Research Article
BibTex RIS Cite

Exponential approximation in variable exponent Lebesgue spaces on the real line

Year 2022, , 214 - 237, 01.12.2022
https://doi.org/10.33205/cma.1167459

Abstract

Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions of finite degree (IFFD) in some variable exponent Lebesgue space of real functions defined on $\boldsymbol{R}:=\left( -\infty ,+\infty \right) $. To do this, we employ a transference theorem which produce norm inequalities starting from norm inequalities in $\mathcal{C}(\boldsymbol{R})$, the class of bounded uniformly continuous functions defined on $\boldsymbol{R}$. Let $B\subseteq \boldsymbol{R}$ be a measurable set, $p\left( x\right) :B\rightarrow \lbrack 1,\infty )$ be a measurable function. For the class of functions $f$ belonging to variable exponent Lebesgue spaces $L_{p\left( x\right) }\left( B\right) $, we consider difference operator $\left( I-T_{\delta }\right)^{r}f\left( \cdot \right) $ under the condition that $p(x)$ satisfies the log-Hölder continuity condition and $1\leq \mathop{\rm ess \; inf} \limits\nolimits_{x\in B}p(x)$, $\mathop{\rm ess \; sup}\limits\nolimits_{x\in B}p(x)<\infty $, where $I$ is the identity operator, $r\in \mathrm{N}:=\left\{ 1,2,3,\cdots \right\} $, $\delta \geq 0$ and
$$
T_{\delta }f\left( x\right) =\frac{1}{\delta }\int\nolimits_{0}^{\delta
}f\left( x+t\right) dt, x\in \boldsymbol{R},
T_{0}\equiv I,
$$
is the forward Steklov operator. It is proved that
$$
\left\Vert \left( I-T_{\delta }\right) ^{r}f\right\Vert _{p\left( \cdot
\right) }
$$
is a suitable measure of smoothness for functions in $L_{p\left( x\right)
}\left( B\right) $, where $\left\Vert \cdot \right\Vert _{p\left( \cdot
\right) }$ is Luxemburg norm in $L_{p\left( x\right) }\left( B\right) .$ We
obtain main properties of difference operator $\left\Vert \left( I-T_{\delta
}\right) ^{r}f\right\Vert _{p\left( \cdot \right) }$ in $L_{p\left( x\right)
}\left( B\right) .$ We give proof of direct and inverse theorems of
approximation by IFFD in $L_{p\left( x\right) }\left( \boldsymbol{R}\right)
. $

References

  • F. Abdullaev, S. Chaichenko, M. Imashgizi and A. Shidlich: Direct and inverse approximation theorems in the weighted Orlicz-type spaces with a variable exponent, Turk. J. Math., 44 (2020), 284-299.
  • F. Abdullaev, A. Shidlich and S. Chaichenko: Direct and inverse approximation theorems of functions in the Orlicz type spaces, Math. Slovaca, 69 (2019), 1367–1380.
  • F. Abdullaev, N. Özkaratepe, V. Savchuk and A. Shidlich: Exact constants in direct and inverse approximation theorems for functions of several variables in the spaces $S_p$, FILOMAT, 33 (2019), 1471–1484.
  • N. I. Ackhiezer: Theory of approximation, Fizmatlit, Moscow, (1965); English transl. of 2nd ed. Frederick Ungar, New York (1956).
  • R. Akgün: Approximation of functions of weighted Lebesgue and Smirnov spaces, Mathematica (Cluj) Tome, 54 (77) (2012), 25–36.
  • R. Akgün: Sharp Jackson and converse theorems of trigonometric approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst., 152 (2010), 1–18.
  • R. Akgün: Inequalities for one sided approximation in Orlicz spaces, Hacet. J. Math. Stat., 40 (2) (2011), 231–240.
  • R. Akgün: Some convolution inequalities in Musielak Orlicz spaces, Proc. Inst. Math. Mech., NAS Azerbaijan, 42 (2) (2016), 279–291.
  • R. Akgün: Approximation properties of Bernstein’s singular integrals in variable exponent Lebesgue spaces on the real axis, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 71 (4) (2022), DOI:10.31801/cfsuasmas.1056890
  • R. Akgün, A. Ghorbanalizadeh: Approximation by integral functions of finite degree in variable exponent Lebesgue spaces on the real axis, Turk. J. Math., 42 (4) (2018), 1887–1903.
  • A. H. Avşar, H. Koç: Jackson and Stechkin type inequalities of trigonometric approximation in $A_{p,q}$, Turk. J. Math., 42 (2018), 2979–2993.
  • A. H. Avşar, Y. E. Yildirir: On the trigonometric approximation of functions in weighted Lorentz spaces using Cesaro submethod, Novi Sad J. Math., 48 (2) (2018), 41–54.
  • C. Bardaro, P. L. Butzer, R. L. Stens and G. Vinti: Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals, J. Math. Anal. Appl., 316 (2006), 269–306.
  • S. N. Bernstein: Sur la meilleure approximation sur tout l’axe reel des fonctions continues par des fonctions entieres de degre n. I, C.R. (Doklady) Acad. Sci. URSS (N.S.) 51 (1946), 331–334.
  • S. N. Bernstein: Collected works, M. Vol. I, Izdat. Akad. Nauk SSSR, Moscow, (1952), 11–104.
  • D. Cruz-Uribe, A. Fiorenza: Variable Lebesgue Spaces, Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis, Birkhauser (2013).
  • R. A. Devore, G. G. Lorentz: Constructive Approximation, Springer-Verlag (1993).
  • L. Diening, P. Harjulehto, P. Hästö and M. Ružiˇcka: Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math., 2017, Springer, Berlin, Heidelberg (2011).
  • L. Diening, M. Ružiˇcka: Calderon–Zymund operators on generalized Lebesgue spaces $L^{p(x)}$ and problems related to fluid dynamics, preprint, Mathematische Fakültat, Albert-Ludwings-Universität Freiburg, 21/2002, 04.07.2002, 1–20, (2002).
  • Z. Ditzian: Inverse theorems for functions in $L^p$ and other spaces, Proc. Amer. Math. Soc., 54 (1976), 80–82.
  • Z. Ditzian, K. G. Ivanov: Strong converse inequalities, J. D’analyse math., 61 (1) (1993), 61–111.
  • A. Dogu, A. H. Avsar and Y. E. Yildirir: Some inequalities about convolution and trigonometric approximation in weighted Orlicz spaces, Proc. Inst. Math. Mech., NAS Azerbaijan, 44 (1) (2018), 107–115.
  • D. Drihem: Restricted boundedness of translation operators on variable Lebesgue spaces, https://doi.org/10.48550/arXiv.1507.08089
  • D. P. Dryanov, M. A. Qazi, and Q. I. Rahman: Entire functions of exponential type in Approximation Theory, In: Constructive Theory of Functions, Varna 2002 (B. Bojanov, Ed.), DARBA, Sofia, (2003), 86–135.
  • X. Fan, D. Zhao: On the spaces $L^{p(x)}(\Omega )$ and $W^{m;p(x)}(\Omega )$, J. Math. Anal. Appl., 263 (2) (2001), 424–446.
  • A. Guven, D. M. Israfilov: Trigonometric approximation in generalized Lebesgue spaces $L^{p(x)}$, J. Math. Inequal., 4 (2) (2010), 285–299.
  • P. Harjulehto, P. Hästö: Orlicz spaces and generalized Orlicz spaces, Lecture Notes in Mathematics, 2236, Springer, (2019).
  • H. Hudzik: On generalized Orlicz-Sobolev space, Funct. Approximatio Comment. Math., 4 (1976), 37–51.
  • I. I. Ibragimov: Teoriya priblizheniya tselymi funktsiyami.(Russian) The theory of approximation by entire functions “Elm" , Baku (1979).
  • S. Z. Jafarov: Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents, Ukr. Math. J., 66 (10) (2015), 1509–1518.
  • S. Z. Jafarov: Approximation by trigonometric polynomials in subspace of variable exponent grand Lebesgue spaces, Global J. Math., 8 (2) (2016), 836–843.
  • S. Z. Jafarov: Ul’yanov type inequalities for moduli of smoothness, Appl. Math. E-Notes, 12 (2012), 221–227.
  • S. Z. Jafarov: S. M. Nikolskii type inequality and estimation between the best approximations of a function in norms of different spaces, Math. Balkanica (N.S.), 21 (1-2) (2007), 173–182.
  • D. M. Israfilov, R. Akgün: Approximation by polynomials and rational functions in weighted rearrangement invariant spaces, J. Math. Anal. Appl., 346 (2008), 489–500.
  • D. M. Israfilov, A. Testici: Approximation problems in the Lebesgue spaces with variable exponent, J. Math. Anal. Appl., 459 (1) (2018), 112–123.
  • D. M. Israfilov, A. Testici: Approximation by Faber–Laurent rational functions in Lebesgue spaces with variable exponent, Indag. Mat., 27 (4) (2016), 914–922.
  • D. M. Israfilov, E. Yirtici: Convolutions and best approximations in variable exponent Lebesgue spaces, Math. Reports, 18 (4) (2016), 497–508.
  • H. Koc: Simultaneous approximation by polynomials in Orlicz spaces generated by quasiconvex Young functions, Kuwait J. Sci., 43 (4) (2016), 18–31.
  • V. Kokilashvili, S. Samko: Singular integrals in weighted Lebesgue spaces with variable exponent, Georgian Math. J., 10 (1) (2003), 145–156.
  • Z. O. Kováˇcik, J. Rákosnik: On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41 (4) (1991), 592–618.
  • F. G. Nasibov: Approximation in $L_2$ by entire functions. (Russian) Akad. Nauk Azerbaidzhan. SSR Dokl., 42 (4) (1986), 3–6.
  • S. M. Nikolskii: Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of several variables, Amer. Math. Soc. Transl. Ser. 2, 80 (1969), 1–38, (Trudy Mat. Inst. Steklov 38 (1951), 211–278).
  • A. A. Ligun, V. G. Doronin: Exact constants in Jackson-type inequalities for the $L_2$-approximation on a straight line. (Russian) Ukraïn. Mat. Zh. 2009; 61 (1): 92–98; translation in Ukrainian Math. J., 61 (1) (2009), 112–120.
  • W. Orlicz: Über konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200–212.
  • R. Paley, N. Wiener: Fourier transforms in the complex domain, Amer. Math. Soc. (1934).
  • V. Yu. Popov: Best mean square approximations by entire functions of exponential type (Russian), Izv. Vysš. Ucebn. Zaved. Matematika, 121 (6) (1972), 65–73.
  • K. R. Rajagopal, M. Ružiˇcka: On the modeling elektroreological materials, Mech. Res. Commun., 23 (4) (1996), 401–407.
  • M. Ružiˇcka: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin (2000).
  • S. Samko: Differentiation and integration of variable order and the spaces Lp(x), in: Operator theory for complex and hypercomplex analysis (Mexico City, 1994), 203–219, Contemp. Math., 212, Amer. Math. Soc., Providence, RI, (1998).
  • I. I. Sharapudinov: The topology of the space $L^{p(t)}([0; 1])$, (Russian), Mat. Zametki, 26 (4) (1979), 613–632.
  • I. I. Sharapudinov: Some questions in the theory of approximation in Lebesgue spaces with variable exponent, Itogi Nauki. Yug Rossii. Mat. Monografiya, vol. 5, Southern Institute of Mathematics of the Vladikavkaz Sceince Centre of the Russian Academy of Sciences and the Government of the Republic of North Ossetia-Alania, Vladikavkaz (2012), 267 pp. Russian.
  • A. F. Timan: Theory of approximation of functions of a real variable. Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, The Macmillan Co., New York: A Pergamon Press Book (1963).
  • M. F. Timan: The approximation of functions defined on the whole real axis by entire functions of exponential type, Izv. Vyssh. Uchebn. Zaved. Mat., 2 (1968), 89–101.
  • R. Taberski: Approximation by entire functions of exponential type, Demonstr. Math., 14 (1981), 151–181 .
  • R. Taberski: Contributions to fractional calculus and exponential approximation, 1986, Funct. Approximatio, Comment. Math., 15 (1986), 81–106.
  • S. S. Volosivets: Approximation of functions and their conjugates in variable Lebesgue spaces, Sbornik: Mathematics, 208 (1) (2017), 44–59.
  • J. Yeh: Real analysis: theory of measure and integration, 2nd ed., (2006).
  • V. V. Zhikov: Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (4) (1986), 675–710 (in Russian).
Year 2022, , 214 - 237, 01.12.2022
https://doi.org/10.33205/cma.1167459

Abstract

References

  • F. Abdullaev, S. Chaichenko, M. Imashgizi and A. Shidlich: Direct and inverse approximation theorems in the weighted Orlicz-type spaces with a variable exponent, Turk. J. Math., 44 (2020), 284-299.
  • F. Abdullaev, A. Shidlich and S. Chaichenko: Direct and inverse approximation theorems of functions in the Orlicz type spaces, Math. Slovaca, 69 (2019), 1367–1380.
  • F. Abdullaev, N. Özkaratepe, V. Savchuk and A. Shidlich: Exact constants in direct and inverse approximation theorems for functions of several variables in the spaces $S_p$, FILOMAT, 33 (2019), 1471–1484.
  • N. I. Ackhiezer: Theory of approximation, Fizmatlit, Moscow, (1965); English transl. of 2nd ed. Frederick Ungar, New York (1956).
  • R. Akgün: Approximation of functions of weighted Lebesgue and Smirnov spaces, Mathematica (Cluj) Tome, 54 (77) (2012), 25–36.
  • R. Akgün: Sharp Jackson and converse theorems of trigonometric approximation in weighted Lebesgue spaces, Proc. A. Razmadze Math. Inst., 152 (2010), 1–18.
  • R. Akgün: Inequalities for one sided approximation in Orlicz spaces, Hacet. J. Math. Stat., 40 (2) (2011), 231–240.
  • R. Akgün: Some convolution inequalities in Musielak Orlicz spaces, Proc. Inst. Math. Mech., NAS Azerbaijan, 42 (2) (2016), 279–291.
  • R. Akgün: Approximation properties of Bernstein’s singular integrals in variable exponent Lebesgue spaces on the real axis, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 71 (4) (2022), DOI:10.31801/cfsuasmas.1056890
  • R. Akgün, A. Ghorbanalizadeh: Approximation by integral functions of finite degree in variable exponent Lebesgue spaces on the real axis, Turk. J. Math., 42 (4) (2018), 1887–1903.
  • A. H. Avşar, H. Koç: Jackson and Stechkin type inequalities of trigonometric approximation in $A_{p,q}$, Turk. J. Math., 42 (2018), 2979–2993.
  • A. H. Avşar, Y. E. Yildirir: On the trigonometric approximation of functions in weighted Lorentz spaces using Cesaro submethod, Novi Sad J. Math., 48 (2) (2018), 41–54.
  • C. Bardaro, P. L. Butzer, R. L. Stens and G. Vinti: Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals, J. Math. Anal. Appl., 316 (2006), 269–306.
  • S. N. Bernstein: Sur la meilleure approximation sur tout l’axe reel des fonctions continues par des fonctions entieres de degre n. I, C.R. (Doklady) Acad. Sci. URSS (N.S.) 51 (1946), 331–334.
  • S. N. Bernstein: Collected works, M. Vol. I, Izdat. Akad. Nauk SSSR, Moscow, (1952), 11–104.
  • D. Cruz-Uribe, A. Fiorenza: Variable Lebesgue Spaces, Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis, Birkhauser (2013).
  • R. A. Devore, G. G. Lorentz: Constructive Approximation, Springer-Verlag (1993).
  • L. Diening, P. Harjulehto, P. Hästö and M. Ružiˇcka: Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math., 2017, Springer, Berlin, Heidelberg (2011).
  • L. Diening, M. Ružiˇcka: Calderon–Zymund operators on generalized Lebesgue spaces $L^{p(x)}$ and problems related to fluid dynamics, preprint, Mathematische Fakültat, Albert-Ludwings-Universität Freiburg, 21/2002, 04.07.2002, 1–20, (2002).
  • Z. Ditzian: Inverse theorems for functions in $L^p$ and other spaces, Proc. Amer. Math. Soc., 54 (1976), 80–82.
  • Z. Ditzian, K. G. Ivanov: Strong converse inequalities, J. D’analyse math., 61 (1) (1993), 61–111.
  • A. Dogu, A. H. Avsar and Y. E. Yildirir: Some inequalities about convolution and trigonometric approximation in weighted Orlicz spaces, Proc. Inst. Math. Mech., NAS Azerbaijan, 44 (1) (2018), 107–115.
  • D. Drihem: Restricted boundedness of translation operators on variable Lebesgue spaces, https://doi.org/10.48550/arXiv.1507.08089
  • D. P. Dryanov, M. A. Qazi, and Q. I. Rahman: Entire functions of exponential type in Approximation Theory, In: Constructive Theory of Functions, Varna 2002 (B. Bojanov, Ed.), DARBA, Sofia, (2003), 86–135.
  • X. Fan, D. Zhao: On the spaces $L^{p(x)}(\Omega )$ and $W^{m;p(x)}(\Omega )$, J. Math. Anal. Appl., 263 (2) (2001), 424–446.
  • A. Guven, D. M. Israfilov: Trigonometric approximation in generalized Lebesgue spaces $L^{p(x)}$, J. Math. Inequal., 4 (2) (2010), 285–299.
  • P. Harjulehto, P. Hästö: Orlicz spaces and generalized Orlicz spaces, Lecture Notes in Mathematics, 2236, Springer, (2019).
  • H. Hudzik: On generalized Orlicz-Sobolev space, Funct. Approximatio Comment. Math., 4 (1976), 37–51.
  • I. I. Ibragimov: Teoriya priblizheniya tselymi funktsiyami.(Russian) The theory of approximation by entire functions “Elm" , Baku (1979).
  • S. Z. Jafarov: Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents, Ukr. Math. J., 66 (10) (2015), 1509–1518.
  • S. Z. Jafarov: Approximation by trigonometric polynomials in subspace of variable exponent grand Lebesgue spaces, Global J. Math., 8 (2) (2016), 836–843.
  • S. Z. Jafarov: Ul’yanov type inequalities for moduli of smoothness, Appl. Math. E-Notes, 12 (2012), 221–227.
  • S. Z. Jafarov: S. M. Nikolskii type inequality and estimation between the best approximations of a function in norms of different spaces, Math. Balkanica (N.S.), 21 (1-2) (2007), 173–182.
  • D. M. Israfilov, R. Akgün: Approximation by polynomials and rational functions in weighted rearrangement invariant spaces, J. Math. Anal. Appl., 346 (2008), 489–500.
  • D. M. Israfilov, A. Testici: Approximation problems in the Lebesgue spaces with variable exponent, J. Math. Anal. Appl., 459 (1) (2018), 112–123.
  • D. M. Israfilov, A. Testici: Approximation by Faber–Laurent rational functions in Lebesgue spaces with variable exponent, Indag. Mat., 27 (4) (2016), 914–922.
  • D. M. Israfilov, E. Yirtici: Convolutions and best approximations in variable exponent Lebesgue spaces, Math. Reports, 18 (4) (2016), 497–508.
  • H. Koc: Simultaneous approximation by polynomials in Orlicz spaces generated by quasiconvex Young functions, Kuwait J. Sci., 43 (4) (2016), 18–31.
  • V. Kokilashvili, S. Samko: Singular integrals in weighted Lebesgue spaces with variable exponent, Georgian Math. J., 10 (1) (2003), 145–156.
  • Z. O. Kováˇcik, J. Rákosnik: On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41 (4) (1991), 592–618.
  • F. G. Nasibov: Approximation in $L_2$ by entire functions. (Russian) Akad. Nauk Azerbaidzhan. SSR Dokl., 42 (4) (1986), 3–6.
  • S. M. Nikolskii: Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of several variables, Amer. Math. Soc. Transl. Ser. 2, 80 (1969), 1–38, (Trudy Mat. Inst. Steklov 38 (1951), 211–278).
  • A. A. Ligun, V. G. Doronin: Exact constants in Jackson-type inequalities for the $L_2$-approximation on a straight line. (Russian) Ukraïn. Mat. Zh. 2009; 61 (1): 92–98; translation in Ukrainian Math. J., 61 (1) (2009), 112–120.
  • W. Orlicz: Über konjugierte Exponentenfolgen, Studia Math., 3 (1931), 200–212.
  • R. Paley, N. Wiener: Fourier transforms in the complex domain, Amer. Math. Soc. (1934).
  • V. Yu. Popov: Best mean square approximations by entire functions of exponential type (Russian), Izv. Vysš. Ucebn. Zaved. Matematika, 121 (6) (1972), 65–73.
  • K. R. Rajagopal, M. Ružiˇcka: On the modeling elektroreological materials, Mech. Res. Commun., 23 (4) (1996), 401–407.
  • M. Ružiˇcka: Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin (2000).
  • S. Samko: Differentiation and integration of variable order and the spaces Lp(x), in: Operator theory for complex and hypercomplex analysis (Mexico City, 1994), 203–219, Contemp. Math., 212, Amer. Math. Soc., Providence, RI, (1998).
  • I. I. Sharapudinov: The topology of the space $L^{p(t)}([0; 1])$, (Russian), Mat. Zametki, 26 (4) (1979), 613–632.
  • I. I. Sharapudinov: Some questions in the theory of approximation in Lebesgue spaces with variable exponent, Itogi Nauki. Yug Rossii. Mat. Monografiya, vol. 5, Southern Institute of Mathematics of the Vladikavkaz Sceince Centre of the Russian Academy of Sciences and the Government of the Republic of North Ossetia-Alania, Vladikavkaz (2012), 267 pp. Russian.
  • A. F. Timan: Theory of approximation of functions of a real variable. Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, The Macmillan Co., New York: A Pergamon Press Book (1963).
  • M. F. Timan: The approximation of functions defined on the whole real axis by entire functions of exponential type, Izv. Vyssh. Uchebn. Zaved. Mat., 2 (1968), 89–101.
  • R. Taberski: Approximation by entire functions of exponential type, Demonstr. Math., 14 (1981), 151–181 .
  • R. Taberski: Contributions to fractional calculus and exponential approximation, 1986, Funct. Approximatio, Comment. Math., 15 (1986), 81–106.
  • S. S. Volosivets: Approximation of functions and their conjugates in variable Lebesgue spaces, Sbornik: Mathematics, 208 (1) (2017), 44–59.
  • J. Yeh: Real analysis: theory of measure and integration, 2nd ed., (2006).
  • V. V. Zhikov: Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 50 (4) (1986), 675–710 (in Russian).
There are 58 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ramazan Akgün 0000-0001-6247-8518

Publication Date December 1, 2022
Published in Issue Year 2022

Cite

APA Akgün, R. (2022). Exponential approximation in variable exponent Lebesgue spaces on the real line. Constructive Mathematical Analysis, 5(4), 214-237. https://doi.org/10.33205/cma.1167459
AMA Akgün R. Exponential approximation in variable exponent Lebesgue spaces on the real line. CMA. December 2022;5(4):214-237. doi:10.33205/cma.1167459
Chicago Akgün, Ramazan. “Exponential Approximation in Variable Exponent Lebesgue Spaces on the Real Line”. Constructive Mathematical Analysis 5, no. 4 (December 2022): 214-37. https://doi.org/10.33205/cma.1167459.
EndNote Akgün R (December 1, 2022) Exponential approximation in variable exponent Lebesgue spaces on the real line. Constructive Mathematical Analysis 5 4 214–237.
IEEE R. Akgün, “Exponential approximation in variable exponent Lebesgue spaces on the real line”, CMA, vol. 5, no. 4, pp. 214–237, 2022, doi: 10.33205/cma.1167459.
ISNAD Akgün, Ramazan. “Exponential Approximation in Variable Exponent Lebesgue Spaces on the Real Line”. Constructive Mathematical Analysis 5/4 (December 2022), 214-237. https://doi.org/10.33205/cma.1167459.
JAMA Akgün R. Exponential approximation in variable exponent Lebesgue spaces on the real line. CMA. 2022;5:214–237.
MLA Akgün, Ramazan. “Exponential Approximation in Variable Exponent Lebesgue Spaces on the Real Line”. Constructive Mathematical Analysis, vol. 5, no. 4, 2022, pp. 214-37, doi:10.33205/cma.1167459.
Vancouver Akgün R. Exponential approximation in variable exponent Lebesgue spaces on the real line. CMA. 2022;5(4):214-37.