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This note is about the observation that the various transition formulas between bases of trigonometric polynomials can be expressed in terms binomial coefficients. More specifically, we write the entries of the Chebyshev matrices $ T$ and $…

We represent a general bilinear Calder\'on-Zygmund operator as a sum of simple dyadic operators. The appearing dyadic operators also admit a simple proof of a sparse bound. In particular, the representation implies a so called sparse T1…

One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is $L^p$ bounded on such a manifold, for $p$ ranging in an open…

We detail an explicit construction of ordinary irreducible representations for the family of finite groups $SL_2({\mathbb Z} /p^n {\mathbb Z})$ for odd primes $p$ and $n\geq 2$. For $n=2$, the construction is a complete set of irreducible…

We introduce a complex q-Fourier transform as a generalization of the (real) one analyzed in [Milan J. Math. {\bf 76} (2008) 307]. By recourse to tempered ultradistributions we show that this complex plane-generalization overcomes all…

We study singular integral operators with kernels that are more singular than standard Calder\'on-Zygmund kernels, but less singular than bi-parameter product Calder\'on-Zygmund kernels. These kernels arise as restrictions to two dimensions…

A family of generalized Erdelyi-Kober type fractional integrals is interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of generalized Cassini ovaloids on $R^N$.…

We present a Riesz integral representation theory in which functions, operators and measures take values in uniform commutative monoids (a commutative monoid with a uniformity making the binary operation of the monoid uniformly continuous).…

A formula which expresses logarithmic energy of Borel measures on R^n in terms of the Fourier transforms of the measures is established and some applications are given. In addition, using similar techniques a (known) formula for Riesz…

We introduce fractional integrals on the $n$-dimensional spherical cap, study their boundednes in weighted $L^p$ spaces and obtain explicit inversion formulas. The results are applied to the inversion problem for Riesz potentials on a…

The aim of this paper is to show that, in various situations, the only continuous linear map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups $\Z/nZ$, the integers $\Z$,…

In this paper, we investigate the trigonometric Heckman-Opdam polynomials of type $A_1$. We establish connections with ultraspherical polynomials and derive an explicit expression for the associated Poisson kernel. Using the product…

In this paper we characterise the optimal pointwise size and regularity estimates for the Dunkl Riesz transform kernel involving both the Euclidean metric and the Dunkl metric, where these two metrics are not equivalent. We further…

We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which the Heisenberg group exhibits the simplest nontrivial example. With the language of the group Fourier Transform, we prove an operator-valued incarnation of the…

The spectral decomposition for an explicit second-order differential operator $T$ is determined. The spectrum consists of a continuous part with multiplicity two, a continuous part with multiplicity one, and a finite discrete part with…

Every polynomial of the form $P=(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable as Schur-Szeg\H{o} composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, where the numbers $a_i$ are unique up to permutation. We give…

Using Kakichev's classical concept and extending Yakubovich-Britvina's approach (\textit{Results. Math.} 55(1-2):175-197, 2009) and (\textit{Integral Transforms Spec. Funct.} 21(4):259--276, 2010) for setting up Kontorovich-Lebedev…

Given a real-analytic function b(x) defined on a neighborhood of the origin with b(0) = 0, we consider local convolutions with kernels which are bounded by |b(x)|^(-a), where a > 0 is the smallest number for which |b(x)|^(-a) is not…

Many areas of science and engineering encounter data defined on spherical manifolds. Modelling and analysis of spherical data often necessitates spherical harmonic transforms, at high degrees, and increasingly requires efficient computation…

We provide a new foundation for combinatorial commutative algebra and Stanley-Reisner theory using the partition complex introduced in [Adi18]. One of the main advantages is that it is entirely self-contained, using only a minimal knowledge…