Periodic functions are functions with a repetitive behaviour. Such a function is often determined by a period #p# and a function rule on an interval of length #p#.
Let #f:\mathbb{R} \rightarrow \mathbb{R}# be a real function. Then #f# is said to be periodic if there exists a positive constant #p# such that, for all #x#,
\[ f(x+p)=f(x)\]
Such a number #p# is called a period of #f#. Instead of saying that #f# is periodic of period #p#, we also say that #f# is #p#-periodic.
If #a# is a period of a periodic function #f#, then so is every element of the form #n\cdot a # for a natural number #n #. The minimum of the set of all periods of #f#, if it exists, is called the fundamental period. For brevity, we often refer to it as the period.
The functions #\sin# and #\cos# both have period #2\pi#, while #\tan# has period #\pi#.
A non-trigonometric example is the function #f(t)=t -\left \lfloor t \right \rfloor#, which has fundamental period #1#.
It is not hard to see that, if a real function #f(x)# is #p#-periodic, then #d\cdot f(c\cdot x)#, where #c# and #d# are real numbers with #c\ne0#, is periodic of period #\frac{p}{c}#.
It is also not hard to see that the translated function #a+f(x-b)# of #x#, where #a# and #b# are real numbers, has period #p# as well.
Thus, for example, the function #3\cos(8\pi x-7)# has period #\frac{2\pi}{8\pi}=\frac{1}{4}#.
The proof of the fact that, if #p# is a period of #f#, then so is #n\cdot p# for every #n \in \mathbb{N}#, follows from induction with respect to #n# and application of the definition of periodicity #n#:
\[f(x+np)=f(x+\underbrace{p + \cdots + p}_{n\ \text{times}})=f(x)\]
A constant function is periodic. Any positive real number is a period of it. Since the infimum of the set of all positive real numbers is #0#, which is not a positive number, the constant function does not have a fundamental period.
If #f# is a real #p#-periodic function, then it also satisfies #f(x) = f(x-p)# for all real numbers #x#. This can be seen by replacing #x# by #x-p# in the definition of periodicity. Therefore, the requirement in the definition of a periodic function that the function have a positive period, it is not an essential limitation.
If #f# is periodic with integer periods #a# and #b#, then also the #\gcd# of #a# and #b# is a period. This follows from the fact that #\gcd(a,b)# can be written as #m\cdot a+ n\cdot b# for two integers #m# and #n#.
We can define a #p#-periodic function #f# by providing a function rule on an interval of the form #\ivco{a}{a+p}# for some number #a# and requiring that #f(x+ p) = f(x)# for all real numbers #x# (that is, that #f# be #p#-periodic). We then say that #f# is defined by periodic extension.
The set of periodic functions with a fixed period has more structure.
The set of periodic functions with a fixed period form a real vector space with the multiplication by a constant as scalar multiplication and the pointwise addition of functions as addition.
It is well known that the set of #F# of real functions, with the multiplication by a constant as scalar multiplication and the pointwise addition of functions, is a vector space. Let #p# be a positive number. We verify that the set of real periodic functions with period #p# is a subspace of #F#. By theorem Linear subspaces are vector spaces, this implies the theorem.
In order to establish that the #p#-periodic functions form a subspace of #F#, it suffices to establish that the set is closed under taking linear combinations and that #0# belongs to it. The latter has already been observed above. As for the former, suppose that #f# and #g# are both periodic functions with period #p# and that #a# and #b# are constants. Then #a\cdot f+b\cdot g# satisfies \[ \begin{array}{rcl}(a\cdot f+b\cdot g)(x+p)&=& a\cdot f(x+p)+b\cdot g(x+p)\\ &&\phantom{xxx}\color{blue}{\text{definition of }a\cdot f+b\cdot g}\\&=&a\cdot f(x)+b\cdot g(x)\\&&\phantom{xxx}\color{blue}{\text{periodicity of } f\text{ and } g}\\&=&(a\cdot f(x)+b\cdot g)(x)\\&&\phantom{xxx}\color{blue}{\text{definition of }a\cdot f+b\cdot g}\end{array}\] for each #x# and so it is a #p#-periodic function as well.
Let #G# be the vector space of continuous real #2\pi#-periodic functions. Then restriction of a function from #G# to the closed interval #\ivcc{-\pi}{\pi}# is a linear map #r: G\to H# of vector spaces, where #H# is the vector space of all continuous functions #h# on #\ivcc{-\pi}{\pi}# with #h(-\pi) = h(\pi)#. If #h# is a function in #H#, then there is a unique way of extending #h# to a member #g# of #G#. After all, #g# is uniquely determined by the function rule #g(x) = h(x+2m\pi)#, where #m=\left\lfloor \frac{\pi-x}{2\pi}\right\rfloor#. The choice of #m# guarantees that #x+2m\pi \in \ivoc{-\pi}{\pi}#. This means that the map #r# is an isomorphism of vector spaces.
Consider, again, the vector space #G# of real continuous #2\pi#-periodic functions, the vector space #H# of all continuous functions #h# on #\ivcc{-\pi}{\pi}# with #h(-\pi) = h(\pi)#, and the restriction map #r:G\to H#. The vector space #H# is an inner product space with respect to the function inner product
\[\dotprod{h}{k}=\dfrac{1}{\pi}\int_{-\pi}^{\pi}h(t)\cdot k(t)\,\dd t\]
Since #r# is an isomorphism of vector spaces, we can set #\dotprod{f}{g} = \dotprod{r(f)}{r(g)}# for #f# and #g# in #G# to obtain an inner product on #G#, with the same mapping rule for #f# and #g# as for #h# and #k#. By use of this inner product, we will regard the vector space #G# as an inner product space. It will be shown later that, for each natural number #n#, the set of functions
\[\basis{\frac{1}{\sqrt{2}},\cos(x),\sin(x),\cos(2x),\sin(2x),\ldots,\cos(nx),\sin(nx)}\]
is an orthonormal system of #G#. The Fourier series of a function #f# we will be looking at later, can be viewed as orthogonal projections of #f# onto the spans of these orthonormal systems. In fact, we will consider a more general class of functions than #G#, namely piecewise continuous functions.
Find the fundamental period of the function #\sin(16 x)#.
#{{\pi}\over{8}}#
The fundamental period of # \sin(x)# is #2\pi#. The function # \sin(16 x)# is this function scaled by # 16#, so the
fundamental period of # \sin(16 x)# is #\frac{2\pi}{16} ={{\pi}\over{8}} #.
Periodic functions
