We present a parallel Fast Legendre Transform (FLT) based on the Driscol{Healy algorithm with computation complexity O(N log 2 N). The parallel FLT is load{ balanced in a hierarchical fashion. We use a load{balanced FFT to deduce a load{ balanced parallel fast cosine transform, which in turn serves as a building block for the Legendre transform engine, from which the parallel FLT is constructed...

We present the role that spectral methods play in the development of the most impressive quantum algorithms, such as the polynomial time number factoring algorithm by Shor. While the fast transform algorithms reduce the number of operations needed in obtaining the transforms from O(2) to O(n2), quantum transforms are in comparison super-fast. The Quantum Fourier Transform can be performed in O(...

A general analytical scheme for computing the n-fold convolution of exponentialsum distribution functions has been developed in this paper. The n-fold convolution is first expressed by multiple sums of recursive integrals. These recursive integrals are then reconstructed with a series of delta functions to avoid separations of integrations. Part of the recursive integrals has been solved analyt...

The Schur transform is a unitary operator that block diagonalizes the action of the symmetric and unitary groups on an n fold tensor product V ⊗n of a vector space V of dimension d. Bacon, Chuang and Harrow [5] gave a quantum algorithm for this transform that is polynomial in n, d and log ǫ, where ǫ is the precision. Following this, it had been an open question whether one can obtain an algorit...

Consider the Vandermonde-like matrix P := (Pk(cos jπ N ))j,k=0, where the polynomials Pk satisfy a three-term recurrence relation. If Pk are the Chebyshev polynomials Tk , then P coincides with CN+1 := (cos jkπ N )j,k=0. This paper presents a new fast algorithm for the computation of the matrixvector product Pa in O(N logN) arithmetical operations. The algorithm divides into a fast transform wh...

Disclaimer: These notes are intended to be an accessible introduction to the subject, with no pretense at completeness. In general, you can find more thorough discussions in Oppenheim's book. Please let me know if you find any typos. In this lecture, we discuss the Discrete Fourier Transform (DFT), which is a fourier representation for finite length signals. The main practical importance of thi...

In many of the numerical methods for pricing American options based on the dynamic programming approach, the most computationally intensive part can be formulated as the summation of Gaussians. Though this operation usually requires O NN ′ work when there are N ′ summations to compute and the number of terms appearing in each summation is N , we can reduce the amount of work to O N +N ′ by usin...

It is demonstrated that the classical Hough transform with shift-elevation parametrization of digital straight lines has additive complexity of at most O(n3/ logn) on a n× n image. The proof is constructive and uses ensemble computation approach to build summation circuits. The proposed method has similarities with the fast Hough transform (FHT) and may be considered a form of the “divide and c...

A class of growth functions u is introduced to construct Hida distributions and test functions. The Legendre transform lu of u is used to define a sequence α(n) = (lu(n)n!), n ≥ 0, of positive numbers. From this sequence we get a CKS-space. Under various conditions on u we show that the associated sequence {α(n)} satisfies those conditions for carrying out the white noise distribution theory on...

The graph Fourier transform (GFT) is in general dense and requires O(n) time to compute and O(n) memory space to store. In this paper, we pursue our previous work on the approximate fast graph Fourier transform (FGFT). The FGFT is computed via a truncated Jacobi algorithm, and is defined as the product of J Givens rotations (very sparse orthogonal matrices). The truncation parameter, J , repres...