调和分析的可计算性
Computability in Harmonic Analysis
摘要：
研究了连通有界域$Ω$的调和测度$ω_x^Ω$关于点$x\inΩ$的构造逼近问题.特别是，使用一个新的概念，可计算的谐波近似，我们表明，对于任意这样的$Ω$，可计算的谐波措施$ω^Ω_x$的单点$x\inΩ$意味着可计算的$ω_y^Ω$的任何$y\in Ω$。对于不同的点，这可能需要不同的算法$y$，这导致我们构造出令人惊讶的连续函数的自然例子，这些函数是作为狄利克雷问题的解而出现的，其值可以在任何点计算，但不能在其所有域上使用相同的算法来计算。我们进一步研究了调和测度可一致计算的条件，即用一个算法，并对具有可计算边界的正则域刻画了它们。
英文摘要：
We study the question of constructive approximation of the harmonic measure $ω_x^Ω$ of a connected bounded domain $Ω$ with respect to a point $x\inΩ$. In particular, using a new notion of computable harmonic approximation, we show that for an arbitrary such $Ω$, computability of the harmonic measure $ω^Ω_x$ for a single point $x\inΩ$ implies computability of $ω_y^Ω$ for any $y\in Ω$. This may require a different algorithm for different points $y$, which leads us to the construction of surprising natural examples of continuous functions that arise as solutions to a Dirichlet problem, whose values can be computed at any point but cannot be computed with the use of the same algorithm on all of their domain. We further study the conditions under which the harmonic measure is computable uniformly, that is by a single algorithm, and characterize them for regular domains with computable boundaries.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
