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Some New Fourier and Jackson-Nikol'skii Type Inequalities In Unbounded Orthonormal Systems

Yıl 2021, , 291 - 304, 16.09.2021
https://doi.org/10.33205/cma.910173

Öz

We consider the generalized Lorentz space L ;q dened via a continuous
and concave function 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.

Kaynakça

  • G. Akishev: An inequality of different metric for multivariate generalized polynomials, East Jour. Approx., 12 (1) (2006), 25–36.
  • G. Akishev: On expansion coefficients in an similar to orthogonal system and the inequality of different metrics, Math Zhurnal, 11 (2) (2011), 22–27.
  • G. Akishev: Similar to orthogonal system and inequality of different metrics in Lorentz–Zygmund space, Math. Zhurnal 13 (1) (2013), 5–16.
  • G. Akishev: An inequality of different metrics in the generalized Lorentz space, Trudy Inst. Mat. Mekh. UrO RAN, 24 (4) (2018), 5–18.
  • 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.
  • 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.
  • 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.
  • 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.
  • G. Alexits: Convergence problems of orthogonal series, International Series of Monographs in Pure and Applied Mathematics, Elsevier, (1961).
  • V.V. Arestov: Inequality of different metrics for trigonometric polynomials, Math. Notes, 27 (4) (1980), 265–269.
  • 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).
  • Z. Ditzian, A. Prymak: Nikol’skii inequalities for Lorentz spaces, Rocky Mountain J. Math., 40 (1) (2010), 209–223.
  • 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.
  • L. R. A. Doktorski, D. Gendler: Nikol’skii inequalities for Lorentz–Zygmund spaces, Bol. Soc. Mat. Mex., 25 (3) (2019), 659–672.
  • B. I. Golubov: On a certain class of complete orthonormal systems, Sib. Mat. Zh., 9 (2) (1968), 297–314.
  • V. I. Ivanov: Certain inequalities in various metrics for trigonometric polynomials and their derivatives, Math. Notes, 18 (4) (1975), 880–885.
  • D. Jackson: Certain problems of closest approximation, Bull. Amer. Math. Soc., 39 (12) (1933), 889–906.
  • A. Kamont: General Haar systems and greedy approximation, Studia Math., 145 (2) (2001), 165–184.
  • 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.
  • A. A. Komissarov: About some properties of functional systems, Manuscript deposited at VINITI. - Dep.VINITI, 5827-83 Dep. Moscow, (1983), 28 pp.
  • 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.
  • J. Marcinkiewicz, A. Zygmund: Some theorems on orthogonal systems, Fund. Math., 28 (1937), 309–335.
  • 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.
  • A. Kh. Myrzagalieva, G. Akishev: Inequality of different metrics for some orthonormal systems, Proceeding 6:th International Conference, 1 (2012), Aktobe, 2012, 279–284.
  • 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.
  • 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.
  • 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.
  • S. M. Nikol’skii: Approximation of classes of functions of several variables and embedding theorems, Nauka, Moscow, (1977).
  • 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.
  • H. Oba , E. Sato and Y. Sato: A note on Lorentz–Zygmund spaces, Georgian Math. J., 18 (2011), 533–548.
  • 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.
  • 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.
  • L.-E. Persson: Relations between summability of functions and their Fourier series, Acta Math. Acad. Scient. Hungar. 27 (3–4) (1976), 267–280.
  • V. A. Rodin: Jackson and Nikol’skii inequalities for trigonometric polynomials in symmetric space, Proceedings of 7-Drogobych Winter School (1974–1976), 133–140.
Yıl 2021, , 291 - 304, 16.09.2021
https://doi.org/10.33205/cma.910173

Öz

Kaynakça

  • G. Akishev: An inequality of different metric for multivariate generalized polynomials, East Jour. Approx., 12 (1) (2006), 25–36.
  • G. Akishev: On expansion coefficients in an similar to orthogonal system and the inequality of different metrics, Math Zhurnal, 11 (2) (2011), 22–27.
  • G. Akishev: Similar to orthogonal system and inequality of different metrics in Lorentz–Zygmund space, Math. Zhurnal 13 (1) (2013), 5–16.
  • G. Akishev: An inequality of different metrics in the generalized Lorentz space, Trudy Inst. Mat. Mekh. UrO RAN, 24 (4) (2018), 5–18.
  • 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.
  • 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.
  • 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.
  • 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.
  • G. Alexits: Convergence problems of orthogonal series, International Series of Monographs in Pure and Applied Mathematics, Elsevier, (1961).
  • V.V. Arestov: Inequality of different metrics for trigonometric polynomials, Math. Notes, 27 (4) (1980), 265–269.
  • 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).
  • Z. Ditzian, A. Prymak: Nikol’skii inequalities for Lorentz spaces, Rocky Mountain J. Math., 40 (1) (2010), 209–223.
  • 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.
  • L. R. A. Doktorski, D. Gendler: Nikol’skii inequalities for Lorentz–Zygmund spaces, Bol. Soc. Mat. Mex., 25 (3) (2019), 659–672.
  • B. I. Golubov: On a certain class of complete orthonormal systems, Sib. Mat. Zh., 9 (2) (1968), 297–314.
  • V. I. Ivanov: Certain inequalities in various metrics for trigonometric polynomials and their derivatives, Math. Notes, 18 (4) (1975), 880–885.
  • D. Jackson: Certain problems of closest approximation, Bull. Amer. Math. Soc., 39 (12) (1933), 889–906.
  • A. Kamont: General Haar systems and greedy approximation, Studia Math., 145 (2) (2001), 165–184.
  • 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.
  • A. A. Komissarov: About some properties of functional systems, Manuscript deposited at VINITI. - Dep.VINITI, 5827-83 Dep. Moscow, (1983), 28 pp.
  • 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.
  • J. Marcinkiewicz, A. Zygmund: Some theorems on orthogonal systems, Fund. Math., 28 (1937), 309–335.
  • 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.
  • A. Kh. Myrzagalieva, G. Akishev: Inequality of different metrics for some orthonormal systems, Proceeding 6:th International Conference, 1 (2012), Aktobe, 2012, 279–284.
  • 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.
  • 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.
  • 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.
  • S. M. Nikol’skii: Approximation of classes of functions of several variables and embedding theorems, Nauka, Moscow, (1977).
  • 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.
  • H. Oba , E. Sato and Y. Sato: A note on Lorentz–Zygmund spaces, Georgian Math. J., 18 (2011), 533–548.
  • 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.
  • 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.
  • L.-E. Persson: Relations between summability of functions and their Fourier series, Acta Math. Acad. Scient. Hungar. 27 (3–4) (1976), 267–280.
  • V. A. Rodin: Jackson and Nikol’skii inequalities for trigonometric polynomials in symmetric space, Proceedings of 7-Drogobych Winter School (1974–1976), 133–140.
Toplam 34 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik
Bölüm Makaleler
Yazarlar

Gabdolla Akishev

Lars Erik Persson 0000-0002-8340-9770

Harpal Singh Bu kişi benim

Yayımlanma Tarihi 16 Eylül 2021
Yayımlandığı Sayı Yıl 2021

Kaynak Göster

APA Akishev, G., Persson, L. E., & Singh, H. (2021). Some New Fourier and Jackson-Nikol’skii Type Inequalities In Unbounded Orthonormal Systems. Constructive Mathematical Analysis, 4(3), 291-304. https://doi.org/10.33205/cma.910173
AMA Akishev G, Persson LE, Singh H. Some New Fourier and Jackson-Nikol’skii Type Inequalities In Unbounded Orthonormal Systems. CMA. Eylül 2021;4(3):291-304. doi:10.33205/cma.910173
Chicago Akishev, Gabdolla, Lars Erik Persson, ve Harpal Singh. “Some New Fourier and Jackson-Nikol’skii Type Inequalities In Unbounded Orthonormal Systems”. Constructive Mathematical Analysis 4, sy. 3 (Eylül 2021): 291-304. https://doi.org/10.33205/cma.910173.
EndNote Akishev G, Persson LE, Singh H (01 Eylül 2021) Some New Fourier and Jackson-Nikol’skii Type Inequalities In Unbounded Orthonormal Systems. Constructive Mathematical Analysis 4 3 291–304.
IEEE G. Akishev, L. E. Persson, ve H. Singh, “Some New Fourier and Jackson-Nikol’skii Type Inequalities In Unbounded Orthonormal Systems”, CMA, c. 4, sy. 3, ss. 291–304, 2021, doi: 10.33205/cma.910173.
ISNAD Akishev, Gabdolla vd. “Some New Fourier and Jackson-Nikol’skii Type Inequalities In Unbounded Orthonormal Systems”. Constructive Mathematical Analysis 4/3 (Eylül 2021), 291-304. https://doi.org/10.33205/cma.910173.
JAMA Akishev G, Persson LE, Singh H. Some New Fourier and Jackson-Nikol’skii Type Inequalities In Unbounded Orthonormal Systems. CMA. 2021;4:291–304.
MLA Akishev, Gabdolla vd. “Some New Fourier and Jackson-Nikol’skii Type Inequalities In Unbounded Orthonormal Systems”. Constructive Mathematical Analysis, c. 4, sy. 3, 2021, ss. 291-04, doi:10.33205/cma.910173.
Vancouver Akishev G, Persson LE, Singh H. Some New Fourier and Jackson-Nikol’skii Type Inequalities In Unbounded Orthonormal Systems. CMA. 2021;4(3):291-304.