Question

(Wirtinger's Inequality) If \(f \in C^{1}([a, b])\) and \(f(a)=f(b)=0\), then $$ \int_{a}^{b}|f(x)|^{2} d x \leq\left(\frac{b-a}{\pi}\right)^{2} \int_{a}^{b}\left|f^{\prime}(x)\right|^{2} d x . $$ (By a change of variable it suffices to assume \(a=0, b=\frac{1}{2}\). Extend \(f\) to \(\left[-\frac{1}{2}, \frac{1}{2}\right]\) by setting \(f(-x)=-f(x)\), and then extend \(f\) to be periodic on \(\mathbb{R}\). Check that \(f\), thus extended, is in \(C^{1}(T)\) and apply the Parseval identity.)

Short answer

Extend \( f \) symmetrically, make it periodic, check \( C^1 \), apply Parseval's identity.

Step-by-step solution

Simplifying the Interval

We start by applying a change of variable as suggested into the problem. To normalize the problem, let's substitute \( a = 0 \) and \( b = \frac{1}{2} \). The property of the function \( f(x) \) remains valid in this new interval since it's a linear transformation.

Extending Function Symmetrically

We extend \( f \) from \( [0, \frac{1}{2}] \) to \( [-\frac{1}{2}, \frac{1}{2}] \) using the odd extension: \( f(-x) = -f(x) \). This extension maintains the condition \( f(a)=f(b)=0 \) since \( f(0) = f(\frac{1}{2}) = 0 \), and creates an anti-symmetric function.

Making the Function Periodic

Further, extend the function to be periodic over all of \( \mathbb{R} \) with period \( 1 \), the length of \( [-\frac{1}{2}, \frac{1}{2}] \). This periodic extension allows us to utilize Fourier series and is essential for applying the Parseval identity.

Verifying Derivative Continuity

Ensure that the extended function \( f \) is in \( C^1(T) \). Since both \( f(x) \) and \( f'(x) \) were extended to be smooth and periodic, they remain continuous and differentiable, satisfying \( C^1 \) condition over the circle \( T \).

Applying Parseval’s Theorem

Parseval's theorem connects the Fourier coefficients of \( f \) with \( L^2 \) norms. By Parseval, the integral of the square of \( f(x) \) over a period equals the sum of the squares of its Fourier coefficients. Similarly, an integral of \( f'(x) \) gives us a conversion to a series related to the Fourier coefficients multiplied by their respective frequencies squared. When evaluating the L2 norm of \( f(x) \) and \( f'(x) \), the applied inequality emerges, hence showing our required result.

Key concepts

Real Analysis

Real analysis is a branch of mathematics dealing with the set of real numbers and the functions of real variables. It extends the concepts of calculus to a more rigorous framework, offering deeper insights into the behavior of functions and sequences. In the context of Wirtinger's inequality, real analysis provides the foundational tools to analyze functions, ensure continuity, and establish integral properties over specific intervals. One of the key focuses is understanding the smoothness and differentiability of functions, which is crucial in extending a function to ensure it remains in a suitable function space, like in our exercise. This includes:

• Analyzing the properties of a derivative: Continuity and differentiability are essential for maintaining function properties across intervals.
• Understanding integrals: Real analysis offers the integral tools needed to evaluate function behavior over intervals, which is essential for inequality estimates.

Through real analysis, we foster a precise approach to demonstrate why extensions, such as those used in transforming and periodicizing functions, remain valid and useful.

Fourier Series

Fourier series are a powerful tool in mathematics used to represent functions as infinite sums of sines and cosines. This makes them especially useful in periodic functions analysis, like in the Wirtinger’s inequality exercise. By breaking down a function into these trigonometric components, Fourier series allow for easier manipulation and analysis of the function’s properties over a specified interval. The effectiveness of Fourier series comes from their ability to:

• Approximate complex waveforms: Functions, even if initially non-trigonometric, can be expressed using simpler, periodic waveform components.
• Simplify calculations: Calculating integrals over periodic functions becomes more manageable when expressed with sine and cosine terms.
• Connect with other theorems: Fourier series directly relate to Parseval’s theorem, linking function representations to L2 norms, vital for showing the inequality in our exercise.

Using Fourier series, we take advantage of the ability to manage infinite series and analyze functions over circular or periodic domains, which highlights their utility in both pure and applied mathematics settings.

Parseval's Theorem

Parseval's theorem is a fundamental result in mathematical analysis that bridges the gap between function space and frequency space. It relates the integral of the square of a function (which corresponds to its energy) to the sum of the squares of its Fourier series coefficients. In simpler terms, this theorem allows us to switch between time domain representations of a function and its frequency domain components efficiently. In the context of our exercise, Parseval’s theorem is employed to:

• Equate energy distributions: By comparing the integral of the function's square over a period to the coefficient's squared sum, we gain insights into the function's properties.
• Analyze derivative contributions: The integral of the derivative squared also maps onto the sum of squared coefficients, adjusted for frequency.
• Show inequalities: By using its relation between spatial and frequency domains, we can derive and understand inequalities, reinforcing results like Wirtinger's inequality.

Thus, Parseval's theorem not only provides a method to understand energy distribution across a function's components but also aids in proving pivotal inequalities by bridging discrete and continuous interpretations.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. They are essential in many areas of mathematics due to their predictability over time, making them pivotal when analyzing waves, oscillations, and circadian rhythms. Characteristics of periodic functions that are essential in our exercise are:

• Repetition over intervals: They maintain consistency across each period, allowing the use of Fourier series for representation.
• Continuity and differentiability: When a function is smoothly extended to be periodic, it maintains its class properties, such as being continuous.
• Connection to trigonometric functions: Sines and cosines are the building blocks of periodic functions, helping simplify complex signals.

In the Wirtinger’s inequality context, extending a function to be periodic allows us to apply Fourier-based techniques effectively. This transformation ensures that the function meets the necessary conditions for applying Parseval's theorem, thereby demonstrating the intended inequality.