Lp-Computability

In this paper we investigate conditions for L^p-computability which are in accordance with the classical Grzegorczyk notion of computability for a continuous function. For a given computable real number p ≥ 1 and a compact computable rectangle I ? ?^q, we show that an L^p function f ∈ L^p(I) is L^P-computable if and only if (i) f is sequentially computable as a linear functional and (ii) the L^p-modulus function of f is effectively continuous at the origin of ?^q.     

Roots in differential rings of ultradifferentiable functions

Let {M _k } be a logconvex sequence satisfying the differentiability condition $$\sup (M_{n + 1} /M_n )^{1/n} < \infty $$ . It is shown that the Carleman class C{k! M _k } contains all C ^∞ roots of its nonflat elements, i.e., if f ∈ C{k! M _k } and α > 0, then $$f^\alpha \in C\{ k!M_k \} whenever f^\alpha \in C^\infty $$ . If {M _k } also satisfies the additional condition M _n ^1/n → ∞, then the Beurling class C(k! M _k ) is also contains all C ^∞ roots of its nonflat elements.     

New Findings on the Bank–Sauer Approach in Oscillation Theory

In 1988, S. Bank showed that if {z _n } is a sparse sequence in the complex plane, with convergence exponent zero, then there exists a transcendental entire A(z) of order zero such that f″+A(z)f=0 possesses a solution having {z _n } as its zeros. Further, Bank constructed an example of a zero sequence {z _n } violating the sparseness condition, in which case the corresponding coefficient A(z) is of infinite order. In 1997, A. Sauer introduced a condition for the density of the points in the zero sequence {z _n } of finite convergence exponent such that the corresponding coefficient A(z) is of finite order.     

Approaches to Effective Semi-Continuity of Real Functions

For semi-continuous real functions we study different computability concepts defined via computability of epigraphs and hypographs. We call a real function f lower semi-computable of type one, if its open hypograph hypo(f) is recursively enumerably open in dom(f) × ?; we call f lower semi-computable of type two, if its closed epigraph Epi(f) is recursively enumerably closed in dom(f) × ?; we call f lower semi-computable of type three, if Epi(f) is recursively closed in dom(f) × ?. We show that type one and type two semi-computability are independent and that type three semi-computability plus effectively uniform continuity implies computability, which is false for type one and type two instead of type three. We show also that the integral of a type three semi-computable real function on a computable interval is not necessarily computable.     

Coincidence of Least Uniform Deviations of Functions from Polynomials and Rational Fractions

For a given nonincreasing vanishing sequence {a _n } _{n = 0} of nonnegative real numbers, we find necessary and sufficient conditions for a sequence {n _k } _{k = 0} to have the property that for this sequence there exists a function f continuous on the interval [0,1] and satisfying the condition that , k = 0,1,2,..., where E _n (f) and R _n,m (f) are the best uniform approximations to the function f by polynomials whose degree does not exceed n and by rational functions of the form r _n,m (x) = p _n (x)/q _m (x), respectively.     

On the connection between weak and strong convergencies in Cm

In this work wome connections are pursued between weak and strong convergence in the spaces C^m (m-times continuously differentiable functions on Rⁿ). Let f_n, f?C^{m + 1}, where n = 1, 2,…, and m is a nonnegative integer. Suppose that the sequence {f_n} converges to f relative to the weak topology of C^{m + 1}. It is shown that this implies the convergence of {f_n} to f with respect to the strong topology of C^m. Several corollaries to this theorem are established; among them is a sufficient condition for uniform convergence. A stronger result is shown to exist when the sequence constitutes an output sequence of a linear weakly continuous operator.     

Approximation by antiderivatives

Let Ω ?C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ ^(?n)} ofn-fold antiderivatives (i.e., we haveφ ⁽⁰⁾(z)∶=φ(z) andφ ^(?n)(z)=dφ ^(?n?1)(z)/dz for alln ∈ N₀ and z ∈ Ω) such that the following properties hold:

1. For any compact setB ?Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n _k} such that {φ^?nk} converges tof uniformly onB.
2. For any open setU ?Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m _k} such that {φ^?mk} converges tof compactly onU.
3. For any measurable setE ?Ω and any functionf that is measurable onE, there exists a sequence {p _k} such that {φ ^(-Pk)} converges tof almost everywhere onE.

     

The Arithmetical Hierarchy of Real Numbers

A real number x is computable iff it is the limit of an effectively converging computable sequence of rational numbers, and x is left (right) computable iff it is the supremum (infimum) of a computable sequence of rational numbers. By applying the operations “sup” and “inf” alternately n times to computable (multiple) sequences of rational numbers we introduce a non‐collapsing hierarchy {Σ_n, Π_n, Δ_n : n ∈ ℕ} of real numbers. We characterize the classes Σ₂, Π₂ and Δ₂ in various ways and give several interesting examples.     

On an Approximate Solution of the Dirichlet Problem for the Generalized Laplacian

For an arbitrary differential operator P of order p on an open set X ? R ⁿ, the Laplacian is defined by Δ = P*P. It is an elliptic differential operator of order 2p provided the symbol mapping of P is injective. Let O be a relatively compact domain in X with smooth boundary, and B_j(j = 0…,p — 1) be a Dirichlet system of order p ? 1 on ?O. By {C_j} we denote the Dirichlet system on ?O adjoint for {B_j} with respect to the Green formula for P. The Hardy space H²(O) is defined to consist of all the solutions f of Δf = 0 in O of finite order of growth near the boundary such that the weak boundary values of the expression {B_jf} and {C_j(Pf)} belong to the Lebesgue space L²(?O). Then the Dirichlet problem consists of finding a solution f ? H²(O) with prescribed data {B_jf} on ?O. We develop the classical Fischer-Riesz equations method to derive a solvability condition of the Dirichlet problem as well as an approximate formula for solutions.     

Orthogonal Laurent polynomials and the strong Hamburger moment problem

This paper is concerned with the strong Hamburger moment problem (SHMP): For a given double sequence of real numbers C = {c_n}^∞_?∞, does there exist a real-valued, bounded, non-decreasing function ψ on (?∞, ∞) with infinitely many points of increase such that for every integer n, c_n = ∝^∞_?∞ (?t)ⁿdψ(t)? Necessary and sufficient conditions for the existence of such a function ψ are given in terms of the positivity of certain Hankel determinants associated with C. Our approach is made through the study of orthogonal (and quasi-orthogonal) Laurent polynomials (referred to here as L-polynomials) and closely related Gaussian-type quadrature formulas. In the proof of sufficiency an inner product for L-polynomials is defined in terms of the given double sequence C. Since orthogonal L-polynomials are believed to be of interest in themselves, some examples of specific systems are considered.     

On the Convergence in Capacity of Rational Approximants

Let {r _n } be a sequence of rational functions deg( r _n ≤ n) that converge rapidly in measure to an analytic function f on an open set in C ^N . We show that {r _n } converges rapidly in capacity to f on its natural domain of definition W _f (which, by a result of Goncar, is an open subset of C ^N ). In particular, for f meromorphic on C ^N and analytic near zero the sequence of Padé approximants {π _n (z, f, λ)} (as defined by Goncar) converges rapidly in capacity to f on C ^N . January 14, 1999. Date revised: October 7, 1999. Date accepted: November 1, 1999.     

Polynomials of Binomial Type and Renewal Sequences

We study polynomials of binomial type that have an exponential generating function of the form { 1 ? f(u)^?x}. They have a close connection with renewal sequences. The asymptotic behavior as n ? ∞ is studied.     

A hierarchy of Turing degrees of divergence bounded computable real numbers

A real number x is f-bounded computable (f-bc, for short) for a function f if there is a computable sequence (x_s) of rational numbers which converges to x f-bounded effectively in the sense that, for any natural number n, the sequence (x_s) has at most f(n) non-overlapping jumps of size larger than 2^-n. f-bc reals are called divergence bounded computable if f is computable. In this paper we give a hierarchy theorem for Turing degrees of different classes of f-bc reals. More precisely, we will show that, for any computable functions f and g, if there exists a constant γ>1 such that, for any constant c, f(nγ)+n+cg(n) holds for almost all n, then the classes of Turing degrees given by f-bc and g-bc reals are different. As a corollary this implies immediately the result of [R. Rettinger, X. Zheng, On the Turing degrees of the divergence bounded computable reals, in: CiE 2005, June 8–15, Amsterdam, The Netherlands, Lecture Notes in Computer Science, vol. 3526, 2005, Springer, Berlin, pp. 418–428.] that the classes of Turing degrees of d-c.e. reals and divergence bounded computable reals are different.     

New iterations with errors for approximating common fixed points for two generalized asymptotically quasi-nonexpansive nonself-mappings

Let X be a real uniformly convex Banach space and C a nonempty closed convex nonexpansive retract of X with P as a nonexpansive retraction. Let T ₁, T ₂: C → X be two uniformly L-Lipschitzian, generalized asymptotically quasi-nonexpansive non-self-mappings of C satisfying condition A′ with sequences {k _n ⁽ⁱ⁾ } and {δ _n ⁽ⁱ⁾ } ? [1, ∞),, i = 1, 2, respectively such that Σ _n=1 ^∞ (k _n ⁽ⁱ⁾ ? 1) < ∞, Σ _n=1 ⁽ⁱ⁾ δ _n ⁽ⁱ⁾ < ∞, and F = F(T ₁) ∩ F(T ₂) ≠ ?. For an arbitrary x ₁ ∈ C, let {x _n } be the sequence in C defined by $$ \begin{gathered} y_n = P\left( {\left( {1 - \beta _n - \gamma _n } \right)x_n + \beta _n T_2 \left( {PT_2 } \right)^{n - 1} x_n + \gamma _n v_n } \right), \hfill \\ x_{n + 1} = P\left( {\left( {1 - \alpha _n - \lambda _n } \right)y_n + \alpha _n T_1 \left( {PT_1 } \right)^{n - 1} x_n + \lambda _n u_n } \right), n \geqslant 1, \hfill \\ \end{gathered} $$ where {α _n }, {β _n }, {γ _n } and {λ _n } are appropriate real sequences in [0, 1) such that Σ _n=1 ^∞ ] γ _n < ∞, Σ _n=1 ^∞ λ _n < ∞, and {u _n }, }v _n } are bounded sequences in C. Then {x _n } and {y _n } converge strongly to a common fixed point of T ₁ and T ₂ under suitable conditions.     

Almost everywhere divergence of lacunary subsequences of partial sums of fourier series

If an increasing sequence {n _m } of positive integers and a modulus of continuity ω satisfy the condition Σ _m=1 ^∞ ω(1/n _m )/m < ∞, then it is known that the subsequence of partial sums \(S_{n_m } \left( {f,x} \right)\) converges almost everywhere to f(x) for any function f ∈ H ₁ ^ω . We show that this sufficient convergence condition is close to a necessary condition for a lacunary sequence {n _m }.     

Approximation of Functions on [−1,1] and Their Derivatives by Polynomial Projection Operators

Pointwise estimates are obtained for the simultaneous approximation of a function f ?C^q[-1,1] and its derivatives f⁽¹⁾, …, f^(q) by means of an arbitrary sequence of bounded linear projection operators L_n which map C[-1,1] into the polynomials of degree at most n, augmented by the interpolation of f at some points near ± 1.     

Some stochastic processes related to random density functions

If { _n } is an orthonormal system and {a _n} is a sequence of random variables such that _n (a _n )²=1 a.s. thenf(t)=| _n a _{n n} (t)|² produces a randomly selcted density function. We study the properties off under the assumptions that |a _n| is decreasing to zero at a geometric rate and { _n } is one of the following four function systems: trigonometric Jacobi, Hermite, or Laguerre. It is shown that, with probability one,f is an analytic function,f has at most a finite number of zeros in any finite interval, and the tail off goes to zero rapidly.     

On the asymptotic convergence of sequences of analytic functions

We consider sequences {f _n } of analytic self mappings of a domain and the associated sequence {Θ _n } of inner compositions given by . The case of interest in this paper concerns sequences {f _n } that converge assymptotically to a function f, in the sense that for any sequence of integers {n _k } with n ₁ < n ₂ < ... one has that locally uniformly in Ω. Most of the discussion concerns the case where the asymptotic limit f is the identity function in Ω. Received: 16 December 2006     

Second degree classical forms

A form (linear functional) u is called regular if there exists a sequence of polynomials {P_n}_n≥0, deg P_n = n which is orthogonal with respect to u. Such a form is said to be of second degree if there are polynomials B and C such that the Stieltjes function satisfies a relation of the form BS²(u) + CS(u) + D = 0.Classical forms correspond to classical orthogonal polynomials: sequences of polynomials whose derivatives also form an orthogonal sequence. In this paper, the authors determine all the classical forms which are of second degree. They show that Hermite, Laguerre and Bessel forms are not of second degree. Only Jacobi forms which satisfy a certain condition possess this property.     

The Wave Equation with Computable Initial Data Whose Unique Solution Is Nowhere Computable

We give a rough statement of the main result. Let D be a compact subset of ?³× ?. The propagation u(x, y, z, t) of a wave can be noncomputable in any neighborhood of any point of D even though the initial conditions which determine the wave propagation uniquely are computable. A precise statement of the result appears below.