The connection between a Taylor series and a continued-fraction involves a nonlinear relation between the Taylor coefficients {an} and the continued-fraction coefficients {bn}. In many instances it turns out that this nonlinear relation transforms a complicated sequence {an} into a very simple one {bn}. We illustrate this simplification in the context of graph combinatorics. PACS numbers: 02.90...

This paper presents an algebraic approach to construct M-band orthonormal wavelet bases with perfect reconstruction. We first derive the system of constraint equations of M-band filter banks, and then an algebraic solution based on matrix decomposition is developed. The structure of the solutions is presented, and practical construction procedures are given. By using this algebraic approach, so...

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform f̂ of continuous or generalized functions f . The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of f̂ evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sens...

The classical Fourier transform Fcl is an isomorphism of the Schwartz space S(Rk) onto itself. The space C∞ c (Rk) of smooth functions with compact support is dense in S(Rk), and the classical Paley-Wiener theorem characterises the image of C∞ c (R k) under Fcl as rapidly decreasing functions having an holomorphic extension to Ck of exponential type. Since Rk is self-dual, the same theorem also...

The Hilbert transform is essentially the only singular operator in dimension 1. This undoubtedly makes it one of the the most important linear operators in harmonic analysis. The Hilbert transform has had a profound bearing on several theoretical and physical problems across a wide range of disciplines; this includes problems in Fourier convergence, complex analysis, potential theory, modulatio...

) ( ) ( 2 q q x x x Tr x f + + = and A = {ai} where ) ( i i f a α = . In reference [1], the sequence A is conjectured to have a two-level autocorrelation for all n > 1 and this conjecture has been verified for 23 3 ≤ ≤ n . In this note, we present an interesting formula for the Hadamard transform of the sequence A or, equivalently, for the function f(x). The Hadamard transform of f(x) is define...

A method for derivation of inversion formulas for the Radon transform of f (x) is given. This method allows one to construct functions ψ(x) which are easy to compute and which have the same wave fronts as f (x). This, in turn, allows one to calculate the singular support S of f (x) by computing ψ(x) given tomographic data. A simple geometrical relation between the singular supports of f andˆf ,...

We give an explicit formula (i.e., an explicit expression for the formal stationary phase formula) for the local Fourier-Laplace transform of a formal germ of meromorphic connection of one complex variable with a possibly irregular singularity. Introduction We will denote by C((t)) (resp. C({t})) the field C[[t]][t] (resp. C{t}[t]) of formal (resp. convergent) Laurent series of the variable t, ...

It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f . However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f and, in particular, identify the wavefront set...