Simple Proof Of A Real Function With Two Incommensurate Periods Is Constant

Mar 2, 2025 by ADMIN 76 views

Introduction

In the realm of real analysis, trigonometry, and Taylor expansion, understanding the properties of periodic functions is crucial. One of the fundamental concepts in this area is the proof of periodicity of functions like $\sin(x)$ and $\cos(x)$ . However, a more intriguing question arises when we consider functions with two incommensurate periods. In this article, we will delve into the proof that a real function with two incommensurate periods is constant.

What are Incommensurate Periods?

Before we dive into the proof, let's first understand what incommensurate periods mean. In mathematics, the period of a function is the distance between two consecutive points on the graph where the function repeats itself. Two periods are said to be incommensurate if they are not multiples of each other, i.e., there is no rational number that can express the ratio of the two periods.

The Problem Statement

We are given a real function $f(x)$ with two incommensurate periods, say $T_1$ and $T_2$ . Our goal is to prove that $f(x)$ is constant, i.e., $f(x) = c$ for some real number $c$ .

The Proof

To prove this statement, we will use a combination of mathematical induction and the properties of periodic functions.

Step 1: Establishing the Base Case

Let's assume that $f(x)$ is a function with two incommensurate periods $T_1$ and $T_2$ . We want to show that $f(x)$ is constant. To do this, we will first establish the base case.

Consider the function $f(x)$ evaluated at $x = 0$ . Let's denote this value as $f(0)$ . Since $f(x)$ is periodic with period $T_1$ , we know that $f(T_1) = f(0)$ . Similarly, since $f(x)$ is periodic with period $T_2$ , we know that $f(T_2) = f(0)$ .

Now, let's consider the value of $f(x)$ at $x = T_1 + T_2$ . Using the periodicity of $f(x)$ with periods $T_1$ and $T_2$ , we can write:

f(T_1 + T_2) = f(T_1) = f(0)

f(T_1 + T_2) = f(T_2) = f(0)

This shows that $f(x)$ has a value of $f(0)$ at $x = T_1 + T_2$ . Since $T_1$ and $T_2$ are incommensurate, we can conclude that $f(x)$ has a value of $f(0)$ at all points in the interval $[T_1, T_1 + T_2]$ .

Step 2: Extending the Result to All Intervals

Now that we have established the base case, we can extend the result to all intervals. Let's consider an arbitrary interval $[a, b]$ . We want to show that $f(x)$ has a constant value on this interval.

Since $T_1$ and $T_2$ are incommensurate, we can find integers $m$ and $n$ such that $mT_1 + nT_2 = b - a$ . This is known as the Frobenius coin problem.

Using the periodicity of $f(x)$ with periods $T_1$ and $T_2$ , we can write:

f(b) = f(mT_1 + nT_2) = f(mT_1) = f(0)

f(a) = f(mT_1 + nT_2) = f(nT_2) = f(0)

This shows that $f(x)$ has a value of $f(0)$ at all points in the interval $[a, b]$ .

Step 3: Conclusion

We have now established that $f(x)$ has a constant value on all intervals. This implies that $f(x)$ is a constant function, i.e., $f(x) = c$ for some real number $c$ .

Conclusion

In this article, we have proved that a real function with two incommensurate periods is constant. This result has far-reaching implications in various areas of mathematics, including real analysis, trigonometry, and Taylor expansion. We hope that this proof will inspire further research in this area and provide a deeper understanding of the properties of periodic functions.

References

[1] Apostol, T. M. (1974). Mathematical Analysis. Addison-Wesley.
[2] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
[3] Taylor, A. E. (1965). Introduction to Functional Analysis. Wiley.

Glossary

Periodic function: A function that repeats itself at regular intervals.
Incommensurate periods: Two periods that are not multiples of each other.
Frobenius coin problem: A problem in number theory that deals with the maximum number of coins that cannot be made by combining two types of coins with different denominations.
Q&A: A Real Function with Two Incommensurate Periods is Constant ===========================================================

Introduction

In our previous article, we proved that a real function with two incommensurate periods is constant. However, we understand that this proof may have left some readers with questions. In this article, we will address some of the most frequently asked questions about this proof and provide additional insights into the properties of periodic functions.

Q: What are incommensurate periods?

A: Incommensurate periods refer to two periods that are not multiples of each other. In other words, there is no rational number that can express the ratio of the two periods.

Q: How do you prove that a function has incommensurate periods?

A: To prove that a function has incommensurate periods, you need to show that the ratio of the two periods is irrational. One way to do this is to use the properties of the function's Fourier series.

Q: What is the significance of the Frobenius coin problem in this proof?

A: The Frobenius coin problem is a problem in number theory that deals with the maximum number of coins that cannot be made by combining two types of coins with different denominations. In the context of this proof, the Frobenius coin problem is used to show that the function has a constant value on all intervals.

Q: Can you provide an example of a function with incommensurate periods?

A: Yes, one example of a function with incommensurate periods is the function $f(x) = \sin(x) + \cos(x)$ . The periods of this function are $2\pi$ and $2\pi\sqrt{2}$ , which are incommensurate.

Q: What are the implications of this proof for other areas of mathematics?

A: This proof has far-reaching implications for other areas of mathematics, including real analysis, trigonometry, and Taylor expansion. It provides a deeper understanding of the properties of periodic functions and has applications in fields such as signal processing and image analysis.

Q: Can you provide a visual representation of the function with incommensurate periods?

A: Yes, one way to visualize the function with incommensurate periods is to plot the function's graph. The graph will show the function's periodic behavior, with the function repeating itself at regular intervals.

Q: How does this proof relate to other proofs of periodicity?

A: This proof is related to other proofs of periodicity, such as the proof of the periodicity of the sine and cosine functions using their series definitions. However, this proof provides a more general result that applies to all functions with incommensurate periods.

Q: Can you provide a summary of the proof?

A: Here is a summary of the proof:

Establish the base case by showing that the function has a constant value on the interval $[0, T_1 + T_2]$ .
Extend the result to all intervals by using the properties of the function's Fourier series and the Frobenius coin problem.
Conclude that the function is constant by showing that it has a constant value on all intervals.

Conclusion

In this article, we have addressed some of the most frequently asked questions about the proof that a real function with two incommensurate periods is constant. We hope that this Q&A article has provided additional insights into the properties of periodic functions and has helped to clarify any confusion about the proof.

References

[1] Apostol, T. M. (1974). Mathematical Analysis. Addison-Wesley.
[2] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
[3] Taylor, A. E. (1965). Introduction to Functional Analysis. Wiley.

Glossary

Periodic function: A function that repeats itself at regular intervals.
Incommensurate periods: Two periods that are not multiples of each other.
Frobenius coin problem: A problem in number theory that deals with the maximum number of coins that cannot be made by combining two types of coins with different denominations.