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 quadratures for analytic functions

For integrals _–1 ¹ w(x)f(x)dx with and with analytic integrands, we consider the determination of optimal abscissasx _i ^o and weightsA _i ^o , for a fixedn, which minimize the errorE _n (f)= _–1 ¹ w(x)f(x)dx – _i ⁼¹ⁿ A _i f(x _i ) over an appropriate Hilbert spaceH ²(E ; w(z)) of analytic functions. Simultaneously, we consider the simpler problem of determining intermediate-optimal weightsA _i *, corresponding to (preassigned) Gaussian abscissasx _i ^G , which minimize the quadrature error. For eachw(x), the intermediate-optimal weightsA _i * are obtained explicitly, and these come out proportional to the corresponding Gaussian weightsA _i ^G . In each case,A _i ^G =A _i *+O( ^–4n ), . For , a complete explicit solution for optimal abscissas and weights is given; in fact, the set {x _i ^G ,A _i *;i=1,...,n} to provides the optimal abscissas and weights. For otherw(x), we study the closeness of the set {x _i ^G ,A _i *;i=1,...,n} to the optimal solution {x _i ^o ,A _i ^o ;i=1,...,n} in terms of _n (), the maximum absolute remainder in the second set ofn normal equations. In each case, _n () is, at least, of the order of ^–4n for large.     

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.     

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

Interpolating sequences for bounded analytic functions

We prove that any sequence in the open ball of a complex Banach space even in that of whose norms are an interpolating sequence for is interpolating for the space of all bounded analytic functions on The construction made yields that the interpolating functions depend linearly on the interpolated values.

     

Schwarz's lemma for vector-valued analytic functions

Let D be a domain in the plane, p a point of D, and X a complex Banach space; let m(D, p, X) = sup{∥f′(p)∥: f is a holomorphic X-valued function on D bounded by one}. Theorem 1 of this paper shows that the number m(D, p, X) is independent of X and shows the existence of X-valued holomorphic functions f with ∥f′(p)∥ = m(D, p). The remainder of the theorems investigate the extremal properties of such functions and give conditions on X which imply that each such function has the form φx where φ is a scalar-valued bounded holomorphic function and x is a point of X with ∥ x ∥ = 1.     

Some starlikeness criterions for analytic functions

We determine a condition on M, α, λ and μ for which

     

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.     

Invariant analytic curves for entire functions

Kharkov. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 3, pp. 3–8, May–June, 1989.     

Extremal functions for functionals on some classes of analytic functions

It is shown that the theorem of Carathéodory and Toeplitz on the characterization of the Taylor coefficients of analytic functions with positive real part can be applied to extremal problems in several classes of analytic functions.