The Trigonometric Orthonormal System On The Circle

The Circle and Its Normalized Measure

Trigonometric functions are naturally organized as an orthonormal system in \(L^2\) on the circle. The normalization of measure matters: it determines the norms of the constant function, the exponentials, and the sine and cosine functions. We fix that normalization first, then compute inner products directly. The goal here is to construct the systems and relate their coefficient conventions; no completeness assertion is needed.

Write \(\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})\) for the circle, represented by \([0,2\pi)\), and let \(\mathcal{B}_{\mathbb{T}}\) be its Borel sigma-algebra. Equip it with the probability measure \(\mu\) given by normalized arc length: integration over the circle means integration over \([0,2\pi)\) with \(d\theta/(2\pi)\). Thus the measure space \((\mathbb{T},\mathcal{B}_{\mathbb{T}},\mu)\) is finite, with \(\mu(\mathbb{T})=1\). We work with complex-valued functions in \(L^2(\mu)\), and also with its real-valued subspace. Functions in \(L^2(\mu)\) are identified when they agree \(\mu\)-almost everywhere.

This inner product is linear in its first argument and conjugate-linear in its second. The coefficient of \(f\) at a unit vector \(u\) is \(\langle f,u\rangle\). It is well-defined for \(f,u\in L^2(\mu)\), since the Cauchy–Schwarz inequality makes the product integrable. In the calculations below, the functions being integrated are continuous, so their integrals can also be evaluated by elementary calculus.

The Complex Exponential System

Proof. For integers \(n,m\), the definition of the inner product and the identity \(\overline{e_m(\theta)}=e^{-im\theta}\) give \(\langle e_n,e_m\rangle=\int_{\mathbb{T}}e^{i(n-m)\theta}\,d\mu(\theta)\). If \(n=m\), the integrand is \(1\), so the integral is \(\mu(\mathbb{T})=1\). If \(n\ne m\), put \(k=n-m\), a nonzero integer. Then \[ \langle e_n,e_m\rangle=\frac{1}{2\pi}\int_0^{2\pi}e^{ik\theta}\,d\theta =\frac{1}{2\pi}\left[\frac{e^{ik\theta}}{ik}\right]_0^{2\pi} =\frac{e^{2\pi ik}-1}{2\pi ik}=0, \] because \(e^{2\pi ik}=1\). Thus the inner product is \(1\) for equal indices and \(0\) for distinct indices, proving orthonormality. \(\square\)

The Real Sine-Cosine System

The corresponding real-valued system uses the constant function and normalized sine and cosine functions. Although these functions are real-valued, they can also be regarded as elements of the complex space \(L^2(\mu)\). Their inner products are then the same real integrals as in the real-valued subspace.

Proof. The constant has squared norm \(\int_{\mathbb{T}}1\,d\mu=1\). For any positive integer \(n\), the identities \(\cos^2(n\theta)=(1+\cos(2n\theta))/2\) and \(\sin^2(n\theta)=(1-\cos(2n\theta))/2\) show that \[ \int_{\mathbb{T}}\cos^2(n\theta)\,d\mu =\int_{\mathbb{T}}\sin^2(n\theta)\,d\mu=\frac12. \] Here the integral of \(\cos(2n\theta)\) is zero, by evaluating its antiderivative over \([0,2\pi]\). Multiplication by \(\sqrt{2}\) therefore gives norm one for both functions. Their product has integral zero because \(2\sin(n\theta)\cos(n\theta)=\sin(2n\theta)\), whose integral over a full period is zero. The constant is orthogonal to each sine and cosine because each has integral zero over \([0,2\pi]\).

It remains to check distinct positive frequencies. The product-to-sum identities give \[ 2\cos(n\theta)\cos(m\theta)=\cos((n-m)\theta)+\cos((n+m)\theta), \] \[ 2\sin(n\theta)\sin(m\theta)=\cos((n-m)\theta)-\cos((n+m)\theta), \] and \[ 2\sin(n\theta)\cos(m\theta)=\sin((n+m)\theta)+\sin((n-m)\theta). \] When \(n\ne m\), every frequency on the right in each relevant identity is a nonzero integer, so each term has integral zero over a full period. Thus the corresponding distinct sine and cosine functions are orthogonal. Each normalized inner product is \(\sqrt{2}\sqrt{2}=2\) times the unnormalized product integral, which is zero for distinct functions and one for each squared norm. This proves the claim. \(\square\)

Converting Between Coefficient Conventions

For \(f\in L^2(\mu)\), define its complex exponential coefficients by \(c_n=\langle f,e_n\rangle\) for \(n\in\mathbb{Z}\). Also define its coefficients against the real sine-cosine system by \[ a_0=\langle f,1\rangle,\qquad a_n=\langle f,\sqrt{2}\cos(n\theta)\rangle,\qquad b_n=\langle f,\sqrt{2}\sin(n\theta)\rangle\quad(n\ge1). \] When \(f\) is complex-valued, \(a_n\) and \(b_n\) may be complex numbers; the word “real” describes the functions in that orthonormal system, not necessarily the coefficients.

Proof. The identity \(e_n(\theta)=\cos(n\theta)+i\sin(n\theta)\) implies \(\overline{e_n(\theta)}=\cos(n\theta)-i\sin(n\theta)\). Using linearity of the integral in its integrand, \[ c_n=\int_{\mathbb{T}}f(\theta)\big(\cos(n\theta)-i\sin(n\theta)\big)\,d\mu(\theta) =\frac{a_n-i b_n}{\sqrt{2}}. \] For the negative index, \(\overline{e_{-n}(\theta)}=\cos(n\theta)+i\sin(n\theta)\), giving \(c_{-n}=(a_n+i b_n)/\sqrt{2}\). At index zero, \(e_0=1\), so \(c_0=a_0\). Adding the two formulas for \(c_n\) and \(c_{-n}\) gives \(a_n=(c_n+c_{-n})/\sqrt{2}\). Subtracting them gives \(c_{-n}-c_n=\sqrt{2}i b_n\); since \(1/i=-i\), this is equivalent to \(b_n=i(c_n-c_{-n})/\sqrt{2}\). Each formula follows from the coefficient definitions, with no assumption that \(f\) is real-valued. \(\square\)

Normalization and a Common Pitfall

The factor \(1/(2\pi)\) in the measure is what makes the constant function have norm one. It also makes \(\cos(n\theta)\) and \(\sin(n\theta)\) have squared norm \(1/2\), so the factor \(\sqrt{2}\) makes their norms one. A common mistake is to switch to unnormalized arc length midway through a calculation while retaining the normalized formulas. For example, with \(d\theta\) on \([0,2\pi)\), the constant has squared norm \(\int_0^{2\pi}1\,d\theta=2\pi\), not \(1\). With the fixed probability measure used here, its squared norm is \((1/(2\pi))\int_0^{2\pi}1\,d\theta=1\). The system and coefficient formulas must always be interpreted with the same measure.

The complex exponentials are convenient when integer frequencies and complex coefficients are natural. The real sine-cosine system makes the real-valued oscillatory components explicit. Their coefficient conversion is an algebraic consequence of \(e^{in\theta}=\cos(n\theta)+i\sin(n\theta)\), together with the chosen normalization. Orthonormality alone does not assert that either system is complete; that is a separate question.