Cell Integrals of the Trigonometric Monomials

We work in the setting of The Real Numbers: Standing Notation and Background. Let $\cos$ and $\sin$ be the cosine and sine functions from $\mathbb{R}$ to $\mathbb{R}$, let $\pi$ be the real number of The Number Pi (i), let $\mathbb{Z}$ be the set of integers, and let $2=1+1$, which is positive and hence invertible by claim 8 of Elementary Order Arithmetic in an Ordered Field. Let $J=\{t\in\mathbb{R}:0\le t<1\}$ and let $(J,\mathcal{B}_{J},\lambda_{J})$ be the measure space introduced in The Integral over the Unit Cell of a Product of One-Variable Functions; integrals with respect to $\lambda_{J}$ are those of Measurable Functions and the Lebesgue Integral: Standing Notation (v), formed for that measure space, and for a map $w:\mathbb{R}\to\mathbb{R}$ we write $\int_{J}w\,d\lambda_{J}$ for the integral of the restriction $w|_{J}$. Differentiability at a point, with a given derivative, is that notion on the interval $\mathbb{R}$, every point of which is an interior point of it by Basic Facts about Intervals of the Real Line (i); continuity means continuity relative to $\mathbb{R}$ formed with the metric $d_{\mathbb{R}}$ of The Real Numbers: Standing Notation and Background (i). For maps $v,w:\mathbb{R}\to\mathbb{R}$ let $vw$ denote the map $t\mapsto v(t)w(t)$, and let $|t|$ denote the absolute value of a real number $t$.

For $m\in\mathbb{Z}$ define $C_{m},S_{m}:\mathbb{R}\to\mathbb{R}$ by

$$C_{m}(t)=\cos(2\pi mt),\qquad S_{m}(t)=\sin(2\pi mt)\qquad(t\in\mathbb{R}).$$

Then the following hold.

1. (Derivatives, continuity and bounds)(i) Let $m\in\mathbb{Z}$. Then $C_{m}$ and $S_{m}$ are differentiable at every $t\in\mathbb{R}$, with

$$C_{m}'(t)=-2\pi m\,S_{m}(t),\qquad S_{m}'(t)=2\pi m\,C_{m}(t),$$

they are continuous on $\mathbb{R}$, and $|C_{m}(t)|\le1$ and $|S_{m}(t)|\le1$ for every $t\in\mathbb{R}$. Moreover $C_{0}(t)=1$ and $S_{0}(t)=0$ for every $t\in\mathbb{R}$. For all $k,m\in\mathbb{Z}$ the products $C_{k}C_{m}$, $S_{k}S_{m}$ and $S_{k}C_{m}$ are continuous on $\mathbb{R}$; consequently, by Zero Extension of a Real-Valued Function, and the Unit-Cell Integral of a Continuous Function (ii), the restrictions to $J$ of $C_{m}$, $S_{m}$, $C_{k}C_{m}$, $S_{k}S_{m}$ and $S_{k}C_{m}$ are all $\lambda_{J}$-integrable, so that every integral below is defined.

2. (Single integrals)(ii) Let $m\in\mathbb{Z}$. Then $\int_{J}C_{m}\,d\lambda_{J}$ equals $1$ if $m=0$ and equals $0$ if $m\ne0$, and $\int_{J}S_{m}\,d\lambda_{J}=0$.

3. (Two cosines)(iii) Let $k,m\in\mathbb{Z}$. Then $\int_{J}C_{k}C_{m}\,d\lambda_{J}$ equals $1$ if $k=0$ and $m=0$; equals $2^{-1}$ if $k\ne0$ and $|k|=|m|$; and equals $0$ if $|k|\ne|m|$.

4. (Two sines)(iv) Let $k,m\in\mathbb{Z}$. Then $\int_{J}S_{k}S_{m}\,d\lambda_{J}$ equals $2^{-1}$ if $k\ne0$ and $k=m$; equals $-2^{-1}$ if $k\ne0$ and $k=-m$; and equals $0$ in every remaining case, that is, when $k=0$ or $|k|\ne|m|$.

5. (A sine and a cosine)(v) For all $k,m\in\mathbb{Z}$ one has $\int_{J}S_{k}C_{m}\,d\lambda_{J}=0$.