 Abstract—Discrete trigonometric transformations, such as the discrete Fourier and cosine/sine transforms, are important in a variety of applications due to their useful properties. For example, one well-known property is the convolution theorem for Fourier transform. In this letter, we derive a discrete Fresnel transform (DFnT) from the infinitely periodic optical gratings, as a linear trigon...

In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a proper étale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a noncommutative Poisson homology whose computation enables us t...

We produce, on general homogeneous groups, an analogue of the usual Hörmander pseudodifferential calculus on Euclidean space, at least as far as products and adjoints are concerned. In contrast to earlier works, we do not limit ourselves to analogues of classical symbols, nor to the Heisenberg group. The key technique is to understand “multipliers” of any given order j, and the operators of con...

We study algorithms for computing the convolution of a private input x with a public input h, while satisfying the guarantees of (ε, δ)-differential privacy. Convolution is a fundamental operation, intimately related to Fourier Transforms. In our setting, the private input may represent a time series of sensitive events or a histogram of a database of confidential personal information. Convolut...

Computing discrete two-dimensional convolutions is an important problem in image processing. In mathematical morphology, an important variant is that of computing binary convolutions, where the kernel of the convolution is a 0{1 valued function. This operation can be quite costly, especially when large kernels are involved. In this paper, we present an algorithm for computing convolutions of th...

Meghanad D. Wagh * By making an example of the earlier proposed cyclic convolution algorithms, the computational efficiency of the multidimensional techniques over finite fields is investigated. It is shown that the multidimensional techniques are inferior to the directly designed algorithms for all lengths except when applied to lengths whose exponents are relatively prime. Relations between t...

This note presents a new algorithm for computing the cyclic convolution of two vectors over a commutative ring. The algorithm requires n(nj+l) •.. (n^+l)^^ multiplications for the convolution of two n-vectors, where n = n^...n^ is a factorization of n into factors which are pairwise relatively prime.

Debbah and Ryan have recently [DR07] proved a result about the limit empirical singular distribution of the sum of two rectangular random matrices whose dimensions tend to infinity. In this paper, we reformulate it in terms of the rectangular free convolution introduced in [BG07b] and then we give a new, shorter, proof of this result under weaker hypothesis: we do not suppose the probability me...

The exact distributions of the different salt transport systems along the human cortical distal nephron are unknown. Immunohistochemistry was performed on serial cryostat sections of healthy parts of tumor nephrectomized human kidneys to study the distributions in the distal convolution of the thiazide-sensitive Na-Cl cotransporter (NCC), the beta subunit of the amiloride-sensitive epithelial N...

A routine dose computation around brachytherapy seeds employing the Fast Fourier Transform (FFT) has been demonstrated and likened to dose computations in a similar field based on the TG43 recommendations. In applying the FFT convolution model to dose computation, a dose convolution kernel which takes into consideration the anisotropy of the dose distribution around a brachytherapy source in th...