We enrich the class of power-constructible functions, introduced in [CCRS23], to a class $\mathcal {C}^{\mathcal {M,F}}$ of algebras of functions which contains all complex powers of subanalytic functions and their parametric Mellin and Fourier transforms, and which is stable under parametric integration. By describing a set of generators of a special prepared form, we deduce information on the asymptotics and on the loci of integrability of the functions of $\mathcal {C}^{\mathcal {M,F}}$. We furthermore identify a subclass $\mathcal {C}^{\mathbb {C},\mathcal {F}}$ of $\mathcal {C}^{\mathcal {M,F}}$, which is the smallest class containing all power-constructible functions and stable under parametric Fourier transforms and right-composition with subanalytic maps. This class is also stable under parametric integration, under taking pointwise and $\text {L}^p$-limits and under parametric Fourier-Plancherel transforms. Finally, we give a full asymptotic expansion in the power-logarithmic scale, uniformly in the parameters, for functions in $\mathcal {C}^{\mathbb {C},\mathcal {F}}$.

This paper focuses on the fundamental aspects of super-resolution, particularly addressing the stability of super-resolution and the estimation of two-point resolution. Our first major contribution is the introduction of two location-amplitude identities that characterize the relationships between locations and amplitudes of true and recovered sources in the one-dimensional super-resolution problem. These identities facilitate direct derivations of the super-resolution capabilities for recovering the number, location, and amplitude of sources, significantly advancing existing estimations to levels of practical relevance. As a natural extension, we establish the stability of a specific $l_{0}$ minimization algorithm in the super-resolution problem.

The second crucial contribution of this paper is the theoretical proof of a two-point resolution limit in multi-dimensional spaces. The resolution limit is expressed as

$$\begin{align*}\mathscr R = \frac{4\arcsin \left(\left(\frac{\sigma}{m_{\min}}\right)^{\frac{1}{2}} \right)}{\Omega} \end{align*}$$

${\frac {\sigma }{m_{\min }}}{\leqslant }{\frac {1}{2}}$

${\frac {\sigma }{m_{\min }}}$

${\mathrm {SNR}}$

$\Omega $

${\frac {\pi }{\Omega }}$

$2$

${\mathscr {R}}$

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

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

$g$

$M$

$M$

$f$

This paper concerns the $L^4$ norm of Littlewood polynomials on the unit circle which are given by

$$ \begin{align*}q_n(z)=\sum_{k=0}^{n-1}\pm z^k;\end{align*} $$

$\{-1,1\}$

$$ \begin{align*}||q_n||_4^4=\frac{1}{2\pi}\int_0^{2\pi}|q_n(e^{i\theta})|^4 d\theta.\end{align*} $$

$||q_n||_4/\sqrt {n}\rightarrow \sqrt [4]{2}$

$n\to \infty $

Let $\{b_n\}_{n=1}^{\infty }$ be a sequence of integers larger than 1. We will study the harmonic analysis of the equal-weighted Moran measures $\mu _{\{b_n\},\{{\mathcal D}_n\}}$ with ${\mathcal D}_n=\{0,1,2,\ldots ,q_n-1\}$, where $q_n$ divides $b_n$ for all $n\geq 1.$ In this paper, we first characterize all the maximal orthogonal sets of $L^2(\mu _{\{b_n\},\{{\mathcal D}_n\}})$ via a tree mapping. By this characterization, we give some sufficient conditions for the maximal orthogonal set to be an orthonormal basis.

In this article, we study the recent development of the qualitative uncertainty principle on certain Lie groups. In particular, we consider that if the Weyl transform on certain step-two nilpotent Lie groups is of finite rank, then the function has to be zero almost everywhere as long as the nonvanishing set for the function has finite measure. Further, we consider that if the Weyl transform of each Fourier–Wigner piece of a suitable function on the Heisenberg motion group is of finite rank, then the function has to be zero almost everywhere whenever the nonvanishing set for each Fourier–Wigner piece has finite measure.

The main purpose of this paper is to prove Hörmander’s $L^p$–$L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the $L^p$–$L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli–Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the $L^p$–$L^q$ norms of the heat kernel for generalised radial Laplacian.

We prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty )$, for certain multi-parameter polynomial ergodic averages by establishing the corresponding multi-parameter maximal and oscillation inequalities. Our result, in particular, gives an affirmative answer to a multi-parameter variant of the Bellow–Furstenberg problem. This paper is also the first systematic treatment of multi-parameter oscillation semi-norms which allows an efficient handling of multi-parameter pointwise convergence problems with arithmetic features. The methods of proof of our main result develop estimates for multi-parameter exponential sums, as well as introduce new ideas from the so-called multi-parameter circle method in the context of the geometry of backwards Newton diagrams that are dictated by the shape of the polynomials defining our ergodic averages.

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ($P_4$): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$, where $A$, $B$, $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$, $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$) into $X$, $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$. We completely characterize the well-posedness of ($P_4$) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

We discuss how countable subadditivity of operators can be derived from subadditivity under mild forms of continuity, and provide examples manifesting such circumstances.

Let $\mathcal {F}$ denote the set of functions $f \colon [-1/2,1/2] \to \mathbb {R}_{\geq 0}$ such that $\int f = 1$. We determine the value of $\inf _{f \in \mathcal {F}} \| f \ast f \|_2^2$ up to a $4 \cdot 10^{-6}$ error, thereby making progress on a problem asked by Ben Green. Furthermore, we prove that a unique minimizer exists. As a corollary, we obtain improvements on the maximum size of $B_h[g]$ sets for $(g,h) \in \{ (2,2),(3,2),(4,2),(1,3),(1,4)\}$.

We prove Lp norm convergence for (appropriate truncations of) the Fourier series arising from the Dirichlet Laplacian eigenfunctions on three types of triangular domains in $\mathbb{R}^2$: (i) the 45-90-45 triangle, (ii) the equilateral triangle and (iii) the hemiequilateral triangle (i.e. half an equilateral triangle cut along its height). The limitations of our argument to these three types are discussed in light of Lamé’s Theorem and the image method.

We consider direct and inverse Jacobi transforms with measures

$$\begin{align*}d\mu(t)=2^{2\rho}(\operatorname{sinh} t)^{2\alpha+1}(\operatorname{cosh} t)^{2\beta+1}\,dt\end{align*}$$

$$\begin{align*}d\sigma(\lambda)=(2\pi)^{-1}\Bigl|\frac{2^{\rho-i\lambda}\Gamma(\alpha+1)\Gamma(i\lambda)} {\Gamma((\rho+i\lambda)/2)\Gamma((\rho+i\lambda)/2-\beta)}\Bigr|^{-2}\,d\lambda,\end{align*}$$

$$\begin{align*}\inf\Lambda((-1)^{m-1}f), \quad m\in \mathbb{N}, \end{align*}$$

$\Lambda (f)=\sup \,\{\lambda>0\colon f(\lambda )>0\}$

$m\ge 2$

$\int _{0}^{\infty }\lambda ^{2k}f(\lambda )\,d\sigma (\lambda )=0$

$k=0,\dots ,m-2$

We prove that admissible functions for this problem are positive-definite with respect to the inverse Jacobi transform. The solution of Logan’s problem was known only when $\alpha =\beta =-1/2$. We find a unique (up to multiplication by a positive constant) extremizer $f_m$. The corresponding Logan problem for the Fourier transform on the hyperboloid $\mathbb {H}^{d}$ is also solved. Using the properties of the extremizer $f_m$ allows us to give an upper estimate of the length of a minimal interval containing not less than n zeros of positive definite functions. Finally, we show that the Jacobi functions form the Chebyshev systems.

We present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr- and Bochner-type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework.

Let $M=(\begin {smallmatrix}\rho ^{-1} & 0 \\0 & \rho ^{-1} \\\end {smallmatrix})$ be an expanding real matrix with $0<\rho <1$, and let ${\mathcal D}_n=\{(\begin {smallmatrix} 0\\ 0 \end {smallmatrix}),(\begin {smallmatrix} \sigma _n\\ 0 \end {smallmatrix}),(\begin {smallmatrix} 0\\ \gamma _n \end {smallmatrix})\}$ be digit sets with $\sigma _n,\gamma _n\in \{-1,1\}$ for each $n\ge 1$. Then the infinite convolution

$$ \begin{align*}\mu_{M,\{{\mathcal D}_n\}}=\delta_{M^{-1}{\mathcal D}_1}\ast\delta_{M^{-2}{\mathcal D}_2}\ast\cdots\end{align*} $$

is called a Moran–Sierpinski measure. We give a necessary and sufficient condition for $L^2(\,\mu _{M,\{{\mathcal D}_n\}})$ to admit an infinite orthogonal set of exponential functions. Furthermore, we give the exact cardinality of orthogonal exponential functions in $L^2(\,\mu _{M,\{{\mathcal D}_n\}})$ when $L^2(\,\mu _{M,\{{\mathcal D}_n\}})$ does not admit any infinite orthogonal set of exponential functions based on whether $\rho $ is a trinomial number or not.

We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

In this paper, we study divergence properties of the Fourier series on Cantor-type fractal measure, also called the mock Fourier series. We give a sufficient condition under which the mock Fourier series for doubling spectral measure is divergent on a set of strictly positive measure. In particular, there exists an example of the quarter Cantor measure whose mock Fourier sums are not almost everywhere convergent.

We prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $\mathbb {T}^n$ or Boolean cubes $\{-1,1\}^N$. Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $\ell _1$-multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.