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Convergence of Classical Fourier Series

  • Toru Maruyama
Chapter
Part of the Monographs in Mathematical Economics book series (MOME, volume 2)

Abstract

We have discussed basic contents of the theory of Fourier series on a general Hilbert space. We now proceed to the classical problem concerning the Fourier series expansion of an integrable function with respect to the trigonometric functions. If we choose \(\mathfrak {L}^2([-\pi , \pi ], \mathbb {C})\) as a Hilbert space and
$$\displaystyle \frac {1}{\sqrt {2\pi } }, \frac {1}{\sqrt {\pi }} \cos x , \; \frac {1}{\sqrt {\pi } } \sin x , \; \cdots ,\; \frac {1}{\sqrt {\pi } } \cos nx , \; \frac {1}{\sqrt {\pi } } \sin nx , \; \cdots ; \; n=1,2, \cdots $$
as a complete orthonormal system, the Fourier series of \(f\in \mathfrak {L}^2([-\pi , \pi ],\mathbb {C})\) is given in the form
$$\displaystyle \frac {a_0}{2} + \sum _{n=1}^\infty (a_n \cos nx + b_n \sin nx) , $$
where
$$\displaystyle a_n = \frac {1}{\pi } \int _{-\pi }^{\pi } f(x) \cos nx \; dx ,\quad b_n = \frac {1}{\pi } \int _{-\pi }^{\pi } f(x) \sin nx \; dx . $$
This Fourier series converges to f in \(\mathfrak {L}^2\)-norm.

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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Toru Maruyama
    • 1
  1. 1.Professor EmeritusKeio UniversityTokyoJapan

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