It is known that the Fourier–Stieltjes coefficients of a nonatomic coin-tossing measure may not vanish at infinity. However, we show that they could vanish at infinity along some integer subsequences, including the sequence ${\{b^{n}\}}_{n\geq 1}$ where $b$ is multiplicatively independent of 2 and the sequence given by the multiplicative semigroup generated by 3 and 5. The proof is based on elementary combinatorics and lower-bound estimates for linear forms in logarithms from transcendental number theory.

The purpose of this note is to construct an example of a discrete non-abelian group G and a subset E of G, not contained in any abelian subgroup, that is a completely bounded $\Lambda (p)$ set for all $p<\infty ,$ but is neither a Leinert set nor a weak Sidon set.

The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with a certain symmetry.

In this paper, we consider the problem of characterizing positive definite functions on compact two-point homogeneous spaces cross locally compact abelian groups. For a locally compact abelian group $G$ with dual group $\widehat{G}$, a compact two-point homogeneous space $\mathbb{H}$ with normalized geodesic distance $\unicode[STIX]{x1D6FF}$ and a profile function $\unicode[STIX]{x1D719}:[-1,1]\times G\rightarrow \mathbb{C}$satisfying certain continuity and integrability assumptions, we show that the positive definiteness of the kernel $((x,u),(y,v))\in (\mathbb{H}\times G)^{2}\mapsto \unicode[STIX]{x1D719}(\cos \unicode[STIX]{x1D6FF}(x,y),uv^{-1})$ is equivalent to the positive definiteness of the Fourier transformed kernels $(x,y)\in \mathbb{H}^{2}\mapsto \widehat{\unicode[STIX]{x1D719}}_{\cos \unicode[STIX]{x1D6FF}(x,y)}(\unicode[STIX]{x1D6FE})$, $\unicode[STIX]{x1D6FE}\in \widehat{G}$, where $\unicode[STIX]{x1D719}_{t}(u)=\unicode[STIX]{x1D719}(t,u)$, $u\in G$. We also provide some results on the strict positive definiteness of the kernel.

In this note we examine Littlewood’s proof of the prime number theorem. We show that this can be extended to provide an equivalence between the prime number theorem and the nonvanishing of Riemann’s zeta-function on the one-line. Our approach goes through the theory of almost periodic functions and is self-contained.

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].

In this paper, we prove some reverse discrete inequalities with weights of Muckenhoupt and Gehring types and use them to prove some higher summability theorems on a higher weighted space $l_{w}^{p}({\open N})$ form summability on the weighted space $l_{w}^{q}({\open N})$ when p>q. The proofs are obtained by employing new discrete weighted Hardy's type inequalities and their converses for non-increasing sequences, which, for completeness, we prove in our special setting. To the best of the authors' knowledge, these higher summability results have not been considered before. Some numerical results will be given for illustration.

We study the inverse boundary value problem for fractional diffusion in a multilayer composite medium. Given data in the right boundary of the second layer, the problem is to recover the temperature distribution in the first layer, which is inaccessible for measurement. The problem is ill-posed and we propose a Fourier spectral approach to achieve Hölder approximations. The convergence analysis is performed in both the $L^{2}$- and $L^{\infty }$-settings.

In this paper we characterize the Fourier transformability of strongly almost periodic measures in terms of an integrability condition for their Fourier–Bohr series. We also provide a necessary and sufficient condition for a strongly almost periodic measure to be the Fourier transform of a measure. We discuss the Fourier transformability of a measure on $\mathbb{R}^{d}$ in terms of its Fourier transform as a tempered distribution. We conclude by looking at a large class of such measures coming from the cut and project formalism.

We study criteria for the uniform convergence of trigonometric series with general monotone coefficients. We also obtain necessary and sufficient conditions for a given rate of convergence of partial Fourier sums of such series.

We prove that, for any finite set $A\subset \mathbb{Q}$ with $|AA|\leqslant K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound

$$\begin{eqnarray}|\{(a_{1}+1)(a_{2}+1)\cdots (a_{k}+1):a_{i}\in A\}|\geqslant \frac{|A|^{k}}{(8k^{4})^{kK}}.\end{eqnarray}$$

$K$

$c\log |A|$

$c=c(k)$

$\unicode[STIX]{x1D6EC}$

In his seminal work on Sidon sets, Pisier found an important characterization of Sidonicity: A set is Sidon if and only if it is proportionally quasi-independent. Later, it was shown that Sidon sets were proportionally “special” Sidon in several other ways. Here, we prove that Sidon sets in torsion-free groups are proportionally $n$-degree independent, a higher order of independence than quasi-independence, and we use this to prove that Sidon sets are proportionally Sidon with Sidon constants arbitrarily close to one, the minimum possible value.

Let $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\in \mathbb{R}$ and $s\in \mathbb{N}$ be given. Let $\unicode[STIX]{x1D6FF}_{x}$ denote the Dirac measure at $x\in \mathbb{R}$, and let $\ast$ denote convolution. If $\unicode[STIX]{x1D707}$ is a measure, $\unicode[STIX]{x1D707}^{\star }$ is the measure that assigns to each Borel set $A$ the value $\overline{\unicode[STIX]{x1D707}(-A)}$. If $u\in \mathbb{R}$, we put $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}=e^{iu(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})/2}\unicode[STIX]{x1D6FF}_{0}-e^{iu(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})/2}\unicode[STIX]{x1D6FF}_{u}$. Then we call a function $g\in L^{2}(\mathbb{R})$ a generalized$(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$-difference of order$2s$ if for some $u\in \mathbb{R}$ and $h\in L^{2}(\mathbb{R})$ we have $g=[\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}+\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},u}^{\star }]^{s}\ast h$. We denote by ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ the vector space of all functions $f$ in $L^{2}(\mathbb{R})$ such that $f$ is a finite sum of generalized $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$-differences of order $2s$. It is shown that every function in ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ is a sum of $4s+1$ generalized $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$-differences of order $2s$. Letting $\widehat{f}$ denote the Fourier transform of a function $f\in L^{2}(\mathbb{R})$, it is shown that $f\in {\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ if and only if $\widehat{f}$  “vanishes” near $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ at a rate comparable with $(x-\unicode[STIX]{x1D6FC})^{2s}(x-\unicode[STIX]{x1D6FD})^{2s}$. In fact, ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ is a Hilbert space where the inner product of functions $f$ and $g$ is $\int _{-\infty }^{\infty }(1+(x-\unicode[STIX]{x1D6FC})^{-2s}(x-\unicode[STIX]{x1D6FD})^{-2s})\widehat{f}(x)\overline{\widehat{g}(x)}\,dx$. Letting $D$ denote differentiation, and letting $I$ denote the identity operator, the operator $(D^{2}-i(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})D-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}I)^{s}$ is bounded with multiplier $(-1)^{s}(x-\unicode[STIX]{x1D6FC})^{s}(x-\unicode[STIX]{x1D6FD})^{s}$, and the Sobolev subspace of $L^{2}(\mathbb{R})$ of order $2s$ can be given a norm equivalent to the usual one so that $(D^{2}-i(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FD})D-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}I)^{s}$ becomes an isometry onto the Hilbert space ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$. So a space ${\mathcal{D}}_{\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},s}(\mathbb{R})$ may be regarded as a type of Sobolev space having a negative index.

We establish the mapping properties of Fourier-type transforms on rearrangement-invariant quasi-Banach function spaces. In particular, we have the mapping properties of the Laplace transform, the Hankel transforms, the Kontorovich-Lebedev transform and some oscillatory integral operators. We achieve these mapping properties by using an interpolation functor that can explicitly generate a given rearrangement-invariant quasi-Banach function space via Lebesgue spaces.

The century-old extremal problem, solved by Carathéodory and Fejér, concerns a non-negative trigonometric polynomial $T(t) = a_0 + \sum\nolimits_{k = 1}^n {a_k} \cos (2\pi kt) + b_k\sin (2\pi kt){\ge}0$ , normalized by a 0=1, where the quantity to be maximized is the coefficient a 1 of cos (2π t). Carathéodory and Fejér found that for any given degree n, the maximum is 2 cos(π/n+2). In the complex exponential form, the coefficient sequence (c k ) ⊂ ℂ will be supported in [−n, n] and normalized by c 0=1. Reformulating, non-negativity of T translates to positive definiteness of the sequence (c k ), and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c: ℤ → ℂ, supported in [−n, n]. Boas and Kac, Arestov, Berdysheva and Berens, Kolountzakis and Révész and, recently, Krenedits and Révész investigated the problem in increasing generality, reaching analogous results for all locally compact abelian groups. We prove an extension to all the known results in not necessarily commutative locally compact groups.

We study function multipliers between spaces of holomorphic functions on the unit disc of the complex plane generated by symmetric sequence spaces. In the case of sequence $\ell ^{p}$ spaces we recover Nikol’skii’s results [‘Spaces and algebras of Toeplitz matrices operating on $\ell ^{p}$ ’, Sibirsk. Mat. Zh.7 (1966), 146–158].

Let $\ell >0$ be arbitrary. We introduce the extremal quantities

$$\begin{eqnarray}G(\ell ):=\sup _{f}\int _{-\ell }^{\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\quad C(\ell ):=\sup _{f}\sup _{a\in \mathbb{R}}\int _{a-\ell }^{a+\ell }f\,dx\,\bigg/\int _{-1}^{1}f\,dx,\end{eqnarray}$$

$\ell =2$

$\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}G(k+\unicode[STIX]{x1D700})$

$\overline{\lim }_{\unicode[STIX]{x1D700}\rightarrow 0+}C(k+\unicode[STIX]{x1D700})$

$(k\in \mathbb{N})$

$\ell$

A necessary and sufficient condition for a continuous function $g$ to be almost periodic on time scales is the existence of an almost periodic function $f$ on $\mathbb{R}$ such that $f$ is an extension of $g$ . Our aim is to study this question for pseudo almost periodic functions. We prove the necessity of the condition for pseudo almost periodic functions. An example is given to show that the sufficiency of the condition does not hold for pseudo almost periodic functions. Nevertheless, the sufficiency is valid for uniformly continuous pseudo almost periodic functions. As applications, we give some results on the connection between the pseudo almost periodic (or almost periodic) solutions of dynamic equations on time scales and of the corresponding differential equations.

We consider the zero-resistivity limit for Hasegawa–Wakatani equations in a cylindrical domain when the initial data are Stepanov almost-periodic in the axial direction. First, we prove the existence of a solution to Hasegawa–Wakatani equations with zero resistivity; second, we obtain uniform a priori estimates with respect to resistivity. Such estimates can be obtained in the same way as for our previous results; therefore, the most important contribution of this paper is the proof of the existence of a local-in-time solution to Hasegawa–Wakatani equations with zero resistivity. We apply the theory of Bohr–Fourier series of Stepanov almost-periodic functions to such a proof.

Let μ λ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ 1/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μ λ shows that μ λ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ 1/(2k). This is surprising because some spectra for the measure μ 1/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ 1/(2k).