Harmonic analysis on Euclidean spaces Basic notions Example

Basel problem via Fourier series

Builds on Basel problem Fourier series L2 convergence of Fourier series

Example

Let $f$ be the $2\pi$ -periodic function with $f(x) = x$ for $-\pi < x \le \pi$ . Its Fourier series is $2 \sum_{n \ge 1} \frac{(-1)^{n+1}}{n} \sin nx$ : indeed $f$ is odd, so the cosine coefficients vanish, and integration by parts gives $b_n = \frac{2}{\pi} \int_0^\pi x \sin nx \, dx = \frac{2(-1)^{n+1}}{n}.$ In complex form, $\hat f(0) = 0$ and $\abs{\hat f(n)} = \frac{1}{\abs{n}}$ for $n \ne 0$ . Parseval's identity, from the L2 convergence of Fourier series, gives $\frac{1}{2\pi} \int_{-\pi}^{\pi} x^2 \, dx = \sum_{n \ne 0} \frac{1}{n^2}, \qquad \text{that is} \qquad \frac{\pi^2}{3} = 2 \sum_{n=1}^{\infty} \frac{1}{n^2}.$ So $\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$ , which is the Basel problem.

Remarks

A second route uses the pointwise convergence of a Fourier series instead of Parseval. The $2\pi$ -periodic function equal to $x^2$ on $[-\pi, \pi]$ is continuous and piecewise smooth, and its Fourier series $\frac{\pi^2}{3} + 4 \sum_{n \ge 1} \frac{(-1)^n}{n^2} \cos nx$ converges to it everywhere. At $x = \pi$ this gives $\pi^2 = \frac{\pi^2}{3} + 4 \sum_{n \ge 1} \frac{1}{n^2}$ , the same value. Parseval applied to $x^2$ gives $\sum_{n \ge 1} \frac{1}{n^4} = \frac{\pi^4}{90}$ in the same way.

Reading path

To understand this page from the ground up, read these 131 pages in order. The foundations come first.

Bernstein polynomial definition
Cesàro summation concept
Compact space concept
Continuous function concept
Dense set concept
Element definition
Extended real number line object
Fejér kernel concept
Fourier coefficient concept
Fourier monomial on the circle concept
Fourier series concept
Harmonic number definition
Hausdorff space concept
Hilbert space concept
Limit point of a set definition
Local maximum definition
Lp space concept
Orthonormal basis concept
Periodic function definition
Periodic function concept
Proposition concept
Real numbers object
Sequence definition
Set bounded above definition
Trigonometric polynomial concept
Uniform convergence concept
Uniform metric on bounded functions example
Set definition
Limit of a function definition
Local minimum definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Real sequence definition
Subsequence definition
Supremum definition
Empty set object
Family of sets definition
Ordered pair definition
Continuity at a point definition
Derivative definition
Intersection definition
Union definition
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Cesàro mean definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Series definition
Cartesian product definition
Continuous function definition
Epsilon-delta definition of continuity statement
Subset definition
Union of a family of sets definition
Fermat's theorem on stationary points statement
Uniqueness quantifier definition
Limit superior definition
Cauchy sequences are bounded statement
Every sequence has a monotone subsequence statement
Monotone convergence theorem for sequences statement
Sum of a finite geometric series statement
Convergent sequence definition
Pointwise convergence definition
Squeeze theorem statement
Partial sum definition
Power series definition
Intermediate value theorem statement
Binary relation definition
Uniform continuity definition
Topological space definition
Bolzano–Weierstrass theorem statement
Radius of convergence definition
The sequence r^n tends to zero for |r| < 1 example
Uniform convergence definition
Convergent series definition
Sine and cosine definition
Uniform continuity on a set definition
Boundedness theorem statement
Function definition
Uniform convergence on a set definition
Fejér's theorem statement
Divergent series definition
Cauchy criterion for sequences statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Sum of a series definition
Absolute convergence definition
Series of functions definition
Extreme value theorem statement
Continuous functions on a closed bounded interval are uniformly continuous statement
Preimage of a set definition
Uniformly Cauchy sequence of functions definition
Comparison test statement
Metric space definition
Geometric series example
Cauchy condensation test statement
Harmonic series example
Rolle's theorem statement
Weierstrass approximation theorem statement
Convergence of a sequence in a metric space definition
Open ball definition
Weierstrass M-test statement
Root test statement
Diameter of a set definition
Harmonic series diverges statement
Mean value theorem statement
Metric topology definition
Epsilon characterisation of Cauchy sequences statement
Cauchy–Hadamard theorem statement
p-series test statement
Power series converge uniformly on smaller discs inside the radius of convergence statement
Open set definition
Complete metric space definition
Uniform convergence of derivatives allows term-by-term differentiation statement
Continuous map of topological spaces definition
Term-by-term differentiation statement
Power series are differentiable inside the radius of convergence statement
Completeness of the space of bounded continuous functions statement
Derivative of cosine statement
Sup norm on bounded continuous functions example
pi object
Stone–Weierstrass theorem statement
Basel problem statement
Density of trigonometric polynomials in continuous functions statement
Fourier basis of L2 of the circle statement
L2 convergence of Fourier series statement
Basel problem via Fourier series example

Tag	`TYJWZDK485`
URI	`m2://TYJWZDK485`
Kind	example
Writing	W0 stub
MSC	42A16, 40A05
Illustrates	L2 convergence of Fourier series
Uses	Basel problem, Fourier series, L2 convergence of Fourier series

Basel problem via Fourier series

Example

Remarks

Reading path

Further reading

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