群扩张上的调和综合
Harmonic Synthesis on Group Extensions
AI总结：
本文研究局部紧 Abel 群的调和综合性，证明可综合群经可综合离散 Abel 群扩张后仍可综合，推广了离散群直和的结论。
摘要：
调和综合描述局部紧 Abel 群上连续复值函数的平移不变线性空间。L. Schwartz 的基本结果表明，实数集上的这类空间由该空间中的指数单项式在拓扑意义下生成——换言之，实数所构成的局部紧 Abel 群是可综合的。D. I. Gurevich 的反例表明，这一结果对多个实变量的连续函数并不成立。另一方面，如果两个离散 Abel 群具有这种可综合性，那么它们的直和也具有该性质。本文证明，如果两个局部紧 Abel 群具有这种可综合性，并且其中至少一个是离散的，那么它们的直和是可综合的。事实上，更一般地，我们证明，一个可综合的局部紧 Abel 群被一个可综合的离散 Abel 群扩张所得的任意扩张都是可综合的。这是迈向完整刻画可综合局部紧 Abel 群的重要一步。
英文摘要：
Harmonic synthesis describes translation invariant linear spaces of continuous complex valued functions on locally compact abelian groups. The basic result due to L. Schwartz states that such spaces on the reals are topologically generated by the exponential monomials in the space -- in other words the locally compact abelian group of the reals is synthesizable. This result does not hold for continuous functions in several real variables as it was shown by D.I. Gurevich's counterexamples. On the other hand, if two discrete abelian groups have this synthesizability property, then so does their direct sum, as well. In this paper we show that if two locally compact abelian groups have this synthesizability property and at least one of them is discrete, then their direct sum is synthesizable. In fact, more generally, we show that any extension of a synthesizable locally compact abelian group by a synthesizable discrete abelian group is synthesizable. This is an important step toward the complete characterisation of synthesizable locally compact abelian groups.
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