Research Article
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Year 2020, Volume: 69 Issue: 2, 1171 - 1183, 31.12.2020
https://doi.org/10.31801/cfsuasmas.710208

Abstract

References

  • Aydın, I., Unal, C., The Kolmogorov--Riesz theorem and some compactness criterions of bounded subsets in weighted variable exponent amalgam and Sobolev spaces, Collectanea Mathematica, (2019), https://doi.org/10.1007/s13348-019-00262-5, 1-19.
  • Aydın, I., Unal, C., On some multipliers of vector-valued amalgam spaces, Int. Journal of Pure and Appl. Math., 116 (2) (2017), 547-557.
  • Aydın, I., On variable exponent amalgam spaces, Analele Stiint. Univ., 20(3) (2012), 5-20.
  • Aydın, I., Weighted variable Sobolev spaces and capacity, J. Funct. Space Appl., Volume 2012, Article ID 132690, 17 pages, doi:10.1155/2012/132690.
  • Aydın, I., Gürkanlı, A.T., Weighted variable exponent amalgam spaces W(L^{p(x)};L_{w}^{q}), Glas. Mat., 47(67) (2012), 165-174.
  • Aydın, I., Unal, C., Birkhoff's ergodic theorem for weighted variable exponent amalgam spaces, Applications and Applied Mathematics: An International Journal (AAM), Special Issue No. 3 (2019), 1-10.
  • Aydın, I., On vector-valued classical and variable exponent amalgam spaces, Commun. Fac. Sci. Univ. Ank. Series A1 , 66 (2) (2017), 100-114.
  • Butzer, P.L., Nessel, R.J., Fourier Analysis and Approximation, Academic Press, Newyork-London, Volume 1, 1971.
  • Cruz-Uribe, D., Fiorenza, A., Approximate identities in variable Lp spaces, Math. Nach., 280 (2007), 256-270.
  • Cruz-Uribe, D., Fiorenza, A., Variable Lebesgue Spaces, Foundations and harmonic analysis, New York, NY, Birkhauser/Springer, 2013.
  • Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.
  • Diening, L., Maximal function on generalized Lebesgue spaces L^{p(.)}, Mathematical Inequalities and Applications, 7 (2004), 245-253.
  • Diening, L., Harjulehto, P., Hästö, P., Ružicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Berlin, 2011.
  • Fan, X., Zhao, D., On the spaces L^{p(x)}(Ω) and W^{k,p(x)}(Ω), J. Math. Anal. Appl., 263 (2) (2001), 424-446.
  • Feichtinger, H. G., Banach convolution algebras of Wiener type. In: Functions, Series, Operators, Proc. Conf. Budapest 38, Colloq. Math. Soc., Janos Bolyai, (1980),509--524.
  • Feichtinger, H. G., Weisz, F., The Segal algebra S₀(ℝ^{d}) and norm summability of Fourier series and Fourier Transforms, Monatshefte Mth., 148 (2006), 333-349.
  • Feichtinger, H. G., Weisz, F., Wiener amalgams and pointwise summability of Fourier Transforms and Fourier series, Math. Proc. Camb. Phil. Soc., 140 (2006), 509-536.
  • Fournier, J.J., Stewart, J., Amalgams of L^{p}and l^{q}, Bull. Amer. Math. Soc., 13 (1985), 1--21.
  • Gröchenig, K., Foundations of Time-Frequency Analysis, Birkhauser, Boston, 2001.
  • Gröchenig, K., Heil, C., Okoudjou, K., Gabor analysis in weighted amalgam spaces, Sampl. Theory Signal Image Process., Int. J., 1 (2002), 225-259.
  • Gürkanlı, A.T., Aydın, I., On the weighted variable exponent amalgam space W(L^{p(x)};L_{m}^{q}), Acta Math. Sci., 4(34B) (2014), 1--13.
  • Gürkanlı, A.T., The amalgam spaces W(L^{p(x)};L^{{p_{n}}}) and boundedness of Hardy-Littlewood maximal operators, Current Trends in Analysis and Its Applications, Proceedings of the 9th ISAAC Congress, Krakow (2013).
  • Hästö, P., Diening, L., Muckenhoupt weights in variable exponent spaces, preprint, http://www.helsinki.fi/~pharjule/varsob/publications.shtml.
  • Heil, C., An introduction to weighted Wiener amalgams. In: Wavelets and Their Applications, Allied Publishers, New Delhi, (2003), 183-216.
  • Holland, F., Harmonic analysis on amalgams of L^{p}and l^{q}, J. London Math. Soc., 2(10) (1975), 295--305.
  • Kokilashvili, V., Meskhi, A., Zaighum, M.A., Weighted kernel operators in variable exponent amalgam spaces, J. Inequal. Appl., (2013), DOI:10.1186/1029-242X-2013-173.
  • Kovacik, O., Rakosnik, J., On spaces L^{p(x)} and W^{k,p(x)}, Czech. Math. J., 41(116) (1991), 592-618.
  • Kulak, O., Gürkanlı, A.T., Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces, J. Inequal. Appl., (2014), 476.
  • Li, K., Sun, W., Pointwise convergence of the Calderon reproducing formula, J. Fourier Anal. Appl., 18 (2012), 439-455.
  • Meskhi, A., Zaighum, M.A., On the boundedness of maximal and potential operators in variable exponent amalgam spaces, Journal of Mathematical Inequalities, 8(1) (2014), 123--152.
  • Ruzicka, M., Electrorheological fluids: modeling and mathematical theory, Springer-Verlag, Berlin, 2000.
  • Simon, P., (C,α) summability of Walsh-Kaczmarz-Fourier Series, J. Approx. Theory, 127(1) (2004), 39-60.
  • Squire, M.L.T., Amalgams of L^{p} and l^{q}, PhD, McMaster University, 1984.
  • Szarvas, K., Weisz, F., Continuous wavelet transform in variable Lebesgue spaces, Stud. Univ. Babeş-Bolyai Math., 59(4) (2014), 497-512.
  • Szarvas, K., Variable Lebesgue spaces and continuous wavelet transforms, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, 32 (2016), 313-325.
  • Trigub, R.M., Bellinsky, E.S., Fourier Analysis and Approximation of Function, Kluwer Academic Publishers, Dordrecht, 2004.
  • Unal, C., Aydın, I., On some properties of the space L_{w}^{p}(ℝⁿ)∩L_{ϑ}^{q(.)}(ℝⁿ), Çankaya University Journal of Science and Engineering, 13(2) (2016), 001-010.
  • Weisz, F., Summability of multi-dimensional Fourier series and Hardy spaces, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 541, 2002.
  • Weisz, F., Summability of multi-dimensional trigonometric Fourier series. Surv Approx Theory, 7 (2012), 1-179.
  • Weisz, F., Convergence of the inverse continuous wavelet transform in Wiener amalgam spaces, Journal of Analysis, 35 (2015), 33--46.
  • Weisz, F., Inversion formulas for the continuous wavelet transform, Acta. Math. Hungar., 138 (2013), 237-258.
  • Weisz, F., Pointwise convergence in Pringsheim's sense of the summability of fourier transforms on Wiener amalgam spaces, Monatsh Math., 175(1) (2014), 143-160.
  • Zhikov, V.V., Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 29(4) (1987), 33-66.

Inverse continuous wavelet transform in weighted variable exponent amalgam spaces

Year 2020, Volume: 69 Issue: 2, 1171 - 1183, 31.12.2020
https://doi.org/10.31801/cfsuasmas.710208

Abstract

The wavelet transform is an useful mathematical tool. It is a mapping of a time signal to the time-scale joint representation. The wavelet transform is generated from a wavelet function by dilation and translation. This wavelet function satisfies an admissible condition so that the original signal can be reconstructed by the inverse wavelet transform. In this study, we firstly give some basic properties of the weighted variable exponent amalgam spaces. Then we investigate the convergence of the θ-means of f in these spaces under some conditions. Finally, using these results the convergence of the inverse continuous wavelet transform is considered in these spaces.

References

  • Aydın, I., Unal, C., The Kolmogorov--Riesz theorem and some compactness criterions of bounded subsets in weighted variable exponent amalgam and Sobolev spaces, Collectanea Mathematica, (2019), https://doi.org/10.1007/s13348-019-00262-5, 1-19.
  • Aydın, I., Unal, C., On some multipliers of vector-valued amalgam spaces, Int. Journal of Pure and Appl. Math., 116 (2) (2017), 547-557.
  • Aydın, I., On variable exponent amalgam spaces, Analele Stiint. Univ., 20(3) (2012), 5-20.
  • Aydın, I., Weighted variable Sobolev spaces and capacity, J. Funct. Space Appl., Volume 2012, Article ID 132690, 17 pages, doi:10.1155/2012/132690.
  • Aydın, I., Gürkanlı, A.T., Weighted variable exponent amalgam spaces W(L^{p(x)};L_{w}^{q}), Glas. Mat., 47(67) (2012), 165-174.
  • Aydın, I., Unal, C., Birkhoff's ergodic theorem for weighted variable exponent amalgam spaces, Applications and Applied Mathematics: An International Journal (AAM), Special Issue No. 3 (2019), 1-10.
  • Aydın, I., On vector-valued classical and variable exponent amalgam spaces, Commun. Fac. Sci. Univ. Ank. Series A1 , 66 (2) (2017), 100-114.
  • Butzer, P.L., Nessel, R.J., Fourier Analysis and Approximation, Academic Press, Newyork-London, Volume 1, 1971.
  • Cruz-Uribe, D., Fiorenza, A., Approximate identities in variable Lp spaces, Math. Nach., 280 (2007), 256-270.
  • Cruz-Uribe, D., Fiorenza, A., Variable Lebesgue Spaces, Foundations and harmonic analysis, New York, NY, Birkhauser/Springer, 2013.
  • Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.
  • Diening, L., Maximal function on generalized Lebesgue spaces L^{p(.)}, Mathematical Inequalities and Applications, 7 (2004), 245-253.
  • Diening, L., Harjulehto, P., Hästö, P., Ružicka, M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math. 2017, Springer, Berlin, 2011.
  • Fan, X., Zhao, D., On the spaces L^{p(x)}(Ω) and W^{k,p(x)}(Ω), J. Math. Anal. Appl., 263 (2) (2001), 424-446.
  • Feichtinger, H. G., Banach convolution algebras of Wiener type. In: Functions, Series, Operators, Proc. Conf. Budapest 38, Colloq. Math. Soc., Janos Bolyai, (1980),509--524.
  • Feichtinger, H. G., Weisz, F., The Segal algebra S₀(ℝ^{d}) and norm summability of Fourier series and Fourier Transforms, Monatshefte Mth., 148 (2006), 333-349.
  • Feichtinger, H. G., Weisz, F., Wiener amalgams and pointwise summability of Fourier Transforms and Fourier series, Math. Proc. Camb. Phil. Soc., 140 (2006), 509-536.
  • Fournier, J.J., Stewart, J., Amalgams of L^{p}and l^{q}, Bull. Amer. Math. Soc., 13 (1985), 1--21.
  • Gröchenig, K., Foundations of Time-Frequency Analysis, Birkhauser, Boston, 2001.
  • Gröchenig, K., Heil, C., Okoudjou, K., Gabor analysis in weighted amalgam spaces, Sampl. Theory Signal Image Process., Int. J., 1 (2002), 225-259.
  • Gürkanlı, A.T., Aydın, I., On the weighted variable exponent amalgam space W(L^{p(x)};L_{m}^{q}), Acta Math. Sci., 4(34B) (2014), 1--13.
  • Gürkanlı, A.T., The amalgam spaces W(L^{p(x)};L^{{p_{n}}}) and boundedness of Hardy-Littlewood maximal operators, Current Trends in Analysis and Its Applications, Proceedings of the 9th ISAAC Congress, Krakow (2013).
  • Hästö, P., Diening, L., Muckenhoupt weights in variable exponent spaces, preprint, http://www.helsinki.fi/~pharjule/varsob/publications.shtml.
  • Heil, C., An introduction to weighted Wiener amalgams. In: Wavelets and Their Applications, Allied Publishers, New Delhi, (2003), 183-216.
  • Holland, F., Harmonic analysis on amalgams of L^{p}and l^{q}, J. London Math. Soc., 2(10) (1975), 295--305.
  • Kokilashvili, V., Meskhi, A., Zaighum, M.A., Weighted kernel operators in variable exponent amalgam spaces, J. Inequal. Appl., (2013), DOI:10.1186/1029-242X-2013-173.
  • Kovacik, O., Rakosnik, J., On spaces L^{p(x)} and W^{k,p(x)}, Czech. Math. J., 41(116) (1991), 592-618.
  • Kulak, O., Gürkanlı, A.T., Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces, J. Inequal. Appl., (2014), 476.
  • Li, K., Sun, W., Pointwise convergence of the Calderon reproducing formula, J. Fourier Anal. Appl., 18 (2012), 439-455.
  • Meskhi, A., Zaighum, M.A., On the boundedness of maximal and potential operators in variable exponent amalgam spaces, Journal of Mathematical Inequalities, 8(1) (2014), 123--152.
  • Ruzicka, M., Electrorheological fluids: modeling and mathematical theory, Springer-Verlag, Berlin, 2000.
  • Simon, P., (C,α) summability of Walsh-Kaczmarz-Fourier Series, J. Approx. Theory, 127(1) (2004), 39-60.
  • Squire, M.L.T., Amalgams of L^{p} and l^{q}, PhD, McMaster University, 1984.
  • Szarvas, K., Weisz, F., Continuous wavelet transform in variable Lebesgue spaces, Stud. Univ. Babeş-Bolyai Math., 59(4) (2014), 497-512.
  • Szarvas, K., Variable Lebesgue spaces and continuous wavelet transforms, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, 32 (2016), 313-325.
  • Trigub, R.M., Bellinsky, E.S., Fourier Analysis and Approximation of Function, Kluwer Academic Publishers, Dordrecht, 2004.
  • Unal, C., Aydın, I., On some properties of the space L_{w}^{p}(ℝⁿ)∩L_{ϑ}^{q(.)}(ℝⁿ), Çankaya University Journal of Science and Engineering, 13(2) (2016), 001-010.
  • Weisz, F., Summability of multi-dimensional Fourier series and Hardy spaces, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 541, 2002.
  • Weisz, F., Summability of multi-dimensional trigonometric Fourier series. Surv Approx Theory, 7 (2012), 1-179.
  • Weisz, F., Convergence of the inverse continuous wavelet transform in Wiener amalgam spaces, Journal of Analysis, 35 (2015), 33--46.
  • Weisz, F., Inversion formulas for the continuous wavelet transform, Acta. Math. Hungar., 138 (2013), 237-258.
  • Weisz, F., Pointwise convergence in Pringsheim's sense of the summability of fourier transforms on Wiener amalgam spaces, Monatsh Math., 175(1) (2014), 143-160.
  • Zhikov, V.V., Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk SSSR Ser. Mat., 29(4) (1987), 33-66.
There are 43 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Öznur Kulak 0000-0003-1433-3159

İsmail Aydın 0000-0001-8371-3185

Publication Date December 31, 2020
Submission Date March 27, 2020
Acceptance Date June 24, 2020
Published in Issue Year 2020 Volume: 69 Issue: 2

Cite

APA Kulak, Ö., & Aydın, İ. (2020). Inverse continuous wavelet transform in weighted variable exponent amalgam spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69(2), 1171-1183. https://doi.org/10.31801/cfsuasmas.710208
AMA Kulak Ö, Aydın İ. Inverse continuous wavelet transform in weighted variable exponent amalgam spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2020;69(2):1171-1183. doi:10.31801/cfsuasmas.710208
Chicago Kulak, Öznur, and İsmail Aydın. “Inverse Continuous Wavelet Transform in Weighted Variable Exponent Amalgam Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69, no. 2 (December 2020): 1171-83. https://doi.org/10.31801/cfsuasmas.710208.
EndNote Kulak Ö, Aydın İ (December 1, 2020) Inverse continuous wavelet transform in weighted variable exponent amalgam spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69 2 1171–1183.
IEEE Ö. Kulak and İ. Aydın, “Inverse continuous wavelet transform in weighted variable exponent amalgam spaces”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 69, no. 2, pp. 1171–1183, 2020, doi: 10.31801/cfsuasmas.710208.
ISNAD Kulak, Öznur - Aydın, İsmail. “Inverse Continuous Wavelet Transform in Weighted Variable Exponent Amalgam Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 69/2 (December 2020), 1171-1183. https://doi.org/10.31801/cfsuasmas.710208.
JAMA Kulak Ö, Aydın İ. Inverse continuous wavelet transform in weighted variable exponent amalgam spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69:1171–1183.
MLA Kulak, Öznur and İsmail Aydın. “Inverse Continuous Wavelet Transform in Weighted Variable Exponent Amalgam Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 69, no. 2, 2020, pp. 1171-83, doi:10.31801/cfsuasmas.710208.
Vancouver Kulak Ö, Aydın İ. Inverse continuous wavelet transform in weighted variable exponent amalgam spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2020;69(2):1171-83.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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