In loving memory of my beloved miniature dachshund Maddie (16 March 2002 – 16 March 2020). We consider nonlocal differential equations with convolution coefficients of the form

\[{-}M\Big(\big(a*(g\circ |u|)\big)(1)\Big)u''(t)=\lambda f\big(t,u(t)\big),\quad t\in(0,1), \]

$g$

$M$

$M$

$f$

Let $M=$ diag $(\rho _1,\rho _2)\in M_{2}({\mathbb R})$ be an expanding matrix and Let $\{D_n\}_{n=1}^{\infty }$ be a sequence of digit sets with $D_n=\left \{(0, 0)^T,\,\,\,(a_n, 0 )^T, \,\,\, (0, b_n )^T \right \}$ , where $a_n, b_n\in \{-1,1\}$ . The associated Borel probability measure

$$ \begin{align*} \mu_{M,\{D_n\}}:=\delta_{M^{-1}D_1}\ast \delta_{M^{-2}D_2}\ast \delta_{M^{-3}D_3}\ast \cdots \end{align*} $$

$\mu _{M, \{D_n\}}$

$3\mid \rho _i$

$i=1, 2$

$a_n=b_n=1$

$n\in {\mathbb N}$

We consider nonlocal equations of the general form

\begin{equation} \left(a*u''\right)(\cdot)+\lambda f\big(\cdot,u(\cdot)\big)=0.\nonumber \end{equation}

We establish the general form of a geometric comparison principle for n-fold convolutions of certain singular measures in ℝd which holds for arbitrary n and d. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in the companion paper [3].

We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $\alpha $ of the form $d\,/\,n,\,n\,=\,2,3,...$ there exist $\alpha $-Salem measures for which the ${{L}^{2}}$ Fourier restriction theorem holds in the range $p\,\le \,\frac{2d}{2d\,-\,\alpha }$. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular $\alpha $-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\,\in \,\mathbb{N}$.

Let ${\it\alpha}\in \mathbb{C}$ in the upper half-plane and let $I$ be an interval. We construct an analogue of Selberg’s majorant of the characteristic function of $I$ that vanishes at the point ${\it\alpha}$. The construction is based on the solution to an extremal problem with positivity and interpolation constraints. Moreover, the passage from the auxiliary extremal problem to the construction of Selberg’s function with vanishing is easily adapted to provide analogous “majorants with vanishing” for any Beurling–Selberg majorant.

For a Lie group $G$, we show that the map $C_{c}^{\infty }\,\left( G \right)\,\times \,C_{c}^{\infty }\,\left( G \right)\,\to \,C_{c}^{\infty }\,\left( G \right),\,\left( \gamma ,\,\eta \right)\mapsto \,\gamma \,*\,\eta $ taking a pair of test functions to their convolution, is continuous if and only if $G$ is $\sigma -$compact. More generally, consider $r,\,s,\,t\,\in {{\mathbb{N}}_{0}}\,\cup \,\left\{ \infty \right\}$ with $t\,\le \,r\,+\,s$, locally convex spaces ${{E}_{1}}\,,\,{{E}_{2}}$ and a continuous bilinear map $b:\,{{E}_{1}}\,\times \,{{E}_{2}}\,\to \,F$ to a complete locally convex space $F$. Let $\beta :\,C_{c}^{r}\,\left( G,\,{{E}_{1}} \right)\,\times \,C_{c}^{S}\,\left( G,\,{{E}_{2}} \right)\,\to$$C_{c}^{t}\,\left( G,\,F \right),\,\left( \gamma ,\,\eta \right)\,\mapsto \,\gamma \,*\,b\,\eta$ be the associated convolution map. The main result is a characterization of those $\left( G,\,r,s,t,b \right)$ for which $\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported ${{L}^{1}}$-functions and convolution of compactly supported Radon measures.

We show that the zeros of a trigonometric polynomial of degree N with the usual (2N + 1) terms can be calculated by computing the eigenvalues of a matrix of dimension 2N with real-valued elements Mjk. This matrix is a multiplication matrix in the sense that, after first defining a vector whose elements are the first 2N basis functions, . This relationship is the eigenproblem; the zeros tk are the arccosine function of λk/2 where the λk are the eigenvalues of . We dub this the “Fourier Division Companion Matrix”, or FDCM for short, because it is derived using trigonometric polynomial division. We show through examples that the algorithm computes both real and complex-valued roots, even double roots, to near machine precision accuracy.

In this paper we consider multipliers on the real Hardy space ${{H}_{2\pi }}$. It is known that the Marcinkiewicz and the Hörmander–Mihlin conditions are sufficient for the corresponding trigonometric multiplier to be bounded on $L_{2\pi }^{p},1<p<\infty$. We show among others that the Hörmander– Mihlin condition extends to ${{H}_{2\pi }}$ but the Marcinkiewicz condition does not.

Suppose that $\gamma \in {{C}^{2}}\left( \left[ 0,\infty \right]) \right)$ is a real-valued function such that $\gamma \left( 0 \right)\,=\,{\gamma }'\left( 0 \right)\,=\,0$, and ${\gamma }''\left( t \right)\,\approx \,{{t}^{m-2}}$, for some integer $m\,\ge \,2$. Let $\Gamma \left( t \right)\,=\,\left( t,\,\gamma \left( t \right) \right),\,t\,>\,0$, be a curve in the plane, and let $d\text{ }\!\!\lambda\!\!\text{ }\,\text{=}\,dt$ be a measure on this curve. For a function $f$ on ${{\mathbf{R}}^{2}}$, let

$$Tf\left( x \right)\,=\,\left( \text{ }\lambda \text{ }*f \right)\left( x \right)=\int_{0}^{\infty }{f\left( x-\Gamma \left( t \right) \right)dt,\,\,x\in {{\mathbf{R}}^{2}}}.$$

An elementary proof is given for the optimal ${{L}^{p}}-{{L}^{q}}$ mapping properties of $T$.

Let ψ be a positive function defined near the origin such lim1 →0+ ψ(t) = 0. We consider the operator Tzƒ, defined as the pricipal value of the convolution of function ƒ and a kernel K(t) = eiy(t)t−z /ψ(t)1−z, where z is a complex number, 0 ≤ Re(z) ≤ 1, 0 < t ≤ 1 and γ is a real function. Assuming certain regularity conditions on ψ and γ and certain relations between ψ and γ we show that Tθ is a bounded operator on Lp (R) for 1/p = (1+ θ) /2 and 0 ≤ θ < 1, and T1 is bounded from H1 (R) to L1 (R).

We study the stability of linear filters associated with certain types of linear difference equations with variable coefficients. We show that stability is determined by the locations of the poles of a rational transfer function relative to the spectrum of an associated weighted shift operator. The known theory for filters associated with constant-coefficient difference equations is a special case.