Let , the partial sum operator of fourier series can be represented by

actually

Since , for , then behavior much like Hilbert transform

(a) ,

(b)

By the fact in this post, in

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when time goes to infinity

Let , the partial sum operator of fourier series can be represented by

actually

Since , for , then behavior much like Hilbert transform

(a) ,

(b)

By the fact in this post, in

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Suppose , consider the fourier series of

and its partial sum

The following statements are equivalent

(a) ,

(b) as

(c) ,

(a)(b). Since trigonometric polynomial are dense in , then we can choose to be a trigonometric polynomial are sufficiently close to in .

(b)(a). Uniform boundedness principle.

(a)(c) and (c)(a) are quite easy.