Related papers: Analytic and Group-Theoretic Aspects of the Cosine…
The Funk, cosine, and sine transforms on the unit sphere are indispensable tools in integral geometry. They are also known to be interesting objects in harmonic analysis. The aim of the paper is to extend basic facts about these transforms…
We introduce a new family of invariant differential operators associated with $\lambda$-cosine and Funk-Radon transforms on Stiefel and Grassmann manifolds. These operators reduce the order of the $\lambda$-cosine transforms and yield new…
We investigate analytic continuation of the matrix cosine and sine transforms introduced in Part I and depending on a complex parameter $\a$. It is shown that the cosine transform corresponding to $\a=0$ is a constant multiple of the…
The cosine transforms of functions on the unit sphere play an important role in convex geometry, the Banach space theory, stochastic geometry and other areas. Their higher-rank generalization to Grassmann manifolds represents an interesting…
We use the classical Fourier analysis to introduce analytic families of weighted differential operators on the unit sphere. These operators are polynomial functions of the usual Beltrami-Laplace operator. New inversion formulas are obtained…
We introduce a new integral transform $T^\lam f$, $\lam \in C^m$, on the Stiefel manifold of orthonormal $m$-frames in $R^n$ which generalizes the $\lam$-cosine transform on the Grassmann manifold of $m$-dimensional linear subspaces of…
We consider functions $f$ of two real variables, given as trigonometric functions over a finite set $F$ of frequencies. This set is assumed to be closed under rotations in the frequency plane of angle $\frac{2k\pi}{M}$ for some integer $M$.…
In this paper we study algebraic structures of the classes of the $L_2$ analytic Fourier-Feynman transforms on Wiener space. To do this we first develop several rotation properties of the generalized Wiener integral associated with Gaussian…
The Fourier transform, known in classical analysis, and generalized in abstract harmonic analysis, can also be considered in the theory of locally compact quantum groups. In this note, I discuss some aspects of this more general Fourier…
The main objective of this work is to develop a framework for Fourier analysis on the group of signatures, $G_N(\mathbb{R}^d)$. Employing Kirillov's orbit method, we define the Fourier transform on this group via irreducible unitary…
The cosine-$\lambda$ transform, denoted $\mathcal{C}^\lambda$, is a family of integral transforms we can define on the sphere and on the Grassmann manifolds $\textrm{Gr}(p, \mathbb{K}^n) =…
On real metric manifolds admitting a co-dimension one foliation, sectorial operators are introduced that interpolate between the generalized Laplacian and the d'Alembertian. This is used to construct a one-parameter family of analytic…
We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point $A$ which lies inside the n-dimensional unit sphere or on the sphere itself.…
The spherical Radon transform on the unit sphere can be regarded as a member of the analytic family of suitably normalized generalized cosine transforms. We derive new formulas for these transforms and apply them to study classes of…
We continue the study of the Hrushovski-Kazhdan integration theory and consider exponential integrals. The Grothendieck ring is enlarged via a tautological additive character and hence can receive such integrals. We then define the Fourier…
The covariantization procedure is usually referred to the translation operator, that is the derivative. Here we introduce a general method to covariantize arbitrary differential operators, such as the ones defining the fundamental group of…
Classical Fourier analysis has an exact counterpart in group theory and in some areas of geometry. Here I'll describe how this goes for nilpotent Lie groups and for a class of Riemannian manifolds closely related to a nilpotent Lie group…
In this paper we introduce the concept of a convolution type operation of functionals on Wiener space. It contains several kinds of the concepts of convolution products on Wiener space, which have been studied by many authors. We then…
In this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple Lie groups by showing that the $Cos^\lamda$ and $Sin^\lambda$-transforms on the Grassmann manifolds…
Many stochastic processes are defined on special geometrical objects like spheres and cones. We describe how tools from harmonic analysis, i.e. Fourier analysis on groups, can be used to investigate probability density functions (pdfs) on…