论文标题
傅立叶变换反转：有界变化，多项式生长，henstock - stieltjes Integration
Fourier transform inversion: Bounded variation, polynomial growth, Henstock--Stieltjes integration
论文作者
论文摘要
在本文中，我们证明了局部变化的局部函数的点上和分布傅立叶变换反转定理，而在Infinity的邻域则是Lebesgue lebesgue commentable Compantable或具有多项式生长。我们还允许在主要价值意义上存在傅立叶变换。如果每个点的左极限和右限制，则调用一个函数。通过求解微分方程$ df（t）-iΩf（t）= g（t）$的主要反转定理可获得调节函数$ f $，其中$ω$是一个复杂的数字，具有积极的假想零件。这是使用Henstock-Stieltjes积分进行的。这是用Riemann和仪表定义的积分。对于此积分，还证明了零件公式集成的某些变体。当函数是多项式增长时，其傅立叶变换以分布含义存在，尽管反转公式仅涉及函数的集成并返回点式值。
In this paper we prove pointwise and distributional Fourier transform inversion theorems for functions on the real line that are locally of bounded variation, while in a neighbourhood of infinity are Lebesgue integrable or have polynomial growth. We also allow the Fourier transform to exist in the principal value sense. A function is called regulated if it has a left limit and a right limit at each point. The main inversion theorem is obtained by solving the differential equation $df(t)-iωf(t)=g(t)$ for a regulated function $f$, where $ω$ is a complex number with positive imaginary part. This is done using the Henstock--Stieltjes integral. This is an integral defined with Riemann sums and a gauge. Some variants of the integration by parts formula are also proved for this integral. When the function is of polynomial growth its Fourier transform exists in a distributional sense, although the inversion formula only involves integration of functions and returns pointwise values.
