Convergence of Newton-Cotes quadratures for analytic functions

We determine (Theorem 3) the smallest closed region, containing the interva of integration, such that the analyticity of the integrand in this closed region implies the convergence of the Newton-Cotes quadratures. By considering, in particular, certain ellipses as regions of analyticity, we obtain (Theorem 4) an improvement of Davis' result on the convergence of Newton-Cotes quadratures for analytic functions.     

Error estimates for Gaussian quadratures of analytic functions

For analytic functions the remainder term of Gaussian quadrature formula and its Kronrod extension can be represented as a contour integral with a complex kernel. We study these kernels on elliptic contours with foci at the points ±1 and the sum of semi-axes ?>1 for the Chebyshev weight functions of the first, second and third kind, and derive representation of their difference. Using this representation and following Kronrod’s method of obtaining a practical error estimate in numerical integration, we derive new error estimates for Gaussian quadratures.     

Hilbert spaces for estimating errors of quadratures for analytic functions

For estimating the error of a Gaussian quadrature by Davis' method, we investigate here the possibility of using a Hilbert space resulting from the weighted line integral inner product with the weight used in the quadrature formula itself.     

Optimal quadrature for analytic functions

Let G be a domain in the complex plane, which is symmetric with respect to the real axis and contains [−1,1]. For a measure τ on [−1,1] satisfying a regularity condition, we determine the geometric rate of the error of integration, measured uniformly on the class of functions analytic in G and bounded by 1, if the τ-integrals are replaced by optimal interpolatory quadrature formulas with n nodes. We show that this rate is obtained for modified Gauss-quadrature formulas with respect to certain varying weights.     

Optimal information for approximating periodic analytic functions

Let be a strip in the complex plane. For fixed integer let denote the class of -periodic functions , which are analytic in and satisfy in . Denote by the subset of functions from that are real-valued on the real axis. Given a function , we try to recover at a fixed point by an algorithm on the basis of the information

where , are the Fourier coefficients of . We find the intrinsic error of recovery

Furthermore the -dimensional optimal information error, optimal sampling error and -widths of in , the space of continuous functions on , are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class , consisting of all -periodic functions, which are analytic in with -integrable boundary values. In the case sampling fails to yield optimal information as well in odd as in even dimensions.

     

Optimal interpolation of analytic functions

We construct an optimal interpolation formula for a particular class of analytic functions, optimization being over a set of interpolation methods which are not necessarily linear. Optimal nodes and the norm of the error are found for the optimal interpolation formula.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 465–476, October, 1972.     

Composite quadratures for functions of dominated variation

We analyze the performance of the closed Newton-Cotes quadratures applied to integrands of low continuity. Order of convergence rates are obtained by dominating with a summable function the total variation of a suitable derivative of the integrand over a sequence of expanding intervals. The analysis given allows any finite number of singularities anywhere in the interval of integration.     

q-Blossoming for analytic functions

Numerical Algorithms - We construct a q-analog of the blossom for analytic functions, the analytic q-blossom. This q-analog also extends the notion of q-blossoming from polynomials to analytic...     

Boundary-value problems for analytic and generalized analytic functions

This paper deals with boundary value problems of linear conjugation with shift for analytic functions in the case of piecewise continuous coefficients. Int main goal is the construction of a canonical matrix for these problems. Boundary value problems with shift for generalized analytic functions and vectors as well as differential boundary value problems are studied. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 59, Algebra and Geometry, 2008.     

Optimal recovery of analytic functions from boundary conditions specified with error

The problem of optimal recovery of an analytic function from its values specified with error on a part of the boundary is solved, together with related extremal problems.     

Gauss-Hermite quadratures for functions from Hilbert spaces with Gaussian reproducing kernels

We study univariate integration with the Gaussian weight for a positive variance α. This is done for the reproducing kernel Hilbert space with the Gaussian kernel for a positive shape parameter γ. We study Gauss-Hermite quadratures, although this choice of quadratures may be questionable since polynomials do not belong to this space of functions. Nevertheless, we provide the explicit formula for the error of the Gauss-Hermite quadrature using n function values. In particular, for 2αγ ²<1 we have an exponential rate of convergence, and for 2αγ ²=1 we have no convergence, whereas for 2αγ ²>1 we have an exponential divergence.     

Weighted norm inequalities for analytic functions

We present a weighted norm inequality involving convolutions of arbitrary analytic functions and certain confluent hypergeometric functions. This result implies a family of weighted norm inequalities both for entire functions of exponential type and for (generalized) hypergeometric series. The approach is based on author's general inequality for continuous functions and some hypergeometric transformations.     

Parabolic analytic functions

Let bev=x+αy where α#0, α² = 0,x andy real are elements of a commutative ring. So inR(1, 3) ife ₀,e ₁,e ₂,e ₃ form a canonical framee ₀²=1,e ₁²=e ₂²=e ₃²=−1 the vector α=e ₀+e ₁ is different from zero while