Proving the Convergence of Fourier Series for Continuous Functions

The convergence of the Fourier series of a continuous function ft to the function itself is a fundamental result in harmonic analysis. This article delves into the proof and importance of this result, providing a detailed explanation of the concepts and theorems involved.

Statement of the Problem

Given a continuous and periodic function ft with period 2π, we want to prove that the Fourier series of ft converges to ft at every point t.

Fourier Series Representation

The Fourier series of a function ft with period 2π is given by the following formula:

SNt a0 Σn1N (ancos(nt) bnsin(nt))

where the coefficients a0, an, and bn are defined as:

a0 1 ft dt

an 1 ft cos(nt) dt

bn 1 ft sin(nt) dt

Dirichlet's Theorem

Dirichlet's theorem is a key result in the theory of Fourier series. It states that if ft is piecewise continuous and has a finite number of discontinuities in the interval [-π, π], then the Fourier series converges to ft at all points where ft is continuous. At points of discontinuity, the Fourier series converges to the average of the left-hand and right-hand limits.

Application to Continuous Functions

Since ft is continuous on [-π, π], it satisfies the conditions of Dirichlet's theorem. Therefore, the Fourier series of ft converges to ft at every point t in the interval.

Pointwise Convergence

For a continuous function, the Fourier series converges pointwise to the function itself. This means that for each t, as N approaches infinity, SNt converges to ft.

Uniform Convergence

While pointwise convergence is sufficient for a continuous function, uniform convergence requires additional conditions. For example, if ft is Lipschitz continuous, then the Fourier series converges uniformly. Without these conditions, the convergence may not be uniform.

Conclusion

The Fourier series of a continuous function ft converges pointwise to ft at every point. This result is based on the properties of trigonometric polynomials and the integral representation of the Fourier coefficients, combined with Dirichlet's theorem on Fourier series convergence.

Final Note:

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