Order-constrained uniform approximations to the exponential based on restricted rationals

LetR _n/m(z∶γ)=P _n(z∶γ)/(1?γz) ^m be a rational approximation to exp (z),z ∈C, of ordern for all real positiveγ. In this paper we show there exists exactly one value ofγ in each of min(n+1,m) interpolation intervals such that the uniform error overR ^? is at a local minimum.     

Universal meromorphic approximation on Vitushkin sets

The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ? ?, there exists a function ?, meromorphic on ?, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ? such that $ \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\} The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ℕ such that converges to f(z) uniformly on K. A similar result is obtained for arbitrary domains G ≠ ℂ. Moreover, in case {λ _n }={n} the function ϕ is frequently universal in terms of Bayart/Grivaux [3]. Original Russian Text ? W.Luh, T.Meyrath, M.Niess, 2008, published in Izvestiya NAN Armenii. Matematika, 2008, No. 6, pp. 66–75.     

Existence Criterion for Estimates of Derivatives of Rational Functions

Suppose that K is a compact set in the open complex plane. In this paper, we prove an existence criterion for an estimate of Markov-Bernstein type for derivatives of a rational function R(z) at any fixed point z ₀ ∈ K. We prove that, for a fixed integer s, the estimate of the form |R ^(s) (z ₀)| ≤ C(K, z ₀, s)n‖R‖ _C(K), where R is an arbitrary rational function of degree n without poles on K and C is a bounded function depending on three arguments K, z ₀, and s, holds if and only if the supremum $$\omega (K,z_0 ,s) = \sup \left\{ {\frac{{\operatorname{dist} (z,K)}}{{\left| {z - z_0 } \right|^{s + 1} }}} \right\}$$ over z in the complement of K is finite. Under this assumption, C is less than or equal to const ·s!ω(K, z ₀, s).     

Congruences of multipartition functions modulo powers of primes

Let p _r (n) denote the number of r-component multipartitions of n, and let S _γ,λ be the space spanned by η(24z) ^γ ?(24z), where η(z) is the Dedekind’s eta function and ?(z) is a holomorphic modular form in \(M_{\lambda}(\mathrm{SL}_{2}(\mathbb{Z}))\) . In this paper, we show that the generating function of \(p_{r}(\frac{m^{k} n +r}{24})\) with respect to n is congruent to a function in the space S _γ,λ modulo m ^k . As special cases, this relation leads to many well known congruences including the Ramanujan congruences of p(n) modulo 5,7,11 and Gandhi’s congruences of p ₂(n) modulo 5 and p ₈(n) modulo 11. Furthermore, using the invariance property of S _γ,λ under the Hecke operator \(T_{\ell^{2}}\) , we obtain two classes of congruences pertaining to the m ^k -adic property of p _r (n).     

A Class of Nonlinear Averaging Integral Operators

LetHbe the class of analytic functions defined in the unit discU, and let coEdenote the convex hull of a setEinC. IfK⊂H, then an operatorI:K→His an averaging operator ifI[f](0) =f(0) andI[f](U) ⊂ cof(U), for allf∈K. The authors show that the operatorI_β,γ[f](z) ≡ [γz^−γ∫^z₀f^β(t)t^γ−1dt]^1/βis an averaging operator on certain subsets ofH.     

Zeros ofA p functions and related classes of analytic functions

The functionf(z), analytic in the unit disc, is inA ^p if \(\int {\int {_{\left| z \right|< 1} \left| {f(z)} \right|^p dxdy< \infty } } \) . A necessary condition on the moduli of the zeros ofA ^p functions is shown to be best possible. The functionf(z) belongs toB ^p if \(\int {\int {_{\left| z \right|< 1} \log ^ + \left| {f(z)} \right|)^p } } \) . Let {z _n } be the zero set of aB ^p function. A necessary condition on |z _n | is obtained, which, in particular, implies that Σ(1?|z _n |)^1+(1/p)+g <∞ for all ε>0 (p≧1). A condition on the Taylor coefficients off is obtained, which is sufficient for inclusion off inB ^p. This in turn shows that the necessary condition on |z _n | is essentially the best possible. Another consequence is that, forq≧1,p<q, there exists aB ^p zero set which is not aB ^q zero set.     

On the maximization of a certain nondifferentiable function

Let us consider a function $$\begin{gathered} \varphi (z) = \min \max f_{ij} (z), \hfill \\ \begin{array}{*{20}c} --- \\ {j \in 1,N} \\ \end{array} \begin{array}{*{20}c} --- \\ {j \in 1,N_j } \\ \end{array} \hfill \\ \end{gathered} $$ where the functionsf _ij(z) are supposed to be continuously differentiable and real-valued on a set Ω ofE _n,z?E _n. The problem is to find max_z?Ω?(z). In this paper, it is proved that ?(z) is directionally differentiable. A necessary condition for a maximum is derived, and some numerical algorithms for maximizing ? are suggested. The results obtained can be applied for solving some problems in mathematical programming and control theory.     

TheT-ideal generated by the standard identitys 3[x 1,x 2,x 3]

LetK = T_o(s₃), {c_n} its codimensions, {l_n} its colengths and {Χ_n} its sequence of co-characters. For 9≦n, c_n =2n - 1 or c_n =n(n + l)/2- 1, 3≦l_n ≦4 and χ_n =[n] + 2[n-1,1] + α[n-2,2] + β[2²,1^n?4] where α + β≦l.     

Weighted rational approximation in the complex plane

Given a triple (G, W, γ) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ(z), which is analytic in G, by weighted rational functions W^m_i+n_i(z)R_mi, n_i(z)_{i = 0}^∞, where R_mi, n_i(z) = P_mi(z)/Q_ni(z) with deg P_mi ≤ m_i and deg Q_ni ≤ n_i for all i ≥ 0 and where m_i + n_i → ∞ as i → ∞ such that lim m_i/[m_i + n_i] = γ. Our main result is a necessary and sufficient condition for such an approximation to be valid. Applications of the result to various classical weights are also included.     

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.

     

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.     

Rational approximation with varying weights I

We investigate two problems concerning uniform approximation by weighted rationals {w ⁿr_n～ _∞ ⁿ⁼¹ }, wherer _n=p_n Namely, forw(x):=e ^x we prove that uniform convergence to 1 ofw ⁿr_n is not possible on any interval [0,a] witha>2π. Forw(x):=x ^?, ?>1, we show that uniform convergence to 1 ofw ⁿr_n is not possible on any interval [b, 1] withb<tan ⁴(π(??1)/4?). (The latter result can be interpreted as a rational analogue of results concerning “incomplete polynomials.”) More generally, for α≥0, β≥0, α+β>0, we investigate forw(x)=e ^x andw(x)=x ^?, the possibility of approximation byw ⁿ p_n/q_n～ _∞ ⁿ⁼¹ , where depp _n≤αn, degq _n≤βn. The analysis utilizes potential theoretic methods. These are essentially sharp results though this will not be established in this paper.     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation

We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz ² ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g⁽²ⁿ⁾:g₀₀,g₀₁,g₁₀,?,g_0,2n,g_1,2n-1,?,g_2n-1,1,g_2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that g_ij=ò[`(z)]<sup>i</sup>z<sup>j</sup> dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ <sup>2</sup>, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number g<sub>n,n+1</sub> \gamma_{n,n+1} satisfying g<sub>n+1,n</sub> o [`(g)]_n,n+1=Ag_n,n-1+Bg_n,n+Cg_n+1,n-1+Dg_n,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z ^k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.     

A hereditarily indecomposable Banach space with rich spreading model structure

We present a reflexive Banach space \(\mathfrak{X}_{usm}\) which is Hereditarily Indecomposable and satisfies the following properties. In every subspace Y of \(\mathfrak{X}_{usm}\) there exists a weakly null normalized sequence {y _n } _n , such that every subsymmetric sequence {z _n } _n is isomorphically generated as a spreading model of a subsequence of {y _n } _n . Also, in every block subspace Y of \(\mathfrak{X}_{usm}\) there exists a seminormalized block sequence {z _n } and \(T:\mathfrak{X}_{usm} \to \mathfrak{X}_{usm}\) an isomorphism such that for every n ∈ ?, T(z _2n?1) = z _2n . Thus the space is an example of an HI space which is not tight by range in a strong sense.     

Padé tables of entire functions of very slow and smooth growth

Letf(z)=σ _j?o ^∞ a _j z ^j be entire with $$|a_{j - 1} a_{j + 1} /a_j^2 | \leqslant \rho _0^2 ,j = 1,2,3, \ldots ,$$ whereρ ₀=0.4559... is the positive root of the equation $$2\sum\limits_{j = 1}^\infty {\rho ^{j^2 } = 1.}$$ . It is shown that the Padé table off is normal, and asL→∞, [L/M _L ](z) converges uniformly in compact subsets ofC tof, for any sequence of nonnegative integers {M _L } _L=1 ^∞. In particular, the diagonal sequence {[L/L]} converges uniformly in compact subsets ofC tof. Furthermore, the constantρ ₀ is shown to be best possible in a strong sense.     

Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences

In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn+1/zn}is increasing(resp.,decreasing),then the sequence {n√zn} is strictly increasing(resp.,decreasing) subject to a certain initial condition. We also give some sufficient conditions when {zn+1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach.     

Iterative solution of nonlinear equations of accretive and pseudocontractive types

Let E be a real uniformly smooth Banach space. Let A:D(A)=E→2E be an accretive operator that satisfies the range condition and A⁻¹(0)≠∅. Let {λ_n} and {θ_n} be two real sequences satisfying appropriate conditions, and for z∈E arbitrary, let the sequence {x_n} be generated from arbitrary x₀∈E by x_n+1=x_n−λ_n(u_n+θ_n(x_n−z)), u_n∈Ax_n, n?0. Assume that {u_n} is bounded. It is proved that {x_n} converges strongly to some x∗∈A−1(0). Furthermore, if K is a nonempty closed convex subset of E and T:K→K is a bounded continuous pseudocontractive map with F(T):={Tx=x}≠∅, it is proved that for arbitrary z∈K, the sequence {x_n} generated from x₀∈K by x_n+1=x_n−λ_n((I−T)x_n+θ_n(x_n−z)), n?0, where {λ_n} and {θ_n} are real sequences satisfying appropriate conditions, converges strongly to a fixed point of T.     

On the limiting case for boundedness of the B-Riesz potential in B-Morrey spaces

We consider the generalized shift operator associated with the Laplace-Bessel differential operator $$ \Delta _B = \sum\limits_{i = 1}^n {\frac{{\partial ^2 }} {{\partial x_j^2 }}} + \sum\limits_{i = 1}^k {\frac{{\gamma _i }} {{x_i }}\frac{\partial } {{\partial x_i }}} $$ , and study the modified B-Riesz potential ? _{α, β} generated by the generalized shift operator acting in the B-Morrey space in the limiting case. We prove that the operator ? _{α, β}, 0 < α < n + |γ|, is bounded from the B-Morrey space L _{(n+|γ|?λ)/α,λ,γ} (? _k,+ ⁿ ) to the B-BMO space BMO _γ (? _k,+ ⁿ ).