Periodic Functions & Orthogonal Functions: Question 8-3.1 & Solution

Question 1

A positive number $T$ is a period of $f$ if $f(x+T)=f(x)$ for every real $x$ . A fundamental period is the smallest positive period, when one exists. Consider $f(x)=\sin 6x+\cos 10x.$ You may use linear independence of sines and cosines at distinct positive integer frequencies; justify that fact here by integrating products on $[0,2\pi]$ .

Tasks

Find a common period of the two summands. Explain why a common period alone does not prove that the sum has that fundamental period.
Expand $f(x+T)-f(x)$ and use independence to characterize every positive period of $f$ . Determine the fundamental period exactly.
Find the fundamental period of $g(x)=3f(2x-1)+7$ . Explain the effects of translation, nonzero amplitude scaling and a constant offset.
Decide whether a nonzero constant function and the zero function have fundamental periods. Explain why the definition cannot simply assign either function period zero as a fundamental period.

Original worksheet page 1: question and worked solution for 8-3-001

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Question 1 – Solution

Strategy. Prove that a period of the sum must preserve each of its distinct frequency components.

Step 1: Find a candidate period. The summands have periods $\pi/3$ and $\pi/5$ , and $\pi$ is a common period. Thus $\pi$ is a period of $f$ . In general, cancellation can give a sum a shorter period or even make it constant, so minimality still needs proof.

Step 2: Prove necessity and minimality. Product-to-sum identities show that distinct positive-integer sine/cosine frequencies have zero product integral on $[0,2\pi]$ , while each squared integral equals $\pi$ . Multiplying a zero linear combination by each component and integrating therefore forces every coefficient to vanish. Expanding the shift of $f$ now gives $\cos 6T=1,\quad\sin 6T=0,\quad\cos 10T=1,\quad\sin 10T=0.$ Equivalently, $6T=2\pi m$ and $10T=2\pi n$ for integers $m,n$ . Then $5m=3n$ , so $m=3j,n=5j$ and $T=j\pi$ . Conversely every such positive shift works. Hence $\boxed{\text{Positive periods: }j\pi\ (j=1,2,\ldots),\qquad T_0=\pi.}$

Step 3: Transform the independent variable. Since $2x-1$ ranges over all real numbers, a shift $T>0$ preserves $g$ exactly when $2T$ preserves $f$ . Thus $\boxed{T_g=\pi/2}$ . Translation, multiplication by a nonzero constant and addition of a constant do not change the set of periods; the argument dilation rescales it.

Step 4: Treat constant functions separately. Every positive number is a period of a constant function, including the zero function. There is no smallest positive number, so neither has a fundamental period. Zero is excluded by the definition and cannot be used to repair this lack of a least positive period.

Original worksheet page 2: question and worked solution for 8-3-001

Periodic Functions & Orthogonal Functions — Question 1

Question 1

Question 1 – Solution