By Anthony W. Knapp

Systematically advance the thoughts and instruments which are important to each mathematician, no matter if natural or utilized, aspiring or established

A finished remedy with a world view of the topic, emphasizing the connections among genuine research and different branches of mathematics

Included all through are many examples and 1000s of problems, and a separate 55-page part offers tricks or entire recommendations for most.

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Additional resources for Basic Real analysis

Sample text

The n=1 n formulation of Parseval’s Theorem as in the remark, but with the interval (0, 2π ) 1 1 2π 1 replacing the interval (−π, π), says that ∞ | 2 (π − x)|2 d x. The n=1 n 2 = π 0 right side is = 1 4π π −π x2 dx = 2π 3 /3 4π ∞ n=1 = π2 6 . Thus 1 π2 . = n2 6 This formula was discovered by Euler by other means before the work of Fourier. 1. The form ( f, g)2 comes close to being one, but it fails to meet all the conditions because ( f, f )2 = 0 is possible without f = 0. 3 The 10. Fourier Series 75 as well as a “norm” deﬁned by f 1/2 2 = ( f, f )2 π 1 2π = −π | f (x)|2 d x 1/2 and a “distance function” deﬁned by d2 ( f, g) = f − g 2 = π 1 2π | f (x) − g(x)|2 d x −π 1/2 .

In the context of Fourier series, it is the periodic extensions that are to have these properties, and then the integration-by-parts formula simpliﬁes. , the integrated term drops out because of the assumed periodicity. The simplest convergence result for Fourier series is that a periodic function (of period 2π) with two continuous derivatives has a uniformly convergent Fourier series. To prove this, we take n = 0 and use the above integration-by-parts formula twice to obtain cn = = 1 2π π −π 1 f (x)e−inx d x = − 2π 1 1 2 π −π 2π −in 1 −in π −π f (x)e−inx d x f (x)e−inx d x.

Let f be in R[−π, π], and let f (x) ∼ Then ∞ inx . n=−∞ cn e ∞ |cn |2 ≤ n=−∞ In particular, ∞ n=−∞ 1 2π π −π | f (x)|2 d x. |cn |2 is ﬁnite. REMARK. In terms of the coefﬁcients an and bn , the corresponding result is ∞ |a0 |2 1 (|an |2 + |bn |2 ) ≤ + 2 π n=1 π −π | f (x)|2 d x. PROOF. The theorem shows that the minimum value of a certain nonnegative π N 1 2 2 quantity depending on N is 2π n=−N |cn | . Thus, for any N , −π | f (x)| d x − π N 1 2 2 n=−N |cn | ≤ 2π −π | f (x)| d x. Letting N tend to inﬁnity, we obtain the corollary.