Completeness of Integer Translates in Function Spaces on

We show that each of the Banach spacesC₀( ) andL^p( ), 2<p<∞, contains a function whose integer translates are complete. This function can also be chosen so that one of the following additional conditions hold: (1) Its non-negative integer translates are already complete. (2) Its integer translates form an orthonormal system inL²( ). (3) Its integer translates form a minimal system. A similar result holds for the corresponding Sobolev space, for certain weightedL²spaces, and in the multivariate setting. We also prove some results in the opposite direction.     

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.     

Sur l'approximation de fonctions intégrables sur [0, 1] par des polynômes de Bernstein modifies

We study here a new kind of modified Bernstein polynomial operators on L¹(0, 1) introduced by J. L. Durrmeyer in [4]. We define for f integrable on [0, 1] the modified Bernstein polynomial M_n f: M_nf(x) = (n + 1) ∑ⁿ_{k = o}P_nk(x)∝¹₀ P_nk(t) f(t) dt. If the derivative d^r f/dx^r with r 0 is continuous on [0, 1], d^r/dx^rM_n f converge uniformly on [0,1] and sup_xε[0,1] ¦M_n f(x) − f(x)¦ 2ω_f(1/trn) if ω_f is the modulus of continuity of f. If f is in Sobolev space W_l,p(0, 1) with l 0, p 1, M_n f converge to f in w^l,p(0, 1).     

A Coding Theory Bound and Zero-Sum Square Matrices

For a code C=C(n,M) the level k code of C, denoted C _k , is the set of all vectors resulting from a linear combination of precisely k distinct codewords of C. We prove that if k is any positive integer divisible by 8, and n=k, M=k2k then there is a codeword in C _k whose weight is either 0 or at most . In particular, if <(4–2)²/48 then there is a codeword in C _k whose weight is n/2–(n). The method used to prove this result enables us to prove the following: Let k be an integer divisible by p, and let f(k,p) denote the minimum integer guaranteeing that in any square matrix over Z _p , of order f(k,p), there is a square submatrix of order k such that the sum of all the elements in each row and column is 0. We prove that lim inf f(k,2)/k<3.836. For general p we obtain, using a different approach, that f(k,p)p( ^{k / ln k )(1+ o} _k ⁽¹⁾⁾.     

Boundary effects on convergence rates for Tikhonov regularization

We consider the Tikhonov regularizer f_λ of a smooth function f ε H^2m[0, 1], defined as the solution (see [1]) to We prove that if f^(j)(0) = f^(j)(1) = 0, J = m, …, k < 2m − 1, then ¦f − f_λ¦_j² Rλ^{(2k − 2j + 3)/2m}, J = 0, …, m. A detailed analysis is given of the effect of the boundary on convergence rates.     

Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on the Unit Circle

We study the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product on the unit circle

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where f(Z)=(f(z₁), …, f^(l₁)(z₁), …, f(z_m), …, f^(l_m)(z_m)), A is a M×M positive definite matrix or a positive semidefinite diagonal block matrix, M=l₁+…+l_m+m, dμ belongs to a certain class of measures, and |z_i|>1, i=1, 2, …, m.     

Optimal broadcast on parallel locality models

In this paper matching upper and lower bounds for broadcast on general purpose parallel computation models that exploit network locality are proven. These models try to capture both the general purpose properties of models like the PRAM or BSP on the one hand, and to exploit network locality of special purpose models like meshes, hypercubes, etc., on the other hand. They do so by charging a cost l(|i−j|) for a communication between processors i and j, where l is a suitably chosen latency function.An upper bound T(p)=∑_i=0^loglogp2ⁱ·l(p^1/2i) on the runtime of a broadcast on a p processor H-PRAM is given, for an arbitrary latency function l(k).The main contribution of the paper is a matching lower bound, holding for all latency functions in the range from l(k)=Ω(logk/loglogk) to l(k)=O(log²k). This is not a severe restriction since for latency functions l(k)=O(logk/log¹⁺log(k)) with arbitrary >0, the runtime of the algorithm matches the trivial lower bound Ω(logp) and for l(k)=Θ(log¹⁺k) or l(k)=Θ(k), the runtime matches the other trivial lower bound Ω(l(p)). Both upper and lower bounds apply for other parallel locality models like Y-PRAM, D-BSP and E-BSP, too.     

On Positive and Copositive Polynomial and Spline Approximation inLp[−1, 1], 0<p<∞

For a functionfL_p[−1, 1], 0<p<∞, with finitely many sign changes, we construct a sequence of polynomialsP_nΠ_nwhich are copositive withfand such that f−P_npCω(f, (n+1)⁻¹)_p, whereω(f, t)_pdenotes the Ditzian–Totik modulus of continuity inL_pmetric. It was shown by S. P. Zhou that this estimate is exact in the sense that if f has at least one sign change, thenωcannot be replaced byω²if 1<p<∞. In fact, we show that even for positive approximation and all 0<p<∞ the same conclusion is true. Also, some results for (co)positive spline approximation, exact in the same sense, are obtained.     

Behavior of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions at the Points s=1 and s=1/2

Let S_k(N)⁺ be the set of primitive cusp forms of even weight k for Γ₀(N) and let L(s, sym ²f) be the symmetric square L-function L(s, f) of a form f ∈ S_k(N)⁺. The moments of the variable L(1, sym ²f), f ∈ S₂(N)⁺, are computed for N = p, and the corresponding limiting distribution is determined in N-aspect. Let f ∈ S_k(1)⁺, g ∈ S_l(1)⁺, and ω_f = Γ(k - 1)/(4π)^k-1 〈f, f〉. Asymptotic formulas for and as k → ∞ are obtained. Bibliography: 17 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 149–167.     

Commutators for Approximation Spaces and Marcinkiewicz-Type Multipliers

It is proved that, under some conditions, weaker than those of the Marcinkiewicz multiplier theorem, the multiplier operator T_μ(∑_k c_ke^ikt)=∑_k μ_kc_ke^ikt satisfies on the Besov space B^σ, q_p the commutator theorem[T, T_μ]_{B^σ, qp, B^σ, qp}c T, where T=max(T_{B^σ0, q0p, B^σ0, q0p}, T_{B^σ1, q1p, B^σ1, q1p} and σ₀>σ>σ₁>0.     

New Upper Bound for the B-Spline Basis Condition Number: II. A Proof of de Boor's 2-Conjecture

For the p-norm condition number κ_k, p of the B-spline basis of order k we prove the upper estimate κ_k, p<k2^k. This proves de Boor's 2^k-conjecture up to a polynomial factor.     

Best Polynomial Approximations in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> and Widths of Some Classes of Functions

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions f(x) ∈ L ₂ ^r (r ∈ ℤ₊) and expressions containing moduli of continuity of the kth order ω_k(f(^r), t). Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from L ₂. For the classes (k, r, Ψ_*) defined by ω _k(f ^(r), t) and the majorant , we determine the exact values of different widths in the space L₂.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1458–1466, November, 2004.     

Completeness of Trigonometric System with Integer Indices {e; x }

Necessary and sufficient conditions are given which ensure the completeness of the trigonometric systems with integer indices; {e^inx; x }^∞_n=−∞ or {e^inx; x }^∞_n=1 in L^α(μ,  ), α1. If there exists a support Λ of the measure μ which is a wandering set, that is, Λ+2kπ, k=0, ±1, ±2, … are mutually disjoint for different k's, then the linear span of our trigonometric system {e^inx; x }^∞_n=−∞ is dense in L^α(μ,  ) α1. The converse statement is also true.     

Comonotone Jackson's Inequality

Let 2s points y_i=−πy_2s<…<y₁<π be given. Using these points, we define the points y_i for all integer indices i by the equality y_i=y_i+2s+2π. We shall write fΔ⁽¹⁾(Y) if f is a 2π-periodic continuous function and f does not decrease on [y_i, y_i−1], if i is odd; and f does not increase on [y_i, y_i−1], if i is even. In this article the following Theorem 1—the comonotone analogue of Jackson's inequality—is proved. 1. If fΔ⁽¹⁾(Y), then for each nonnegative integer n there is a trigonometric polynomial τ_n(x) of order n such that τ_nΔ⁽¹⁾(Y), and |f(x)−π_n(x)|c(s) ω(f; 1/(n+1)), x , where ω(f; t) is the modulus of continuity of f, c(s)=const. Depending only on s.     

A Remez-Type Inequality for Non-dense Müntz Spaces with Explicit Bound

LetΛ :=(λ_k)^∞_k=0be a sequence of distinct nonnegative real numbers withλ₀ :=0 and ∑^∞_k=1 1/λ_k<∞. Let(0, 1) and(0, 1−) be fixed. An earlier work of the authors shows that [formula]is finite. In this paper an explicit upper bound forC(Λ, , ) is given. In the special caseλ_k :=k^α,α>1, our bounds are essentially sharp.     

On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle

Let u(r,θ) be biharmonic and bounded in the circular sector ¦θ¦ < π/4, 0 < r < ρ (ρ > 1) and vanish together with δu/δθ when ¦θ¦ = π/4. We consider the transform û(p,θ) = ∝₀¹r^{p − 1}u(r,θ)dr. We show that for any fixed θ₀ u(p,θ₀) is meromorphic with no real poles and cannot be entire unless u(r, θ₀) ≡ 0. It follows then from a theorem of Doetsch that u(r, θ₀) either vanishes identically or oscillates as r → 0.     

Weighted simultaneous Chebyshev approximation

In this paper we discuss the problem of weighted simultaneous Chebyshev approximation to functions f₁,…f_m ε C(X) (1 m ∞), i.e., we wish to minimize the expression {∑^m_{j = 1} λ_j¦f_j − q¦^p}^1/p∞, where λ_j > 0, ∑^m_{j = 1} λ_j = 1, p 1. For this problem we establish the main theorems of the Chebyshev theory, which include the theorems of existence, alternation, de La Vallée Poussin, uniqueness, strong uniqueness, as well as that of continuity of the best approximation operator, etc.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

On binary-exponent alternating sums

We investigate binary-exponent alternating sums, i.e. sums with the general appearance , whereb _k is the sum of the digits in the binary representation ofk. Whenf(k)=k ^s ,s integer, the sum cancels ifs<p, and, whensp, takes values which can be recursively calculated. This recursion is studied more in detail. The casef(k)=sin(2(x+k)/2 ^s ) illustrates the properties of the current summation operator.     

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).