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Some new Fourier and Jackson-Nikol’skii type inequalities in unbounded orthonormal systems. (English) Zbl 07528442

Summary: We consider the generalized Lorentz space \(L_{\psi,q}\) defined via a continuous and concave function \(\psi\) and the Fourier series of a function with respect to an unbounded orthonormal system. Some new Fourier and Jackson-Nikol’skii type inequalities in this frame are stated, proved and discussed. In particular, the derived results generalize and unify several well-known results but also some new applications are pointed out.

MSC:

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42B05 Fourier series and coefficients in several variables
42C15 General harmonic expansions, frames
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
26D15 Inequalities for sums, series and integrals
26D20 Other analytical inequalities
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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[1] G. Akishev: An inequality of different metric for multivariate generalized polynomials, East Jour. Approx., 12 (1) (2006), 25-36.
[2] G. Akishev: On expansion coefficients in an similar to orthogonal system and the inequality of different metrics, Math Zhurnal, 11 (2) (2011), 22-27. · Zbl 1488.42006
[3] G. Akishev: Similar to orthogonal system and inequality of different metrics in Lorentz-Zygmund space, Math. Zhurnal 13 (1) (2013), 5-16. · Zbl 1488.42009
[4] G. Akishev: An inequality of different metrics in the generalized Lorentz space, Trudy Inst. Mat. Mekh. UrO RAN, 24 (4) (2018), 5-18. · Zbl 1488.42009
[5] G. Akishev: On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz space, Trudy Inst. Mat. Mekh. UrO RAN, 25 (2) (2019), 9-20.
[6] G. Akishev L.-E. Persson and A. Seger: Some Fourier inequalities for unbounded orthogonal systems in Lorentz-Zygmund spaces, J. Inequal. Appl., 2019:171 (2019), 18 pp. · Zbl 07459199
[7] G. Akishev, D. Lukkassen and L.-E. Persson: Some new Fourier inequalities for unbounded orthogonal systems in Lorentz-Zygmund spaces, J. Inequal. Appl., 2020:77 (2020), 12pp. · Zbl 07460852
[8] G. Akishev, L.E. Persson and H. Singh: Inequalities for the Fourier coefficients in unbounded orthogonal systems in generalized Lorentz spaces, Nonlinear Studies, 27 (4) (2020), 1-19. · Zbl 1479.42006
[9] G. Alexits: Convergence problems of orthogonal series, International Series of Monographs in Pure and Applied Mathematics, Elsevier, (1961). · Zbl 0098.27403
[10] V.V. Arestov: Inequality of different metrics for trigonometric polynomials, Math. Notes, 27 (4) (1980), 265-269. · Zbl 0508.42001
[11] S.V. Bochkarev: The Hausdorff-Young-Riesz theorem in Lorentz spaces and multiplicative inequalities, Tr. Mat. Inst. Steklova 219 (1997), 103-114 (Translation in Proc. Steklov Inst. Math., 219 (4) (1997), 96 - 107). · Zbl 0922.46024
[12] Z. Ditzian, A. Prymak: Nikol’skii inequalities for Lorentz spaces, Rocky Mountain J. Math., 40 (1) (2010), 209-223. · Zbl 1190.41005
[13] L. R. Ya. Doktorski: An application of limiting interpolation to Fourier series theory, In: A. Buttcher , D. Potts , P. Stollmann and D. Wenzel (eds). The Diversity and Beauty of Applied Operator Theory. Operator Theory: Advances and Applications, 268 (2018), Birkhäuser. 179-191.
[14] L. R. A. Doktorski, D. Gendler: Nikol’skii inequalities for Lorentz-Zygmund spaces, Bol. Soc. Mat. Mex., 25 (3) (2019), 659-672. · Zbl 1428.41013
[15] B. I. Golubov: On a certain class of complete orthonormal systems, Sib. Mat. Zh., 9 (2) (1968), 297-314. · Zbl 0183.06301
[16] V. I. Ivanov: Certain inequalities in various metrics for trigonometric polynomials and their derivatives, Math. Notes, 18 (4) (1975), 880-885. · Zbl 0328.42019
[17] D. Jackson: Certain problems of closest approximation, Bull. Amer. Math. Soc., 39 (12) (1933), 889-906. · Zbl 0008.11104
[18] A. Kamont: General Haar systems and greedy approximation, Studia Math., 145 (2) (2001), 165-184. · Zbl 0980.41017
[19] E. A. Kochetkova: Embedding theorems and inequalities of different metrics for best approximations in complete orthogonal systems, In: Functional analysis, spectral theory. Ulyanovsk. (1984), 46-54. · Zbl 0603.41017
[20] A. A. Komissarov: About some properties of functional systems, Manuscript deposited at VINITI. - Dep.VINITI, 5827-83 Dep. Moscow, (1983), 28 pp.
[21] A. N. Kopezhanova, L.-E. Persson: On summability of the Fourier coefficients in bounded orthonormal systems for functions from some Lorentz type spaces, Eurasian Math. J., 1 (2) (2010), 76-85. · Zbl 1226.46022
[22] J. Marcinkiewicz, A. Zygmund: Some theorems on orthogonal systems, Fund. Math., 28 (1937), 309-335. · JFM 63.0202.01
[23] V. M. Mustakhaeva, G. Akishev: Inequality of different metrics for polynomials in orthonormal systems, Youth and science in the modern world: Materials of the 2nd Republic. Scientific Conference - Taldykorgan, (2010), 95-97.
[24] A. Kh. Myrzagalieva, G. Akishev: Inequality of different metrics for some orthonormal systems, Proceeding 6:th International Conference, 1 (2012), Aktobe, 2012, 279-284.
[25] R. J. Nessel, G. Wilmes: Nikol’skii-type inequalities for trigonometric polynomials and entire functions of exponential type, J. Austral. Math. Soc., 25 (1) (1978), 7-18. · Zbl 0376.42001
[26] R. J. Nessel, G. Wilmes: Nikol’skii-type inequalities in connection with regular spectral measures, Acta Math. Acad. Scient. Hungar., 33 (1-2) (1979), 169-182. · Zbl 0398.42019
[27] S. M. Nikol’skii: Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables, Trudy Mat. Inst. Steklov., 38 (1951), 244-278. · Zbl 0049.32301
[28] S. M. Nikol’skii: Approximation of classes of functions of several variables and embedding theorems, Nauka, Moscow, (1977).
[29] E. D. Nursultanov: Nikol’skii inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space, Proc. Steklov Inst. Math., 255 (2006), 185-202. · Zbl 1351.42012
[30] H. Oba , E. Sato and Y. Sato: A note on Lorentz-Zygmund spaces, Georgian Math. J., 18 (2011), 533-548. · Zbl 1237.43002
[31] A. M. Olevskii: An orthonormal system and its applications, Mat. Sb., 71 (3) (1966), 297-336; English transl. in Amer.Math. Soc. Transl., 76 (2) (1968), 217-263.
[32] V. I. Ovchinnikov, V. D. Raspopova and V. A. Rodin: Sharp estimates of the Fourier coefficients of summable functions and K-functionals, Mathematical Notes of the Academy of Sciences of the USSR. 32 (1982), 627-631. · Zbl 0514.42037
[33] L.-E. Persson: Relations between summability of functions and their Fourier series, Acta Math. Acad. Scient. Hungar. 27 (3-4) (1976), 267-280. · Zbl 0337.42002
[34] V. A. Rodin: Jackson and Nikol’skii inequalities for trigonometric polynomials in symmetric space, Proceedings of 7-Drogobych Winter School (1974-1976), 133-140.
[35] G. Akishev: An inequality of different metric for multivariate generalized polynomials, East Jour. Approx., 12 (1) (2006), 25-36.
[36] G. Akishev: On expansion coefficients in an similar to orthogonal system and the inequality of different metrics, Math Zhurnal, 11 (2) (2011), 22-27. · Zbl 1488.42006
[37] G. Akishev: Similar to orthogonal system and inequality of different metrics in Lorentz-Zygmund space, Math. Zhurnal 13 (1) (2013), 5-16. · Zbl 1488.42009
[38] G. Akishev: An inequality of different metrics in the generalized Lorentz space, Trudy Inst. Mat. Mekh. UrO RAN, 24 (4) (2018), 5-18. · Zbl 1488.42009
[39] G. Akishev: On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz space, Trudy Inst. Mat. Mekh. UrO RAN, 25 (2) (2019), 9-20.
[40] G. Akishev L.-E. Persson and A. Seger: Some Fourier inequalities for unbounded orthogonal systems in Lorentz-Zygmund spaces, J. Inequal. Appl., 2019:171 (2019), 18 pp. · Zbl 07459199
[41] G. Akishev, D. Lukkassen and L.-E. Persson: Some new Fourier inequalities for unbounded orthogonal systems in Lorentz-Zygmund spaces, J. Inequal. Appl., 2020:77 (2020), 12pp. · Zbl 07460852
[42] G. Akishev, L.E. Persson and H. Singh: Inequalities for the Fourier coefficients in unbounded orthogonal systems in generalized Lorentz spaces, Nonlinear Studies, 27 (4) (2020), 1-19. · Zbl 1479.42006
[43] G. Alexits: Convergence problems of orthogonal series, International Series of Monographs in Pure and Applied Mathematics, Elsevier, (1961). · Zbl 0098.27403
[44] V.V. Arestov: Inequality of different metrics for trigonometric polynomials, Math. Notes, 27 (4) (1980), 265-269. · Zbl 0508.42001
[45] S.V. Bochkarev: The Hausdorff-Young-Riesz theorem in Lorentz spaces and multiplicative inequalities, Tr. Mat. Inst. Steklova 219 (1997), 103-114 (Translation in Proc. Steklov Inst. Math., 219 (4) (1997), 96 - 107). · Zbl 0922.46024
[46] Z. Ditzian, A. Prymak: Nikol’skii inequalities for Lorentz spaces, Rocky Mountain J. Math., 40 (1) (2010), 209-223. · Zbl 1190.41005
[47] L. R. Ya. Doktorski: An application of limiting interpolation to Fourier series theory, In: A. Buttcher , D. Potts , P. Stollmann and D. Wenzel (eds). The Diversity and Beauty of Applied Operator Theory. Operator Theory: Advances and Applications, 268 (2018), Birkhäuser. 179-191.
[48] L. R. A. Doktorski, D. Gendler: Nikol’skii inequalities for Lorentz-Zygmund spaces, Bol. Soc. Mat. Mex., 25 (3) (2019), 659-672. · Zbl 1428.41013
[49] B. I. Golubov: On a certain class of complete orthonormal systems, Sib. Mat. Zh., 9 (2) (1968), 297-314. · Zbl 0183.06301
[50] V. I. Ivanov: Certain inequalities in various metrics for trigonometric polynomials and their derivatives, Math. Notes, 18 (4) (1975), 880-885. · Zbl 0328.42019
[51] D. Jackson: Certain problems of closest approximation, Bull. Amer. Math. Soc., 39 (12) (1933), 889-906. · Zbl 0008.11104
[52] A. Kamont: General Haar systems and greedy approximation, Studia Math., 145 (2) (2001), 165-184. · Zbl 0980.41017
[53] E. A. Kochetkova: Embedding theorems and inequalities of different metrics for best approximations in complete orthogonal systems, In: Functional analysis, spectral theory. Ulyanovsk. (1984), 46-54. · Zbl 0603.41017
[54] A. A. Komissarov: About some properties of functional systems, Manuscript deposited at VINITI. - Dep.VINITI, 5827-83 Dep. Moscow, (1983), 28 pp.
[55] A. N. Kopezhanova, L.-E. Persson: On summability of the Fourier coefficients in bounded orthonormal systems for functions from some Lorentz type spaces, Eurasian Math. J., 1 (2) (2010), 76-85. · Zbl 1226.46022
[56] J. Marcinkiewicz, A. Zygmund: Some theorems on orthogonal systems, Fund. Math., 28 (1937), 309-335. · JFM 63.0202.01
[57] V. M. Mustakhaeva, G. Akishev: Inequality of different metrics for polynomials in orthonormal systems, Youth and science in the modern world: Materials of the 2nd Republic. Scientific Conference - Taldykorgan, (2010), 95-97.
[58] A. Kh. Myrzagalieva, G. Akishev: Inequality of different metrics for some orthonormal systems, Proceeding 6:th International Conference, 1 (2012), Aktobe, 2012, 279-284.
[59] R. J. Nessel, G. Wilmes: Nikol’skii-type inequalities for trigonometric polynomials and entire functions of exponential type, J. Austral. Math. Soc., 25 (1) (1978), 7-18. · Zbl 0376.42001
[60] R. J. Nessel, G. Wilmes: Nikol’skii-type inequalities in connection with regular spectral measures, Acta Math. Acad. Scient. Hungar., 33 (1-2) (1979), 169-182. · Zbl 0398.42019
[61] S. M. Nikol’skii: Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables, Trudy Mat. Inst. Steklov., 38 (1951), 244-278. · Zbl 0049.32301
[62] S. M. Nikol’skii: Approximation of classes of functions of several variables and embedding theorems, Nauka, Moscow, (1977).
[63] E. D. Nursultanov: Nikol’skii inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space, Proc. Steklov Inst. Math., 255 (2006), 185-202. · Zbl 1351.42012
[64] H. Oba , E. Sato and Y. Sato: A note on Lorentz-Zygmund spaces, Georgian Math. J., 18 (2011), 533-548. · Zbl 1237.43002
[65] A. M. Olevskii: An orthonormal system and its applications, Mat. Sb., 71 (3) (1966), 297-336; English transl. in Amer.Math. Soc. Transl., 76 (2) (1968), 217-263.
[66] V. I. Ovchinnikov, V. D. Raspopova and V. A. Rodin: Sharp estimates of the Fourier coefficients of summable functions and K-functionals, Mathematical Notes of the Academy of Sciences of the USSR. 32 (1982), 627-631. · Zbl 0514.42037
[67] L.-E. Persson: Relations between summability of functions and their Fourier series, Acta Math. Acad. Scient. Hungar. 27 (3-4) (1976), 267-280. · Zbl 0337.42002
[68] V. A. Rodin: Jackson and Nikol’skii inequalities for trigonometric polynomials in symmetric space, Proceedings of 7-Drogobych Winter School (1974-1976), 133-140.
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