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.

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 develop a new analytical solution of a three-dimensional atmospheric pollutant dispersion. The main idea is to subdivide vertically the planetary boundary layer into sub-layers, where the wind speed and eddy diffusivity assume average values for each sub-layer. Basically, the model is assessed and validated using data obtained from the Copenhagen diffusion and Prairie Grass experiments. Our findings show that there is a good agreement between the predicted and observed crosswind-integrated concentrations. Moreover, the calculated statistical indices are within the range of acceptable model performance.

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

Using some formulas of S. Ramanujan, we compute in closed form the Fourier transform of functions related to Riemann zeta function $\zeta (s)=\sum \nolimits _{n=1}^{\infty } {1}/{n^{s}}$ and other Dirichlet series.

Truncating the Fourier transform averaged by means of a generalized Hausdorff operator, we approximate functions and the adjoint to that Hausdorff operator of the given function. We find estimates for the rate of approximation in various metrics in terms of the parameter of truncation and the components of the Hausdorff operator. Explicit rates of approximation of functions and comparison with approximate identities are given in the case of continuous functions from the class $\text {Lip }\alpha $ .

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.

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.

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

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

$\ell =2$

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

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

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

$\ell$

We show a pointwise estimate for the Fourier transform on the line involving the number of times the function changes monotonicity. The contrapositive of the theorem may be used to find a lower bound to the number of local maxima of a function. We also show two applications of the theorem. The first is the two weight problem for the Fourier transform, and the second is estimating the number of roots of the derivative of a function.

The aim of this paper is to exhibit a real Paley–Wiener space sitting inside the Schwartz space, and to give a quick and simple proof of a Paley–Wiener-type theorem. A simple and elementary proof of a theorem postulated by H. H. Bang is also given.

Let $\ast$ denote convolution and let $\delta_x$ denote the Dirac measure at a point $x$. A function in $L^2({\mathbb{R}})$ is called a difference of order $1$ if it is of the form $ g- \delta_{x} \ast g$ for some $x \in {\mathbb{R}}$ and $g \in L^2 ({\mathbb{R}})$. Also, a difference of order $2$ is a function of the form $ g- 2^{-1}( \delta_{x} \ast g + \delta_{-x} \ast g)$ for some $x \in {\mathbb{R}}$ and $g \in L^2 ({\mathbb{R}})$. In fact, the concept of a ‘difference of order $s$’ may be defined in a similar manner for each $s > 0$. If $\widehat{f}$ denotes the Fourier transform of $f$, it is known that a function $f$ in $L^2 ({\mathbb{R}})$ is a finite sum of differences of order $s$ if and only if $\int_{-\infty}^{\infty} |\widehat {f}(x)|^2 |x|^{-2s} dx < \infty$, and the vector space of all such functions is denoted by ${\mathcal{D}}_{s} (L^2 ({\mathbb{R}}) )$. Every function in ${\mathcal{D}}_{s} (L^2 ({\mathbb{R}}) )$ is a sum of ${\mathrm{int}} (2s) + 1$ differences of order $s$, where ${\mathrm{int}} (t)$ denotes the integer part of $t$. Thus, every function in ${\mathcal{D}}_{1} (L^2 ({\mathbb{R}}) )$ is a sum of three first order differences, but it was proved in 1994 that there is a function in ${\mathcal{D}}_1 (L^2 ({\mathbb{R}}) )$ which is never the sum of two first order differences. This complemented, for the group ${\mathbb{R}}$, the corresponding result for first order differences obtained by Meisters and Schmidt in 1972 for the circle group. The results show that there is a function in $L^2 ({\mathbb{R}})$ such that, for each $s \ge 1/2$, this function is a sum of ${\mathrm{int}} (2s) + 1$ differences of order $s$ but it is never the sum of ${\mathrm{int}} (2s)$ differences of order $s$. The proof depends upon extending to higher dimensions the following result in two dimensions obtained by Schmidt in 1972 in connection with Heilbronn's problem: if $x_1, \ldots, x_n$ are points in the unit square,

$$\sum_{1 \le i < j \le n} |x_i - x_j|^{-2} \ge 200^{-1} n^2 \ln n.$$

Following on from the work of Meisters and Schmidt, this work further develops a connection between certain estimates in combinatorial geometry and some questions of sharpness in harmonic analysis.

It was shown by Edgar and Rosenblatt that f ∈ Lp (ℝn), 1 ≤ p < 2n/ (n-1), and f ≠0, then f has linearly independent translates. Using a result of Hömander, it is shown here that the same theorem holds if p = 2n / (n−1). This gives a sharp result because for n ≥2, there exists f ∈C0 (ℝn), f ≠0, which is simultaneously in all Lp (ℝn), p > 2n/(n−1), that has a linear dependence relation among its translates. References and some discussion are included.

We define an iterative interpolation process for data spread over a closed discrete subgroup of the Euclidean space. We describe the main algebraic properties of this process. This interpolation process, under very weak assumptions, is always convergent in the sense of Schwartz distributions. We find also a convenient necessary and sufficient condition for continuity of each interpolation function of a given iterative interpolation process.