论文标题
谐波分析中的可计算性
Computability in Harmonic Analysis
论文作者
论文摘要
我们研究了连接有限域$ω$相对于点$ x \inΩ$的谐波度量$ω_x^ω$的建设性近似问题。特别是，使用可计算的谐波近似的新概念，我们表明，对于单个点$ x \inΩ$的谐波度量$ω^ω_x$的可计算性，表示任何$ y \ y \ y y \ incom y \ incom y \。这可能需要针对不同点$ y $的不同算法，这使我们构建了令人惊讶的自然示例连续函数的自然示例，这些函数作为解决方案的解决方案，可以在任何时候计算，但不能通过在其所有域上使用相同的算法来计算其值。我们进一步研究了谐波度量是可计算的条件，即单个算法，并为具有可计算边界的常规域而表征它们。
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.
