Abstract
Let \(A^q_s({{\mathbf {T}}})\) denote the space of all Lebesgue integrable functions f on the torus \({\mathbf {T}}\) such that \(\sum _{m \in {{\mathbf {Z}}}} |\widehat{f}(m)|^q \langle m \rangle ^{sq} < \infty \), where \(\{ \widehat{f}(m) \}_{m \in {{\mathbf {Z}}}}\) denote the Fourier coefficients of f. We consider necessary and sufficient conditions for all functions \(F \in A^1_\beta ({{\mathbf {T}}})\) to operate on all real-valued functions in \(A^q_s({{\mathbf {T}}})\).
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Bényi, Á., Oh, T.: Modulation spaces, Wiener amalgam spaces, and Brownian motions. Adv. Math. 228, 2943–2981 (2011)
Bhimani, D.G.: Composition operators on Wiener amalgam spaces. Nagoya Math. J. 240, 257–274 (2020)
Bhimani, D.G., Ratnakumar, P.-K.: Functions operating on modulation spaces and nonlinear dispersive equations. J. Funct. Anal. 270, 621–648 (2016)
Bourdaud, G., Sickel, W.: Composition operators on function spaces with fractional order of smoothness. In: Ozawa, T., Sugimoto, M. (eds.) Harmonic Analysis and Nonlinear Partial Differential Equations, RIMS Kokyuroku Bessatsu, B26, pp. 93–132. Res. Inst. Math. Sci. (RIMS), Kyoto (2011)
Brandenburg, L.H.: On identifying the maximal ideals in Banach algebras. J. Math. Anal. Appl. 50, 489–510 (1975)
Devinatz, A., Hirschman, I.I., Jr.: Multiplier transformations on \(l^{2,\alpha }\). Ann. Math. 2(69), 575–587 (1959)
Duoandikoetxea, J.: Fourier Analysis. Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence, RI (2001)
Helson, H., Kahane, J.-P., Katznelson, Y., Rudin, W.: The functions which operate on Fourier transforms. Acta Math. 102, 135–157 (1959)
Hörmander, L.: The Analysis of Linear Partial Differential Operators. II. Differential operators with constant coefficients. Classics in Mathematics. Springer, Berlin (2005)
Kahane, J.-P.: Sur certaines classes de séries de Fourier absolument convergentes. J. Math. Pures Appl. 9(35), 249–259 (1956)
Kahane, J.-P.: Séries de Fourier Absolument Convergentes. Springer, Berlin (1970)
Kato, T., Sugimoto, M., Tomita, N.: Nonlinear operations on a class of modulation spaces. J. Funct. Anal. 278, 108447 (2020)
Katznelson, Y.: Sur le calcul symbolique dans quelques algèbres de Banach. Ann. Sci. École Norm. Sup. 3(76), 83–123 (1959)
Katznelson, Y.: An Introduction to Harmonic Analysis, Second corrected Dover Publications Inc, New York (1976)
Kobayashi, M., Sato, E.: Operating functions on modulation and Wiener amalgam spaces. Nagoya Math. J. 230, 72–82 (2018)
Leblanc, N.: Calcul symbolique dans les algèbres de fonctions sphériques zonales. C. R. Acad. Sci. Paris Sér. A-B 264, A672–A674 (1967)
Leblanc, N.: Les fonctions qui opèrent dans certaines algèbres à poids. Math. Scand. 25, 190–194 (1969)
Lévy, P.: Sur la convergence absolue des séries de Fourier. Compos. Math. 1, 1–14 (1935)
Meyer, Y.: Remarques sur un théorème de J. M. Bony. Rend. Circ. Mat. Palermo 1, 1–20 (1981)
Okoudjou, K.A.: A Beurling–Helson type theorem for modulation spaces. J. Funct. Spaces Appl. 7, 33–41 (2009)
Reich, M., Sickel, W.: Multiplication and composition in weighted modulation spaces. In: Qian, T., Rodino, L.G. (eds.) Mathematical Analysis, Probability and Applications-Plenary Lectures. Springer Proceedings in Mathematics & Statistics 177, pp. 103–149. Springer, Cham (2016)
Rudin, W.: Fourier Analysis on Groups. Dover Publications, New York (2017)
Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order. Nemytskij Operators and Nonlinear Partial Differential Equations. de Gruyter, Berlin (1996)
Ruzhansky, M., Sugimoto, M., Wang, B.: Modulation Spaces and Nonlinear Evolution Equations. Evolution Equations of Hyperbolic and Schrodinger Type, pp. 267–283. Birkhäuser/Springer Basel AG, Basel (2012)
Wiener, N.: Tauberian theorems. Ann. Math. 2(33), 1–100 (1932)
Zygmund, A.: Trigonometric Series, vol. I. II. Cambridge University Press, Cambridge (2002)
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We would like to thank Professor Hans Georg Feichtinger for several valuable comments and suggestions which improve our paper. The authors also wish to thank the anonymous referees for providing some helpful comments.
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Communicated by Ferenc Weisz.
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This work was supported by JSPS KAKENHI Grant Numbers JP19K03533, JP17K05289.
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Kobayashi, M., Sato, E. Operating Functions on \(A^q_s({{\mathbf {T}}})\). J Fourier Anal Appl 28, 42 (2022). https://doi.org/10.1007/s00041-022-09925-7
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DOI: https://doi.org/10.1007/s00041-022-09925-7