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## Limits of starshaped sets

### Archiv der Mathematik (1987-03-01) 48: 241-249 , March 01, 1987

## More General Step-Weight Functions ϕ

### Moduli of Smoothness (1987-01-01) 9: 46-54 , January 01, 1987

In Section 1.2 three conditions were imposed on *ϕ* which were the basis of all the proofs up to now. In this chapter we will relax Conditions I and II and will show that in a way III cannot be relaxed without changing the definition of *ω*_{ϕ}^{r}
(*f,t*)_{p}. For *ϕ*(*x*) ~ *x*^{β(0)} where *β*(0) < 0 and *ϕ*(*x*) ~ *x*^{β(∞)} where *β*(∞) > 1 we will provide such a change.

## Front Matter - Rational Homotopy Type

### Rational Homotopy Type (1987-01-01): 1264 , January 01, 1987

## The cardinality of FM(1+1+n)

### Order (1987-03-01) 4: 15-30 , March 01, 1987

The aim of this note is to develop a counting formula for the modular lattice *FM*(1+1+*n*) freely generated by two single elements and an *n*-element chain. This answers Problem 44 in Birkhoff [1] which asks one to determine *FM*(1+1+*n*). The proof of our recursive formula is based on the scaffolding method developed by R. Wille.

## A generalization of Chernoff inequality via stochastic analysis

### Probability Theory and Related Fields (1987-05-01) 75: 149-157 , May 01, 1987

### Summary

Let μ be a probability measure on *R*^{d}with density *c*(exp(-2*U(x)*), where *U∈C*^{2}(R^{d}),
$$\left| {\nabla U(x)} \right|^2 - \Delta U(x) \to \infty $$
and *U(x)*→∞ as *|x|*→∞. By using stochastic analysis and theorems in Schrödinger operators we have the following result: there exists a constant *c*>0 such that
(1)
$$Var_\mu f\underline \leqslant c E_\mu \left| {\nabla f} \right|^2 $$
for any *f∈L*^{1}(μ) with a well-defined distributional gradient ∇*f*. Under our condition, the operator
$$ - \frac{1}{2}\Delta + \nabla U \cdot \nabla $$
in *L*^{2}(μ) has discrete spectrum 0 = λ_{1} < λ_{2} = ... = λ_{m} < λ_{m + 1} ≦ ... with corresponding eigenfunctions {*φ*_{n}} which form a C.O.N.S. (complete orthonormal system). If the R.H.S. of (1) is finite then equality holds iff
$$f = \sum\limits_{i = 1}^m {b_i \phi _i } $$
for some *b*_{1},...,b_{m}∈R. Moreover, the constant *c* can be taken as (2λ_{2})^{−1}.

When *U* is a quadratic form, (1) is the Chernoff inequality (Chernoff 1981; Chen 1982). The approach here can be generalized to infinite dimensional Gaussian measures, or the case with μ being a probability measure in a bounded domain of *R*^{d}or some discrete cases.

## Derivation and implementation of an algorithm for singular integrals

### Computing (1987-09-01) 38: 235-245 , September 01, 1987

A numerical construction of extended Gaussian quadrature rules for weight functions with algebraic and logarithmic singularities is presented. A computer program is described and numerical examples are given.

## Some results about smoothing methods of fourier series

### Annali di Matematica Pura ed Applicata (1987-12-01) 149: 207-216 , December 01, 1987

### Summary

In this paper we compare the Fourier polynomials approximation, in C(T^{n}) or in L^{p}(T^{n}), with that one obtained by a class of smoothing methods, which naturally arise in solving ill posed problems. It is also given a sharp evaluation of the above approximation in the space Lip (α, C(T)), 0<α<1.

## Möbius Inversion in Lattices

### Classic Papers in Combinatorics (1987-01-01): 403-415 , January 01, 1987

In the development of computational techniques for combinatorial theory, attention has lately centered on R*OTA’S* theory of Möbius inversion [6]. The main theorem of R*OTA’S* paper, concerning the computation of the Möbius invariant across a Galois connection, is a prerequisite to the use of lattice-theoretic methods in combinatorics.

## Polynomial growth estimates for multilinear singular integral operators

### Acta Mathematica (1987-12-01) 159: 51-80 , December 01, 1987

## Fitting Cosines: Some Procedures and Some Physical Examples

### Advances in the Statistical Sciences: Applied Probability, Stochastic Processes, and Sampling Theory (1987-01-01) 34: 75-100 , January 01, 1987

The paper is concerned with a variety of time series models that in some sense lead to the fitting of a cosine function of unknown frequency. Both linear and nonlinear models are considered, including both decaying cosines and sustained ones. The discussion is illustrated with examples from seismology (free oscillations of the Earth), geophysics (the Chandler wobble), nuclear magnetic resonance, laser Doppler velocimetry and oceanography (dispersion). The paper ends by surveying a variety of results developed for specific models by various authors. A variety of open problems are indicated.