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

The self‐affine measure corresponding to a upper or lower triangle expanding matrix M and the digit set in the space is supported on the generalized spatial Sierpinski gasket, where are the standard basis of unit column vectors in . We consider in this paper the existence of orthogonal exponentials on the Hilbert space , i.e., the spectrality of . 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 . 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.     

Spectrality of a class of self-affine measures with decomposable digit sets

The self-affine measure associated with an expanding matrix and a finite digit set is uniquely determined by the self-affine identity with equal weight.The spectral and non-spectral problems on the selfaffine measures have some surprising connections with a number of areas in mathematics,and have been received much attention in recent years.In the present paper,we shall determine the spectrality and non-spectrality of a class of self-affine measures with decomposable digit sets.We present a method to deal with such case,and clarify the spectrality and non-spectrality of a class of self-affine measures by applying this method.     

Spectrality of planar self-affine measures with two-element digit set

The iterated function system with two element digit set is the simplest case and the most important case in the study of self affine measures.The one dimensional case corresponds to the Bernoulli convolution whose spectral property is understandable.The higher dimensional analogue is not known,for which two conjectures about the spectrality and the non spectrality remain open.In the present paper,we consider the spectrality and non spectrality of planar self affine measures with two element digit set.We give a method to deal with the two dimensional case,and clarify the spectrality and non spectrality of a class of planar self affine measures.The result here provides some supportive evidence to the two related conjectures.     

Wightman axiomatic approach in noncommutative field theory

The axiomatic approach based on Wightman functions is developed in noncommutative field theory. We prove that the main results of the axiomatic approach remain valid if the noncommutativity affects only the spatial variables.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 403#x2013;416, February, 2005.     

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

The self‐affine measure corresponding to an expanding matrix and the digit set in the space is supported on the spatial Sierpinski gasket, where are the standard basis of unit column vectors in and . In the case and , it is conjectured that the cardinality of orthogonal exponentials in the Hilbert space 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 . In the present paper, we construct a class of the eight‐element orthogonal exponentials in the corresponding Hilbert space to disprove the conjecture. We also illustrate that the constructed sets of orthogonal exponentials are maximal.     

Duality relation on spectra of self‐affine measures

The present paper establishes a duality relation for the spectra of self‐affine measures. This is done under the condition of compatible pair and is motivated by a duality conjecture of Dutkay and Jorgensen on the spectrality of self‐affine measures. For the spectral self‐affine measure, we first obtain a structural property of spectra which indicates that one can get new spectra from old ones. We then establish a duality property for the spectra which confirms the conjecture in a certain case.     

Analysis of μM,D‐orthogonal exponentials for the planar four‐element digit sets

The self‐affine measure is a unique probability measure satisfying the self‐affine identity with equal weight. It only depends upon an expanding matrix M and a finite digit set D. In this paper we study the question of when the ‐space has infinite families of orthogonal exponentials. Such research is necessary to further understanding the spectrality of . For a class of planar four‐element digit sets, we present several methods to deal with this question. The application of each method is also given, which extends the known results in a simple manner.