Related papers: The Arithmetic Cosine Transform: Exact and Approxi…
Approximate methods have been considered as a means to the evaluation of discrete transforms. In this work, we propose and analyze a class of integer transforms for the discrete Fourier, Hartley, and cosine transforms (DFT, DHT, and DCT),…
Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained. No series commutations are involved in the procedure; nevertheless the…
The discrete cosine transform (DCT) is a relevant tool in signal processing applications, mainly known for its good decorrelation properties. Current image and video coding standards -- such as JPEG and HEVC -- adopt the DCT as a…
An orthogonal approximation for the 8-point discrete cosine transform (DCT) is introduced. The proposed transformation matrix contains only zeros and ones; multiplications and bit-shift operations are absent. Close spectral behavior…
The Fourier transform is approximated over a finite domain using a Riemann sum. This Riemann sum is then expressed in terms of the discrete Fourier transform, which allows the sum to be computed with a fast Fourier transform algorithm more…
Code translation is a crucial process in software development and migration projects, enabling interoperability between different programming languages and enhancing software adaptability and thus longevity. Traditional automated…
This paper introduces a new fast algorithm for the 8-point discrete cosine transform (DCT) based on the summation-by-parts formula. The proposed method converts the DCT matrix into an alternative transformation matrix that can be decomposed…
Arithmetic complexity has a main role in the performance of algorithms for spectrum evaluation. Arithmetic transform theory offers a method for computing trigonometrical transforms with minimal number of multiplications. In this paper, the…
We provide a basis transformation that inverts the coordinate Bethe Ansatz. It is widely believed that the Bethe Ansatz is complete, based on numerical evidence and combinatorial arguments. We present a constructive and comprehensive…
We present efficient methods to interpolate data with a quantum computer that complement uploading techniques and quantum post-processing. The quantum algorithms are supported by the efficient Quantum Fourier Transform (QFT) and classical…
We express the Riemann zeta function $\zeta\left(s\right)$ of argument $s=\sigma+i\tau$ with imaginary part $\tau$ in terms of three absolutely convergent series. The resulting simple algorithm allows to compute, to arbitrary precision,…
Due to its remarkable energy compaction properties, the discrete cosine transform (DCT) is employed in a multitude of compression standards, such as JPEG and H.265/HEVC. Several low-complexity integer approximations for the DCT have been…
This paper studies the cosine as basis function for the approximation of univariate and continuous functions without memory. This work studies a supervised learning to obtain the approximation coefficients, instead of using the Discrete…
We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually…
The Arithmetic Fourier Transform is a numerical formulation for computing Fourier series and Taylor series coefficients. It competes with the Fast Fourier Transform in terms of speed and efficiency, requiring only addition operations and…
Starting from the classical integral representation of the $\zeta(s)$ function introduced by Riemann in 1859, this paper reexamines its analytic symmetry structure. By performing a geometric decomposition of the integral representation, we…
A practical method to compute the Riemann zeta function is presented. The method can compute $\zeta(1/2+it)$ at any $\lfloor T^{1/4} \rfloor$ points in $[T,T+T^{1/4}]$ using an average time of $T^{1/4+o(1)}$ per point. This is the same…
We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the second author's fast algorithm for numerically evaluating…
With the integration of communication and computing, it is expected that part of the computing is transferred to the transmitter side. In this paper we address the general problem of Frequency Modulation (FM) for function approximation…
This paper introduces Adaptive Computation Time (ACT), an algorithm that allows recurrent neural networks to learn how many computational steps to take between receiving an input and emitting an output. ACT requires minimal changes to the…