This arose from Sandeep's problem Do you know its property - 9

Be warned that this note requires significant mathematical maturity, which is why I set it at level 5.

Note: The LCM of 2 real numbers is defined in the same way. Find the smallest positive number (if it exists) such that for some integers \(n,m \), we have \( n \alpha = m \beta \).

Claim 1: If the fundamental period of \(f\) is \( \alpha \) and the fundamental period of \( g\) is \( \beta \), then the fundamental period of \( f + g \) is the LCM of \(\alpha, \beta \)?

Is this claim true? No, not necessarily. We only know that their LCM must be a period, but it need not be the fundamental period. There could be a smaller period due to cancellation (see Krishna's comment). What is true, is the following:

Fact: If the period of \(f\) is \( \alpha \) and the period of \( g\) is \( \beta \), and \( \frac{\alpha} { \beta } \) is rational, then a period of \( f + g \) is the LCM of \(\alpha, \beta \).

That leaves us to consider the irrational case.

Claim 2: If the period of \(f\) is \( \alpha \) and the period of \( g\) is \( \beta \), and \( \frac{\alpha} { \beta } \) is irrational, then \(f+g\) is never periodic.

It is easy to find examples where \( f + g\) is not periodic, like \( \sin x + \sin \pi x \). However, must this statement be always true?

We know that there is no obvious candidate for a period, but can't there be some "weird" functions which cancel out, like in the rational case? The following 4 hints will guide us through the rest of this investigation. You should try working on these hints first, before reading further

Hint 1: Prove that a continuous periodic function is bounded.

Hint 2: Prove that if \(f\) and \(g\) are both continuous, periodic, non-constant functions with irrational ratios, then \(f+g\) is not periodic.

Hint 3: Find a function that is not constant, whose periods is dense in \( \mathbb{R} \) but the set of periods is not \( \mathbb{R} \). Note: The function is not continuous.

Hint 4: Now find \(f\) and \(g\) with irrational ratio of periods, such that \(f+g\) is periodic (and non-constant).

Note: I used the Axiom of Choice (which I believe in).

Proof of hint 1. Let \(f\) be a continuous, periodic function with period \( \alpha \). Suppose that \(f\) is not bounded. That means that for any \( N \), there exists an \( x_N \) such that \( | f ( x_N )| > N \). WLOG, we may assume that \( 0 \leq x_N \leq \alpha \), as we translate it to the first period.

Since the points \(x_N\) lie in the closed interval \( [0, \alpha ] \), there exists an accumulation point, which means that there is a certain subsequence \( x_{N_k} \) which converges to \( x^* \). Since \(f\) is a continuous function, we have \( f( x^*) = \lim_{k\rightarrow \infty} f(x_{N_k} ) \). But the RHS tends to infinity, which contradicts the assumption that \( f(x^*) \) is finite. Hence, the assumption is false, and \( f \) is bounded. \(_\square\)

Proof of hint 2. We first prove the following claim:

*Claim* If a function \(f\) is continuous, periodic, and it's periods are dense in \( \mathbb{R} \), then it is the constant function.

This follows immediately by considering \( f( \alpha ) \), where \( \alpha\) is a period of \(f\). It is dense in \( \mathbb{R} \), and their values are all equal to \( 0 \). By continuity, it is the constant function. \(_\square\)

Now, on to the proof. Suppose that \( f\) and \(g\) are continuous, periodic, non-constant functions with irrational period ratio, and \( f+ g \) is periodic. If \( f+g \) is the constant function, then we can show that \(f\) and \(g\) must have the same period, which contradicts the assumption. Hence, we may assume that \( f + g \) has a smallest period \( \gamma > 0 \).

We are given that \( f( x+ \gamma) + g( x + \gamma ) = f(x) + g(x) \). Define \( h(x) = f( x + \gamma) - f(x) = g(x) - g( x + \gamma) \), which is a continuous function (since it is the difference of 2 continuous functions).

For any pair of integers \(n, m \), observe that \[ \begin{align} & h ( x + n \alpha + m \beta) \\ = & f( x + n \alpha + m\beta + \gamma) - f( x + n \alpha + m \beta ) \\ = & f ( x + m \beta + \gamma) - f ( x + m \beta ) = h ( x + m \beta ) \\ = & g ( x + m \beta ) - g ( x + m\beta + \gamma)\\ =& g ( x ) - g ( x + \gamma ) \\ =& h ( x ) \end{align} \]

This implies that the function \( h (x) \) is periodic with period \( n \alpha + m \beta \). But since the ratio \( \frac{ \alpha } { \beta } \) is irrational, this implies that \( \mathbb{Z} \alpha + \mathbb{Z} \beta \) is dense in \( \mathbb{R} \), and thus \( h(x) \) must be a constant, which we will denote by \(h \).

If \( h \neq 0 \), then \( f( x + n \gamma) = f(x) + n h \) which is an unbounded sequence, contradicting Hint 1.

If \( h = 0 \), then \( f \) and \(g\) are both periodic with period \( gamma \). Now, at least one of \( \frac{ \alpha } { \gamma } \) or \( \frac { \beta } { \gamma} \) is irrational (WLOG, let it be the first ). Once again, this implies that \( \mathbb{Z} \alpha + \mathbb{Z} \gamma \) is dense in \( \mathbb{R} \), and thus that \(f\) is a constant function. However, this contradicts the assumption. \(_\square\)

Hence, what we can conclude at this point is

Fact: If \(f\) is a continuous periodic function, then either \(f\) is constant, or \(f\) has a fundamental period (smallest).

Fact: If \(f\) and \(g\) are continuous, periodic, non-constant functions with an irrational ratio of fundamental periods, then \( f + g \) is not periodic.

The claim is true, **if** we restrict out attention to only continuous functions.

Now, we move on to the fun and interesting part. The world is extremely well-behaved (relatively) when functions are continuous. Let's observe what happens when things go crazy.

Proof of Hint 3. Consider the function \[ f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & x \not \in \mathbb{Q} \\ \end{cases}. \]

This function is periodic with period and rational \(q \). Then set of periods is dense in \( \mathbb{R} \), but is not \( \mathbb{R} \). \(_\square\)

In particular, this shows us that we might not be able to talk about the fundamental period of a non-continuous periodic function.

Proof of Hint 4. We know that \( 1, e, \pi \) are rational-linearly independent. What this means is that if \( q_1 \times 1 + q_2 \times e + q_3 \times \pi = 0 \), where \( q_i \) are rational numbers, then \( q _ i = 0 \).

Define the rational span of \( (a,b) \) to be the set of all numbers of the form \( \mathbb{Q} a + \mathbb{Q} b \).

Let \( \chi_{1, e } \) be the characteristic function of the rational span of \( (1,e) \), and \( \chi_{e, \pi} \) and \( \chi_{1, \pi} \) similarly defined.

Then, set \( f = \chi_{ 1, e } + \chi_{e, \pi} \) and \( g = - \chi{ e, \pi} + \chi { 1 , \pi } \). Then, we have \( e\) is a period of \(f\), \( \pi \) is a period of \(g\) and \(1\) is a period of \( f + g \). \(_ \square\)

Thus, we have 2 functions with irrational ratio of periods, but sum to give a periodic function.

## Comments

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TopNewestWhat an amazing observation you've just shared!! I'm astonished!! Let me think.. – Sanjeet Raria · 1 year, 12 months ago

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– Calvin Lin Staff · 1 year, 12 months ago

I've added the details on how to proceed. It would still be worthwhile to consider and work through the hints yourself.Log in to reply

This claim is

notalways true.If both the functions are

evenand complimentary to each other(differ by phase difference of \(\displaystyle \frac{\pi}{2}\)) then the period isL.C.M of \(\frac{1}{2}\)( \(\alpha\) & \(\beta\))

Example:-

f = \(|\sin x|\)

g = \(|\cos x|\)

f + g = \(|\sin x| + |\cos x|\)

Here period of 'f + g' is \(\displaystyle \frac{\pi}{2}\) – Krishna Sharma · 1 year, 12 months ago

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– Danny Kills · 1 year, 12 months ago

'Time' Period???? :-)Log in to reply

– Krishna Sharma · 1 year, 12 months ago

Aah! Fixed itLog in to reply

– Sanjeet Raria · 1 year, 12 months ago

But Calvin sir here has brought a very interesting question. That the sum function is still periodic even when their LCM doesn't exist.Log in to reply

– Calvin Lin Staff · 1 year, 12 months ago

For rational ratios, what would be the correct claim?Log in to reply

Ex:-

f = \(\sin^{2} x\)

g = \(\cos^{2} x\)

f + g = \(\sin^{2} x\) + \(\cos^{2} x\) = 1

Function is periodic but fundamental period is not defined

Similarly

f + g = \(\sec^{2} x\) - \(\tan^{2} x\) = 1 – Krishna Sharma · 1 year, 11 months ago

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As mentioned in "Fact", all that we know is "a period of \(f+g\) is the LCM". As you brought up, we are not guaranteed that \(f+g\) must have a fundamental period. – Calvin Lin Staff · 1 year, 11 months ago

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– Sanjeet Raria · 1 year, 12 months ago

Let LCM of \(T_1\) & \(T_2\) be \(T\) then \(T\) will be the period of \((f+g)\), provided there does not exist a positive number \(K(<T)\) for which \(f(x+K)+g(x+K)=f(x)+g(x)\) else \(K\) will be the period. Generally the situation arises due to the interchanging of \(f(x)\) and \(g(x)\) by addition of the positive number \(K\). In case of \((|\sin x|+|\cos x|)\) \(|\sin x|\) & \(|\cos x|\) interchanged by adding \(\frac{π}{2}\) & the result being\(|\cos x| +|\sin x|=|\sin x|+|\cos x|\) hence the period is \(\frac{π}{2}\) instead of \(π\).Log in to reply

Verrryyy nice mr calvin – Ishak Tahir · 1 year, 5 months ago

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Find the period of Y=tanx/sin (2/3) x and y=sinx*cos3x – Lengheng Nim · 1 year, 7 months ago

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If only period is concerned one such example is \(f(x)=\sin x\) & \(g(x)=0\). But if you talk about fundamental period i can't think of any example. Do you have any?? @Calvin Lin – Sanjeet Raria · 1 year, 12 months ago

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So, the interesting case occurs when \( \frac { \alpha } { \beta } \) is irrational.

Question: Does there exist 2 functions \(f \) and \(g \), with fundamental periods \( \alpha , \beta \) and \( \frac{ \alpha } { \beta } \neq \mathbb{Q} \), such that \( f + g \) is periodic?

Hint: Prove that if \(f\) and \(g\) are bounded, then \( f+g \) is not periodic.

Hint: Prove that if \( f\) and \(g\) are continuous and periodic, then they are bounded.

Hint: Consider what happens if \(f\) and \(g\) are not continuous. – Calvin Lin Staff · 1 year, 12 months ago

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L.C.M of irrational with

similarirrational is definedBut L.C.M of rational with irrational is

notdefined(called as aperiodic or not periodic)Example :-

f + g = \(\sin x\) + {x} (fractional part of 'x')

Is aperiodic – Krishna Sharma · 1 year, 12 months ago

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It is true that in a lot of cases, \( f + g \) is aperiodic. But, can we create a scenario where \( f + g \) is periodic?? This is interesting, because I currently believe that the answer is "There is such a function!"

Do not simply repeat what you believe, or what you recall someone else saying is true. Instead, find a justification of your statements. – Calvin Lin Staff · 1 year, 12 months ago

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Is this a high level of mathematical maturity?? Cause all these things were taught to me by my mathematics teacher recently. I'm currently in class 12th. – Danny Kills · 1 year, 12 months ago

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In order to work through and appreciate the hints, you need a certain level of mathematical maturity. See the proof of Hint 1 as an example. – Calvin Lin Staff · 1 year, 12 months ago

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– Danny Kills · 1 year, 11 months ago

Yes! I still have a really long way to go... I got a little hint of what you were doing but I couldn't understand everything. I can imagine graphically that those hints are true, but would be unable to prove them mathematically.Log in to reply

I started this note thinking that it was easy / straightforward, given how a lot of people state it as if it was fact. But now you can tell your teacher that it's not true! You can even show him the counter examples!

As I explored further, I found that I had difficulty accounting for the "cancellation", and it slowly began to cross my mind that the statement might not be true. I tried to detail how I explored this topic, starting with the simplification of continuous functions which is well loved, and then moving on to discontinuous functions which are not well-understood. I tried to motivate / justify the counter-example, but this can be hard to do. – Calvin Lin Staff · 1 year, 11 months ago

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@Calvin Lin sir in claim 2 can you please explain how can the ratio of two numbers be an irrational number.........according to defination an irrational number can't be expressed as the ratio of two numbers – Aman Sharma · 1 year, 12 months ago

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Note that every number can be expressed as the ratio of two numbers, namely \( x = \frac{x}{1} \). – Calvin Lin Staff · 1 year, 12 months ago

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– Sanjeet Raria · 1 year, 12 months ago

\(\frac{π}{1}=π\)Log in to reply

– Aman Sharma · 1 year, 12 months ago

Ohhh okk thanksLog in to reply