Dependencies

\(\boldsymbol {\Psi }\) is an injective function on \(\mathbb {I}^n\) for any \(\lambda {\gt} 0\)

Let \(m {\gt} 0\) be an integer. The \(2^m\)-th cyclotomic polynomial is the irreducible polynomial, \(\Phi _{2^m}\left(r\right) \equiv r^{2^{m - 1}} - 1\).

A cyclotomic field is an extension of the field of rational numbers, \(\mathbb {Q}\), that is formed by adjoining some complex root, \(\zeta \), of a cyclotomic polynomial.

For some \(\lambda {\gt} 0\), \(\boldsymbol {\Psi }\) is an injective function on \(\mathbb {I}^n\).

\(\varphi _k\left(r\right) \equiv \frac{3}{7}+ \frac{8^{-k}}{14}+ \frac{1}{2} \sum \limits _{m = 0}^{k} 8^{-m} T_{2^{m}}\left(r\right)\), where \(k \in \mathbb {N}\).

\(\varphi \left(r\right) \equiv \frac{3}{7}+ \frac{1}{2} \sum \limits _{m = 0}^{\infty } 8^{-m} T_{2^{m}}\left(r\right)\).

Let \(\mathbb {F}_\tau \equiv \mathbb {R} \bigcap \bigcup \limits _{k = 1}^\infty \mathbb {Q}\left(\zeta _{2^k}\right)\), where \(\zeta _{2^k}\) is some root of \(\Phi _{2^k}\).

Let \(\gamma = -\cos \frac{\left(3j + 1\right)\pi }{2^{k - 1} \times 3}\) for some \(0 \leq j \leq 2^{k - 1}\) and \(k {\gt} 0\).

“[A] Helson set in a locally compact abelian group \(G\) is a closed set \(E\) such that the algebra \(A(E)\) of restrictions to \(E\) of functions in \(A(G)\) (Fourier transforms of summable functions on the dual group) coincides with \(C(E)\), the algebra of all continuous functions on \(E\)." (translated from page 91 of Kahane 1978)

\(\psi _{p,q}\left(x\right) \equiv \lambda _p \varphi \left(x - \frac{q}{2n}\right)\), where \(x \in \mathbb {I}\) and \(q \in \{ 0, 1, \cdots , 2n\} \).

Let \(\mathbf{J}\left(\mathbf{x}\right)\) be a matrix with \(2n + 1\) rows and \(n\) columns such that each element is \(J_{q,p} \equiv \lambda _p \dot{\varphi }_{\infty }\left(x_p - \frac{q}{2n}\right) = \frac{\lambda ^p}{2\Lambda } \sum \limits _{m = 0}^\infty 4^{-m} U_{-1 + 2^m}\left(x_p - \frac{q}{2n}\right)\) per Lemma 2.4.5.

\(\Lambda \equiv \sum \limits _{p = 0}^{n - 1} \lambda ^p\), where \(\lambda {\gt} 0\) and \(n \geq 2\).

\(\lambda _p \equiv \frac{\lambda ^p}{\Lambda } \in \left(0,1\right)\).

\(\chi _q\left(\Psi _q\left(x_0, x_1, \cdots , x_{n - 1}\right)\right) \equiv \chi \left(\Psi _q\left(x_0, x_1, \cdots , x_{n - 1}\right) + q\right)\).

Let \(\boldsymbol {\Psi }\left(\mathbf{x}\right) \equiv \begin{bmatrix} \Psi _0\left(\mathbf{x} - 0\right) \\ \Psi _1\left(\mathbf{x} - \frac{1}{2n}\right) \\ \vdots \\ \Psi _q\left(\mathbf{x} - \frac{q}{2n}\right) \\ \vdots \\ \Psi _{2n}\left(\mathbf{x} - 1\right) \end{bmatrix}\) be a column vector of size \(2n + 1\), where \(\mathbf{x}^\top = \begin{bmatrix} x_0 \\ x_1 \\ \vdots \\ x_p \\ \vdots \\ x_{n - 1} \end{bmatrix}\) is a column vector of size \(n\) that holds the arguments to \(f\), and \(\Psi _q\left(\mathbf{x} - \frac{q}{2n}\right) \equiv \sum \limits _{p = 0}^{n - 1} \frac{\lambda ^p}{\Lambda } \varphi \left(x_p - \frac{q}{2n}\right)\) in our construction from Chapter 2.

Let \(r\) be a real number. The first-kind Chebyshev polynomial in \(r\) with index \(j\) is defined via a three-term recursion as

The second-kind Chebyshev polynomial in \(r\) with index \(j \geq -1\) is defined recursively as

\(T_{2^{m}}\left(r\right)^2 \leq 1\) if and only if \(r^2 \leq 1\).

\(\varphi \in C\left(\mathbb {T}\right)\).

\(\varphi \) is a weak contraction over \(\mathbb {T}\).

Let \(\mathbb {T} = \left[-1,1\right]\). If \(t \in \mathbb {T}\), then the sequence of partial sums, \(\{ \varphi _k\left(t\right)\} \) forms a uniformly Cauchy sequence, but if \(r \notin \mathbb {T}\), then the infinite sum in \(\varphi \left(r\right)\) diverges.

If \(t \in \mathbb {T}\), then \(2^{-k - 1} \leq \dot{\varphi }_{k}\left(t\right) \leq 1 - 2^{-k - 1}\).

If \(t \in \mathbb {T}\), then the sequence of partial sums, \(\{ \dot{\varphi }_{k}\left(t\right)\} \) forms a uniformly Cauchy sequence.

If \(q = 0\) or \(q = 2n\), then \(\psi _{p,q} \in C\left(\mathbb {I}\right)\).

\(\varphi \) is not differentiable at \(-1\) and \(1\), which are essential logarithmic singularities.

If \(\tau = -\cos \frac{j\pi }{2^k}\) for some \(0 {\lt} j {\lt} 2^k\), then \(\tau \) is a root of \(U_{-1 + 2^k}\).

If \(k {\gt} 0\), then \(U_{-1 + 2^k}\left(r\right) = \prod \limits _{m = 1}^k \Phi _{2^m}\left(r\right)\).

The only two fixed points of \(2 T_{2^{m}}\left(r\right)^2 - 1 \mapsto T_{2^{m + 1}}\left(r\right)\) are \(-\frac{1}{2}\) and \(1\).

\(\mathbf{J}\left(\mathbf{x}\right)\) has full column rank for all \(\mathbf{x} \in \mathbb {I}^n\).

\(\varphi _k\left(-r\right) = \varphi _k\left(r\right) - r\).

\(\dot{\varphi }_{k}\left(-r\right) = 1 - \dot{\varphi }_{k}\left(r\right)\).

\(\ddot{\varphi }_{k}\left(-r\right) = \ddot{\varphi }_{k}\left(r\right)\).

If \(b {\gt} 1\), then \(\sum \limits _{m = j}^k b^{-m} = \frac{b^{1 - j} - b^{-k}}{b - 1}\).

\(\varphi \) is a strictly increasing function over \(\mathbb {T}\).

If \(\lambda \notin \mathbb {F}_\tau \), then \(\Psi _q\) is an injective function on \(\left(\mathbb {F}_\tau \bigcap \mathbb {I}\right)^n\).

If \(\tau = -\cos \frac{j\pi }{2^k}\), then \(\varphi _k\left(\tau \right) = \varphi \left(\tau \right)\).

If \(t \in \left(-1,1\right)\), then the sequence of partial sums, \(\{ \ddot{\varphi }_{k}\left(t\right)\} \) forms a uniformly Cauchy sequence.

The \(1 + 2^k\) extrema of \(T_{2^k}\) are called (Chebyshev-Lobatto) nodes and occur at \(\tau = -\cos \frac{j\pi }{2^k}\) with \(0 \leq j \leq 2^k\) and \(k \in \mathbb {N}\).

“If \(E\) contains the algebraic sum of two infinite sets, \(E\) is not a Helson set.”

\(\left(1 - r^2\right) \ddot{T}_{j}\left(r\right) - t \dot{T}_{j}\left(r\right) + j^2 T_j\left(r\right) = 0\).

If \(\gamma \) coincides on any subinterval of \(\mathbb {I}\) with a polynomial of degree less than \(n\), then \(\{ \gamma \left(\Psi _1\right), \gamma \left(\Psi _2\right), \dots , \gamma \left(\Psi _{2n}\right)\} \) is not a Helson set.

If \(t \in \mathbb {T}\), then \(0 \leq \varphi _k\left(t\right) \leq 1\).

If \(t \in \mathbb {T}\), then \(0 \leq \varphi \left(t\right) \leq 1\).

\(\varphi _k\left(r\right) = \frac{5}{14} + \frac{8^{-k}}{7} + \frac{r}{2} + \frac{1}{8} \sum \limits _{m = 0}^{k - 1} 8^{-m} T_{2^{m}}\left(r\right)^2\).

If \(k {\gt} 0\), then \(\varphi _k\) is a strictly convex function over \(\mathbb {T}\), and if \(k = 0\), then \(\varphi _0\) is weakly convex over \(\mathbb {T}\).

\(T_{2^{m}}\left(-r\right) = \left(-1\right)^{2^m} T_{2^{m}}\left(r\right)\).

If \(j \geq 0\), then \(\dot{T}_j\left(r\right) = j U_{j - 1}\left(r\right)\).

If \(t \in \mathbb {T}\), then \(T_j\left(t\right) = \cos \left(j \cos ^{-1}t\right)\).

\(T_{m + 1} = 2T_{m}^2 - 1\), where \(m \in \mathbb {N}\).

\(U_{-1 + 2^{m + 1}}\left(r\right) = 2 T_{2^{m}}\left(r\right) U_{-1 + 2^m}\left(r\right)\).

\(\dot{\varphi }_{k}\left(r\right) = \frac{1}{2} \sum \limits _{m = 0}^k 4^{-m} U_{-1 + 2^m}\left(r\right)\).

If \(t \in \left(-1,1\right)\), then \(\ddot{\varphi }_{k}\left(t\right) = \frac{t}{1 - t^2} \dot{\varphi }_{k}\left(t\right) - \frac{1}{2\left(1 - t^2\right)} \sum \limits _{m = 0}^k 2^{-m} T_{2^{m}}\left(t\right)\).

\(\lim \limits _{k \uparrow \infty } \dot{\varphi }_k\) is a weak derivative of \(\varphi \) that coincides with \(\dot{\varphi }\) on \(\left(-1,1\right)\).

\(\lim \limits _{k \uparrow \infty } \ddot{\varphi }_k\) is a weak second derivative of \(\varphi \) that coincides with \(\ddot{\varphi }\) on \(\left(-1,1\right)\).

\(\varphi \) is a weakly convex function over \(\mathbb {T}\).

If \(0 {\lt} q {\lt} 2n\), then \(\psi _{p,q} \in C^2\left(\mathbb {I}\right)\).

If \(\gamma \) is an increasing homeomorphism of \(\mathbb {I}\) that does not coincide with a polynomial of degree less than \(n\) on any subinterval of \(\mathbb {I}\), then \(\gamma \left(E\right)\) is quasi-surely a Helson set. Moreover, the condition on \(\gamma \) is necessary and sufficient, and \(\chi _q \in A^+\left(\mathbb {I}\right)\) where \(A^+\left(\mathbb {I}\right)\) is a class of functions \(\chi \left(z\right) = \sum \limits _{j = 0}^\infty \widehat{a}_j e^{2\pi \imath j z}\).

If \(q = 0\) or \(q = 2n\), then \(\psi _{p,q} \notin C^1\left(\mathbb {I}\right)\).