On approximation of discontinuous functions inL-metric

В работе рассматрива ются аппроксимативн ые свойства функцииf (x)=signx при ее приближения алгебра ическими многочлена ми в метрикеL[?1,1]. Исследован вопрос о единственности мно гочлена наилучшего п риближения и в явном виде указываю тся сами такие многочлены. Под считано точное значе ние наилучших приближен ийE _n (f) _L . Результаты обобщаются на сдвину тые функцииf(x)=sign (х-х ₀), гдех ₀ - корень многочлена Чебышева второго рода, х₀ε (-1,1).     

On Kellog's theorem for discontinuous green functions

Translated from Matematicheskie Zametki, Vol. 53, No. 3, pp. 151–153, January, 1993.     

On optimal polynomial interpolation of analytic functions

This paper discusses the problem of choosing the Lagrange interpolation points T = (t₀, t₁,…, t_n) in the interval −1 t 1 to minimize the norm of the error, considered as an operator from the Hardy space H²(R) of analytic functions to the space C[−1, 1]. It is shown that such optimal choices converge for fixed n, as R → ∞, to the zeros of a Chebyshev polynomial. Asymptotic estimates are given for the norm of the error for these optimal interpolations, as n → ∞ for fixed R. These results are then related to the problem of choosing optimal interpolation points with respect to the Eberlein integral. This integral is based on a probability measure over certain classes of analytic functions, and is used to provide an average interpolation error over these classes. The Chebyshev points are seen to be limits of optimal choices in this case also.     

On the Lebesgue constants for interpolation of analytic functions

Дль сИстЕМы РАжлИЧНы х тОЧЕкΤ=(t ₁,...,t _n ) Иж ОтРЕ жкА [?1,1] Иk?[0,1) ВВОДИтсь ВЕлИЧ ИНА $$L_n (\tau ,p,k) = \mathop {\max }\limits_{t \in [ - 1,1]} (\mathop \Sigma \limits_{j = 1}^n |D_j (t)|^p )^{1/p} ,$$ где $$D_j (t) = \frac{{\omega _j (t)}}{{\omega _j (t_j )}}[1 - kW_j^2 (t)],{\mathbf{ }}\omega _j (t) = \mathop \prod \limits_{\begin{array}{*{20}c} {m = 1} \\ {m \ne 1} \\ \end{array} }^n W_m (t),{\mathbf{ }}W_m (t) = \frac{{t - t_m }}{{1 - kt_m t}}.$$ пРИk=0 ОНА сОВпАДАЕт с кОНс тАНтОИ лЕБЕгА, сВьжАН НОИ с ИНтЕРпОльцИЕИ МНОгО ЧлЕНОМ лАгРАНжА. пОкАжАНА сВ ьжь ВЕлИЧИНыL _n (Τ, p, k) с жАД АЧАМИ ИНтЕРпОльцИИ АНАлИт ИЧЕскИх ФУНкцИИ. Дль сИстЕМы $$Z = \left\{ {sn\left[ {\left( {\frac{{2j - 1}}{n} - 1} \right)K,k} \right]} \right\}_{j = 1}^n ,$$ ьВльУЩЕИсь АНАлОгОМ ЧЕБышЕВскОИ сИстЕМы, пОлУЧЕНы ОцЕНкИL _n (Z, p, k) пРИp≧2 Иp≧1.     

On the convergence of double Fourier series of discontinuous functions

In the paper we obtain a formula for the computation of the rectangular partial sums of a double Fourier series of a function of two variables at the point (0, 0), having a discontinuity along any ray at this point. We also give an estimate of the remainder in the abovementioned formula.     

On the Nevanlinna-Pick interpolation problem for generalized stieltjes functions

The solutions of the Nevanlinna-Pick interpolation problem for generalized Stieltjes matrix functions are parametrized via a fractional linear transformation over a subset of the class of classical Stieltjes functions. The fractional linear transformation of some of these functions may have a pole in one or more of the interpolation points, hence not all Stieltjes functions can serve as a parameter. The set of excluded parameters is characterized in terms of the two related Pick matrices.Dedicated to the memory of M. G. Kreîn     

ON INTERPOLATION OF A CERTAIN CLASS OF FUNCTIONS

I.StatementofResultsForarealaleta~ma-c(1,Ial).Lets21.WeuseG.todenotethes-dimensionalunitcube,and(:,')todenotethescalarproductofvectors:and,inH.Form(ml,',m.)EZsand^~(Al,',A.)EHwedenotel'II^=mtl...m:a.Inparticular,ifA=(A,',A),thenwewritellmJI'=IImll^.Fulthermore,forf=(TI,'.)r.)EH,letiff=lrlI ... Ir.].Letf~(n,',r.)ERswithfi20(i~1,',s),andletf(:)beasingle--valuedfunctionsuchthatf(xl,'sxit',x.)~f(xl,')xi 1,',x.),i~1,')s.Writefi=pi fi(i~1,',s),wherepi~fi~0iffi=0an…     

On algebras generated by convolutions and discontinuous functions

Banach algebras generated by Fourier and Mellin convolution operators with discontinuous presymbols and by discontinuous functions in L_p (IR⁺, x) spaces with weight are investigated. The Fredholm properties are characterized by a symbol calculus and an index formula for such operators is presented. These results were obtained by H. O. Cordes in [3] for the case p=2, =0 and presymbols, which are discontinuous only at infinity and generalized in [20] for 1相似文献   

On interpolation with products of positive definite functions

In this paper we consider the problem of scattered data interpolation for multivariate functions. In order to solve this problem, linear combinations of products of positive definite kernel functions are used. The theory of reproducing kernels is applied. In particular, it follows from this theory that the interpolating functions are solutions of some varational problems.     

On the Lagrange interpolation for a subset ofC k functions

We study the optimal order of approximation forC ^k piecewise analytic functions (cf. Definition 1.2) by Lagrange interpolation associated with the Chebyshev extremal points. It is proved that the Jackson order of approximation is attained, and moreover, ifx is away from the singular points, the local order of approximation atx can be improved byO(n ^?1). Such improvement of the local order of approximation is also shown to be sharp. These results extend earlier results of Mastroianni and Szabados on the order of approximation for continuous piecewise polynomial functions (splines) by the Lagrange interpolation, and thus solve a problem of theirs (about the order of approximation for |x|³) in a much more general form.     

On optimal extrapolation and interpolation of fuzzy analytic functions

Для класса ? аналитич еских в единичном кру ге функций, ограниченны х по модулю единицей, погрешност ью наилучшего прибли жения в точкеz ₀ по значениям в точкахz ₁,..., z_n, заданным с погрешнос тьюδ, называется вели чинаr(z ₀, z₁ z..., z_n, α)=inf sup sup ¦f(z₀)-S(f₁, ...f_n)¦, где нижняя грань бере тся по всевозможным ф ункциям S: Сⁿ→С. ДляE～((?1,1) иz ₀∈ ∈(-1,1)Е рассматривается задача о нахождении п орядка информативности мно жестваЕ, т.е. минимальногоп, на котором достигается нижняя грань в равенстве $$R(z_0 ,\delta ,E) = \mathop {\inf }\limits_n {\text{ }}\mathop {\inf }\limits_{z_1 , \ldots ,z_n \in E} {\text{ }}r(z_0 ,z_1 , \ldots ,z_n ,\delta ).$$ Кроме того, приδ, близ ких к 1, решена задача о нахождении величины $$r_n (\delta ,E) = \mathop {\inf }\limits_{z_1 , \ldots ,z_n \in Ez_0 \in E} \sup r(z_0 ,z_1 , \ldots ,z_n ,\delta )$$ и найдены узлы, на кото рых достигается нижн яя грань.     

On the minimization of possibly discontinuous functions by means of pointwise approximations

A general approach for the solution of possibly discontinuous optimization problems by means of pointwise (perhaps smooth) approximations will be proposed. It will be proved that sequences generated by pointwise approximation techniques eventually satisfy well justified stopping criteria. Numerical examples will be given.     

On a class of discontinuous vector-valued functions and the associated quasi-variational inequalities

We study in detail a class of discontinuous vector-valued functions defined on a closed convex subset of Rn, which was introduced by B. Ricceri [7] and which is very useful in the theory of variational inequalities. The results are used to give a new proof for the existence theorem due to P. Cubiotti [3]. The proof allows us to have a better understanding of quasi-variational inequalities associated with the abovementioned class of functions.     

On the approximation of the continuous functions by some Hermite-Fejer interpolation polynomials

The object of this note is to improve Some wellknown results, which are related with the approximation problems of the continuous functions by Hermite-Fejér interpolation which based on the zeros of Chebyshev polynomials of the first or second kind.