The necessary conditions to transform hydroxyamide (1) and 2-oxazolines (2, 3) into 3- or 4-methyl benzo[f] isoquinolines (4, 5) has been established. The mechanism of these reactions has been proposed claiming the existence of the protonated 2-oxazoline (6) and the styrylamides (7) as intermediates.

[08.3] Compute e−πx 2 ∗ e−πx2 and sinx x ∗ sinx x . (Be careful what you assert: sin x x is not in L (R).) Discussion: The idea is to invoke f ∗ g = (f̂ · ĝ)̂ for even functions f, g ∈ L, since for even functions the inverse Fourier transform is the same as the forward Fourier transform. Conveniently, Gaussians are in L ∩ L, and have Fourier transforms which are again Gaussians: ̂ e−πax(ξ) = 1 √ a...

Harmonic analysis is at the heart of much of signal and image processing. This is mainly the harmonic analysis of abelian groups, and ultimately, after sampling, and quantizing, finite abelian groups. Nevertheless, this general group theoretic viewpoint has proved fruitful, yielding a natural group-based multiresolution framework as well as some new and potentially useful nonabelian examples. T...

Let (M, g) be a simple Riemannian manifold. Under the assumption that the metric g is real-analytic, it is shown that if the geodesic ray transform of a function f ∈ L(M) vanishes on an appropriate open set of geodesics, then f = 0 on the set of points lying on these geodesics. Using this result, a version of Helgason’s support theorem for the geodesic ray transform is proven. The approach is b...

The need to evaluate expressions of the form f(A) or f(A)b, where f is a nonlinear function, A is a large sparse n × n matrix, and b is an n-vector, arises in many applications. This paper describes how the Faber transform applied to the field of values of A can be used to determine improved error bounds for popular polynomial approximation methods based on the Arnoldi process. Applications of ...

The Cayley transform of A, F = (I+A)−1 (I−A), is studied when A is a P -matrix, an M -matrix, an inverse M-matrix, a positive definite matrix, or a totally nonnegative matrix. Given a matrix A in each of these positivity classes and using the fact that the Cayley transform is an involution, properties of the ensuing factorization A = (I+F )−1 (I−F ) are examined. Specifically, it is investigate...

This paper extends the Meijer transformation, M, given by I /2K 2p f(t) (pt) (2 p)dt, (Mf) (p) (i+) 0 where f belongs to an appropriate function space, e (-1,) and K is the modified Bessel function of third kind of order , to certain generalized functions. A testing space is constructed so as to contain the (pt)/2K(2p), of the transformation. Some properties of the kernel, function space and it...

A new approach has been proposed to reduce the computations involved in the ISAR imaging, which uses Harmonic Wavelet (HW) based Time-Frequency Representation (TFR). Since the HW based TFR falls into a category of non-parametric Time-Frequency (T-F) analysis tool, it is computationally efficient compared to parametric T-F analysis tools such as Adaptive Joint Time-Frequency Transform (AJTFT), A...

Since Boehmians were introduced, extensions of the Fourier transform to spaces of Boehmians attracted a lot of attention (see [2, 3, 4, 5, 6, 7, 8, 9]). In some cases, the range of the extended Fourier transform is a space of functions. In other constructions, the range is a space of distributions or a space of Boehmians. In this paper, we would like to consider the space of tempered Boehmians ...