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Abstract
We study general Franklin systems, i.e. systems of orthonormal piecewise linear functions corresponding to quasi-dyadic sequences of partitions of [0,1]. The following problems are treated: unconditionality of the general Franklin basis in $L^p$, 1 < p < ∞, and $H^p$, 1/2 < p ≤ 1; equivalent conditions for the unconditional convergence of the Franklin series in $L^p$ for 0< p ≤ 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces BMO and Lip(α), 0 < α < 1, in terms of the Fourier-Franklin coefficients.
We study general Franklin systems, i.e. systems of orthonormal piecewise linear functions corresponding to quasi-dyadic sequences of partitions of [0,1]. The following problems are treated: unconditionality of the general Franklin basis in $L^p$, 1 < p < ∞, and $H^p$, 1/2 < p ≤ 1; equivalent conditions for the unconditional convergence of the Franklin series in $L^p$ for 0< p ≤ 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces BMO and Lip(α), 0 < α < 1, in terms of the Fourier-Franklin coefficients.
CONTENTS
1. Introduction.....................................................................................5
1.1. Notation......................................................................................7
2. Definition and properties of general Franklin systems..................10
2.1. Piecewise linear functions.........................................................10
2.2. Franklin functions.....................................................................11
2.3. Sequences of partitions and Franklin functions........................13
2.3.1. Regularity of sequences of partitions...................................14
2.4. Sequences of partitions and general Haar systems.................16
2.5. Technical lemmas.....................................................................17
3. Franklin series in $L^p$, 1 < p < ∞...............................................21
4. Franklin series in $L^p$, 0 < p ≤ 1, and $H^p$, 1/2 < p ≤ 1..........27
5. The necessity of strong regularity in $H^p$, 1/2 < p ≤ 1...............42
6. Haar and Franklin series with identical coefficients......................46
7. Characterization of the spaces BMO and Lip(α), 0 < α < 1...........51
References.......................................................................................58
1. Introduction.....................................................................................5
1.1. Notation......................................................................................7
2. Definition and properties of general Franklin systems..................10
2.1. Piecewise linear functions.........................................................10
2.2. Franklin functions.....................................................................11
2.3. Sequences of partitions and Franklin functions........................13
2.3.1. Regularity of sequences of partitions...................................14
2.4. Sequences of partitions and general Haar systems.................16
2.5. Technical lemmas.....................................................................17
3. Franklin series in $L^p$, 1 < p < ∞...............................................21
4. Franklin series in $L^p$, 0 < p ≤ 1, and $H^p$, 1/2 < p ≤ 1..........27
5. The necessity of strong regularity in $H^p$, 1/2 < p ≤ 1...............42
6. Haar and Franklin series with identical coefficients......................46
7. Characterization of the spaces BMO and Lip(α), 0 < α < 1...........51
References.......................................................................................58
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
374
Liczba stron
59
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXXIV
Daty
wydano
1998
otrzymano
1997-11-06
Twórcy
autor
- Erevan State University, Department of Mathematics, Alex Manoukian St. 1, 375049 Erevan, Armenia, ggg@instmath.sci.am
autor
- Instytut Matematyczny PAN, ul. Abrahama 18, 81-825 Sopot, Poland, ak@impan.gda.pl
Bibliografia
- [1] S. V. Bochkarev, Some inequalities for the Franklin series, Anal. Math. 1 (1975), 249-257.
- [2] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494-1504.
- [3] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249-304.
- [4] S.-Y. A. Chang and Z. Ciesielski, Spline characterizations of H¹, Studia Math. 75 (1983), 183-192.
- [5] Z. Ciesielski, Properties of the orthonormal Franklin system, Studia Math. 23 (1963), 141-157.
- [6] Z. Ciesielski, Properties of the orthonormal Franklin system II, Studia Math. 27 (1966), 289-323.
- [7] Z. Ciesielski, Equivalence, unconditionality and convergence a.e. of the spline bases in $L^p$ spaces, in: Approximation Theory, Banach Center Publ. 4, PWN, Warszawa 1979, 55-68.
- [8] Z. Ciesielski, The Franklin orthogonal system as unconditional basis in Re H¹ and VMO, in: Functional Analysis and Approximation, Internat. Ser. Numer. Math. 60, Birkhäuser, Basel, 1981, 117-125.
- [9] Z. Ciesielski and A. Kamont, Projections onto piecewise linear functions, Funct. Approx. Comment. Math. 25 (1997), 129-143.
- [10] Z. Ciesielski and S. Kwapień, Some properties of the Haar, Walsh-Paley, Franklin and the bounded polygonal orthonormal bases in Lₚ spaces, Comment. Math. Prace Mat., Tomus Specialis in honorem Ladislai Orlicz II (1979), 37-42.
- [11] Z. Ciesielski, P. Simon and P. Sjölin, Equivalence of Haar and Franklin bases in $L^p$ spaces, Studia Math. 60 (1977), 195-210.
- [12] R. R. Coifman, A real variable characterization of $H^p$, Studia Math. 51 (1974), 268-274.
- [13] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
- [14] J. L. Doob, Stochastic Processes, Wiley, New York, 1953.
- [15] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137-193.
- [16] Ph. Franklin, A set of continuous orthogonal functions, Math. Ann. 100 (1928), 522-529.
- [17] G. G. Gevorkyan, Unboundedness of the shift operator in the Franklin system in the space L¹, Mat. Zametki 38 (1985), 523-533 (in Russian).
- [18] G. G. Gevorkyan, Theorems on the modified Franklin-Strömberg system, Izv. Akad. Nauk Armyan. SSR Ser. Mat. 24 (1) (1989), 69-92 (in Russian).
- [19] G. G. Gevorkyan, On Haar and Franklin series with the same coefficients, Erevan. Gos. Univ. Uchen. Zap. Estestv. Nauki 3 (1989), 3-9 (in Russian).
- [20] G. G. Gevorkyan, Theorems on the modified Franklin-Strömberg system II, Izv. Akad. Nauk Armenii Mat. 26 (1) (1991), 31-51 (in Russian).
- [21] G. G. Gevorkyan, Some theorems on unconditional convergence and the majorant of Franklin series and their application to the spaces $Re(H^p)$, Proc. Steklov Inst. Math. (1992), 49-76.
- [22] G. G. Gevorkyan, Franklin and Haar series in Lorentz spaces $L_{p,q}[0,1]$, 0 < q ≤ p < 1, East J. Approx. 3 (1997), 419-444.
- [23] A. Kamont, Spline spaces and weighted moduli of smoothness, J. Approx. Theory, submitted.
- [24] B. Kashin and A. Saakyan, Orthogonal Series, Nauka, Moscow, 1984 (in Russian).
- [25] P. Simon, Remarks on the shift operators with respect to the Haar and Franklin systems, Acta Math. Acad. Sci. Hungar. 39 (1982), 251-254.
- [26] P. Sjölin, The Haar and Franklin systems are not equivalent bases in L¹, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 1099-1100.
- [27] P. Sjölin and J. O. Strömberg, Basis properties of Hardy spaces, Ark. Mat. 21 (1983), 111-125.
- [28] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, 1993.
- [29] P. Wojtaszczyk, The Franklin system is an unconditional basis in H¹, Ark. Mat. 20 (1982), 293-300.
- [30] P. Wojtaszczyk, Banach Spaces for Analysts, Cambridge Univ. Press, Cambridge, 1991.
- [31] A. Zygmund, Trigonometric Series, Vol. 1 (2nd ed.), Cambridge Univ. Press, Cambridge, 1959.
Języki publikacji
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Uwagi
1991 Mathematics Subject Classification: Primary 42C10.
Identyfikator YADDA
bwmeta1.element.zamlynska-6c3e6ab7-dc6a-49e6-92ab-8168c32ef1a6
Identyfikatory
ISSN
0012-3862
Kolekcja
DML-PL
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