moment preserving spline approximation on finite intervals continuing previous wotk we discuss the problem of approximating a functionf on the interval 0 1 by a spline function of degreem withn variable knots matching as many of the initial moments off as possible additional constraints on the derivatives of the approximation at one endpoint of 0 1 may also be imposed we show that if the approximations exist they can be represented in terms of generalized gauss lobatto and gauss radau quadrature rules relative to appropriate moment functionals or measures depending off pointwise convergence asn for fixedm 0 is shown for functionsf that are completely monotonic on 0 1 among others numerical examples conclude the paper

on orthogonal polynomials for certain nondefinite linear functionals abstract we consider the non definite linear functionals l n f r w x f n x d x and prove the nonexistence of orthogonal polynomials with all zeros real in several cases the proofs are based on the connection with moment preserving spline approximation

on spectral n bernoulli measures abstract for 0 1 and n 1 an integer let be the self similar measure defined by i 0 n 1 1 n 1 i we prove that l 2 has an exponential orthonormal basis if and only if 1 q for some q 0 and n divides q the special case is the cantor measure with 1 2 k and n 2 16 which was proved recently to be the only spectral measure among the bernoulli convolutions with 0 1 4

calculation of gauss quadratures with multiple free and fixed knots algorithms are derived for the evaluation of gauss knots in the presence of fixed knots by modification of the jacobi matrix for the weight function of the integral simple gauss knots are obtained as eigenvalues of symmetric tridiagonal matrices and a rapidly converging simple iterative process based on the merging of free and fixed knots of quadratic convergence is presented for multiple gauss knots the procedures also allow for the evaluation of the weights of the quadrature corresponding to the simple gauss knots a new characterization of simple gauss knots as a solution of a partial inverse eigenvalue problem is derived

stopping functionals for gaussian quadrature formulas abstract gaussian formulas are among the most often used quadrature formulas in practice in this survey an overview is given on stopping functionals for gaussian formulas which are of the same type as quadrature formulas i e linear combinations of function evaluations in particular methods based on extended formulas like the important gauss kronrod and patterson schemes and methods which are based on gaussian nodes are presented and compared

on the determinacy problem for measures we study the general moment problem for measures on the real line with polynomials replaced by more general spaces of entire functions as a particular case we describe measures that are uniquely determined by a restriction of their fourier transform to a finite interval we apply our results to prove an extension of a theorem by eremenko and novikov on the frequency of oscillations of measures with a spectral gap high pass signals near infinity

spectral gaps for sets and measures if x is a closed subset of the real line denote by gx the supremum of the size of the gap in the fourier spectrum of a measure taken over all non trivial finite complex measures supported on x in this paper we attempt to find gx in terms of x

nonincremental proper generalized decomposition solution of parametric uncoupled models defined in evolving domains this work addresses the recurrent issue related to the existence of reduced bases related to the solution of parametric models defined in evolving domains in this first part of the work we address the case of decoupled kinematics that is models whose solution does not affect the domain in which they are defined the chosen framework considers an updated lagrangian description of the kinematics solved by using natural neighbor galerkin methods within a nonincremental space time framework that can be generalized for addressing parametric models examples showing the performance and potentialities of the proposed methodology are included

discrete approximations to spherically symmetric distributions we consider the approximation of spherically symmetric distributions in d by linear combinations of heaviside step functions or dirac delta functions the approximations are required to faithfully reproduce as many moments as possible we discuss stable methods of computing such approximations taking advantage of the close connection with gauss christoffel quadrature numerical results are presented for the distributions of maxwell bose einstein and fermi dirac

shape preserving approximations by polynomials and splines a review of shape preserving approximation methods and algorithms for approxi mating univariate functions or discrete data is given the notion of shape refers to the geometrical behavior of a function s or approximant s graph and usually includes positivity monotonicity and or convexity but in the recent literature the broader concept of shape also includes symmetry gen eralized convexity unimodality lipschitz property possessing peaks or discontinuities etc special stress is put on shape preserving interpolation methods by polynomials and splines of course this text has no pretensions to be complete geywords shape preserving approximation approximation of univariate functions approxima tion of discrete data shape preserving interpolation polynomial spline positivity monotonicity convexity generalized convexity unimodality lipschitz property

weighted exponential approximation and non classical orthogonal spectral measures abstract a long standing open problem in harmonic analysis is given a non negative measure on r find the infimal width of frequencies needed to approximate any function in l 2 we consider this problem in the perturbative regime and characterize asymptotic smallness of perturbations of measures which do not change that infimal width then we apply this result to show that there are no local restrictions on the structure of orthogonal spectral measures of one dimensional schrodinger operators on a finite interval this answers a question raised by v a marchenko

type alternative for frostman measures abstract for a finite positive borel measure on r its exponential type t is defined as the infimum of a 0 such that finite linear combinations of complex exponentials with frequencies between 0 and a are dense in l 2 the definition can be easily extended from finite to broader classes of measures in this paper we prove a new formula for t and use it to study growth and additivity properties of measures with finite positive type as one of the applications we show that frostman measures on r may only have type zero or infinity

spectrality of infinite bernoulli convolutions let bk ak k 1 be a sequence of positive integers with upper bound and let e be the uniformly discrete probability measure on the finite set e for 0 1 the infinite convolution ak bk a1 b1 2 a2 b2 is called an infinite bernoulli convolution in this paper we investigate whenever there exists such that e 2 i x is an orthonormal basis for l2 ak bk

polya sequences toeplitz kernels and gap theorems abstract a separated sequence on the real line is called a polya sequence if any entire function of zero exponential type bounded on is constant in this paper we solve the problem by polya and levinson that asks for a description of polya sets we also show that the polya levinson problem is equivalent to a version of the so called beurling gap problem on fourier transforms of measures the solution is obtained via a recently developed approach based on the use of toeplitz kernels and de branges spaces of entire functions

on the fourier orthonormal bases of cantor moran measures abstract let d n p n n 1 be a sequence of integers so that 0 d n p n for n 1 the infinite convolution of probability measures with finite support and equal distribution p n d n p 1 1 0 d 1 p 1 p 2 1 0 d 2 is a borel probability measure cantor moran measure in this paper we study the existence of fourier basis for l 2 p n d n i e find a discrete set such that e e 2 i x is an orthonormal basis for l 2 p n d n we give some sufficient conditions for this aim and some examples to explain the theory

spectrality of the planar sierpinski family abstract let be a borel probability measure with compact support in r 2 is called a spectral measure if there exists a countable set r 2 such that e e 2 i x is an orthonormal basis for l 2 in this note we prove that the integral sierpinski measure a d is a spectral measure if and only if a d is admissible this completely settles the spectrality of integral sierpinski measures in r 2

beurling malliavin theory for toeplitz kernels we consider the family of toeplitz operators t j bar s a acting in the hardy space h2 in the upper halfplane j and s are given meromorphic inner functions and a is a real parameter in the case where the argument of s has a power law type behavior on the real line we compute the critical value c j s inf left a mathop mathrm ker t j bar s a ne0 right the formula for c j s generalizes the beurling malliavin theorem on the radius of completeness for a system of exponentials

a problem on completeness of exponentials let be a nite positive measure on the real line for a 0 denote by ea the family of exponential functions ea fe ist j s2 0 a g

scaling of spectra of a class of random convolution on r abstract given a borel probability measure on r and a real number p we call p a spectral eigenvalue of the measure if there exists a discrete set such that the sets e e 2 i x and e p e 2 i p x are both orthonormal basis for hilbert space l 2 in the present paper we determine the spectral eigenvalues of a class of random convolution on r

sierpinski type spectral self similar measures abstract we consider the sierpinski type self similar measure on r 2 with contraction ratio 0 1 we show that is a spectral measure if and only if 1 3 p for some integer p 0 a similar characterization for bernoulli convolution is due to dai 2 over which 1 2 p

quadratures with multiple nodes power orthogonality and moment preserving spline approximation quadrature formulas with multiple nodes power orthogonality and some applications of such quadratures to moment preserving approximation by defective splines are considered an account on power orthogonality s and orthogonal polynomials and generalized gaussian quadratures with multiple nodes including stable algorithms for numerical construction of the corresponding polynomials and cotes numbers are given in particular the important case of chebyshev weight is analyzed finally some applications in moment preserving approximation of functions by defective splines are discussed