# Fast Discrete Polynomial Transforms with Applications to Data Analysis for Distance Transitive Graphs

@article{Driscoll1997FastDP, title={Fast Discrete Polynomial Transforms with Applications to Data Analysis for Distance Transitive Graphs}, author={James R. Driscoll and Dennis M. Healy and Daniel N. Rockmore}, journal={SIAM J. Comput.}, year={1997}, volume={26}, pages={1066-1099} }

Let $\poly = \{P_0,\dots,P_{n-1}\}$ denote a set of polynomials with complex coefficients. Let $\pts = \{z_0,\dots,z_{n-1}\}\subset \cplx$ denote any set of {\it sample points}. For any $f = (f_0,\dots,f_{n-1}) \in \cplx^n$, the {\it discrete polynomial transform} of $f$ (with respect to $\poly$ and $\pts$) is defined as the collection of sums, $\{\fhat(P_0),\dots,\fhat(P_{n-1})\}$, where $\fhat(P_j) = \langle f,P_j \rangle = \sum_{i=0}^{n-1} f_iP_j(z_i)w(i)$ for some associated weight function…

## Figures from this paper

## 107 Citations

### Sparse Recovery for Orthogonal Polynomial Transforms

- Computer Science, MathematicsICALP
- 2020

Sublinear-time algorithms are given for solving the sparse recovery problem for orthogonal transforms $\vF$ that arise from Orthogonal polynomials, and the algorithm works for any $\vf$ that is an orthogona polynomial transform derived from Jacobi polynmials.

### Complexity of the Fourier transform on the Johnson graph

- Mathematics, Computer Science
- 2017

The proof is based on a restricted version of the Robinson-Schensted algorithm based on the construction of $n-1$ intermediate bases, each one parametrized by certain pairs composed by a standard Young tableau and a word.

### A Two Pronged Progress in Structured Dense Matrix Multiplication

- Computer Science, Mathematics
- 2016

The notion of recurrence width of matrices is introduced, which is finer than all the above classes of structured matrices and thus it can compute multiplication for all of them using the same core algorithm.

### Calcul formel dans la base des polynômes unitaires de Chebyshev

- Mathematics
- 2015

Nous proposons des methodes simples et efficaces pour manipuler des expressions trigonometriques de la forme $F=\sum_{k} f_k\cos\tfrac{k\pi}{n}, f_k\in Z$ ou $d<n$ fixe. Nous utilisons les polynomes…

### Computing with Expansions in Gegenbauer Polynomials

- Computer ScienceSIAM J. Sci. Comput.
- 2009

Efficient methods are obtained for the evaluation of expansions at prescribed nodes, and for the projection onto a sequence of Gegenbauer polynomials from given function values, respectively, using (nonequispaced) discrete cosine transforms.

### Algebraic Signal Processing Theory: Cooley-Tukey-Type Algorithms for Polynomial Transforms Based on Induction

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2011

Novel O(nlogn) general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4 are derived by decomposing the regular modules of these algebras as a stepwise induction.

### Rapid Evaluation of Nonreflecting Boundary Kernels for Time-Domain Wave Propagation

- Computer ScienceSIAM J. Numer. Anal.
- 2000

The main technical result in this paper is that the logarithmic derivative of the Hankel function H_\nu(1)(z) can be approximated in the upper half of the z-plane with relative error $\varepsilon$ by a rational function of degree d.

### The Algebraic Approach to the Discrete Cosine and Sine Transforms and Their Fast Algorithms

- Computer Science, MathematicsSIAM J. Comput.
- 2003

This paper presents an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated with four series of Chebyshev polynomials.

## References

SHOWING 1-10 OF 51 REFERENCES

### Classical Orthogonal Polynomials of a Discrete Variable

- Mathematics
- 1991

The basic properties of the polynomials p n (x) that satisfy the orthogonality relations
$$ \int_a^b {{p_n}(x)} {p_m}(x)\rho (x)dx = 0\quad (m \ne n) $$
(2.0.1)
hold also for the polynomials…

### Computing Fourier Transforms and Convolutions on the 2-Sphere

- Computer Science, Mathematics
- 1994

Convolution theorems generalizing well known and useful results from the abelian case are used to develop a sampling theorem on the sphere, which reduces the calculation of Fourier transforms and convolutions of band-limited functions to discrete computations.

### On the construction of Gaussian quadrature rules from modified moments.

- Mathematics, Computer Science
- 1970

An algorithm for solving the problem of constructing Gaussian quadrature rules from 'modified moments' is derived, which generalizes one due to Golub and Welsch.

### Fast Fourier transform and convolution algorithms

- Computer Science
- 1981

This book explains the development of the Fast Fourier Transform Algorithm and its applications in Number Theory and Polynomial Algebra, as well as some examples of its application in Quantization Effects.

### Fast transforms and sampling for compact groups

- Mathematics
- 1993

In this thesis we first develop a sampling theory for functions on compact groups and sections of vector bundles associated with these groups. We construct specific finitely supported distributions…

### Eecient Computation of Isotypic Projections for the Symmetric Group

- Mathematics
- 1993

Spectral analysis on the symmetric group Sn calls for computing projections of functions deened on Sn and its homogeneous spaces, onto invariant subspaces. In particular, for the analysis of…

### Algorithms for Discrete Fourier Transform and Convolution

- Mathematics
- 1989

Contents: Introduction to Abstract Algebra.- Tensor Product and Stride Permutation.- Cooley-Tukey FFF Algorithms.- Variants of FFT Algorithms and Their Implementations.- Good-Thomas PFA.- Linear and…

### Addition Requirements for Matrix and Transposed Matrix Products

- Computer Science, MathematicsJ. Algorithms
- 1988