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# Приближение целыми функциями экспоненциального типа и обобщенная гладкость

2019

Approximation Theory is one of the well developed sciences having a lot of applications in other fields of both the fundamental and applied mathematics, such as the theory of differential equations, data and signal processing, mathematics of finance, computational mathematics, programing and algorithms, etc. Within the present project we mainly deal with approximation of non-periodic functions by bandlimited functions in the standard scale of the Lp-spaces, where 0<p<=+infty, in both the one-dimensional and multivariate cases.
Approximation Theory of functions defined on a finite-dimensional Euclidean spaces has its origins in the works of Bernstein (S.N. Bernstein: {\it Collection of works.} Academy of Sciences of USSR. 1954. V.2, p. 371-375). Since the non-periodic function can be represented by a Fourier integral and has a continuous frequency spectrum, then the class of trigonometric polynomials of order $\leq\sigma$ is no longer valid as a class of approximating functions and it is replaced by a class of functions $B^p_\sigma$, which are restrictions to the real line of entire functions of exponential type of order $\leq\sigma$, belonging to $L_p (\R)$, which by virtue of the Paley-Wiener-Schwartz Theorem (H\"ormander, L.: {\it The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis.} Berlin, Heidelberg, New-York, Tokio: Springer-Verlag, 1983., page 181) are called bandlimited functions.
This field of research was actively developed by Aachen school under the supervision of P.L. Butzer (Butzer, P. and R. Nessel: {\it Fourier Analysis and Approximation}. Vol. 1. New-York and London: Academic Press 1971., Butzer, P., Splettst\"osser W., Stens R.:{\it The Sampling Theorem and Linear Prediction in Signal Analysis.} Jahresberichte der Dt. Math-Verein, 90 (1988), p. 1--70.). Note that the theory of approximation by bandlimited functions has a number of features. On one hand the domain of the functions has  infinite measure, the finite sums in the construction of approximating methods are replaced by infinite sums, the Fourier coefficients are replaced by the Fourier transform. Hence we face with problems of the correctness of such definitions, the solution of which requires the involvement of the theory of generalized functions. On the other hand, the theory of approximation by bandlimited functions has many applications in the theory of data and signal processing, defining its own specific problems.
From the very beginning of the researches in approximation theory it has become clear that the quality of approximation depends on smoothness. For the first time such a dependence has been established by D. Jackson and S. Bernstein. Nowadays, the finding relations between approximation properties and smoothness is one of the central problems. Afterwards, a quite considerable period has been devoted to studying some special approximation methods, such as the Fejer means, the Vallee-Poussin means, the Rogosinski means, the Bochner-Riesz means, etc. Each approximation procedure were studied by using its own special method with taking into account its specific features. That time one dealt with the classical moduli of smoothness and the usual derivatives as smoothness characteristics. The description of this development stage can be found in numerous literature (see e. g. R. DeVore, G. Lorenz, Constructive Approximation, Grundlehren Math. Wiss., 303, Springer-Verlag, Berlin, 1993).
As is known, the Jackson's and Bernstein's type estimates are not sharp on order in many cases. By this reason one always tried to establish explicit relations between approximation and smoothness in the form of equivalences of approximation errors and smoothness moduli. For these purposes one applied the based on the Fourier transform criteria for multipliers in the spaces of integrable and continuous functions related to the cases p=1 and p=+infty, respectively (see e. g. E. M. Stein and G.Weiss, Introduction to Fourier Analysis on Euclidean Spaces, in Princeton Math. Ser., 32, Princeton Univ. Press, Princeton, NJ, 1971). Sometimes it worked well. For example, in the one-dimensional case the errors of approximation by the Rogosinski means and the Bochner-Riesz means have been proved to be equivalent to the second order smoothness modulus (R. M. Trigub, Absolute convergence of Fourier integrals, summability of Fourier series and approximation by polynomial functions on the torus, Izv. Akad. Nauk SSSR, ser. matem., 44 (1980), 1378-1409 [in Russian]). The moduli of smoothness in the multivariate case related to the Bochner-Riesz means in the sense of equivalences have been constructed in Z. Ditzian, Measure of smoothness related to the Laplacian, Trans. AMS, 326 (1991), 407-422 and https://link.springer.com/article/10.1007%2Fs00041-014-9373-y.
The general construction uniting approximation methods and smoothness goes back to the works by J. Boman and H. S. Shapiro (see e. g. J. Boman, H. S. Shapiro, Comparison theorems for a generalized modulus of continuity, Ark. Mat., 9:1–2 (1971), 91–116, J. Boman, Equivalence of generalized moduli of continuity, Ark. Mat. 18:1–2 (1980), 73–100. Some necessary and sufficient conditions providing an equivalence of various generalized moduli by Boman and Shapiro were found in terms of some ideals in the Banach algebra of multipliers described with the help of the Fourier transform. It turned out that these conditions could be verifed in practice in some exeptional cases. This is why this general approach has only been applied to moduli of smoothness of natural order or to slight modifications of them. Until recently, the only smoothness values, which significantly differed from the classical ones, were the moduli of smoothness of arbitrary positive orders and the Weyl derivatives (see e. g. P. L. Butzer, H. Dyckhoff, E. G'orlich, Best triginometric approximation, fractional order derivatives and Lipschitz classes, Can. J. Math. 29 (1977), 781–793; R. Taberski, Trigonometric approximation in the norms and seminorms, Studia Math., 80 (1984), 197-217; M. K. Potapov and B. V. Simonov, Positive-order moduli of smoothness of functions in the spaces Lp, 1<=p<=+infty, Sovrem. Probl. Mat. Mekh., 7 (1) (2011), Moscow Univ. Publ. House, Moscow, 100–109. [in Russian]).
Some significant progress has been made due to introduction of the generalized modulus of smoothness generated by an arbitrary periodic function. This construction proves to be the most adequate one for applications to approximation theory in the following sense: on the one hand it is explicit enough to underlie a substantive theory for all 0<p<=+infinity, and on the other hand it is general enough to reach the level achieved for approximation methods and K-functionals, to encompass all the cases considered previously, to augment the objective environment of approximation theory by new ‘missing’ constructions, and to set out links between structural and constructive characteristics in the form of equivalences (lower and upper estimates with the same order). At the present time, the concept of generalized smoothness modulus and the relasted topics of approximation theory have been well developed at least for the generators which are close in a certain sense to homogeneous functions (see e. g.  S. Artamonov, K. Runovski, H.-J. Schmeisser, Approximation by bandlimited functions, K-functionals and generalized moduli of smoothness, Analysis Mathematica, 2018, Vol. 46. P. 1-24 ; K. Runovski and H.-J. Schmeisser, General moduli of smoothness andapproximation by families of linear polynomial operators, New perspectives on approximation and sampling theory, Appl. Numer. Harmonic. Analesis, Birkh¨auser/Springer, Cham 2014, 269–298).

#### Публикации по проекту:

Artamonov S., Runovski K., Schmeisser H. Approximation by families of generalized sampling series, realizations of generalized K-functionals and generalized moduli of smoothness // Journal of Mathematical Analysis and Applications. 2020. Vol. 489. No. 1. P. 1-19. doi
Артамонов С. Ю., Руновский К., Шмайссер Х. Периодические пространства Бесова и обобщенные модули гладкости // Математические заметки. 2020. Т. 108. № 4. C. 617-621. doi