Trigonometric polynomials and the heat operators on a torus

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

This course proves a theorem of T. Austin [Austin]: a function of many variables that is Lipschitz for the sup distance is close to a function of a bounded number of the variables, and the bound does not depend on the number of the variables. The proof uses the heat operators on a torus and one inequality for them, hypercontractivity, which the course proves from a logarithmic Sobolev inequality on the circle.

This first lesson collects the tools: trigonometric polynomials on a torus, their approximation of continuous functions, the heat operators, and three inequalities for the heat operators that are proved with Fourier coefficients.

The lesson needs the Lebesgue integral with Fubini's theorem and Jensen's inequality (core course Measure and Integration) and calculus. The plan of the proof and the two lemmas of Section 4 follow [Austin, Section 2].

1. Functions on a torus

Let \(\mathbb T=\mathbb R/\mathbb Z\) and \(N\ge1\). A function on \(\mathbb T^N\) is the same as a function on \(\mathbb R^N\) that has the period \(1\) in every coordinate. Its integral \(\int f\) is the integral over the cube \([0,1)^N\); it does not change if \(f\) is translated: \(\int f(x+y)\,dx=\int f(x)\,dx\). For \(1\le p<\infty\) put \[ \begin{gathered} \|f\|_p=\Bigl(\int|f|^p\Bigr)^{1/p},\\ \langle f,h\rangle=\int f\,\overline h . \end{gathered} \] Since the total mass is \(1\), \(\|f\|_p\le\|f\|_q\) for \(p\le q\). The sup distance of two points of \(\mathbb T^N\) is \(d_\infty(x,y)=\max_n\|x_n-y_n\|\), where \(\|u\|\) is the distance of a real number \(u\) to the nearest integer.

Characters. For \(k\in\mathbb Z^N\) let \[ e_k(x)=e^{2\pi i(k_1x_1+\dots+k_Nx_N)} . \] Then \(e_ke_l=e_{k+l}\), \(\overline{e_k}=e_{-k}\), and \(\int e_k\) is \(1\) for \(k=0\) and \(0\) otherwise. So \(\langle e_k,e_l\rangle\) is \(1\) for \(k=l\) and \(0\) otherwise.

A trigonometric polynomial is a finite sum \(f=\sum_kc_ke_k\) with complex numbers \(c_k\). The coefficients are determined by \(f\): \(c_k=\langle f,e_k\rangle\). The function \(f\) is real if and only if \(c_{-k}=\overline{c_k}\) for all \(k\). Moreover \[ \begin{gathered} \|f\|_2^2=\sum_k|c_k|^2,\\ \partial_nf=\sum_k2\pi i\,k_n\,c_k\,e_k , \end{gathered} \] where \(\partial_n\) is the partial derivative in the coordinate \(x_n\). The derivatives of a real trigonometric polynomial are real trigonometric polynomials.

Means over coordinates. For a bounded Borel function \(f\) and a coordinate \(n\) let \(E_nf\) be the mean of \(f\) over the coordinate \(x_n\): with \(x^{w}\) for the point \(x\) with \(w\) in place of \(x_n\), \[ (E_nf)(x)=\int_0^1f(x^{w})\,dw . \] For a set \(S\subset\{1,\dots,N\}\) let \(E_S\) be the product of the operators \(E_n\) with \(n\notin S\), in any order. The function \(E_Sf\) depends only on the coordinates in \(S\); a function with this property is a junta on \(S\). By Jensen's inequality and Fubini's theorem, \(\|E_nf\|_p\le\|f\|_p\), and so \(\|E_Sf\|_p\le\|f\|_p\).

For a character, \(E_ne_k=e_k\) if \(k_n=0\), and \(E_ne_k=0\) otherwise. So for a trigonometric polynomial \[ E_Sf=\sum_{k:\ k_n=0\text{ for all }n\notin S}c_ke_k . \]

Lemma 1.1 (Poincaré inequality). For a trigonometric polynomial \(f\) on \(\mathbb T^N\) and a set \(S\) of coordinates, \[ \|f-E_Sf\|_2^2\le\frac1{4\pi^2}\sum_{n\notin S}\|\partial_nf\|_2^2 . \]

Proof. \(f-E_Sf\) is the sum of the terms \(c_ke_k\) over the set \(Z\) of the \(k\) with \(k_n\ne0\) for some \(n\notin S\). For \(k\in Z\), \(\sum_{n\notin S}k_n^2\ge1\). So \[ \begin{aligned} \|f-E_Sf\|_2^2&=\sum_{k\in Z}|c_k|^2\\ &\le\sum_k\sum_{n\notin S}k_n^2|c_k|^2 , \end{aligned} \] and \(\|\partial_nf\|_2^2=\sum_k4\pi^2k_n^2|c_k|^2\). \(\blacksquare\)

2. Approximation by trigonometric polynomials

For \(m\ge1\) and \(u\in\mathbb T\) the Fejér kernel is \[ F_m(u)=\sum_{|k|<m}\Bigl(1-\frac{|k|}m\Bigr)e_k(u) . \]

Lemma 2.1. (1) \(F_m(u)=\tfrac1m\bigl|\sum_{j=0}^{m-1}e_j(u)\bigr|^2\ge0\). (2) \(\int F_m=1\). (3) If \(0<\delta\le\tfrac12\) and \(\|u\|\ge\delta\), then \(F_m(u)\le\dfrac1{m\sin^2(\pi\delta)}\).

Proof. (1) \(\bigl|\sum_{j<m}e_j\bigr|^2=\sum_{j,l<m}e_{j-l}\), and \(k=j-l\) occurs \(m-|k|\) times. (2) The coefficient of \(e_0\) is \(1\). (3) For \(u\notin\mathbb Z\), the sum of the geometric progression is \[ \Bigl|\sum_{j=0}^{m-1}e^{2\pi iju}\Bigr|=\frac{|e^{2\pi imu}-1|}{|e^{2\pi iu}-1|}\le\frac{2}{2|\sin\pi u|} , \] and \(|\sin\pi u|\ge\sin\pi\delta\) for \(\|u\|\ge\delta\). \(\blacksquare\)

On \(\mathbb T^N\) let \(K_m(y)=F_m(y_1)\cdots F_m(y_N)\). For a bounded Borel function \(f\) on \(\mathbb T^N\), its \(m\)-th Fejér mean is \[ f_m(x)=\int f(x-y)\,K_m(y)\,dy . \]

Lemma 2.2. Let \(f\) be a bounded Borel function on \(\mathbb T^N\).

  1. \(f_m\) is a trigonometric polynomial; it is real if \(f\) is real.
  2. If \(a\le f\le b\) for real numbers \(a,b\), then \(a\le f_m\le b\).
  3. If \(f\) is \(L\)-Lipschitz for the sup distance, so is \(f_m\).
  4. \((E_Sf)_m=E_S(f_m)\) for every set \(S\) of coordinates.
  5. If \(f\) is continuous, then \(f_m\to f\) uniformly as \(m\to\infty\).

Proof. \(K_m\ge0\) and \(\int K_m=1\) by Lemma 2.1. This gives (2). It also gives (3): \(|f(x-y)-f(x'-y)|\le L\,d_\infty(x,x')\) for every \(y\), and \(f_m(x)-f_m(x')\) is the mean of these differences with the weight \(K_m\). (1) \(K_m=\sum_k\kappa_ke_k\) with \(\kappa_k=\prod_n(1-|k_n|/m)\) for the \(k\) with all \(|k_n|<m\). The substitution \(z=x-y\) gives \(\int f(x-y)e_k(y)\,dy=\langle f,e_k\rangle\,e_k(x)\). So \[ f_m=\sum_k\kappa_k\,\langle f,e_k\rangle\,e_k . \] If \(f\) is real, \(f_m\) is real because \(K_m\) is real. (4) By Fubini's theorem, \(\langle E_nf,e_k\rangle\) is \(\langle f,e_k\rangle\) if \(k_n=0\) and \(0\) otherwise. With the formula in (1) for \(E_Sf\) and the formula for \(E_S\) on trigonometric polynomials, both sides of (4) are \(\sum\kappa_k\langle f,e_k\rangle e_k\) over the \(k\) with \(k_n=0\) for all \(n\notin S\).

(5) Let \(\epsilon>0\). Since \(f\) is uniformly continuous, there is \(\delta\in(0,\tfrac12]\) with \(|f(x-y)-f(x)|\le\epsilon\) whenever \(d_\infty(y,0)\le\delta\). The set of the \(y\) with \(d_\infty(y,0)>\delta\) is contained in the union of the \(N\) sets \(\{\|y_n\|>\delta\}\), and by Lemma 2.1 the integral of \(K_m\) over each of them is at most \(1/(m\sin^2\pi\delta)\). Hence \[ \begin{aligned} |f_m(x)&-f(x)|\\ &\le\int|f(x-y)-f(x)|\,K_m(y)\,dy\\ &\le\epsilon+\frac{2N\sup|f|}{m\sin^2(\pi\delta)} , \end{aligned} \] which is at most \(2\epsilon\) for large \(m\). \(\blacksquare\)

3. The heat operators

Let \(g(z)=(2\pi)^{-1/2}e^{-z^2/2}\) be the density of the standard Gaussian law on \(\mathbb R\). It is even, and \[ \begin{gathered} \int g=1,\qquad \int z\,g(z)\,dz=0,\\ \int z^2g(z)\,dz=1 . \end{gathered} \] Write \(g_N(z)=g(z_1)\cdots g(z_N)\) for \(z\in\mathbb R^N\).

Definition. For \(t\ge0\) and a bounded Borel function \(f\) on \(\mathbb T^N\), put \[ (P_tf)(x)=\int_{\mathbb R^N}f\bigl(x+\sqrt{2t}\,z\bigr)\,g_N(z)\,dz . \] So \(P_tf(x)\) is the mean of \(f\) at a point that is moved away from \(x\) by a Gaussian vector, each coordinate with the variance \(2t\). \(P_0f=f\). The operators \(P_t\) are the heat operators.

Lemma 3.1. Let \(f,h\) be bounded real Borel functions on \(\mathbb T^N\) and \(t\ge0\).

  1. \(P_t\) is linear; \(P_tf\ge0\) if \(f\ge0\); \(P_t1=1\); \(|P_tf|\le P_t|f|\); and \(P_tf\) is continuous if \(f\) is.
  2. \(\int P_tf=\int f\), \(\ \|P_tf\|_p\le\|f\|_p\) for \(1\le p<\infty\), and \(\sup|P_tf|\le\sup|f|\).
  3. \(\langle P_tf,h\rangle=\langle f,P_th\rangle\).
  4. Let \(P_t'\) be the heat operator in the first coordinate alone and \(P_t''\) the heat operator in the other \(N-1\) coordinates, each applied with the remaining coordinates fixed. Then \(P_t=P_t'P_t''=P_t''P_t'\). Also \(P_tE_S=E_SP_t\) for every set \(S\) of coordinates.

Proof. (1) is clear from the definition; continuity follows from the theorem on dominated convergence. (2) By Fubini's theorem and the invariance of the integral under translations, \[ \begin{aligned} \int P_tf&=\int g_N(z)\int f(x+\sqrt{2t}\,z)\,dx\,dz\\ &=\int f . \end{aligned} \] By Jensen's inequality for the probability density \(g_N\), \(|P_tf|^p\le P_t(|f|^p)\); integrate. (3) By Fubini's theorem and the substitution \(x'=x+\sqrt{2t}\,z\), the left side is \(\int g_N(z)\int f(x')\,h(x'-\sqrt{2t}\,z)\,dx'\,dz\), and \(g_N(-z)=g_N(z)\). (4) is Fubini's theorem. \(\blacksquare\)

Lemma 3.2. \(P_te_k=e^{-4\pi^2|k|^2t}\,e_k\) for every \(k\in\mathbb Z^N\), where \(|k|^2=k_1^2+\dots+k_N^2\).

Proof. The definition (applied to the real and the imaginary part) gives \(P_te_k(x)=e_k(x)\prod_n\varphi(2\pi k_n\sqrt{2t})\) with \[ \begin{aligned} \varphi(\xi)&=\int e^{i\xi z}g(z)\,dz\\ &=\int\cos(\xi z)\,g(z)\,dz ; \end{aligned} \] the part with the sine vanishes because \(g\) is even. Differentiate under the integral, which is allowed since \(|z|g(z)\) is integrable, use \(g'(z)=-z\,g(z)\), and integrate by parts: \[ \begin{aligned} \varphi'(\xi)&=-\int z\sin(\xi z)\,g(z)\,dz\\ &=\int\sin(\xi z)\,g'(z)\,dz=-\xi\,\varphi(\xi). \end{aligned} \] With \(\varphi(0)=1\) this gives \(\varphi(\xi)=e^{-\xi^2/2}\), and \(\tfrac12(2\pi k_n)^2\cdot2t=4\pi^2k_n^2t\). \(\blacksquare\)

Corollary 3.3. Let \(f=\sum_kc_ke_k\) be a trigonometric polynomial.

  1. \(P_tf=\sum_ke^{-4\pi^2|k|^2t}c_ke_k\). It is a trigonometric polynomial, real if \(f\) is real, and \(P_sP_tf=P_{s+t}f\).
  2. \(\partial_nP_tf=P_t\partial_nf\) for every coordinate \(n\).
  3. The function \(u(t,x)=P_tf(x)\) is infinitely differentiable in \((t,x)\), and it solves the heat equation \[ \partial_tu=\partial_1^2u+\dots+\partial_N^2u . \]
  4. \(P_tf\to f\) uniformly as \(t\to0\).

Proof. (1) is Lemma 3.2 and linearity. The rest is read from the finite sum in (1): \(\partial_n\) multiplies the term of \(e_k\) by \(2\pi ik_n\), and \(\partial_t\) by \(-4\pi^2|k|^2\). \(\blacksquare\)

4. Two inequalities

Lemma 4.1 (\(P_t\) moves a function slowly). For a real trigonometric polynomial \(f\) on \(\mathbb T^N\) and \(t\ge0\), \[ \|f-P_tf\|_1\le\sqrt{2t}\,\sum_{n=1}^N\|\partial_nf\|_1 . \]

Proof. Let \(c=\sqrt{2t}\), and for \(z\in\mathbb R^N\) let \(d_z=\sum_nz_n\,\partial_nf\), a real trigonometric polynomial. For \(x\) and \(z\), \[ f(x+cz)-f(x)=c\int_0^1d_z(x+scz)\,ds . \] The function \(P_tf-f\) is the mean of the left side over \(z\). For fixed \(z\) and \(s\), the integral of \(|d_z(x+scz)|\) over \(x\) is \(\|d_z\|_1\), because the integral does not change under translations. So \[ \begin{aligned} &\|f-P_tf\|_1\le c\int g_N(z)\,\|d_z\|_1\,dz\\ &\quad=c\int\Bigl(\int|d_z(x)|\,g_N(z)\,dz\Bigr)dx . \end{aligned} \] Fix \(x\) and write \(a_n=\partial_nf(x)\), so that \(d_z(x)=\sum_nz_na_n\). By the Cauchy–Schwarz inequality, and because \(\int z_nz_{n'}g_N(z)\,dz\) is \(1\) for \(n=n'\) and \(0\) otherwise, \[ \begin{aligned} &\int|d_z(x)|\,g_N(z)\,dz\\ &\quad\le\Bigl(\int d_z(x)^2\,g_N(z)\,dz\Bigr)^{1/2}\\ &\quad=\Bigl(\sum_na_n^2\Bigr)^{1/2}\le\sum_n|a_n| . \end{aligned} \] Integrate over \(x\). \(\blacksquare\)

Lemma 4.2 (\(P_t\) makes a function smooth). For a trigonometric polynomial \(f\) on \(\mathbb T^N\) and \(t>0\), \[ \sum_{n=1}^N\|\partial_nP_tf\|_2^2\le\frac{\|f\|_2^2}{2e\,t} . \]

Proof. By Corollary 3.3, the coefficient of \(e_k\) in \(\partial_nP_tf\) is \(2\pi ik_ne^{-4\pi^2|k|^2t}c_k\). So, with \(v_k=4\pi^2|k|^2\), the left side is \[ \sum_kv_k\,e^{-2tv_k}\,|c_k|^2 . \] The function \(v\mapsto ve^{-2tv}\) on \([0,\infty)\) has its largest value at \(v=\tfrac1{2t}\), where it is \(\tfrac1{2et}\). \(\blacksquare\)

The two lemmas work against each other in the proof of the main theorem: for a small \(t\), the function \(P_tf\) is still close to \(f\), and its derivatives are already controlled in the norm \(\|\cdot\|_2\).

5. Exercises

5.1. Let \(N=1\). Write \(F_2\) and \(F_3\) as real trigonometric polynomials, and compute the Fejér mean \(f_2\) of \(f(u)=\cos(2\pi u)+\cos(4\pi u)\).

5.2. Compute \(P_tf\) for \(f(x)=\cos(2\pi x_1)\cos(2\pi x_2)+\sin(6\pi x_1)\) on \(\mathbb T^2\), and \(E_{\{1\}}P_tf\).

5.3. Let \(f\) be a trigonometric polynomial. Show that \(\|P_tf-\int f\|_2\le e^{-4\pi^2t}\|f-\int f\|_2\).

5.4. Let \(f\) be a real trigonometric polynomial with \(f\ge\delta\) for a number \(\delta\). Show that \(P_tf\ge\delta\) for all \(t\ge0\).

5.5. Let \(f\) be a real trigonometric polynomial on \(\mathbb T^N\) that is \(L\)-Lipschitz for the sup distance. Show that \(\sum_n|\partial_nf(x)|\le L\) for every \(x\). (This is used in the last lesson.)

6. Solutions

5.1. \(F_2=1+\tfrac12(e_1+e_{-1})\), that is \(F_2(u)=1+\cos(2\pi u)\), and \(F_3(u)=1+\tfrac43\cos(2\pi u)+\tfrac23\cos(4\pi u)\). By Lemma 2.2(1), \(f_2\) multiplies the coefficient of \(e_k\) by \(1-|k|/2\) for \(|k|<2\) and removes the others: \(f_2(u)=\tfrac12\cos(2\pi u)\).

5.2. \(\cos(2\pi x_1)\cos(2\pi x_2)\) is a sum of the four characters with \(k=(\pm1,\pm1)\), \(|k|^2=2\), and \(\sin(6\pi x_1)\) of the two with \(k=(\pm3,0)\), \(|k|^2=9\). So \[ \begin{aligned} P_tf={}&e^{-8\pi^2t}\cos(2\pi x_1)\cos(2\pi x_2)\\ &+e^{-36\pi^2t}\sin(6\pi x_1), \end{aligned} \] and \(E_{\{1\}}\) keeps the characters with \(k_2=0\): \(E_{\{1\}}P_tf=e^{-36\pi^2t}\sin(6\pi x_1)\).

5.3. \(P_tf-\int f=\sum_{k\ne0}e^{-4\pi^2|k|^2t}c_ke_k\), and \(|k|^2\ge1\) for \(k\ne0\). So the square of its norm is at most \(e^{-8\pi^2t}\sum_{k\ne0}|c_k|^2\).

5.4. \(f-\delta\ge0\), so \(P_t(f-\delta)\ge0\) by Lemma 3.1(1), and \(P_t\delta=\delta\).

5.5. Fix \(x\), and let \(\sigma_n\) be \(1\) if \(\partial_nf(x)\ge0\) and \(-1\) otherwise. For \(0<h<\tfrac12\) the point \(x+h\sigma\) has the sup distance \(h\) from \(x\), so \(f(x+h\sigma)-f(x)\le Lh\). Divide by \(h\) and let \(h\to0\): the left side tends to \(\sum_n\sigma_n\partial_nf(x)=\sum_n|\partial_nf(x)|\).

References