周脈東

(1.陜西師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院,陜西 西安 710062;2.陜西教育學(xué)院數(shù)學(xué)系,陜西 西安 710061)

一類四元素?cái)?shù)字集的正交函數(shù)的個(gè)數(shù)

周脈東1,2

(1.陜西師范大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院,陜西 西安 710062;2.陜西教育學(xué)院數(shù)學(xué)系,陜西 西安 710061)

自仿測度的譜與非譜問題近年引起了很大的關(guān)注,關(guān)于自仿測度的非譜問題,其中之一就是要估算它在L2空間上的正交指數(shù)的個(gè)數(shù).通過對μM,D傅里葉變換的零點(diǎn)集性質(zhì)的分析和討論,對現(xiàn)有的結(jié)論進(jìn)行了改進(jìn),確定了相應(yīng)四元素?cái)?shù)字集的平面自仿測度在L2的空間上正交指數(shù)函數(shù)的最大個(gè)數(shù)為3.

自仿測度;譜測度;正交指數(shù)函數(shù);迭代函數(shù)系

1 引言

關(guān)于非譜測度μM,D,在L2(μM,D)上存在至多有限個(gè)正交指數(shù),主要的問題是證明在L2(μM,D)上正交指數(shù)的個(gè)數(shù)并找出它們來.文獻(xiàn) [3]證明了在廣義平面 Sierpinski墊在 L2(μM,D)上的正交指數(shù)的個(gè)數(shù)最多是3個(gè),文獻(xiàn)[4-5]在此基礎(chǔ)上給出了更一般的結(jié)論.文獻(xiàn)[6] (定理2)對于給出的擴(kuò)張矩陣和四個(gè)元素的數(shù)字集,證明了最多存在4個(gè)相互正交的指數(shù)函數(shù),本文改進(jìn)了文獻(xiàn)[6]中定理2的結(jié)論,證明了相互正交的指數(shù)函數(shù)最多為3個(gè).

2 定理及證明

3 結(jié)論

[1]Fuglede B.Commuting self-adjoint partial di ff erential operators and a group theoretic problem[J].Funt. Anal.,1974,16:101-121.

[2]Hutchinson J E.Fractals and self-similarity[J].Indiana Univ.Math.,1981,30:713-747.

[3] Li Jianlin. Orthogonal exponential on the generalized plane Sierpinski gasket[J].J.Approx Theory, 2008,153(2):161-169.

[4]Li Jianlin.Non-spectral problem for a class of planar self-affine measure[J].J.Funt.Anal.,2008,255(11)11: 3125-3148.

[5]Li Jianlin.Non-spectrality of planar self-affine measure with three-element digit set[J].J.Funt.Anal., 2009,257(2):537-552.

[6]Li Jianlin.The cardinality of certainμM,Dorthogonal exponentials[J].J.Math.Anal.Appl.,2010,362:514-522.

The cardinality of a class of orthogonal exponential function with four-element digit set

Zhou Maidong1,2
(1.College of Mathematics and Information Science,Shaanxi Normal University,Xi′an 710062,China;
2.Department of Mathematics,Shaanxi Institute of Education,Xi′an 710061,China)

The problem of the spectrality or non-spectrality of a self-affine measure have been

much attention in recent years.One of non-spectrality problem on a self-affine measure is to estimate the number of orthogonal exponentials in L2.By the analysis of characterization of the zero set of the Fourier transform of μM,D,Improve the result and get the biggest number of the orthogonal exponential functions of planar self-affine measure with four-elements digit set for the L2space is 3.

self-affine measure,spectral measure,orthogonal exponential functions,iterated function system

O174.12

A

1008-5513(2012)01-0080-05

2011-02-10.

陜西省科學(xué)技術(shù)研究發(fā)展計(jì)劃（二）項(xiàng)目(2011JE002).

周脈東(1969-),碩士,講師,研究方向：譜測度及其理論.

2010 MSC:42B10