ORDER EQUALITIES IN DIFFERENT METRICS FOR MODULI OF SMOOTHNESS OF VARIOUS ORDERS

Niyazi A. Il'yasov     (Baku State University, Baku, AZ 1148, Azerbaijan)

Abstract


In this paper, we obtain order equalities for the \(k\)th order \(L_{q}(T)\)-moduli of smoothness \(\omega_{k}(f;\delta)_{q}\) in terms of expressions that contain the \(l\)th order \(L_{p}(T)\)-moduli of smoothness \(\omega_{ l }(f;\delta)_{p}\) on the class of periodic functions \(f\in L_{p}(T)\) with monotonically decreasing Fourier coefficients, where \(1<p<q<\infty,\) \(k,l \in \mathbb{N},\) and \(T=(-\pi,\pi].\)


Keywords


Inequalities of different metrics for moduli of smoothness, Order equality, Trigonometric Fourier series with monotonic coefficients

Full Text:

PDF

References


  1. Bary N.K. A Treatise on Trigonometric Series. Vols. I, II. Oxford, New York: Pergamon Press, 1964, Vol. I, 533 p; Vol. II, 508 p. Original Russian text published in Trigonometricheskie ryady , Moscow: Fiz.-Mat. Giz. Publ., 1961, 936 p.
  2. Gol’dman M.L. An imbedding criterion for different metrics for isotropic Besov spaces with arbitrary moduli of continuity. Proc. Steklov Inst. Math., 1994. No. 2. P. 155–181.
  3. Il’yasov N.A. On the inequality between modulus of smoothness of various orders in different metrics. Math. Notes , 1991. Vol. 50, No. 2. P. 877–879. DOI: 10.1007/BF01157580
  4. Il’yasov N.A. On the direct theorem of approximation theory of periodic functions in different metrics. Proc. Steklov Inst. Math., 1997. Vol. 219. P. 215–230.
  5. Il’yasov N.A. The inverse theorem in various metrics of approximation theory for periodic functions with monotone Fourier coefficients. Trudy Inst. Mat. i Mekh. UrO RAN [Proc. of Krasovskii Institute of Mathematics and Mechanics of the UB RAS], 2016. Vol. 22, No. 4. P. 153–162. (in Russian) DOI: 10.21538/0134-4889-2016-22-4-153-162
  6. Il’yasov N.A. The direct theorem of the theory of approximation of periodic functions with monotone Fourier coefficients in different metrics. Proc. Steklov Inst. Math., 2018. Vol. 303, Suppl. 1. P. S92–S106. DOI: 10.1134/S0081543818090109
  7. Kolyada V.I. On relations between moduli of continuity in different metrics. Proc. Steklov Inst. Math., 1989. Vol. 181. P. 127–148.
  8. Timan M.F. Best approximation and modulus of smoothness of functions defined on the entire real axis. Izv. Vyssh. Ucheb. Zaved. Mat., 1961. No. 6. P. 108–120. (in Russian)
  9. Timan M.F. Some embedding theorems for \(L_p\) -classes of functions. Dokl. Akad. Nauk SSSR, 1970. Vol. 193, No. 6, P. 1251–1254. (in Russian)
  10. Timan M.F. The imbedding of the \(L_{p}^{(k)}\) -classes of functions. Izv. Vyssh. Ucheb. Zaved. Mat., 1974. No. 10(149). P. 61–74. (in Russian)
  11. Ul’yanov P.L. The imbedding of certain function classes \(H_{p}^{\omega}\). Math. USSR–Izv., 1968. Vol. 2, No. 3. P. 601–637.
  12. Ul’yanov P.L. Imbedding theorems and relations between best approximations (moduli of continuity) in different metrics. Math. USSR–Sb., 1970. Vol. 10, No. 1. P. 103–126.



DOI: http://dx.doi.org/10.15826/umj.2018.2.004

Article Metrics

Metrics Loading ...

Refbacks

  • There are currently no refbacks.