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PDF(443 KB)
PDF(443 KB)
空间Sierpinski垫上的五元素正交指数系
The Five-element Orthogonal Exponentials on the Spatial Sierpinski Gasket
设p1，p2，p3∈Z\{0，±1}，e1，e2，e3是R3上标准的单位正交基.由扩张矩阵M=diag[p1，p2，p3]和数字集D={0，e1，e2，e3}确定的自仿测度μM，D是支撑在空间Sierpinski垫T（M，D）上，其对应的Hilbert空间L2}（μM，D）上正交指数系的有限性与无限性问题已经解决.在有限的情形下，空间L2}（μM，D）上正交指数系基数的最佳上界为"4"的猜测还未完全解决.本文构造出了此空间上一列五元素正交指数函数系，说明上述最佳上界为"4"的猜测是错误的.
Let p1, p2, p3 ∈ Z\{0, ±1} and e1, e2, e3 be the standard basis of unit column vectors in R3. The self-affine measure μM,D associated with an expanding matrix M=diag[p1, p2, p3] and a digit set D={0, e1, e2, e3} is supported on the spatial Sierpinski gasket T (M, D). It is known that the finiteness or infiniteness of orthogonal exponentials in the corresponding Hilbert space L2(μM,D) has been solved completely. In the finite case, it is conjectured that the cardinality of orthogonal exponentials in L2(μM,D) is at most "4", where the number 4 is the best upper bound. That is, all the four-element orthogonal exponentials are the maximum. In the present paper, we construct a class of the five-element orthogonal exponentials in the Hilbert space L2(μM,D), which shows that the above conjecture is false.
自仿测度 / 正交指数系 / 非谱性 / 数字集 {{custom_keyword}} /
self-affine measure / orthogonal exponentials / non-spectrality / digit set {{custom_keyword}} /
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国家自然科学基金资助项目（11571214）；中央高校基本科研业务费专项基金（GK201601004）
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