Introduction to Fourier series
Introduction to Fourier series
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傅立叶级数简介
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发表时间：
2019
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通讯作者：
Charles L. Epstein
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作者：
Charles L. Epstein
另一方面，1876年P. du Bois-Reymond发现了一个连续函数，其傅里叶级数在单点发散。通过一致有界性定理，我们将在后面证明存在连续函数，其傅立叶级数在任意给定的可数点集合上发散。A. Kolmogorov（1923/26）给出了L(T)函数的傅里叶级数处处都是点发散的例子。连续函数C(T)空间中三角多项式（有限傅立叶级数）的密度可由Weierstraß近似定理推导。我们给出了C(T)中三角多项式密度的一个稍微不同的证明，引入并使用了fejsamr核。1904年，L. fejsamir给出了L(T)中三角多项式密度的更直接的证明，实际上使用了一个直接由三角多项式组成的近似恒等式。我们复制这个证明。
From the other side, in 1876 P. du Bois-Reymond found a continuous function whose Fourier series diverges at a single point. Via the uniform boundedness theorem, we will show later that there are continuous functions whose Fourier series diverges at any given countable collection of points. A. Kolmogorov (1923/26) gave an example of an L(T) function whose Fourier series diverges pointwise everywhere. The density of trigonometric polynomials (finite Fourier series) in the space of continuous function C(T) can be made to follow from Weierstraß’ aproximation theorem. We give a somewhat different proof of density of trigonometric polynomials in C(T), introducing and using the Fejér kernel. In 1904, L. Fejér gave an even more direct proof of the density of trigonometric polynomials in L(T), in effect using an approximate identity made directly in terms of trigonometric polynomials. We reproduce this proof.