} true(μ_{M, D} true)

has been solved completely in . In the finite case, we may assume (without loss of generality) that

p_{1} \in 2 Z ∖ false{0 false}, p_{2} \in false(2 boldZ + 1 false) ∖ false{\pm 1 false}, p_{3} \in false(2 boldZ + 1 false) ∖ false{\pm 1 false} .

In the case

p_{2} = p_{3}

, Li proved that the maximal cardinality of

μ_{M, D}

‐orthogonal exponentials is “4”. That is, for any

μ_{M, D}

‐orthogonal exponentials

E (Λ)

with

| Λ | = 4

, Λ is maximal in the sense of following Definition .…”

Section: Introductionmentioning

confidence: 99%

There are eight‐element orthogonal exponentials on the spatial Sierpinski gasket

Mathematische Nachrichten

2018

Self Cite

The self‐affine measure μM,D corresponding to an expanding matrix M=diagtrue[p1,p2,p3true] and the digit set D=0,e1,e2,e3 in the space R3 is supported on the spatial Sierpinski gasket, where e1,e2,e3 are the standard basis of unit column vectors in R3 and p1,p2,p3∈Z∖false{0,±1false}. In the case p1∈2Z and p2,p3∈2Z+1, it is conjectured that the cardinality of orthogonal exponentials in the Hilbert space L2true(μM,Dtrue) is at most “4”, where the number 4 is the best upper bound. That is, all the four‐element sets of orthogonal exponentials are maximal. This conjecture has been proved to be false by giving a class of the five‐element orthogonal exponentials in L2true(μM,Dtrue). In the present paper, we construct a class of the eight‐element orthogonal exponentials in the corresponding Hilbert space L2true(μM,Dtrue) to disprove the conjecture. We also illustrate that the constructed sets of orthogonal exponentials are maximal.