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## Studia Mathematica

1993 | 107 | 1 | 33-59
Tytuł artykułu

### Integrability theorems for trigonometric series

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We show that, if the coefficients (a_n) in a series $a_0/2+∑_{n=1}^∞ a_n cos(nt)$ tend to 0 as n → ∞ and satisfy the regularity condition that $∑_{m=0}^∞ {∑_{j=1}^∞ [∑_{n=j2^m}^{(j+1)2^m-1} |a_n - a_{n+1}|]²}^{1/2} < ∞$, then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (b_n) in a series $∑_{n=1}^∞ b_n sin(nt)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if $∑_{n=1}^∞ |b_n|/n < ∞$. These conclusions were previously known to hold under stronger restrictions on the sizes of the differences $Δa_n = a_n - a_{n+1}$ and $Δb_n = b_n - b_{n+1}$. We were led to the mixed-norm conditions that we use here by our recent discovery that the same combination of conditions implies the integrability of Walsh series with coefficients (a_n) tending to 0. We also show here that this condition on the differences implies that the cosine series converges in L¹-norm if and only if $a_n log n → 0$ as n → ∞. The corresponding statement also holds for sine series for which $∑_{n=1}^∞ |b_n|/n < ∞$. If either type of series is assumed a priori to represent an integrable function, then weaker regularity conditions suffice for the validity of this criterion for norm convergence.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
33-59
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-10-29
poprawiono
1993-04-20
Twórcy
autor
• Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
autor
• Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2
Bibliografia
• [1] B. Aubertin and J. J. F. Fournier, An integrability theorem for Walsh series, Boll. Un. Mat. Ital., to appear.
• [2] A. E. Baernstein III and E. Sawyer, Embedding and multiplier theorems for $H^p(R^n)$, Mem. Amer. Math. Soc. 318 (1985).
• [3] L. A. Balashov and S. A. Telyakovskiĭ, Some properties of lacunary series and the integrability of trigonometric series, Trudy Mat. Inst. Steklov. Akad. Nauk SSSR 143 (1977), 32-41 (in Russian); English transl.: Proc. Steklov Inst. Math. 1980 (1), 33-43.
• [4] N. Bari, A Treatise on Trigonometric Series, 2 vols., translated by Margaret F. Mullins, MacMillan, New York, 1964.
• [5] F. F. Bonsall, Boundedness of Hankel matrices, J. London Math. Soc. (2) 29 (1984), 289-300.
• [6] W. O. Bray and V. Stanojević, On the integrability of complex trigonometric series, Proc. Amer. Math. Soc. 93 (1985), 51-58.
• [7] M. Buntinas, Some new multiplier theorems for Fourier series, Proc. Amer. Math. Soc. 101 (1987), 497-502.
• [8] M. Buntinas and N. Tanović-Miller, New integrability and L¹-convergence classes for even trigonometric series II, in: Approximation Theory, J. Szabados and K. Tandori (eds.), Colloq. Math. Soc. János Bolyai 58, North-Holland, Amsterdam, 1991, 103-125.
• [9] D. Ćeranić and N. Tanović-Miller, An integrability and L¹-convergence class for general trigonometric series, to appear.
• [10] C.-P. Chen, L¹-convergence of Fourier series, J. Austral. Math. Soc. Ser. A 41 (1986), 376-390.
• [11] R. A. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
• [12] J. R. Dorronsoro, Mean oscillation and Besov spaces, Canad. Math. Bull. 28 (1985), 474-480.
• [13] G. A. Fomin, A class of trigonometric series, Mat. Zametki 23 (1978), 213-222 (in Russian); English transl.: Math. Notes 23 (1978), 117-123.
• [14] J. J. F. Fournier and W. M. Self, Some sufficient conditions for uniform convergence of Fourier series, J. Math. Anal. Appl. 126 (1987), 355-374.
• [15] J. J. F. Fournier and J. Stewart, Amalgams of $L^p$ and $ℓ^q$, Bull. Amer. Math. Soc. (N. S.) 13 (1985), 1-21.
• [16] D. Grow and V. B. Stanojević, Representations of Fourier coefficients in tauberian L¹-convergence classes, J. Math. Anal. Appl. 160 (1991), 47-50.
• [17] C. S. Herz, Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms, J. Math. Mech. 18 (1968), 283-323.
• [18] F. Holland, Harmonic analysis on amalgams of $L^p$ and $ℓ^q$, J. London Math. Soc. (2) 10 (1975), 295-305.
• [19] F. Holland and D. Walsh, Boundedness criteria for Hankel operators, Proc. R. Irish Acad. 84A (1984), 141-154.
• [20] A. S. Kolmogorov, Sur l'ordre de grandeur des coefficients de la série de Fourier-Lebesgue, Bull. Internat. Acad. Polon. Sci. Lettres (A) Sci. Math. 1923, 83-86.
• [21] F. Móricz, On the integrability and L¹-convergence of complex trigonometric series, Proc. Amer. Math. Soc. 113 (1991), 53-64.
• [22] F. Móricz, On L¹-convergence of Walsh series. II, Acta Math. Hungar. 58 (1991), 203-210.
• [23] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Ser., Durham, N.C., 1976.
• [24] M. Pepić and N. Tanović-Miller, to appear.
• [25] M. Plancherel et G. Pólya, Fonctions entières et intégrales de Fourier multiples, Parties $1^e$ et $2^e$, Comment. Math. Helv. 9 (1936-1937), 224-248; 10 (1937-38), 110-163.
• [26] F. Ricci and M. Taibleson, Boundary values of harmonic functions in mixed-norm spaces and their atomic structure, Ann. Scuola Norm. Sup. Pisa (4) 10 (1983), 1-54.
• [27] S. Sidon, Hinreichende Bedingungen für den Fourier-Charakter einer trigonometrischen Reihe, J. London Math. Soc. 14 (1939), 158-160.
• [28] W. T. Sledd and D. A. Stegenga, An H¹ multiplier theorem, Ark. Mat. 19 (1981), 265-270.
• [29] Č. V. Stanojević, Classes of L¹-convergence of Fourier and Fourier-Stieltjes series, Proc. Amer. Math. Soc. 82 (1981), 209-215.
• [30] Č. V. Stanojević, Structure of Fourier coefficients and Fourier-Stieltjes coefficients of series with slowly varying convergence moduli, Bull. Amer. Math. Soc. 19 (1988), 283-286.
• [31] S. J. Szarek and T. Wolniewicz, A proof of Fefferman's theorem on multipliers, preprint 209, Institute of Mathematics, Polish Academy of Sciences, 1980.
• [32] P. Szeptycki, On functions and measures whose Fourier transforms are measures, Math. Ann. 179 (1968), 31-41.
• [33] N. Tanović-Miller, On integrability and L¹-convergence of cosine series, Boll. Un. Mat. Ital. (7) 4-B (1990), 499-516.
• [34] S. A. Telyakovskiĭ, Integrability conditions of trigonometric series and their applications to the study of linear methods of summing Fourier series, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1209-1236 (in Russian).
• [35] S. A. Telyakovskiĭ, Concerning a sufficient condition of Sidon for the integrability of trigonometric series, Mat. Zametki 14 (1973), 317-328 (in Russian); English transl.: Math. Notes 14 (1973), 742-748.
• [36] W. H. Young, On the Fourier series of bounded functions, Proc. London Math. Soc. (2) 12 (1913), 41-70.
• [37] A. Zygmund, Trigonometric Series, 2 volumes, Cambridge University Press, Cambridge 1959.
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Bibliografia
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