The notion of indicator of an analytic function, that describes the function’s growth along rays, was introduced by Phragmen and Lindelöf. Trigonometric convexity is a defining property of the indicator. For multivariate cases, an analogous property of trigonometric convexity was not known so far. We prove the property of trigonometric convexity for the indicator of multivariate analytic functions, introduced by Ivanov. The results that we obtain are sharp. Derivation of a multidimensional analogue of the inverse Fourier transform in a sector and obtaining estimates on its decay is an important step of our proof.

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.

For the kernel $B_{\kappa ,a}(x,y)$ of the $(\kappa ,a)$-generalized Fourier transform $\mathcal {F}_{\kappa ,a}$, acting in $L^{2}(\mathbb {R}^{d})$ with the weight $|x|^{a-2}v_{\kappa }(x)$, where $v_{\kappa }$ is the Dunkl weight, we study the important question of when $\|B_{\kappa ,a}\|_{\infty }=B_{\kappa ,a}(0,0)=1$. The positive answer was known for $d\ge 2$ and $\frac {2}{a}\in \mathbb {N}$. We investigate the case $d=1$ and $\frac {2}{a}\in \mathbb {N}$. Moreover, we give sufficient conditions on parameters for $\|B_{\kappa ,a}\|_{\infty }>1$ to hold with $d\ge 1$ and any a.

We also study the image of the Schwartz space under the $\mathcal {F}_{\kappa ,a}$ transform. In particular, we obtain that $\mathcal {F}_{\kappa ,a}(\mathcal {S}(\mathbb {R}^d))=\mathcal {S}(\mathbb {R}^d)$ only if $a=2$. Finally, extending the Dunkl transform, we introduce nondeformed transforms generated by $\mathcal {F}_{\kappa ,a}$ and study their main properties.

Let $S \subset \mathbb {R}^{n}$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\mathbb {R}^{n}$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^{q}(X)} \lesssim \| f \|_{L^{p}(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \lesssim R^{\alpha }$ for all balls $B_R$ in $\mathbb {R}^{n}$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^{q}$ against the measure $\chi _X \,{\textrm {d}}x$. Our approach consists of replacing the characteristic function $\chi _X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted Hölder-type inequality that holds for general non-negative Lebesgue measurable functions on $\mathbb {R}^{n}$ and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du–Zhang theorem in the range $0 < \alpha < n/2$.

Eigenfunctions of the Fourier transform with prescribed zeros played a major role in the proof that the E8 and the Leech lattice give the best sphere packings in respective dimensions 8 and 24 by Cohn, Kumar, Miller, Radchenko and Viazovska. The functions used for a linear programming argument were constructed as Laplace transforms of certain modular and quasimodular forms. Similar constructions were used by Cohn and Gonçalves to find a function satisfying an optimal uncertainty principle in dimension 12. This paper gives a unified view on these constructions and develops the machinery to find the underlying forms in all dimensions divisible by 4. Furthermore, the positivity of the Fourier coefficients of the quasimodular forms occurring in this context is discussed.

The purpose of this paper is to characterize the entire solutions of the homogeneous Helmholtz equation (solutions in ℝd) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^\alpha (\mathbb {S}^{d-1}),$ with α ∈ ℝ. We present two characterizations. The first one is written in terms of certain L2-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For α > 0 this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for α < 0 it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.

The Friedgut–Kalai–Naor (FKN) theorem states that if ƒ is a Boolean function on the Boolean cube which is close to degree one, then ƒ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice.

As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.

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 exhibit the first explicit examples of Salem sets in ℚp of every dimension 0 < α < 1 by showing that certain sets of well-approximable p-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the p-adic setting.

Let $\ell \in \mathbb{N}$ and $p\in (1,\infty ]$. In this article, the authors establish several equivalent characterizations of Sobolev spaces $W^{2\ell +2,p}(\mathbb{R}^{n})$ in terms of derivatives of ball averages. The novelty in the results of this article is that these equivalent characterizations reveal some new connections between the smoothness indices of Sobolev spaces and the derivatives on the radius of ball averages and also that, to obtain the corresponding results for higher order Sobolev spaces, the authors first establish the combinatorial equality: for any $\ell \in \mathbb{N}$ and $k\in \{0,\ldots ,\ell -1\}$, $\sum _{j=0}^{2\ell }(-1)^{j}\binom{2\ell }{j}|\ell -j|^{2k}=0$.

This paper studies the uncertainty principle for spherical $h$ -harmonic expansions on the unit sphere of ${{\mathbb{R}}^{d}}$ associated with a weight function invariant under a general finite reflection group, which is in full analogy with the classical Heisenberg inequality. Our proof is motivated by a new decomposition of the Dunkl–Laplace–Beltrami operator on the weighted sphere.

The problem of pricing arithmetic Asian options is nontrivial, and has attracted much interest over the last two decades. This paper provides a method for calculating bounds on option prices and approximations to option deltas in a market where the underlying asset follows a geometric Lévy process. The core idea is to find a highly correlated, yet more tractable proxy to the event that the option finishes in-the-money. The paper provides a means for calculating the joint characteristic function of the underlying asset and proxy processes, and relies on Fourier methods to compute prices and deltas. Numerical studies show that the lower bound provides accurate approximations to prices and deltas, while the upper bound provides good though less accurate results.

We establish estimates for restrictions to certain curves in ${{\mathbb{R}}^{2}}$ of the Fourier transforms of some fractal measures.

We prove that if the Fourier transform of a compactly supported measure is in ${{L}^{2}}$ of a half-space, then the measure is absolutely continuous to Lebesgue measure. We then show how this result can be used to translate information about the dimensionality of a measure and the decay of its Fourier transform into geometric information about its support.

Weconsider the Fourier restriction operators associated to certaindegenerate curves in ℝd for which the highest torsion vanishes. We prove estimates with respect to affine arclength and with respect to the Euclidean arclength measure on the curve. The estimates have certain uniform features, and the affine arclength results cover families of flat curves.

We prove optimal radially weighted L2-norm inequalities for the Fourier extension operator associated to the unit sphere in ℝn. Such inequalities valid at all scales are well understood. The purpose of this short paper is to establish certain more delicate single-scale versions of these.

For a wide family of multivariate Hausdorff operators, the boundedness of an operator from this family i s proved on the real Hardy space. By this we extend and strengthen previous results due to Andersen and Móricz.

The following problem has been suggested by Paul Turán. Let $\Omega $ be a symmetric convex body in the Euclidean space ${{\mathbb{R}}^{d}}$ or in the torus ${{\mathbb{T}}^{d}}$. Then, what is the largest possible value of the integral of positive definite functions that are supported in $\Omega $ and normalized with the value 1 at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in $\mathbb{R}$. That is, under the same conditions and normalizations, the supremum of possible function values at $z$ is to be found for any given point $z\,\in \,\Omega $. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to ${{\mathbb{R}}^{d}}$ and to non-convex domains as well.

Here we present another approach to the problem, giving the solution in ${{\mathbb{R}}^{d}}$ and for several cases in ${{\mathbb{T}}^{d}}$. Actually, we elaborate on the fact that the problem is essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for ${{\mathbb{R}}^{d}}$ and that for ${{\mathbb{T}}^{d}}$ is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition.

We extend uncertainty principles which are valid for the Fourier transform to the setting of the ambiguity function. A general result is established for annihilating sets: strongly/weakly annihilating sets for the Fourier transform yield such sets for the ambiguity function, extending a result known for sets of finite measure. We apply this to sublevel sets of nondegenerate quadratic forms. Our main result is a sharp version of Beurling's uncertainty principle for the ambiguity function.

We establish a sharp Fourier restriction estimate for a measure on a $k$-surface in ${{\mathbb{R}}^{n}}$, where $n\,=\,k\left( k\,+\,3 \right)/2$