4 Analysis prerequisites

In this chapter, we prove some results related to the analysis of polynomial and rational functions, as well as compactly supported distributions.

Throughout, the set of real polynomials will be denoted as \(\mathcal{P}\), and the set of real polynomials of degree at most \(q\) by \(\mathcal{P}_q\).

We will denote as \(\mathbb {T}= \mathbb {R}/ 2 \pi \mathbb {Z}\) the additive circle of length \(2 \pi \).

4.1 Markov inequalities

4.2 Chebyshev polynomials

For \(j \geq 0\), we denote as \(T_j\) the Chebyshev polynomial of the first kind of degree \(j\), defined by the relation \(T_j(\cos \theta ) = \cos (j \theta )\) for \(\theta \in \mathbb {R}\).

Let us fix a real number \(K {\gt} 0\) and an integer \(q \geq 0\).

Lemma 4.2.1

Let \(h \in \mathcal{P}_q\). There exists a unique family of real numbers \((a_j)_{0 \leq j \leq q}\) such that, for all \(x \in \mathbb {R}\),

\begin{equation*} h(Kx) = \sum _{j=0}^q a_j T_j (x). \end{equation*}

Proof ▶

This follows from the fact that \(h(Kx) \in \mathcal{P}_q\) and \((T_j)_{0 \leq j \leq q}\) is a basis of \(\mathcal{P}_q\), since for all \(0 \leq j \leq q\), \(\deg T_j = j\).

It will be useful to view the coefficients \((a_j)_{0 \leq j \leq q}\) as trigonometric Fourier coefficients, which we do by introducing the following function \(f\).

Lemma 4.2.2

Let \(h \in \mathcal{P}_q\) and \((a_j)_{0 \leq j \leq q}\) be the coefficients from Lemma 4.2.1. Let \(f : \mathbb {T}\rightarrow \mathbb {R}\) be the function defined by \(f(\theta ) = h(K \cos \theta )\). Then, \(f\) is a trigonometric polynomial equal to

\begin{equation*} f(\theta ) = \sum _{j=0}^q a_j \cos (j \theta ). \end{equation*}

Proof ▶

The function \(\cos : \mathbb {T}\rightarrow [-1,1]\) is a smooth function, so by composition \(f(\theta ) = h(K \cos \theta )\) also is. By definition of \(f\) and \((a_j)_{0 \leq j \leq q}\), for all \(\theta \in \mathbb {T}\),

\begin{equation*} f(\theta ) = h(K \cos \theta ) = \sum _{j=0}^q a_j T_j(\cos \theta ). \end{equation*}

By definition of \((T_j)_{j \geq 0}\), \(T_j(\cos \theta ) = \cos (j \theta )\).

Lemma 4.2.3

Let \(h \in \mathcal{P}_q\) and \((a_j)_{0 \leq j \leq q}\) be the coefficients from Lemma 4.2.1. Then,

\begin{equation*} |a_0| \leq \| h \| _{C^0[-K,K]}. \end{equation*}

Proof ▶

By the expression of the Fourier coefficients of \(f\), we have

\begin{equation*} a_0 = \frac{1}{2 \pi } \int _0^{2 \pi } f(\theta ) \d\theta . \end{equation*}

Hence by the triangle inequality

\begin{equation*} |a_0| \leq \frac{1}{2 \pi } \int _0^{2 \pi } |f(\theta )| \d\theta = \frac{1}{2 \pi } \int _0^{2 \pi } |h(K \cos \theta )| \d\theta \leq \| h \| _{C^0[-K,K]}. \end{equation*}

We will now provide some bounds on the coefficients \((a_j)_{0 \leq j \leq q}\) in terms of \(L^\beta \)-bounds on derivatives of \(f\). This will be done using the Hausdorff-Young inequality. Recall that, for a function \(f \in L^1(\mathbb {T})\), the Fourier coefficients are defined as

\begin{equation*} c_j = \frac{1}{2 \pi } \int _0^{2 \pi } f(\theta ) e^{-i j \theta } \d\theta . \end{equation*}

Theorem 4.2.4

Let \(1 {\lt} \beta \leq 2\) and \(1/\beta _* = 1 - 1/\beta \). Let \(f \in L^\beta (\mathbb {T})\) of Fourier coefficients \((c_j)_{j \in \mathbb {Z}}\). Then,

\begin{equation*} \left(\sum _{j \in \mathbb {Z}} |c_j|^{\beta _*} \right)^{1/\beta _*} \leq \left(\frac{1}{2 \pi } \int _0^{2 \pi } |f(\theta )|^\beta \d\theta \right)^{1/\beta }. \end{equation*}

Proof ▶

I think this might be in Mathlib? If not this is a classic result and should be added.

We deduce the following lemma.

Lemma 4.2.5

Let \(\beta {\gt} 1\) and \(1/\beta _* = 1-1/\beta \). Let \(f \in L^1(\mathbb {T})\) of Fourier coefficients \((c_j)_{j \in \mathbb {Z}}\). Assume \(f' \in L^\beta (\mathbb {T})\). Then,

\begin{equation*} \sum _{j \neq 0} |c_j| \leq 4 \beta _* \| f’ \| _{L^\beta (\mathbb {T})}. \end{equation*}

Proof ▶

Assume \(1 {\lt} \beta \leq 2\). By the Hölder inequality,

\begin{equation*} \sum _{j \neq 0} |c_j| = \sum _{j \neq 0} \frac{1}{j} \times |j c_j| \leq \left(\sum _{j \neq 0} \frac{1}{j^\beta } \right)^{1/\beta } \left(\sum _{j \neq 0} |j c_j|^{\beta _*} \right)^{1/\beta _*}. \end{equation*}

We shall bound these two sums successively.

Since \(\beta {\gt}1\) the first sum converges. By parity it is equal to twice the sum for \(j \geq 1\). Since \(t \mapsto 1/t^\beta \) is decreasing on \([1,\infty )\), we can compare the sum and integral which yields

\begin{equation*} \sum _{j =1}^\infty \frac{1}{j^\beta } \leq 1 + \int _1^\infty \frac{\d t}{t^\beta } = 1 - \left[ \frac{1}{(\beta -1)t^{\beta -1}} \right]_1^\infty = 1 + \frac{1}{\beta - 1} = \frac{\beta }{\beta - 1} = \frac{1}{1 - 1/\beta } = \beta _*. \end{equation*}

As a conclusion,

\begin{equation*} \left(\sum _{j \neq 0} \frac{1}{j^\beta } \right)^{1/\beta } \leq 2 \beta _*^{1/\beta } \leq 2 \beta _* \end{equation*}

since \(0 \leq 1/\beta \leq 1\) and \(\beta _* \geq 1\).
For the second sum, we observe that the Fourier coefficients of \(f'\) are \((i j c_j)_{j \in \mathbb {Z}}\). Since \(1 {\lt} \beta \leq 2\), we can apply Theorem 4.2.4 to \(f'\), which yields

\begin{equation*} \left(\sum _{j \neq 0} |j c_j|^{\beta _*} \right)^{1/\beta _*} \leq \left(\frac{1}{2 \pi } \int _0^{2 \pi } |f’(\theta )|^\beta \d\theta \right)^{1/\beta } \leq \frac{1}{(2 \pi )^{1/\beta }} \| f’\| _{L^\beta (\mathbb {T})} \leq \frac{1}{\sqrt{2 \pi }} \| f’\| _{L^\beta (\mathbb {T})} \end{equation*}

since \(\beta \leq 2\).

Putting everything together we obtain

\begin{equation*} \sum _{j \neq 0} |c_j| \leq \sqrt{\frac{2}{\pi }} \beta _* \| f’\| _{L^\beta (\mathbb {T})} \end{equation*}

which implies the claim in the case \(1 {\lt} \beta \leq 2\) since \(\sqrt{2/\pi } \leq 4\).

Now assume \(\beta \geq 2\). By the Hölder inequality, since \(\mathbb {T}\) has finite measure \(2 \pi \), \(L^\beta (\mathbb {T}) \subseteq L^2(\mathbb {T})\) and

\begin{equation*} \| f’\| _{L^2(\mathbb {T})}^2 \leq \left(\int _0^{2 \pi } 1 \d\theta \right)^{1-\frac{2}{\beta }} \left(\int _0^{2 \pi } |f’(\theta )|^{\beta } \d\theta \right)^{\frac{2}{\beta }} = (2 \pi )^{1-\frac{2}{\beta }} \| f’\| _{L^\beta (\mathbb {T})}^2 \leq 2 \pi \| f’\| _{L^\beta (\mathbb {T})}^2. \end{equation*}

In particular, we can apply the result for \(\beta =2\) (in which case \(\beta _*=2\)), which yields

\begin{equation*} \sum _{j \neq 0} |c_j| \leq 2 \sqrt{\frac{2}{\pi }} \| f’\| _{L^2(\mathbb {T})} \leq 4 \| f’\| _{L^\beta (\mathbb {T})} \leq 4 \beta _* \| f’\| _{L^\beta (\mathbb {T})} \end{equation*}

since \(\beta _* \geq 1\).

Now we go back to our initial problem.

Lemma 4.2.6

Let \(h \in \mathcal{P}_q\), and \(f\) and \((a_j)_{0 \leq j \leq q}\) be as in Lemma 4.2.2. For any \(m \geq 0\), any \(\beta {\gt} 1\), if we denote \(1/\beta _* := 1 - 1/\beta \), we have

\begin{equation*} \sum _{j=1}^q j^m |a_j| \leq 4 \beta _* \| f^{(m+1)}\| _{L^\beta (\mathbb {T})}. \end{equation*}

Proof ▶

First, we notice that the Fourier coefficients of \(f\) are exactly given by

\begin{equation*} c_j = \begin{cases} \frac{1}{2} a_{|j|} & 0 {\lt} |j| \leq q \\ a_0 & j=0 \\ 0 & |j| {\gt} q. \end{cases}\end{equation*}

Let \(m \geq 0\). By differentiation, the Fourier coefficients of \(f^{(m)}\) satisfy, for all \(j\), \(c_j^{(m)} = i^m j^m c_j\). In particular,

\begin{equation*} \sum _{j \neq 0} |c_j^{(m)}| = \sum _{j \neq 0} j^m |c_j| = \sum _{j =1}^q j^m |a_j|. \end{equation*}

The conclusion follows by applying Lemma 4.2.5 to the function \(f^{(m)}\), which is in \(L^\beta (\mathbb {T})\) as it is a trigonometric polynomial and in particular bounded.

4.3 Compactly supported distributions

In this section, we will prove a few useful elementary results on distributions. We denote as \(C^\infty (I)\) the space of smooth functions \(\mathbb {R}\rightarrow \mathbb {R}\), and \(C^\infty _c(\mathbb {R})\) those with compact support. For \(h \in C^\infty _c(\mathbb {R})\), we denote as \(\mathrm{supp} \, h\) its support. For real numbers \(a {\lt} b\) and \(h \in C^\infty (\mathbb {R})\), we write

\begin{equation*} \| h\| _{C^m[a,b]} = \sup _{0 \leq k \leq m} \sup _{a \leq x \leq b} |h^{(k)}(x)| \end{equation*}

where \(h^{(k)}\) is the \(k\)-th order derivative of \(h\). We denote as \(\mathcal{D}(\mathbb {R})\) the space of classical distributions on \(\mathbb {R}\), i.e. continuous linear functionals \(\nu : C^\infty _c(\mathbb {R}) \rightarrow \mathbb {R}\). Here continuous means that, for any real number \(K \geq 0\), there exists a constant \(C \geq 0\) and an integer \(m \geq 0\) (both depending on \(K\)) such that

\begin{equation*} \forall h \in C^\infty _c(\mathbb {R}), \quad \mathrm{supp} \, h \subseteq [-K,K] \Rightarrow |\nu (h)| \leq C \| h\| _{C^m[-K,K]}. \end{equation*}

We recall the definition of support, which should be added quite soon to Mathlib.

Definition 4.3.1

Let \(A\) be a subset of \(\mathbb {R}\). We say \(\nu \in \mathcal{D}(\mathbb {R})\) vanishes on \(A\) if, for all \(h \in C_c^\infty (\mathbb {R})\), if \(\mathrm{supp} \, h \subseteq A\), then \(\nu (h)=0\).

Definition 4.3.2

Let \(\nu \in \mathcal{D}(\mathbb {R})\). The support of \(\nu \) is the intersection of all closed sets \(A \subseteq \mathbb {R}\) such that \(\nu \) vanishes on the complement of \(A\). We denote it as \(\mathrm{supp} \, \nu \).

Definition 4.3.3

A distribution \(\nu \in \mathcal{D}(\mathbb {R})\) is a compactly supported distribution if its support is compact.

Lemma 4.3.4

Let \(\nu \in \mathcal{D}(\mathbb {R})\) be a compactly supported distribution. Then there is a unique continuous linear extension \(\nu : C^\infty (\mathbb {R}) \rightarrow \mathbb {R}\) of \(\nu \). Furthermore, there exist real numbers \(K \geq 0\), \(C{\gt}0\) and an integer \(m \geq 0\) such that

\begin{equation*} \forall h \in C^\infty (\mathbb {R}), \quad |\nu (h)| \leq C \| h\| _{C^m[-K,K]}. \end{equation*}

Proof ▶

In Rudin Functional Analysis according to Wikipedia.

This allows us to make sense of \(\nu (h)\) for \(\nu \) a compactly supported distribution and \(h\) a polynomial, an essential component of the proof.

Definition 4.3.5

Let \(\nu \in \mathcal{D}(\mathbb {R})\) be a compactly supported distribution. We define

\[ \rho (\nu ) := \limsup _{p \to \infty } |\nu (x^p)|^{1/p} . \]

Lemma 4.3.6

Let \(\nu \in \mathcal{D}(\mathbb {R})\) be a compactly supported distribution. Then \(\rho (\nu ) {\lt} \infty \).

Proof ▶

By Lemma 4.3.4, there exists \(K \geq 0\), \(C \geq 0\) and an integer \(m\) such that, for all \(h \in C^\infty (\mathbb {R})\), \(|\nu (h)| \leq C \| h\| _{C^m[-K,K]}\). Let \(p \geq 0\) be an integer. Then

\[ |\nu (x^p)| \leq C \| x^p\| _{C^m[-K,K]} = C \sup _{0 \leq k \leq m} \sup _{|x| \leq K} |p(p-1) \ldots (p-m+1) x^{p-m}| \leq C p^m (K+1)^p \]

and hence \((|\nu (x^p)|^{1/p})_{p \geq 0}\) is bounded.

Lemma 4.3.7

Let \(\nu \in \mathcal{D}(\mathbb {R})\) be a compactly supported distribution. Let \(\nu \in \mathcal{D}(\mathbb {R})\) be a compactly supported distribution. Then \(\mathrm{supp} \, \nu \subseteq [-\rho (\nu ), \rho (\nu )]\).

Proof ▶

Lemma 4.9 in [