In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of either a function defined on a finite abelian group, or a bandlimited function on a compact abelian group. We discuss generalizations of the FFT to arbitrary finite groups and compact Lie groups.

We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log(1/ε)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2 n with error bounded by ε. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n/ε)). We also give an upper bound of O(n(log n) 2 log log n) on the circuit size of the exact QFT modulo 2 n, for which the best previous bound was O(n 2). As an application of the above depth bound, we show that Shor’s factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomial-size, in combination with classical polynomial-time pre- and post-processing. In the language of computational complexity, this implies that factoring is in the complexity class ZPP BQNC, where BQNC is the class of problems computable with bounded-error probability by quantum circuits with polylogarithmic depth and polynomial size. Finally, we prove an Ω(log n) lower bound on the depth complexity of approximations of the

This paper presents a chronological overview of the developments in interpolation theory, from the earliest times to the present date. It brings out the connections between the results obtained in different ages, thereby putting the techniques currently used in signal and image processing into historical perspective. A summary of the insights and recommendations that follow from relatively recent theoretical as well as experimental studies concludes the presentation. Keywords—Approximation, convolution-based interpolation, history, image processing, polynomial interpolation, signal processing, splines. “It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it. ” 1 I.

. Generalized FFTs are efficient algorithms for computing a Fourier transform of a function defined on finite group, or a bandlimited function defined on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applications. This paper will attempt to survey some of these applications. Appendices include some more detailed examples. 1. A brief history The now "classical" Fast Fourier Transform (FFT) has a long and interesting history. Originally discovered by Gauss, and later made famous after being rediscovered by Cooley and Tukey [21], it may be viewed as an algorithm which efficiently computes the discrete Fourier transform or DFT. In between Gauss and Cooley-Tukey others developed special cases of the algorithm, usually motivated by the need to make efficient data analysis of one sort or another. To cite but a few examples, Gauss was interested in efficiently interpolating the orbits of asteroids [43...

Abstract: We discuss the fundamental role of entanglement as the essential nonclassical feature providing the computational speedup in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform algorithm. We describe the implementation of the FFT algorithm for the group of integers modulo 2 n in the quantum context, showing how the group-theoretic formalism leads to the standard quantum network and identifying the property of entanglement that gives rise to the exponential speedup (compared to the classical FFT). Finally we outline the use of the Fourier transform in extracting periodicities, which underlies its utility in the known quantum algorithms.

We discuss the fundamental role of entanglement as the essential non-classical feature providing the computational speed-up in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform (FFT) algorithm. We describe the implementation of the FFT algorithm for the group of integers modulo 2 n in the quantum context, showing how the group-theoretic formalism leads to the standard quantum network, and identify the property of entanglement that gives rise to the exponential speed-up (compared to the classical FFT). Finally, we outline the use of the Fourier transform in extracting periodicities, which underlies its utility in the known quantum algorithms.

The Fast Fourier Transform (FFT) is one of the most widely used digital signal processing algorithms. While advances in semiconductor processing technology have enabled the performance and integration of FFT processors to increase steadily, these advances have also caused the power consumed by processors to increase as well. This power increase has resulted in a situation where the number of potential FFT applications limited by maximum power budgets - not performance - is significant and growing.

This paper attracted much attention while a similar result obtained by William Karush in his Master's Thesis in 1939 [154] under the supervision of Lawrence M. Graves at the University of Chicago and by Fritz John (1910-1995) in 1948 [147] were almost totally ignored (John's paper was even rejected)

An announcement of the library was given in [7] in 1994. Since then the functionality of the library has been substantially extended. In this paper we will refer only briefly to the topics covered in [7] and concentrate on more recent developments. A series of working notes is available on the WWW at