Approximation of unbounded functions via compactification

Approximants to functions f(s) that are allowed to possess infinite limits on their interval of definition, are constructed.To this end a compactification of Rⁿ is developed which is based on the projection of Rⁿ on a bowl-shaped subset of a parabolic surface. This compactification induces a bijection and a metric with several desirable properties that make it a useful tool for rational approximation of unbounded functions.Roughly speaking this compactification enables us to show that unbounded functions can be approximated by rational functions on a closed interval; thus we also obtain an extension to Weierstrass’ celebrated theorem. An extension to a Fourier-type theorem is also obtained. Roughly speaking, our result states that unbounded periodic functions can be approximated by quotients of certain trigonometric sums. The characteristics of the main results are the following. The approximations do not require the original approximated function to possess a restricted rate of growth. Neither do they require that the approximated function possess any amount of smoothness. Moreover, the numerator and denominator, in an approximating quotient are guaranteed not to vanish simultaneously. Furthermore, some of the proposed approximations are guaranteed to be bounded at every point at which the original approximated function is bounded. Beside the tool of compactification we also employ Bernstein polynomials and Cesaro means of “trigonometric sums”.     

The b-transform

The b-transform is used to convert entire functions into “primary b-functions” by replacing the powers and factorials in the Taylor series of the entire function with corresponding “generalized powers” (which arise from a polynomial function with combinatorial applications) and “generalized factorials.” The b-transform of the exponential function turns out to be a generalization of the Euler partition generating function, and partition generating functions play a key role in obtaining results for the b-transforms of the elementary entire transcendental functions. A variety of normal-looking results arise, including generalizations of Euler's formula and De Moivre's theorem. Applications to discrete probability and applied mathematics (i.e., damped harmonic motion) are indicated. Also, generalized derivatives are obtained by extending the concept of a b-transform.     

Bergman polynomials on an archipelago: Estimates, zeros and shape reconstruction

Growth estimates of complex orthogonal polynomials with respect to the area measure supported by a disjoint union of planar Jordan domains (called, in short, an archipelago) are obtained by a combination of methods of potential theory and rational approximation theory. The study of the asymptotic behavior of the roots of these polynomials reveals a surprisingly rich geometry, which reflects three characteristics: the relative position of an island in the archipelago, the analytic continuation picture of the Schwarz function of every individual boundary and the singular points of the exterior Green function. By way of explicit example, fine asymptotics are obtained for the lemniscate archipelago |z^m−1|<r^m, 0<r<1, which consists of m islands. The asymptotic analysis of the Christoffel functions associated to the same orthogonal polynomials leads to a very accurate reconstruction algorithm of the shape of the archipelago, knowing only finitely many of its power moments. This work naturally complements a 1969 study by H. Widom of Szegő orthogonal polynomials on an archipelago and the more recent asymptotic analysis of Bergman orthogonal polynomials unveiled by the last two authors and their collaborators.     

Classical and approximate sampling theorems; studies in the and the uniform norm

The approximate sampling theorem with its associated aliasing error is due to J.L. Brown (1957). This theorem includes the classical Whittaker–Kotel’nikov–Shannon theorem as a special case. The converse is established in the present paper, that is, the classical sampling theorem for , 1p<∞, w>0, implies the approximate sampling theorem. Consequently, both sampling theorems are fully equivalent in the uniform norm.Turning now to -space, it is shown that the classical sampling theorem for , 1<p<∞ (here p=1 must be excluded), implies the -approximate sampling theorem with convergence in the -norm, provided that f is locally Riemann integrable and belongs to a certain class Λ^p. Basic in the proof is an intricate result on the representation of the integral as the limit of an infinite Riemann sum of |f|^p for a general family of partitions of ; it is related to results of O. Shisha et al. (1973–1978) on simply integrable functions and functions of bounded coarse variation on . These theorems give the missing link between two groups of major equivalent theorems; this will lead to the solution of a conjecture raised a dozen years ago.     

Estimates of Best Approximation and Fourier Transforms in Integral Metrics

Under some assumptions on a function F and its Fourier transform we prove new estimates of best approximation of F by entire functions of exponential type σ in L_p( ), 1 ≤ p < 2. The proof is based on some inequalities for in L₁( ) which may be treated as generalizations of results of Bausov and Telyakovskii. As an application we obtain exact estimates of best approximation of some infinitely differentiable functions.     

Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains

In this paper we prove new results for p harmonic functions, p≠2, 1<p<∞, in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of the boundary of a Lipschitz domain, with constants only depending on p,n and the Lipschitz constant of the domain. For p capacitary functions, in starlike Lipschitz ring domains, we prove an even stronger result, Theorem 2, showing that the ratio is Hölder continuous up to the boundary. Moreover, for p capacitary functions in starlike Lipschitz ring domains we prove, Theorems 3 and 4, appropriate extensions to p≠2, 1<p<∞, of famous results of Dahlberg [12] and Jerison and Kenig [25] on the Poisson kernel associated to the Laplace operator (i.e. p=2).     

A Chebyshev Quadrature Rule for One Sided Finite Part Integrals

This paper is concerned with a Chebyshev quadrature rule for approximating one sided finite part integrals with smooth density functions. Our quadrature rule is based on the Chebyshev interpolation polynomial with the zeros of the Chebyshev polynomial T_N+1(τ)−T_N−1(t). We analyze the stability and the convergence for the quadrature rule with a differentiable function. Also we show that the quadrature rule has an exponential convergence when the density function is analytic.     

On the L convergence of Lagrange interpolating entire functions of exponential type

Let f: be a continuous, 2π-periodic function and for each n ε let t_n(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that lim_{n → ∞} ∝₀^2π¦f(θ) − t_n(f θ)¦^p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz.     

Asymptotic stability and smooth equivalence op ordinary equations : PMM vol.40, n 6, 1976, pp. 1048–1057

Under certain specified conditions the asymptotic stability is a coarse property [1],(i.e. addition of fairly smooth functions to the right-hand sides of equations, does not disturb the asymptotic stability). It is shown below that in this cage the unperturbed system is coarse in a more general sense, namely, any smooth system acted upon by fairly small smooth perturbations, can be returned to its unperturbed state by a smooth reversible transformation. The value and order of the perturbations and the domain of existence of the transformation are all estimated explicitly. The condition required for the above assertion to hold, is that of the existence of a Liapunov function admitting, together with its derivative, specified estimates. This requirement holds, in particular, in the case when the right-hand sides of the unperturbed system are homogeneous functions, the position of equilibrium is asymptotically stable, and its neighborhood contains no solutions bounded when −∞ <t < ∞ (see [1]). If the system is analytic, the requirement will hold in at least all critical cases investigated in which the asymptotic stability with t → ∞ or t → −∞ is fixed, since in these cases the Liapunov function will be analytic, or simply polynomial. It follows therefore from the theorem which we prove, that in all the cases in question, the system is reduced by a smooth transformation, to the polynomial form. If the unperturbed system is linear, then from the theorem proved follows a theorem on linearization appearing in [2]; if the system is nonlinear but of second order, a theorem from [3] ensues. The results obtained in this paper for the nonlinear autonomous systems are extended to the case when the perturbations are continuous and bounded functions of time. This makes possible the investigation of the dynamics of the process in the neighborhood of asymptotically stable equilibria and of periodic modes, ignoring a wide range of external perturbations.     

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

Orthogonal Rational Functions and Nested Disks

In Akhiezer's book [“The Classical Moment Problem and Some Related Questions in Analysis,” Oliver & Boyd, Edinburghasol;London, 1965] the uniqueness of the solution of the Hamburger moment problem, if a solution exists, is related to a theory of nested disks in the complex plane. The purpose of the present paper is to develop a similar nested disk theory for a moment problem that arises in the study of certain orthogonal rational functions. Let {α_n}^∞_n=0be a sequence in the open unit disk in the complex plane, let

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( /|α_k|=−1 whenα_k=0), and let

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We consider the following “moment” problem: Given a positive-definite Hermitian inner product ·, · on × , find a non-decreasing functionμon [−π, π] (or a positive Borel measureμon [−π,π)) such that

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In particular we give necessary and sufficient conditions for the uniqueness of the solution in the case that If this series diverges the solution is always unique.     

Analytic continuation of local representations of symmetric spaces

We prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space (G, K, σ). The problem of passing from the Euclidean group to the Poincaré group appears first to have been addressed and solved this way by Klein and Landau. Let G be a Lie group, K a closed subgroup, and σ an involutive automorphism with K as fixed-point subgroup. If = + is the corresponding symmetric Lie algebra, we form ^* = + , and let G^* denote the simply connected Lie group with ^* as Lie algebra. We consider virtual representations π of G on a fixed complex Hilbert space , adopting the definitions due to J. Fröhlich, K. Osterwalder, and E. Seiler; in particular, π(g⁻¹) π(σ(g))^* (possibly unbounded operators) for g in a neighborhood of e in G. We prove that every such π continues analytically to a strongly continuous unitary representation of G^* on . Our theorem extends results due to Klein-Landau, Fröhlich et al., and others, earlier, for special cases. Previous results were known only for special (G, K, σ), and then only for certain π.     

On Chebyshev–Markov Rational Functions over Several Intervals

Chebyshev–Markov rational functions are the solutions of the following extremal problem

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withKbeing a compact subset of andω_n(x) being a fixed real polynomial of degree less thann, positive onK. A parametric representation of Chebyshev–Markov rational functions is found forK=[b₁, b₂]…[b_2p−1, b_2p], −∞<b₁b₂<…<b_2p−1b_2p<+∞ in terms of Schottky–Burnside automorphic functions.     

Binomial posets, Möbius inversion, and permutation enumeration

A unified method is presented for enumerating permutations of sets and multisets with various conditions on their descents, inversions, etc. We first prove several formal identities involving Möbius functions associated with binomial posets. We then show that for certain binomial posets these Möbius functions are related to problems in permutation enumeration. Thus, for instance, we can explain “why” the exponential generating function for alternating permutations has the simple form (1 + sin x)/(cos x). We can also clarify the reason for the ubiquitous appearance of e^x in connection with permutations of sets, and of ξ(s) in connection with permutations of multisets.     

Strong Asymptotic Behavior and Weak Convergence of Polynomials Orthogonal on an Arc of the Unit Circle

Let σ be a finite positive Borel measure supported on an arc γ of the unit circle, such that σ′>0 a.e. on γ. We obtain a theorem about the weak convergence of the corresponding sequence of orthonormal polynomials. Moreover, we prove an analogue of the Szeg –Geronimus theorem on strong asymptotics of the orthogonal polynomials on the complement of γ, which completes to its full extent a result of N. I. Akhiezer. The key tool in the proofs is the use of orthogonality with respect to varying measures.     

Rates of Best Rational Approximation of Analytic Functions

Let E be a compact set in the extended complex plane C and let f be holomorphic on E. Denote by ρ_n the distance from f to the class of all rational functions of order at most n, measured with respect to the uniform norm on E. We obtain results characterizing the relationship between estimates of lim inf_n→∞ ρ^1/n_n and lim sup_n→∞ ρ^1/n_n.     

Signal Analytic Proofs of Two Basic Results on Lattice Expansions

We present new and short proofs of two theorems in the theory of lattice expansions. These proofs are based on a necessary and sufficient condition, found by Wexler and Raz, for biorthogonality. The first theorem is the Lyubarskii–Seip–Wallstén theorem for lattices, according to which the set of Gaussians 2^1/4 exp(-π(t - na)² + 2πimbt), n, m , constitutes a frame when a > 0,b > 0,ab < 1. In addition, we display dual functions for this case. The second theorem is the result that a set g_na,mb(t) = g(t - na) exp(2πimbt), n, m of time–frequency translates of a g L²( ) cannot be a frame when a > 0,b > 0,ab > 1.     

Some approximation results involving the q‐Szász–Mirakjan–Kantorovich type operators via Dunkl's generalization

The purpose of this paper is to introduce a family of q‐Szász–Mirakjan–Kantorovich type positive linear operators that are generated by Dunkl's generalization of the exponential function. We present approximation properties with the help of well‐known Korovkin's theorem and determine the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K‐functional, and the second‐order modulus of continuity. Furthermore, we obtain the approximation results for bivariate q‐Szász–Mirakjan–Kantorovich type operators that are also generated by the aforementioned Dunkl generalization of the exponential function. Copyright © 2017 John Wiley & Sons, Ltd.     

Random walks generated by area preserving maps with zero Lyapounov exponents

We study the asymptotic limit distributions of Birkhoff sums S_n of a sequence of random variables of dynamical systems with zero entropy and Lebesgue spectrum type. A dynamical system of this family is a skew product over a translation by an angle α. The sequence has long memory effects. It comes that when α/π is irrational the asymptotic behavior of the moments of the normalized sums S_n/f_n depends on the properties of the continuous fraction expansion of α. In particular, the moments of order k, , are finite and bounded with respect to n when α/π has bounded continuous fraction expansion. The consequences of these properties on the validity or not of the central limit theorem are discussed.     

Curve Fitting, Differential Equations And The Riemann Hypothesis

It is known that the Riemann zeta function ζ (s) in the critical strip 0 < Re(s) < 1, may be represented as the Mellin transform of a certain function φ (x) which is related to one of the theta functions. The function φ (x) satisfies a well known functional equation, and guided by this property we deduce a family of approximating functions involving an arbitrary parameter α. The approximating function corresponding to the value of α = 2 gives rise to a particularly accurate numerical approximation to the function φ (x). Another approximation to φ (x), which is based upon the first one, is obtained by solving a certain differential equation. Yet another approximating function may be determined as a simple extension of the first. All three approximations, when used in conjunction with the Mellin transform expression for ζ (s) in the critical strip, give rise to an explicit expression from which it is clear that Re(s) = 1/2 is a necessary and sufficient condition for the vanishing of the imaginary part of the integral, the real part of which is non-zero. Accordingly, the analogy with the Riemann hypothesis is only partial, but nevertheless Re(s) = 1/2 emerges from the analysis in a fairly explicit manner. While it is generally known that the imaginary part of the Mellin transform must vanish along Re(s) = 1/2, the major contribution of this paper is the presentation of the actual calculation for three functions which approximate φ (x). The explicit nature of these calculation details may facilitate progress towards the corresponding calculation for the actual φ (x), which may be necessary in a resolution of the Riemann hypothesis.2000 Mathematics Subject Classification: Primary—11M06, 11M26