Wirtinger inequality

For

f (0) = f (T) = 0

,

\int_{0}^{T} ∣ f^{'} ∣^{2} d t \geq \frac{π ^{2}}{T ^{2}} \int_{0}^{T} ∣ f ∣^{2} d t,

with equality for a multiple of

sin (π t / T)

.

Table of contents
- Minimum Dirichlet energy at fixed L2 norm on an interval Wirtinger inequality
- Periodic Wirtinger inequality Wirtinger inequality

Minimum Dirichlet energy at fixed L2 norm on an interval ( $min_{y \in H_{0}^{1} (0, L), ∥ y ∥_{2} = 1} \int_{0}^{L} ∣ y^{'} ∣^{2} = π^{2} / L^{2}$ )

Expand a real function with zero endpoint values in its Fourier sine series,

y = \sum_{n \geq 1} a_{n} sin (nπ x / L)

. The Parseval identity gives

\int y^{2} = (L /2) \sum a_{n}^{2}

. For its weak derivative, the Fourier cosine series has coefficients

(nπ / L) a_{n}

and zero constant term, so

\int_{0}^{L} ∣ y^{'} ∣^{2} d x = \frac{L}{2} \sum_{n \geq 1} (\frac{nπ}{L})^{2} a_{n}^{2} \geq \frac{π ^{2}}{L ^{2}} \int_{0}^{L} y^{2} d x .

The derivative coefficient identity follows by integration by parts, rather than by assuming pointwise differentiation of an arbitrary series. This proves the Wirtinger inequality and its equality case. Under unit L2 norm, the only real minimizers are

y = \pm 2/ L sin (π x / L)

.

Periodic Wirtinger inequality

If a continuously differentiable

L

-periodic function

f

has mean zero, then

\int_{0}^{L} f (s)^{2} d s \leq \frac{L ^{2}}{4 π ^{2}} \int_{0}^{L} f^{'} (s)^{2} d s .

Equality holds exactly for

f (s) = A cos (2 π s / L) + B sin (2 π s / L)

.

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