Recovery of band limited functions via cardinal splines

The main result of this paper asserts that if a function f is in the class B_π,p, 1 <p < ∞; that is, those p-integrable functions whose Fourier transforms are supported in the interval [ - π, π], then f and its derivatives f^(j) j = 1, 2, …, can be recovered from its sampling sequence{f(k)} via the cardinal interpolating spline of degree m in the metric ofL _q(ℝ)), 1 <p=q < ∞, or 11 <p=q < ⩽ ∞.     

A Cohen type inequality for Laguerre-Sobolev expansions

Let introduce the Sobolev type inner product

     

Cardinal splines in nonparametric regression

We discuss a nonparametric regression model on an equidistant grid of the real line. A class of kernel type estimates based on the so-called fundamental cardinal splines will be introduced. Asymptotic optimality of these estimates will be established for certain functional classes. This model explains the often mentioned heuristic fact that cubic splines are adequate for most practical applications.      

A Liapunov type inequality for Sugeno integrals

The classical Liapunov inequality shows an interesting upper bound for the Lebesgue integral of the product of two functions. This paper proposes a Liapunov type inequality for Sugeno integrals. That is, we show that holds for some constant H_s,t,r where 0<t<s<r,f:[0,1]→[0,∞) is a non-increasing concave function, and μ is the Lebesgue measure on R. We also present two interesting classes of functions for which the classical Liapunov type inequality for Sugeno integrals with H_s,t,r=1 holds. Some examples are provided to illustrate the validity of the proposed inequality.     

Riesz basis, Paley-Wiener class and tempered splines

The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ∮2m(T,R)∩L2(R) which is the space of polynomial splines with irregularly distributed nodes T={tj}j∈Z, where {tj}j∈Z is a real sequence such that {eitξ}j∈Z constitutes a Riesz basis for L2([-π,π]). From these results, the asymptotic relation E(f,Bπ,2)2=lim E(f,∮2m(T,R)∩L2(R))2 is proved, where Bπ,2 denotes the set of all functions from L2(R) which can be continued to entire functions of exponential type ≤π, i.e. the classical Paley-Wiener class.     

On Bernstein's inequality for entire functions of exponential type

It was shown by S.N. Bernstein that if f is an entire function of exponential type τ such that |f(x)|?M for −∞<x<∞, then |f^′(x)|?Mτ for −∞<x<∞. If p is a polynomial of degree at most n with |p(z)|?M for |z|=1, then f(z):=p(eiz) is an entire function of exponential type n with |f(x)|?M on the real axis. Hence, by the just mentioned inequality for functions of exponential type, |p^′(z)|?Mn for |z|=1. Lately, many papers have been written on polynomials p that satisfy the condition znp(1/z)≡p(z). They do form an intriguing class. If a polynomial p satisfies this condition, then f(z):=p(eiz) is an entire function of exponential type n that satisfies the condition f(z)≡einzf(−z). This led Govil [N.K. Govil, Lp inequalities for entire functions of exponential type, Math. Inequal. Appl. 6 (2003) 445-452] to consider entire functions f of exponential type satisfying f(z)≡eiτzf(−z) and find estimates for their derivatives. In the present paper we present some additional observations about such functions.     

Whittle type inequality for demisubmartingales

A Whittle type inequality for demisubmartingales is derived and a strong law of large numbers for functions of a demisubmartingale is obtained.

     

Optimal smoothing and interpolating splines with constraints

This paper considers the problem for designing optimal smoothing and interpolating splines with equality and/or inequality constraints. The splines are constituted by employing normalized uniform B-splines as the basis functions, namely as weighted sum of shifted B-splines of degree k. Then a central issue is to determine an optimal vector of the so-called control points. By employing such an approach, it is shown that various types of constraints are formulated as linear function of the control points, and the problems reduce to quadratic programming problems. We demonstrate the effectiveness and usefulness by numerical examples including approximation of probability density functions, approximation of discontinuous functions, and trajectory planning.     

Weak type inequality for noncommutative differentially subordinated martingales

In the paper we focus on self-adjoint noncommutative martingales. We provide an extension of the notion of differential subordination, which is due to Burkholder in the commutative case. Then we show that there is a noncommutative analogue of the Burkholder method of proving martingale inequalities, which allows us to establish the weak type (1,1) inequality for differentially subordinated martingales. Moreover, a related sharp maximal weak type (1,1) inequality is proved. Research supported by MEN Grant 1 PO3A 012 29.     

Sizer for smoothing splines

Smoothing splines are an attractive method for scatterplot smoothing. The SiZer approach to statistical inference is adapted to this smoothing method, named SiZerSS. This allows quick and sure inference as to “which features in the smooth are really there” as opposed to “which are due to sampling artifacts”, when using smoothing splines for data analysis. Applications of SiZerSS to mode, linearity, quadraticity and monotonicity tests are illustrated using a real data example. Some small scale simulations are presented to demonstrate that the SiZerSS and the SiZerLL (the original local linear version of SiZer) often give similar performance in exploring data structure but they can not replace each other completely. Marron’s research was supported by the Dept. of Stat. and Appl. Prob., National Univ. of Singapore, and by the National Science Foundation Grant DMS-9971649. Zhang’s research was supported by the National Univ. of Singapore Academic Research grant R-155-000-023-112. The Editor, the Associate Editor, and the referees are appreciated for their invaluable comments and suggestions that help improve the article significantly.     

Approximation of smooth functions by polyharmonic cardinal splines inL p (ℝn) spacespace

The remainders and the convergence of cardinal polyharmonic spline interpolation are studied, and the asymptotic behavior of the best approximation by polyharmonic spline and the averageK-width of some class of smooth functions defined on the Euclidean spaceR ⁿ are determined. This project is supported by the National Natural Science Foundation of China (No. 19671012) and by the Doctoral Programme Foundation of Institution of Higher Education of Country Education Committee of China.     

A gap 1 cardinal transfer theorem

We extend the gap 1 cardinal transfer theorem (κ ⁺, κ ) → (λ ⁺, λ ) to any language of cardinality ≤λ, where λ is a regular cardinal. This transfer theorem has been proved by Chang under GCH for countable languages and by Silver in some cases for bigger languages (also under GCH). We assume the existence of a coarse (λ, 1)‐morass instead of GCH. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Cotlar type inequality for the multilinear singular integral operators and its applications

A Cotlar type inequality is established for the multilinear singular integral operators. As applications, some two-weight norm inequalities are obtained for the maximal operator corresponding to the multilinear singular integral operators.     

Can a large cardinal be forced from a condition implying its negation?

In this note, we provide an affirmative answer to the title question by giving two examples of cardinals satisfying conditions implying they are non-Rowbottom which can be turned into Rowbottom cardinals via forcing. In our second example, our cardinal is also non-Jonsson.

     

Strong type of Steckin inequality for the linear combination of Bernstein operators

In this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein operators, which gives the optimal approximation rate. Moreover, a method to prove lower estimates for linear operators is introduced. As a result the lower estimate for the linear combinations of Bernstein operators is obtained by using the Ditzian–Totik modulus of smoothness.     

A generalized Gronwall-Bellman-Ou-Iang type inequality for discontinuous functions and applications to BVP

In this paper we investigate a generalized Gronwall-Bellman-Ou-Iang type inequality for piecewise continuous functions. By studying the unknown function in different continuous regions Ω_kj of Ω, we give the complete estimate for the unknown function. We apply our result to a boundary value problem of a partial impulsive differential equation for estimation of its solution.     

Bernstein–Nikolskii and Plancherel–Polya inequalities in Lp ‐norms on non‐compact symmetric spaces

By using Bernstein‐type inequality we define analogs of spaces of entire functions of exponential type in L_p (X), 1 ≤ p ≤ ∞, where X is a symmetric space of non‐compact. We give estimates of L_p ‐norms, 1 ≤ p ≤ ∞, of such functions (the Nikolskii‐type inequalities) and also prove the L_p ‐Plancherel–Polya inequalities which imply that our functions of exponential type are uniquely determined by their inner products with certain countable sets of measures with compact supports and can be reconstructed from such sets of “measurements” in a stable way (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)