The methods for solving optimal filtering problems in the case of the classical stationary processes have been well known since the late 1940s. Practice often gives rise to what is not a classical stationary process but a generalized one, and white noise is one simple example. Hence, it is of interest to describe the system action on the generalized stationary processes, and then to carry over filtering methods to them. For arbitrary generalized stochastic processes this seems to be a challenging problem. In this correspondence, we identify a rather general class of S/sub J/-generalized stationary processes for which the desired extension can be done for matched filters. This class can be considered as a model of colored noise, and it is wide enough to include white noise, positive frequencies white noise, as well as certain generalized processes occurring in practice, namely, when the smoothing effect gives rise to the situation in which the distribution of probabilities may not exist at some time instances. One advantage of the suggested model is that it connects optimal filter design with inverting of integral operators; the methods for the latter can be found in the extensive literature.

Many important problems in pure and applied mathematics and engineering can be reduced to linear algebra on dense structured matrices. The structure of these dense matrices is understood in the sense that their n2 entries can be "compressed" to a smaller number O(n) of parameters. Operating directly on these parameters allows one to design efficient fast algorithms for these matrices. One of the most prominent matrix problems is that of multiplying a (structured) matrix with a vector. Many fundamental algorithms like convolution, Fast Fourier Transform, Fast Cosine/Sine Transform, and polynomial and rational multipoint evaluation and interpolation can be interpreted as superfast multiplication of a vector by structured matrices (like Toeplitz, DFT, Vandermonde, Cauchy). In this paper, we introduce a novel and fairly general class of structured matrices, which we call confluent Cauchy- like matrices, that contains all the above classes as a special case, and we will des a confluent Cauchy-like matrix by a vector. This is precisely what has been done in this paper

Using methods originating in numerical analysis, we will develop a unified framework for derivation of efficient list decoding algorithms for algebraicgeometric codes.We will demonstrate our method by accelerating Sudan's list decoding algorithm for Reed-Solomon codes [22], its generalization to algebraic-geometric codes by Shokrollahi and Wasserman [21], and the recent improvement of Guruswami and Sudan [8] in the case of Reed-Solomon codes.The basic problem we attack in this paper is that of efficiently finding nonzero elements in the kernel of a structured matrix.The structure of such an n x n-matrix allows it to be "compressed" to an parameters for some (Y which is usually a constant in applications.The concept of structure is formalized using the displacement operator.The displacement operator allows to perform matrix op erations on the compressed version of the matrix.In particular, we can find a PLU-decomposition of the original matrix in time O(cun'), which is quadratic in n for constant (Y.We will derive appropriate displacement operators for matrices that occur in the context of list decoding, and apply our general algorithm to them.For example, we will obtain algorithms that use O(&) and O(n7/3!) operations over the base field for list decoding of Reed-Solomon codes and algebraic-geometric codes from certain plane curves, respectively, where e is the length of the list.Assuming that e is constant, this gives algorithms of running time O(n') and O(n7/s), which is the same as the running time of conventional decoding algorithms.We will also sketch methods lo parallelize our algorithms.

The classical scalar Nevanlinna-Pick interpolation problem has a long and distinguished history, appearing in a variety of applications in mathematics and electrical engineering. There is a vast literature on this problem and on its various far reaching generalizations. It is widely known that the now classical algorithm for solving this problem proposed by Nevanlinna in 1929 can be seen as a way of computing the Cholesky factorization for the corresponding Pick matrix. Moreover; the classical Nevanlinna algorithm takes advantage of the special structure of the Pick matrix to compute this triangular factorization in only O(n/sup 2/) arithmetic operations, where n is the number of interpolation points, or equivalently, the size of the Pick matrix. Since the structure-ignoring standard Cholesky algorithm [though applicable to the wider class of general matrices] has much higher complexity O(n/sup 3/), the Nevanlinna algorithm is an example of what is now called fast algorithms. In this paper we use a divide-and-conquer approach to propose a new superfast O(n log/sup 3/ n) algorithm to construct solutions for the more general boundary tangential Nevanlinna-Pick problem. This dramatic speed-up is achieved via a new divide-and-conquer algorithm for factorization of rational matrix functions; this superfast algorithm seems to have a practical and theoretical significance itself. It can be used to solve similar rational interpolation problems [e.g., the matrix Nehari problem], and a variety, of engineering problems. It can also be used for inversion and triangular factorization of matrices with displacement structure, including Hankel-like, Vandermonde-like, and Cauchy-like matrices.