Interpolation series for fractional derivatives and iterates of functions and operators

For a class of complex valued functions on the real line a fractional derivative is defined which is an entire function of exponential type of the order. It is shown that these derivatives can be found by a Newton interpolation series. For a class of linear operators, a fractional derivative for their resolvents also is defined. These fractional derivatives and the fractional iterates of these operators are related and both can be found by a Newton interpolation series on the nth-order iterates of the operators.     

q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q-analogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C_p. Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.     

Uniform approximation of continuous functions by probability functions of Boolean bases

Random Boolean expressions obtained by random and independent substitution of the constants 1, 0 with probabilities p, 1 ? p, respectively, into random non-iterated formulas over a given basis are considered. The limit of the probability of appearance of expressions with the value 1 under unrestricted growth of the complexity of expressions, which is called the probability function, is considered. It is shown that for an arbitrary continuous function f(p) mapping the segment [0, 1] into itself there exists a sequence of bases whose probability functions uniformly approximate the function f(p) on the segment [0, 1].     

A function with prescribed initial derivatives in different Banach spaces

Let X₀ ? X₁ ? ··· ? X_p be Banach spaces with continuous injection of X_k into X_{k + 1} for 0 ? k ? p ? 1, and with X₀ dense in X_p. We seek a function u: [0, 1] → X₀ such that its kth derivative u^(k), k = 0, 1,…, p, is continuous from [0, 1] into x_k, and satisfies the initial condition u^(k)(0) = a_k?X_k. It is shown that such a function exists if and only if the initial values a₀, a₁, …, a_p satisfy a certain condition reminiscent of interpolation theory. This condition always holds when p = 1; when p ? 2, the spaces X_k (k = 0, 1, …, p) may or may not be such that the desired function exists for any given initial values a_k?X_k.     

The Perron method for p-harmonic functions in metric spaces

We use the Perron method to construct and study solutions of the Dirichlet problem for p-harmonic functions in proper metric measure spaces endowed with a doubling Borel measure supporting a weak (1,q)-Poincaré inequality (for some 1?q<p). The upper and lower Perron solutions are constructed for functions defined on the boundary of a bounded domain and it is shown that these solutions are p-harmonic in the domain. It is also shown that Newtonian (Sobolev) functions and continuous functions are resolutive, i.e. that their upper and lower Perron solutions coincide, and that their Perron solutions are invariant under perturbations of the function on a set of capacity zero. We further study the problem of resolutivity and invariance under perturbations for semicontinuous functions. We also characterize removable sets for bounded p-(super)harmonic functions.     

Minimal-norm-derivative spline function in interpolation and approximation

The paper is concerned with applications of quadratic splines with minimal derivative to approximation of functions in approximation and interpolation problems. A smooth spline is constructed on a uniform mesh so as the norm of the spline derivative is minimal; the nodes of the spline and the nodes of interpolations coincide. This approach allows construction of a spline from given values of the function on the mesh without additional assignment of the value of the function derivative at the initial point, because the derivative can be determined from the minimality condition for the norm of the spline derivative in L ₂.     

Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation

We consider the Hermite trigonometric interpolation problem of order 1 for equidistant nodes, i.e., the problem of finding a trigonometric polynomial t that interpolates the values of a function and of its derivative at equidistant points. We give a formula for the Fourier coefficients of t in terms of those of the two classical trigonometric polynomials interpolating the values and those of the derivative separately. This formula yields the coefficients with a single FFT. It also gives an aliasing formula for the error in the coefficients which, on its turn, yields error bounds and convergence results for differentiable as well as analytic functions. We then consider the Lagrangian formula and eliminate the unstable factor by switching to the barycentric formula. We also give simplified formulae for even and odd functions, as well as consequent formulae for Hermite interpolation between Chebyshev points.     

Comparative exponential dichotomies and column diagonal dominance

It is shown that the dichotomy obtained by Lazer for diagonally dominant linear systems is a weaker condition than that of exponential dichotomy but that the two conditions are equivalent in the case of bounded coefficients. Thus Fink's result that exponential decay of all solutions implies exponential dichotomy does not extend to the case of mixed growth and decay. It is also shown that column dominant systems of mixed sign admit exponential dichotomies when the coefficients are bounded.     

Refined theorems of approximation theory in the space of p-absolutely continuous functions

In this paper, we prove direct and inverse theorems of approximation theory in the space of p-absolutely continuous functions which generalize Terekhin’s results in the same way as Timan’s results in L _p generalize the classical theorems of approximation theory. The main theorems are refined for functions with quasimonotone Fourier coefficients and, in a number of cases, the resulats are shown to be sharp.     

Mean convergence of Lagrange interpolation, II

The purpose of the paper is to investigate weighted L^p convergence of Lagrange interpolation taken at the zeros of Hermite polynomials. It is shown that if a continuous function satisfies some growth conditions, then the corresponding Lagrange interpolation process converges in every L^p (1 < p < ∞) provided that the weight function is chosen in a suitable way.     

Absolutely continuous Hardy–Sobolev functions in the unit disc

Let f(z) be an analytic function defined in the unit disc whose fractional derivative of order belongs to H^p, 0<p1. We show that as a consequence of a monotonicity condition on the decay of the Taylor coefficients, it is possible to improve the usual radial boundary growth estimate for H^p functions by a logarithmic factor. As a consequence we show that under certain regularity conditions imposed on the decay and oscillations of the absolute values of the function's Taylor coefficients, it is possible to estimate the function's modulus of continuity and modulus of absolute continuity and that a consequence of this is that as p→0, these functions will be generally smoother. Examples are also given of Hardy–Sobolev functions having modulus of absolute continuity different than modulus of continuity.     

A two-dimensional Minkowski ?(x) function

A function from a triangle to itself is defined that has both interesting number theoretic and analytic properties. This function is shown to be a natural generalization of the classical Minkowski ?(x) function. It is shown there exists a natural class of pairs of cubic irrational numbers in the same cubic number field that are mapped to pairs of rational numbers, in analog to ?(x) mapping quadratic irrationals on the unit interval to rational numbers on the unit interval. It is also shown that this new function satisfies an analog to the fact that ?(x), while increasing and continuous, has derivative zero almost everywhere.     

A Riesz basis for Bargmann-Fock space related to sampling and interpolation

It is shown that the Bargmann-Fock spaces of entire functions, A^p (C),p≧1 have a bounded unconditional basis of Wilson type [DJJ] which is closely related to the reproducing kernel. From this is derived a new sampling and interpolation result for these spaces.     

Differentiability of p-Harmonic Functions on Metric Measure Spaces

We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger’s measurable differentiable structures.     

Free Interpolation in the Spaces of Analytic Functions With Derivative of Order <Emphasis Type="Italic">s</Emphasis> From the Hardy Space

We consider the problem of free interpolation for the spaces of analytic functions with derivative of order s in the Hardy space H^p. For the sets that satisfy the Stolz condition, we obtain a condition necessary for interpolation: if 1 ≤ p < ∞, then the set must be a union of s sparse sets. For p = ∞, we obtain a necessary and sufficient condition for interpolation: the set must be a union of s + 1 sparse sets. In this case, we construct an extension operator. Bibliography 11 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 169–202.     

The best quadrature based on given Hermite information for the Sobolev class KW r[a, b]

As usual, denote by KW ^r[a, b] the Sobolev class consisting of every function whose (r ? 1)th derivative is absolutely continuous on the interval [a, b] and rth derivative is bounded by K a.e. in [a, b]. For a function f ∈ KW ^r [a, b], its values and derivatives up to r ? 1 order at a set of nodes x are known. These values are said to be the given Hermite information. This work reports the results on the best quadrature based on the given Hermite information for the class KW ^r [a, b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product, the best interpolation formula for the class KW ^r [a, b] is also obtained.     

Schur coefficients in several variables

In the paper we study weakly continuous Schur-class-valued maps and their associated Schur coefficient families, that we call functional Schur coefficients. A case of special interest is the family of the “slices” through the polytorus of an n-variable function in the unit ball of H^∞(Dn), which is shown to be a weakly continuous map from the polytorus into the Schur class. The continuity properties of its functional Schur coefficients are used to characterize the rational inner functions in the polydisk algebra. As a consequence we obtain extensions in several variables of the Schur-Cohn test on zeroes of polynomials. This provides in particular a necessary and sufficient condition of stability for multi-dimensional AR filters.     

Bemerkungen zur glatten Interpolation

Summary The usual interpolation formulae for equidistant abscissae (Newton-Gregory, Bessel) define a piecewise polynomial approximation function whose first derivative is generally discontinuous at every meshpoint. It is shown how these formulae can be modified (without using higher derivatives of the given function) such that the piecewise polynomial approximation function hass continuous derivatives (wheres is a given integer).     

Viscoelastic Timoshenko Beams with Occasionally Constant Relaxation Functions

For a prescribed desirable arbitrary decay suitable viscoelastic materials are determined through their relaxation functions. It is shown that if we wish to have a decay of order ??(t) then the kernels should be of the same order. That is their product with this function should be summable.