Question

Assume that \(f\) is continuous on \([-\pi, \pi]\) and \(f(-\pi)=f(\pi)\). Prove that \(f\) can be uniformly approximated by trigonometric polynomials. That is, given \(\epsilon>0\) there exists a trigonometric polynomial \(T\) of some degree such that $$ |f(x)-T(x)|<\epsilon $$ for all \(x \in[-\pi, \pi]\).

Short answer

Continuous functions on \([- pi, \pi]\) with \(f(-\pi) = f(\pi)\) can be uniformly approximated by trigonometric polynomials.

Step-by-step solution

Understand the Problem

We need to prove that any continuous function \(f\) on the interval \([-pi, pi]\) which satisfies \(f(-\pi) = f(\pi)\) can be uniformly approximated by a trigonometric polynomial. This means for every given \(\epsilon > 0\), we need to find a trigonometric polynomial \(T(x)\) such that for all \(x\), \(|f(x) - T(x)| < \epsilon\).

Define Trigonometric Polynomials

Trigonometric polynomials are functions of the form \(T_n(x) = a_0 + \sum_{k=1}^{n} (a_k \cos(kx) + b_k \sin(kx))\). Our task is to find such a polynomial that approximates \(f\) within the given \(\epsilon\) tolerance.

Use Weierstrass Approximation Theorem for Trigonometric Polynomials

The Weierstrass approximation theorem states that any continuous periodic function defined on an interval can be uniformly approximated by polynomials (in this case, trigonometric polynomials) as closely as desired. Since \(f\) is continuous on \([-pi, pi]\) and periodic with period \(2\pi\), we can employ this theorem.

Construct the Desired Trigonometric Polynomial

Based on the Weierstrass theorem, for the given \(\epsilon > 0\), there exists a trigonometric polynomial \(T(x)\) such that the maximum deviation between \(f(x)\) and \(T(x)\) over \([-pi, pi]\) is less than \(\epsilon\). Thus, \(|f(x) - T(x)| < \epsilon\) for all \(x \in [-\pi, \pi]\).

Conclusion and Verification

Since \(T(x)\) is constructed to be less than \(\epsilon\) away from \(f(x)\) uniformly over the interval \([-pi, pi]\), we have shown that \(f(x)\) can indeed be uniformly approximated by trigonometric polynomials. The condition \(f(-\pi) = f(\pi)\) ensures periodicity, which is crucial for the applicability of the theorem.

Key concepts

Trigonometric Polynomials

Trigonometric polynomials are special mathematical functions expressed as sums of sine and cosine terms. These functions take the form \[ T_n(x) = a_0 + \sum_{k=1}^{n} (a_k \cos(kx) + b_k \sin(kx)) \]where \(a_0, a_k,\) and \(b_k\) are coefficients defining the polynomial, and \(n\) is the degree of the polynomial.
These polynomials are useful in approximating functions due to their flexibility and periodic nature. They're excellent tools when dealing with problems that have periodic conditions, like those with intervals that repeat or cycle.
Since trigonometric polynomials can precisely capture periodic behavior, they help us approximate more complex functions, often simplifying analysis and computation. They are widely used in fields such as signal processing, Fourier analysis, and solving differential equations.
In essence, they let us transform complex periodic signals into manageable components.

Weierstrass Approximation Theorem

The Weierstrass approximation theorem is a fundamental concept in mathematical analysis. It states that every continuous function defined on a closed interval can be uniformly approximated by polynomial functions to any desired precision.
This theorem is incredibly powerful, providing a foundation for proving that trigonometric polynomials can approximate periodic functions efficiently. In particular, when dealing with functions that are continuous and periodic on an interval like \([-\pi, \pi]\), it ensures that there exists a trigonometric polynomial that can approximate these functions as closely as needed.
What makes this theorem so impactful is its application to real-world problems. It allows mathematicians and scientists to approximate complex continuous functions with simple, finite terms. This simplification is crucial in computing and numerical analysis, especially when exact solutions are hard to derive.
Through the use of the Weierstrass theorem, we can confidently construct trigonometric polynomials that closely mimic continuous periodic functions within any set tolerance.

Continuous Functions

Continuous functions are a type of function with no breaks, jumps, or discontinuities over their domain. In practical terms, if you can draw the graph of a function without lifting your pencil, that function is continuous.
These functions are crucial in mathematics as they allow the application of powerful theorems and results, such as the Weierstrass approximation theorem. For instance, when a function \(f\) is continuous on a closed interval, it exemplifies smooth behavior and predictability, making it ideal for approximation.
Continuous functions ensure that small changes in the input yield small changes in the output, a property that is vital for both theoretical proofs and practical applications in engineering, physics, and economics. It's also this nature of continuous functions that makes approximation through methods like trigonometric polynomials feasible and effective.
Whether it's calculating motion in physics or solving complex equations in numerical analysis, continuous functions provide reliable solutions that can be approximated and computed efficiently.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. Essentially, if \(f(x) = f(x + P)\) for some period \(P\) and for all \(x\), then \(f(x)\) is considered periodic.
The simplest examples of periodic functions are the sine and cosine functions, which have a period of \(2\pi\). These functions are foundational in trigonometry and play a significant role in modeling cyclical phenomena like sound waves, tides, and mechanical vibrations.
In the context of the Weierstrass approximation theorem and trigonometric polynomials, periodicity is vital. For instance, if a function \(f\) is continuous and periodic, trigonometric polynomials can be employed to approximate \(f\) over its period with high precision.
This property simplifies many problems, allowing us to leverage the repeating pattern of periodic functions to find solutions or perform simulations over time. It is crucial in various applications, including electrical engineering and computer graphics, where repeated patterns are essential.