On the real line, let the Fourier transform of kn be k̂n(ξ) = k̂(ξ−n) where k̂(ξ) is a smooth compactly supported function. Consider the bilinear operators Sn(f, g)(x) = ∫ f(x + y)g(x − y)kn(y) dy. If 2 ≤ p, q ≤ ∞, with 1/p + 1/q = 1/2, I prove that ∞ ∑ n=−∞ ‖Sn(f, g)‖2 ≤ C‖f‖p‖g‖q . The constant C depends only upon k.

where f is an unknown function, and the ei's are noise. Our goal is to recover f using wavelet methods. We shall assume that n = 2 is an integer power of 2, and consider both the cases of independent and correlated noise. Let f = (f0; : : : ; fn 1) T , y = (y0; : : : ; yn 1) T and W be the discrete wavelet transform (DWT) matrix (e.g., see Donoho & Johnstone 1994). A wavelet{based estimate f̂ of...

This document provides proofs of the following: • The binomial distribution can be approximated with a Gaussian distribution at large values of N . • The arcsin square-root transform is the variance stabilising transform for the binomial distribution. • The Gaussian approximation for the binomial distribution is more accurate at smaller values of N after the variance stabilising transform has b...

The fundamental concept in the theory of fuzzy transform (F-transform) is that of fuzzy partition. The original definition assumes that each two fuzzy subsets overlap in such a way that sum of membership degrees in each point is equal to 1. However, this condition can be generalized to obtain a denser fuzzy partition that leads to improvement of approximation properties of Ftransform. However, ...

Fourier transform Consider the interval [0, λ] and suppose that we identify the point λ with 0 (i.e., think of it as a Torus and work modulo λ). Another way to think about this is as the basic cell of the lattice λZ, whose dual is the lattice (1/λ)Z. A periodic function on this torus has to period length of the form λ/n for an integer n. Thus, the Fourier transform of a function on this torus i...

As well known, spacelike proton form factors expressed in the Breit frame may be interpreted as the Fourier transform of static space distributions of electric charge and current. In particular, the electric form factor is simply the Fourier transform of the charge distribution F (q) = ∫ e ρ(r)dr. We don’t have an intuitive interpretation of the same level of simplicity for the proton timelike ...

The tensor representation is an effective way to reconstruct the image from a finite number of projections, especially, when projections are limited in a small range of angles. The image is considered in the image plane and reconstruction is in the Cartesian lattice. This paper introduces a new approach for calculating the splitting-signals of the tensor transform of the discrete image f(xi, yj...

Fourier transform Consider the interval [0, λ] and suppose that we identify the point λ with 0 (i.e., think of it as a Torus and work modulo λ). Another way to think about this is as the basic cell of the lattice λZ, whose dual is the lattice (1/λ)Z. A periodic function on this torus has to period length of the form λ/n for an integer n. Thus, the Fourier transform of a function on this torus i...

It was shown [ 21 that any SFF can be decomposed in this manner. Thus, F(x) is an SFF if, and only if, it can be expressed as the sum of four functions in the form of the above equation. Additional SFF studies are reported in refs. [ 3-51. Another issue that has been recently investigated is the fractional Fourier transform [ 6-91. Two distinct definitions of the fractional Fourier transform ha...

We propose a matrix approach to the computation of BattleLemarié's wavelets. The Fourier transform of the scaling function is the product of the inverse F(x) of a square root of a positive trigonometric polynomial and the Fourier transform of a B-spline of order m . The polynomial is the symbol of a bi-infinite matrix B associated with a B-spline of order 2m . We approximate this bi-infinite ma...