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.     

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.

     

Log-concavity and LC-positivity

A triangle {a(n,k)}_0?k?n of nonnegative numbers is LC-positive if for each r, the sequence of polynomials is q-log-concave. It is double LC-positive if both triangles {a(n,k)} and {a(n,n−k)} are LC-positive. We show that if {a(n,k)} is LC-positive then the log-concavity of the sequence {xk} implies that of the sequence {zn} defined by , and if {a(n,k)} is double LC-positive then the log-concavity of sequences {xk} and {yk} implies that of the sequence {zn} defined by . Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Pemantle on characteristics of negative dependence.     

A Note on the Asymptotic Normality of Sample Autocorrelations for a Linear Stationary Sequence

We consider a stationary time series {X_t} given byX_t=∑^∞_k=−∞ ψ_kZ_t−k, where {Z_t} is a strictly stationary martingale difference white noise. Under assumptions that the spectral densityf(λ) of {X_t} is squared integrable andm^τ ∑_|k|?m ψ²_k→0 for someτ>1/2, the asymptotic normality of the sample autocorrelations is shown. For a stationary long memoryARIMA(p, d, q) sequence, the conditionm^τ ∑_|k|?m ψ²_k→0 for someτ>1/2 is equivalent to the squared integrability off(λ). This result extends Theorem 4.2 of Cavazos-Cadena [5], which were derived under the conditionm ∑_|k|?m ψ²_k→0.     

Some Convergence Properties of Descent Methods

In this paper, we discuss the convergence properties of a class of descent algorithms for minimizing a continuously differentiable function f on R ⁿ without assuming that the sequence { x _k } of iterates is bounded. Under mild conditions, we prove that the limit infimum of is zero and that false convergence does not occur when f is convex. Furthermore, we discuss the convergence rate of { } and { f(x _k )} when { x _k } is unbounded and { f(x _k )} is bounded.     

Biorthogonal bases and multiple interpolation in weighted Hardy spaces

Let {z_k=x_k+iy_k} be a sequence on upper half plane and {s_i} be the number of appearence of z_k in {z₁,z₂,...,z_k}. Suppose sup s_i<+∞. Let ω(x) be a weight belonging to A^∞ and . We Consider the weighted Hardy space and operator T_p mapping f(z)∈H _+w ^p into a sequence defined by , 0<p≤+∞, j=1,2,.... Then T_p(H _+w ^p )=l^p if and only if {z_k} is uniformly separated. Besides the effective solution for interpolation is obtained. Supported by National Science Foundation of China and Shanghai Youth Science Foundation     

Representation of measurable functions by series in Walsh subsystems

For any sequence {ω(n)} _n∈ℕ tending to infinity we construct a “quasiquadratic” representation spectrum Λ = {n ² + o(ω(n))} _n∈ℕ: for any almost everywhere (a. e.) finite measurable function f(x) there exists a series in the form $ \mathop \sum \limits_{k \in \Lambda } $ \mathop \sum \limits_{k \in \Lambda } α _k ω _k (x) that converges a. e. to this function, where {w _k (x)} _k∈ℕ is the Walsh system. We find representation spectra in the form {n ^l + o(n ^l )} _n∈ℕ, where l ∈ {2 ^k } _k∈ℕ.     

Interpolation in certain Hilbert spaces of analytic functions

In this article we study, for a Hilbert spaceB of analytic functions in the open unit disk, the dependence of the structure of the space of sequencesB(Z)={{f(z_k)} _k=1 ^∞ :f∈B} on the choice of the sequence Z={z_k} _k=1 ^∞ of distinct points of the unit disk [6].     

Convergence of greedy approximation for the trigonometric system

Summary We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form G_m(f ) := ∑_kЄΛ f^(k) e ^{(i k,x)}, where ΛˆZ^d is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients f^ (k) of function f . Note that G_m(f ) gives the best m-term approximant in the L₂-norm and, therefore, for each f ЄL₂, ║f-G_m(f )║₂→0 as m →∞. It is known from previous results that in the case of p ≠2 the condition f ЄL_p does not guarantee the convergence ║f-G_m(f )║_p→0 as m →∞.. We study the following question. What conditions (in addition to f ЄL_p) provide the convergence ║f-G_m(f )║_p→0 as m →∞? In our previous paper [10] in the case 2< p ≤∞ we have found necessary and sufficient conditions on a decreasing sequence {A_n}_n=1^∞ to guarantee the L_p-convergence of {G_m(f )} for all f ЄL_p , satisfying a_n (f ) ≤A_n , where {a_n (f )} is a decreasing rearrangement of absolute values of the Fourier coefficients of f. In this paper we are looking for necessary and sufficient conditions on a sequence {M (m)} such that the conditions f ЄL_p and ║G_M(m)(f ) - G_m(f )║_p →0 as m →∞ imply ║f - G_m(f )║_p →0 as m →∞. We have found these conditions in the case when p is an even number or p = ∞.     

A survey of truncation error analysis for Padé and continued fraction approximants

To compute the value of a functionf(z) in the complex domain by means of a converging sequence of rational approximants {f _n(z)} of a continued fraction and/or Padé table, it is essential to have sharp estimates of the truncation error ¦f(z)–f _n(z)¦. This paper is an expository survey of constructive methods for obtaining such truncation error bounds. For most cases dealt with, {f _n(z)} is the sequence of approximants of a continued fractoin, and eachf _n(z) is a (1-point or 2-point) Padé approximant. To provide a common framework that applies to rational approximantf _n(z) that may or may not be successive approximants of a continued fraction, we introduce linear fractional approximant sequences (LFASs). Truncation error bounds are included for a large number of classes of LFASs, most of which contain representations of important functions and constants used in mathematics, statistics, engineering and the physical sciences. An extensive bibliography is given at the end of the paper.Research supported in part by the U.S. National Science Foundation under Grants INT-9113400 and DMS-9302584.     

Distribution of roots of quasianalytic functions

For functions of certain quasianalytic classes C{m_n} on (?∞, ∞) we determine a function ξ (x), depending on {m_n}, which is such that a sequence {x_k} is a sequence of the roots off(x) ε C{m_n} if and only if for somea $$\int_a^\infty {\tfrac{{dn(x)}}{{\xi (x - a}}< \infty ,} $$ where n(x) is a distribution function of the sequence {x_k}.     

Rate of decay of weakly lacunary series

Let function f(z) ≠ 0 be analytic in the unit disk and have sparse nonzero Taylor coefficients. Then the rate of decay of the function f as x → 1 − 0 depends on the rate of sparseness of its nonzero Taylor coefficients. In this paper, we consider the case f(z) = $ \sum\nolimits_{k = 0}^\infty {a_k z^{n_k } } $ \sum\nolimits_{k = 0}^\infty {a_k z^{n_k } } , where n _k ≥ A ₀(k + 2) ^p logb(k + 2).     

On almost sure convergence of Cesaro averages of subsequences of vector-valued functions

A remarkable theorem proved by Komlòs [4] states that if {f_n} is a bounded sequence in L¹(R), then there exists a subsequence {f_nk} and f L¹(R) such that f_nk (as well as any further subsequence) converges Cesaro to f almost everywhere. A similar theorem due to Révész [6] states that if {f_n} is a bounded sequence in L²(R), then there is a subsequence {f_nk} and f L²(R) such that Σ_k=1^∞ a_k(f_nk − f) converges a.e. whenever Σ_k=1^∞ | a_k |² < ∞. In this paper, we generalize these two theorems to functions with values in a Hilbert space (Theorems 3.1 and 3.3).     

Nonvanishing derivatives and normal families

We consider the differential operators Ψ _k , defined by Ψ₁(y) =y and Ψ _k+1(y)=yΨ _k y+d/dz(Ψ _k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ _k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z ²+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ _k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ _k (f ^′/f) =f ^(k)/f, we deduce in particular that iff andf ^(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f ^′/f :f ∈F} is normal. The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999, and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank Günter Frank for helpful discussions.     

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {r_n(z)}_n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {r_n(z)}_n∈N in the complement CP¹?D_f, where D_f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r_n(z)}_n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.     

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.     

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Extensions of the D(?k2)-triples {k2, k2 ± 1, 4k2 ± 1}

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k ², k ² + 1, 4k ² + 1, d} is a D(−k ²)-quadruple, then d = 1, and that if {k ² − 1, k ², 4k ² − 1, d} is a D(k ²)-quadruple, then d = 8k ²(2k ² − 1).     

On Classes of Functions Related to Starlike Functions with Respect to Symmetric Conjugate Points Defined by a Fractional Differential Operator

Let A denote the class of analytic functions f, in the open unit disk E = {z : |z| < 1}, normalized by f(0) = f′(0) − 1 = 0. In this paper, we introduce and study the class ST^n,a_l,m(h){ST^{n,\alpha}_{\lambda,m}(h)} of functions f ? A{f\in A}, with \fracD^n,a_l f_m(z)z 1 0{\frac{D^{n,\alpha}_\lambda f_m(z)}{z}\neq 0}, satisfying

Property $${(\hbar)}$$ and cellularity of complete Boolean algebras

A complete Boolean algebra \mathbbB{\mathbb{B}}satisfies property ((^h/_2p)){(\hbar)}iff each sequence x in \mathbbB{\mathbb{B}}has a subsequence y such that the equality lim sup z _n = lim sup y _n holds for each subsequence z of y. This property, providing an explicit definition of the a posteriori convergence in complete Boolean algebras with the sequential topology and a characterization of sequential compactness of such spaces, is closely related to the cellularity of Boolean algebras. Here we determine the position of property ((^h/_2p)){(\hbar)}with respect to the hierarchy of conditions of the form κ-cc. So, answering a question from Kurilić and Pavlović (Ann Pure Appl Logic 148(1–3):49–62, 2007), we show that ${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}${``\mathfrak{h}{\rm -cc}\Rightarrow (\hbar)"}is not a theorem of ZFC and that there is no cardinal \mathfrakk{\mathfrak{k}}, definable in ZFC, such that ${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}${``\mathfrak{k} {\rm -cc} \Leftrightarrow (\hbar)"}is a theorem of ZFC. Also, we show that the set { k: each k-cc c.B.a. has ((^h/_2p) ) }{\{ \kappa : {\rm each}\, \kappa{\rm -cc\, c.B.a.\, has}\, (\hbar ) \}}is equal to [0, \mathfrakh){[0, \mathfrak{h})}or [0, \mathfrak h]{[0, {\mathfrak h}]}and that both values are consistent, which, with the known equality {k: each c.B.a. having  ((^h/_2p) ) has the k-cc } = [\mathfrak s, ￥){{\{\kappa : {\rm each\, c.B.a.\, having }\, (\hbar )\, {\rm has\, the}\, \kappa {\rm -cc } \} =[{\mathfrak s}, \infty )}}completes the picture.