Solutions of Riccati-Abel equation in terms of third order trigonometric functions

Solutions of the generalized Riccati equations with third order nonlinearity, named as Riccati-Abel equation, are expressed via third order trigonometric functions. It is shown, as the ordinary Riccati equation, also the Riccati-Abel equation has a relationship with a linear differential equations. A summation formula for solutions of Riccati-Abel equation is established. Possible applications of this formula in the generalized dynamics is outlined. The method admits an extension to the case of generalized Riccati equations with any order of nonlinearity     

Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines

Summary A method is presented for fitting a function defined on a general smooth spherelike surfaceS, given measurements on the function at a set of scattered points lying onS. The approximating surface is constructed by mapping the surface onto a rectangle, and using a tensor-product of polynomial splines with periodic trigonometric splines. The use of trigonometric splines allows a convenient solution of the problem of assuring that the resulting surface is continuous and has continuous tangent planes at all points onS. Two alternative algorithms for computing the coefficients of the tensor fit are presented; one based on global least-squares, and the other on the use of local quasi-interpolators. The approximation order of the method is established, and the numerical performance of the two algorithms is compared.Supported in part by the National Science Foundation under Grant DMS-8902331 and by the Alexander von Humboldt Foundation     

Approximation of differentiable functions by polynomial splines in domains with external peaks

For unbounded domains with external power-type peaks, we propose a method for the approximation of functionsf(x) w _p ^r () by polynomial splines in the metricw _p ^r (), 1pq, and present the corresponding estimates.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1224–1233, September, 1994.     

Local properties of functions and approximation by trigonometric polynomials

Suppose Φ_{p, E} (p>0 an integer, E ?[0, 2π]) is a family of positive nondecreasing functions? _x(t) (t>0, x∈ E) such that? _x(nt)≤n^P ? _x(t) (n=0,1,...), t_n is a trigonometric polynomial of order at most n, and Δ _h ^l (f, x) (l>0 an integer) is the finite difference of orderl with step h of the functionf.THEOREM. Supposef (x) is a function which is measurable, finite almost everywhere on [0, 2π], and integrable in some neighborhood of each point xε E,? _X εΦ_p,E and $$\overline {\mathop {\lim }\limits_{\delta \to \infty } } |(2\delta )^{ - 1} \smallint _{ - \delta }^\delta \Delta _u^l (f,x)du|\varphi _x^{ - 1} (\delta ) \leqslant C(x)< \infty (x \in E).$$ . Then there exists a sequence {t _n } _n=1 ^∞ which converges tof (x) almost everywhere, such that for x ε E $$\overline {\mathop {\lim }\limits_{n \to \infty } } |f(x) - l_n (x)|\varphi _x^{ - 1} (l/n) \leqslant AC(x),$$ where A depends on p andl.     

Monotonicity of order preserving operator functions

We discuss monotonicity of order preserving operator functions and related order preserving operator inequalities.LetA?B?0withA>0,t∈[0,1]andp?1. Let

     

On the best approximation of periodic functions of two variables by polynomial splines

We consider the problem of the best approximation of periodic functions of two variables by a subspace of splines of minimal defect with respect to a uniform partition.     

Characteristic polynomial and higher order traces of third order three dimensional tensors

Eigenvalues of tensors play an increasingly important role in many aspects of applied mathematics. The characteristic polynomial provides one of a very few ways that shed lights on intrinsic understanding of the eigenvalues. It is known that the characteristic polynomial of a third order three dimensional tensor has a stunning expression with more than 20000 terms, thus prohibits an effective analysis. In this article, we are trying to make a concise representation of this characteristic polynomial in terms of certain basic determinants. With this, we can successfully write out explicitly the characteristic polynomial of a third order three dimensional tensor in a reasonable length. An immediate benefit is that we can compute out the third and fourth order traces of a third order three dimensional tensor symbolically, which is impossible in the literature.     

Alternative proofs for inequalities of some trigonometric functions

By using an identity relating to Bernoulli's numbers and power series expansions of cotangent function and logarithms of functions involving sine function, cosine function and tangent function, four inequalities involving cotangent function, sine function, secant function and tangent function are established.     

Concerning the order of approximation of periodic continuous functions by trigonometric interpolation polynomials

Letx _kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR _n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R _n(x_kn)=f(x_kn),R _n ^(j)(x_kn)=0,j=1,2,4,k=0,1…,n−1. We setw ₂(t,f) as the second modulus of continuity off(x). Then we prove that |R _n(x)-f(x)|=0(nw₂(1/nf)). We also examine the question of lower estimate of ‖R _n-f‖. This generalizes an earlier work of the author.     

Best one-sided approximation of some classes of smooth functions on the whole real axis by cardinal splines of higher order

The objective of this paper is to study the quantity . In this paper we establish a duality theorem for the quantity for m≥r. By virtue of the duality theroem we obtain the exact estimation for for all m≥r.     

Accelerated polynomial approximation of finite order entire functions by growth reduction

Let be an entire function of positive order and finite type. The subject of this note is the convergence acceleration of polynomial approximants of by incorporating information about the growth of for . We consider ``near polynomial approximation' on a compact plane set , which should be thought of as a circle or a real interval. Our aim is to find sequences of functions which are the product of a polynomial of degree and an ``easy computable' second factor and such that converges essentially faster to on than the sequence of best approximating polynomials of degree . The resulting method, which we call Reduced Growth method (-method) is introduced in Section 2. In Section 5, numerical examples of the -method applied to the complex error function and to Bessel functions are given.

     

Approximation of trigonometric functions by trigonometric polynomials with interpolation

The paper studies the approximation order of periodic functions by trigonometric polynomials with interpolation in arbitrary set of nodes. A method of construction of Hermite interpolation polynomials is pointed out.     

Generalization of some extremal properties of splines

We generalize well-known inequalities for the norms of the derivatives of periodic splines with minimal defect, perfect splines, and monosplines.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 403–407, March, 1995.