Plancherel formula for the attenuated Radon transform

The distribution-theoretic version of the Plancherel formula ― know as the Penney-Fujiwara Plancherel Formula ― for the decomposition of the quasi-regular representation of a Lie group G on L 2 (G/H) is considered. Attention is focused on the case that the spectrum consists of irreducible representations induced from a finite-dimensional representation. This happens with great regularity for Strichartz homogeneous spaces wherein G and H are semidirect products of normal abelian subgroups by a reductive Lie group. The results take an especially simple form if G/H is symmetric. Criteria for finite multiplicity and for multiplicity-free spectrum are developed

Ten series of matrix integrals (over non-compact Riemannian symmetric spaces) imitating the standard beta function are constructed. This is a broad generalization of Hua Loo Keng's integrals (1958) and Gindikin's B-integrals (1964). As a consequence Plancherel's formula for the Berezin kernel representations of classical groups is obtained in explicit form. Matrix models of non-compact Riemannian symmetric spaces are also discussed.

A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpinski gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blowups of a Sierpinski gasket, where spectrum is pure point of infinite multiplicity. Both works used the method of spectral decimations to obtain explicit description of the eigenvalues and eigenfunctions. In this paper we combine the ideas from these earlier works to obtain a description of the spectral resolution of the Laplacian for noncompact fractafolds. Our main abstract results enable us to obtain a completely explicit description of the spectral resolution of the fractafold Laplacian. For some specific examples we turn the spectral resolution into a "Plancherel formula". We also present such a formula for the graph Laplacian on the 3-regular tree, which appears to be a new result of independent interest. In the end we discuss periodic fractafolds and fractal fields.

The Plancherel formula for line bundles on complex hyperbolic spaces

Harmonic analysis as spectral theory of Laplacians

La formule de Plancherel pour les groupes de Lie presque algébriques réels

The Plancherel formula for countable groups

Kalgebra ~ , and G is the corresponding compact form. We shall not discuss this here. In the present article we shall study unitary representations of the group ~ : we shall calculate their character (section 3 below), the Plancherel measure on the group (section 4) and on a symmetric space (section 5), we shall study zonal harmonics for r epresentations of class 1 (section 6), and we shall consider a problem of integral geometry in Euclidean space which arises from representations of the group ~ (section 7). The formulas for the representations of the groups 5 , G, G K have much in common. The questions listed above for the groups G K are studied in the classical works of G. Weyl; for the groups G many of them have not yet been completely answered(Planche rel's formula integral geometry on symmetric spaces of negative curvature).

An Algebraic Framework for Group Duality

In this work, we introduce the two-sided Quaternion Fourier transform and main properties. Using the decomposition of quaternion, we propose a new form of Plancherel's formula related to transformation. We also obtain a form of convolution theorem associated with the two-sided Quaternion Fourier Trans-form.

For the space of all lattices in an -dimensional -adic linear space an analogue of the matrix beta function is constructed; this beta function can degenerate to the Tamagawa zeta function. An analogue of Berezin kernels for is proposed. Conditions for the positive-definiteness of these kernels and an explicit Plancherel's formula are obtained.

The quadratic‐phase Fourier transform (QPFT) is a neoteric addition to the class of integral transforms and embodies a variety of signal processing tools like the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this paper, we generalize the quadratic‐phase Fourier transform to quaternion‐valued signals, known as the quaternion quadratic‐phase Fourier transform (Q‐QPFT). We initiate our investigation by studying the QPFT of 2D quaternionic signals, and later on, we introduce the Q‐QPFT of 2D quaternionic signals. Using the fundamental relationship between the Q‐QPFT and quaternion Fourier transform (QFT), we derive the inversion, Parseval's, and Plancherel's formulae associated with the Q‐QPFT. Some other properties including linearity, shift, and modulation of the Q‐QPFT are also studied. Finally, we formulate several classes of uncertainty principles (UPs) for the Q‐QPFT like Heisenberg‐type UP, logarithmic UP, Hardy's UP, Beurling's UP, and Donoho–Stark's UP. This study can be regarded as the first step in the applications of the Q‐QPFT in the real world.

We show that for any f square-integrable on a separable unimodular locally compact group G (relative to Haar measure), the integral over G of the square of the absolute value of f equals the integral over a certain measure space of the square of the (at each point of the space) of the Fourier transform of f. By relative norm we mean that defined by von Neumann [5] for factors, and for the present we define the Fourier transform thru the use of the von Neumann reduction theory [6]. The formula which is obtained here has as instances, though not as direct corollaries, the formula of Plancherel, its generalization to separable locally compact abelian groups, the Peter-Weyl theorem, and a formula recently obtained by Gelfand and Neumark [2] for the case of the Lorentz group. The measure space in question has its measure ring isomorphic to the Boolean ring B of closed linear manifolds in L2(G) invariant under both left and right translations, and the corresponding measure is uniquely determined, modulo normalization of the at each point of the space. This measure ring acts as a kind of measure-theoretic to the group. For example, making the so-called standard normalization of the norm, if G is abelian, then B as a measure ring is abstractly identical with the measure ring of the character group of G, under Haar measure; if G is compact, then B is abstractly the measure ring of a discrete set of points, the set being in one-to-one correspondence with the collection of equivalence classes of continuous irreducible representations of G, the measure of a point being proportional to the degree of the corresponding representation. Basic in our work is a certain countably-additive non-negative function, which we call the dual of G, defined on the lattice of all projections in the weak closure V of the algebra generated by left translations in L2(G),-or alternatively, on the lattice of closed linear manifolds in L2(G) invariant under right translations. The gage, which is defined in an intrinsic fashion, is invariant under transformation by unitary operators in V, and is in fact a weight function in the terminology of von Neumann [6]. His reduction theory consequently yields a representation of the gage as an integral over the measure space described above of constituents (relative dimension functions) arising from factors. This representation together with the facts that the convolution operator on L2(G) defined by a self-adjoint element of L2(G) is hypermaximal symmetric (Ambrose [1]) and that every bounded linear operator on L2(G) which commutes with all right translations is in V (Segal [4]), are the principal known results used in our derivation of the generalized Plancherel formula.

For the space Lat{sub n} of all lattices in an n-dimensional p-adic linear space an analogue of the matrix beta function is constructed; this beta function can degenerate to the Tamagawa zeta function. An analogue of Berezin kernels for Lat{sub n} is proposed. Conditions for the positive-definiteness of these kernels and an explicit Plancherel's formula are obtained.

One of the difficult points in the proof of Harish-Chandra's Plancherel formula for spherical functions on a semisimple Lie group is to show that an appropriate inversion formula exists for the Fourier transform f f, and that this inversion formula holds for sufficiently many functions in the space of spherical square-integrable functions on G. Briefly, let f be a square integrable spherical function on G for which the Fourier transform f(X)= 5f(x)cp_(x)dx is well-defined. Here, as in [5], qA is the elementary (zonal) G spherical function corresponding to the parameter X. The problem is to show that for f in a L2-dense subspace of square-integrable spherical functions, the following inversion formula holds: