Direct and inverse theorems of approximation theory in Lebesgue spaces with Muckenhoupt weights

O. L. Vinogradov

Ufa mathematical journal, Tome 15 (2023) no. 4, pp. 42-61

Résumé

In this work we establish direct and inverse theorems of approximation theory in Lebesgue spaces $L_{p,w}$ with Muckenhoupt weights $w$ on the axis and on a period. The classical definition of the modulus of continuity can be meaningless in weighted spaces. Therefore, as the modules of continuity, including non-integer order, we use the norms of powers of deviation of Steklov means. The properties of these quantities are derived, some of which are similar to the properties of usual modules of continuity. In addition to the direct and inverse theorems, we obtain equivalence relations between the modules of continuity and the $K$- and $R$-functionals. The proofs are based on estimates for the norms of convolution operators and they do not employ a maximal function. This allows us to establish the results for all $p\in[1,+\infty)$ not excluding the case $p=1$. Previously used methods that employed the maximal function in one form or another are unsuitable for $p\to1$. In addition, by the convolution-based approach we can obtain results simultaneously in the periodic and non-periodic case. With rare exceptions, constants are not specified explicitly, but their dependence on parameters is always tracked. All constants in the estimates depend on $[w]_p$ (Muckenhoupt characteristics of weight $w$), and there is no other dependence on $w$ and $p$. The norms of convolution operators are estimated explicitly in terms of $[w]_p$. The methods of this work can be applied to prove direct and inverse theorems in more general functional spaces.

Détail

Keywords: best approximations, modules of continuity, Muckenhoupt weights
Mots-clés : convolution.

Bibliographie

[1] A.F. Timan, Theory of approximation of functions of a real variable, The Macmillan Company, New York, 1960 | MR

[2] E.A. Gadzhieva, Study of properties of functions with quasimonotone Fourier coefficients in generalized weighted Nikolskii-Besov spaces, PhD thesis, Baku, 1986 (in Russian)

[3] N.X. Ky, “Moduli of mean smoothness and approximation with $A_p$-weights”, Annales Univ. Sci. Budapest, 40 (1997), 7–48 | MR

[4] R. Akgün, “Sharp Jackson and converse theorems of trigonometric approximation in weighted Lebesgue spaces”, Proc. A. Razmadze Math. Inst., 152 (2010), 1–18 | MR | Zbl

[5] R. Akgün, “Polynomial approximation in weighted Lebesgue spaces”, East J. on Appr., 17:3 (2011), 253–266 | MR | Zbl

[6] R. Akgün, “Realization and characterization of modulus of smoothness in weighted Lebesgue spaces”, St.-Petersburg Math. J., 26:5 (2015), 741–756 | DOI | MR | Zbl

[7] R. Akgün, “Gadjieva's conjecture, $K$-functionals, and some applications in weighted Lebesgue spaces”, Turk. J. Math., 42:3 (2018), 1484–1503 | MR | Zbl

[8] Y.E. Yildirir, D.M. Israfilov, “Approximation theorems in weighted Lorentz spaces”, Carpat. J. Math., 26:1 (2010), 108–119 | MR | Zbl

[9] R. Akgün, “Jackson type inequalities for differentiable functions in weighted Orlicz spaces”, St.-Petersburg Math. J., 34:1 (2023), 1–24 | DOI | MR | MR

[10] M. Rosenblum, “Summability of Fourier series in $L^p(d\mu)$”, Trans. Amer. Math. Soc., 105:1 (1962), 32–42 | MR | Zbl

[11] B. Muckenhoupt, “Two weight function norm inequalities for the Poisson integral”, Trans. Amer. Math. Soc., 210 (1975), 225–231 | DOI | MR | Zbl

[12] A.D. Nakhman, B.P. Osilenker, “Estimates of weighted norms of some operators generated by multiple trigonometric Fourier series”, Soviet Math. (Iz. VUZ), 26:4 (1982), 46–59 | MR | Zbl

[13] A.D. Nakhman, “Theorems of Rosenblum-Muckenhoupt type for multiple Fourier series of vector-valued functions”, Soviet Math. (Iz. VUZ), 28:4 (1984), 31–39 | MR | Zbl

[14] A.D. Nakhman, “Elementary estimates for weighted norms of convolution operators”, Soviet Math. Iz. VUZ, 33:10 (1989), 103–106 | MR | Zbl

[15] A.D. Nakhman, “Weighted norm inequalities for the convolution operators”, Trans. Tambov State Techn. Univ., 15:3 (2009), 653–660 | MR

[16] E.M. Dyn'kin, B.P. Osilenker, “Weighted estimates for singular integrals and their applications”, J. Soviet Math., 30:3 (1985), 2094–2154 | DOI | Zbl

[17] E.M. Stein, Harmonic analysis: real-variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton, 1993 | MR | Zbl

[18] S.M. Buckley, “Estimates for operator norms on weighted spaces and reverse Jensen inequalities”, Trans. Amer. Math. Soc., 340:1 (1993), 253–272 | DOI | MR

[19] G. Wilmes, “On Riesz-type inequalities and $K$-functionals related to Riesz potentials in $\mathbb R^N$”, Numer. Funct. Anal. and Optimiz., 1:1 (1979), 57–77 | DOI | MR | Zbl

[20] R.M. Trigub, E.S. Belinsky, Fourier analysis and approximation of functions, Kluwer Academic Publishers, Boston – Dordrecht – London, 2004 | MR | Zbl

[21] D.S. Lubinsky, “Weighted Markov – Bernstein inequalities for entire functions of exponential type”, Publ. de l'institut Mathématique. Nouvelle série, 96 (110) (2014), 181–192 | MR | Zbl

[22] G.I. Natanson, M.F. Timan, “Geometric means of a sequence of best approximations”, Vestn. Leningr. Univ. Mat. Mekh. Astron., 19 (1979), 50–52 (in Russian) | Zbl

[23] O.L. Vinogradov, “On the constants in abstract inverse theorems of approximation theory”, St.-Petersburg Math. J., 34:4 (2023), 573–589 | DOI | MR

[24] R. Hunt, B. Muckenhoupt, R. Wheeden, “Weighted norm inequalities for the conjugate function and Hilbert transform”, Trans. Amer. Math. Soc., 176 (1973), 227–251 | DOI | MR | Zbl

[25] Z. Ditzian, V.H. Hristov, K.G. Ivanov, “Moduli of smoothness and $K$-functionals in $L_p$, $0

1$”, Constr. Approx., 11:1 (1995), 67–83 | DOI | MR | Zbl

[26] R. Akgün, “Direct theorems of trigonometric approximation for variable exponent Lebesgue spaces”, Rev. de la Union Mat. Arg., 60:1 (2019), 121–135 | MR | Zbl

[27] R. Akgün, “Exponential approximation of functions in Lebesgue spaces with Muckenhoupt weight”, Issues Anal., 12(30):1 (2023), 3–24 | DOI | MR