1,720,999 research outputs found
The rate of convergence of expansions in Freud polynomials
No full textAbstractFor a function f of bounded variation on compact intervals, satisfying certain growth conditions, we estimate the rate of convergence of its expansion in a series of polynomials orthogonal on the whole real axis with respect to a weight function, now known as a Freud weight. The case where f has higher order derivatives of bounded variation is also studied. The principal techniques include the finite-infinite range inequalities due to the author and Saff, and Freud's theorems on one-sided weighted L1-approximation. Our theorem holds, in particular, when the weight function is exp( −xm), m a positive even integer
Polynomial operators and local smoothness classes on the unit interval
No full textAbstractWe obtain a characterization of local Besov spaces of functions on [-1,1] in terms of algebraic polynomial operators. These operators are constructed using the coefficients in the orthogonal polynomial expansions of the functions involved. The example of Jacobi polynomials is studied in further detail. A by-product of our proofs is an apparently simple proof of the fact that the Cesàro means of a sufficiently high integer order of the Jacobi expansion of a continuous function are uniformly bounded
Weighted polynomial approximation of entire functions, II
No full textAbstractNecessary and sufficient conditions are given for a function f defined almost everywhere on the real line to have an extension to the complex plane as an entire function of specified order and finite type. These conditions are in terms of the degree of approximation of f by polynomials in weighted Lp norms
A tribute to Géza Freud
No full textAbstractWe discuss some of the recent work in approximation theory motivated by the research of Géza Freud (1922–1979)
On the Representation of Band Limited Functions Using Finitely Many Bits
No full textAbstractIn this paper, we consider the question of representing an entire function of finite order and type in terms of finitely many bits, and reconstructing the function from these. Instead of making any further assumptions about the function, we measure the error in reconstruction in a suitably weighted Lp norm. The optimal number of bits in order to obtain a given accuracy is given by the Kolmogorov entropy. We determine this entropy in the case of certain compact subsets of these weighted Lp spaces and obtain constructive algorithms to determine the asymptotically optimal bit representation from finitely many samples of the function. Our theory includes both equidistant and non-uniform sampling. The reconstructions are polynomials, having several other optimality properties
Approximation in certain intermediate spaces
No full textAbstractA theorem of Bojanic gives a precise estimate on the rate of convergence of the Fourier series of a function of boundend variation. While the method of K-functionals is not directly applicable to obtain similar estimates for functions in classes intermediate to BV[−1, 1] and C[−1, 1]. we obtain such an estimate in the case of a general class of operators. The result is given in terms of an expression, which for continuous functions, is equivalent to the K-functional. As particular cases, we study the expansions in certain (general) orthogonal polynomials, Lagrange interpolation at the zeros of (general) orthogonal polynomials, and Hermite-Fejér interpolation at the zeros of generalized Jacobi polynomials. When applicable, our result (essentially) includes the previously known results, while many corollaries are new
Weighted quadrature formulas and approximation by zonal function networks on the sphere
No full textAbstractLet q≥1 be an integer, Sq be the unit sphere embedded in the Euclidean space Rq+1. A zonal function (ZF) network with an activation function φ:[-1,1]→R and n neurons is a function on Sq of the form x↦∑k=1nakφ(x·ξk), where ak's are real numbers, ξk's are points on Sq. We consider the activation functions φ for which the coefficients {φ^(ℓ)} in the appropriate ultraspherical polynomial expansion decay as a power of (ℓ+1)-1. We construct ZF networks to approximate functions in the Sobolev classes on the unit sphere embedded in a Euclidean space, yielding an optimal order of decay for the degree of approximation in terms of n, compared with the nonlinear n-widths of these classes. Our networks do not require training in the traditional sense. Instead, the network approximating a function is given explicitly as the value of a linear operator at that function. In the case of uniform approximation, our construction utilizes values of the target function at scattered sites. The approximation bounds are used to obtain error bounds on a very general class of quadrature formulas that are exact for the integration of high degree polynomials with respect to a weighted integral. The bounds are better than those expected from a straightforward application of the Sobolev embeddings
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