GABOR ANALYSIS OF THE SPACES {\boldmath M( p, q,  w) ({Rd) AND S( p, q, r, w, ω) (Rd)

doi:10.1016/S0252-9602(11)60216-6

Abstract

Abstract:

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define
the space M( p, q, w) ( R^d) to be the subspace of tempered distributions f ∈S´(R^d) such that the Gabor transform V_g(f) of f is in the weighted Lorentz space L( p, q, wdμ) (R^2d) . We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1≤p, q≤∞. We also investigate the embeddings between these spaces and the dual space of M( p, q, w) (R^d) . Later we define the space S( p, q, r, w, ω)(R^d) for 1<p<∞, 1≤q ≤∞. We endow it with a sum norm and show that it becomes a Banach convolution algebra. We also discuss some properties of S( p, q, r, w, ω) (R^d) . At the end of this article, we characterize the multipliers of
the spaces M( p, q, w) (R^d) and S( p, q, r, w, ω) ( R^d) .

Key words: Gabor transform, weigted Lorentz space, multiplier

CLC Number:

43A15

Ayse Sandikci A. Turan G¨urkanli. GABOR ANALYSIS OF THE SPACES {\boldmath M( p, q, w) ({R^d) AND S( p, q, r, w, ω) (R^d)[J].Acta mathematica scientia,Series B, 2011, 31(1): 141-158.

Trendmd

References

[1] Bennet C, Sharpley R. Interpolation of Operators. Academic Press, Inc, 1998

[2] Blozinski A P. On a convolution theorem for L( p, q) spaces. Trans of the Amer Math Society, 1972, 164: 255--265

[3] Chen Y K, Lai H C. Multipliers of Lorentz spaces. Hokkaido Math J, 1975, 4: 247--260

[4] Chung H M, Hunt R A, Kurtz D S. The Hardy-Littlewood maximal function on L( p, q) spaces with weights. Indiana University
Mathematics Journal, 1982, 31(1): 109--120

[5] Cigler J. Normed ideals in L¹( G) . Indag Math, 1969, 31: 273--282

[6] Comisky C V. Multipliers of Banach modules. Indagationes Mathematicae, 1971, 33: 32--38

[7] Dogan M, G\"{u}rkanl\i\ A T. Multipliers of the space S_ω(G) . Mathematica Balkanica, 2001, 15(3/4): 199--212

[8] Doran R S, Wichmann J. Approximate Identities and Factorization in Banach Modules//Lecture Notes in Mathematics 768. Springer-Verlag, 1979

[9] Duyar C, G\"{u}rkanl\i\ A T. Multipliers and relative completion in weighted Lorentz spaces. Acta Mathematica Scienta, 2003, 23B(4): 467--476

[10] Edmunds D E, Evans W D. Hardy Operators Function Spaces and Embeddings. Berlin Heidelberg New York: Springer, 2004

[11] Feichtinger H. Modulation spaces on locally compact Abelian groups. Technical report, University of Vienna, 1983

[12] Feichtinger H, G\"{u}rkanl\i\ A T. On a family of weighted convolution algebras\textit{. }Internat J Math and Math Sci, 1990, 13(3): 517--526

[13] Fischer R H, G\"{u}rkanl\i\ A T, Liu T S. On a family of weighted spaces.\textit{\ }Math Slovaca, 1996, 46(1): 71--82

[14] Gaudry G I. Multipliers of weighted Lebesgue and measure spaces. Proc London Math Soc, 1969, 19(3): 327--340

[15] G\"{u}rkanl\i\ A T. Multipliers of some Banach ideals and Wiener-Ditkin sets\textit{. }Math Slovaca, 2005, 55(2): 237--248

[16] G\"{u}rkanl\i\ A T. Time frequency analysis and multipliers of the spaces M( p, q) (R^d) and S( p, q) (R^d) . J Math Kyoto Univ, 2006, 46(3): 595--616

[17] G\"{u}rkanl\i\ A T. Compact embeddings of the spaces A_w_, ω^p(R^d). Taiwanese Journal of Mathematics, 2008, 12(7): 1757--1767

[18] Gr\"{o}chenig K. Foundation of Time-Frequency Analysis. Boston: Birkhäuser, 2001

[19] Hunt R A. On L( p, q) Spaces. Extrait de L'Enseignement Mathematique, 1966, 12(4): 249--276

[20] Liu T S, Van Rooij A. Sums and intersections of normed linear spaces. Mathematische Nachrichten, Band 42: 29--42

[21] O'Neil R. Convolution operators and L( p, q) spaces. Duke Math J, 1963, 30: 129--142

[22] Reiter H. Classical Harmonic Analysis and Locally Compact Groups. Oxford: Oxford University Press, 1968

[23] Rieffel M A. Induced Banach representation of Banach algebras and locally compact groups. J Funct Anal, 1967, (1): 443--491

[24] Rieffel M A. Multipliers and tensor products of L^p-spaces of locally compact groups. Studia Math, 1969, 33: 71--82

[25] Sand\i k\c{c}\i\ A, G\"{u}rkanl\i\ A T. The space Ω_m^p(R^d) and some properties. Ukrainian Mathematical Journal, 2006, 58(1): 139--145

[26] Treves F. Topological Vector Spaces, Distributions and Kernels. New York: Academic Press, 1967

[27] Wang H C. Homogeneous Banach Algebras. New York and Basel: Marcel Dekker Inc, 1977

[28] Yap L H Y. Some remarks on convolution operators and L(p, q) spaces. Duke Math J, 1969, 36: 647--658

[29] Yap L H Y. On two classes of subalgebras of L¹(G). Proc Japan Acad, 1972, 48: 315--319

GABOR ANALYSIS OF THE SPACES {\boldmath M( p, q, w) ({R^d) AND S( p, q, r, w, ω) (R^d)

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Ziyao LIU, Dashan FAN. CERTAIN OSCILLATING OPERATORS ON HERZ-TYPE HARDY SPACES [J]. Acta mathematica scientia,Series B, 2025, 45(2): 310-326.
[2]	Fatemeh Abtahi, Ali Rejali, Farshad Sayaf. THE $\rm BSE$ PROPERTY FOR SOME VECTOR-VALUED BANACH FUNCTION ALGEBRAS^* [J]. Acta mathematica scientia,Series B, 2024, 44(5): 1945-1954.
[3]	Shangquan BU, Gang CAI. THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES^∗ [J]. Acta mathematica scientia,Series B, 2023, 43(4): 1603-1617.
[4]	Wei WANG, Bin ZHAO. CHARACTERIZATION OF RESIDUATED LATTICES VIA MULTIPLIERS [J]. Acta mathematica scientia,Series B, 2022, 42(5): 1902-1920.
[5]	Qianjun HE, Xinfeng WU, Dunyan YAN. BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES [J]. Acta mathematica scientia,Series B, 2022, 42(3): 1191-1208.
[6]	Xiangyu ZHOU, Langfeng ZHU. L² EXTENSIONS WITH SINGULAR METRICS ON KÄHLER MANIFOLDS [J]. Acta mathematica scientia,Series B, 2021, 41(6): 2021-2038.
[7]	Yongjian LIU, Stanis law MIGORSKI, Van Thien NGUYEN, Shengda ZENG. EXISTENCE AND CONVERGENCE RESULTS FOR AN ELASTIC FRICTIONAL CONTACT PROBLEM WITH NONMONOTONE SUBDIFFERENTIAL BOUNDARY CONDITIONS [J]. Acta mathematica scientia,Series B, 2021, 41(4): 1151-1168.
[8]	Pablo GALINDO, Mikael LINDSTRÖM. THE BLOCH SPACE ON THE UNIT BALL OF A HILBERT SPACE: MAXIMALITY AND MULTIPLIERS [J]. Acta mathematica scientia,Series B, 2021, 41(3): 899-906.
[9]	Naiqi SONG, Heping LIU, Jiman ZHAO. BILINEAR SPECTRAL MULTIPLIERS ON HEISENBERG GROUPS [J]. Acta mathematica scientia,Series B, 2021, 41(3): 968-990.
[10]	Yong CHEN, Ya LIU, Chuntao QIN. REDUCIBILITY FOR A CLASS OF ANALYTIC MULTIPLIERS ON SOBOLEV DISK ALGEBRA [J]. Acta mathematica scientia,Series B, 2021, 41(2): 361-370.
[11]	Adam NOWAK, Krzysztof STEMPAK. POTENTIAL OPERATORS AND LAPLACE TYPE MULTIPLIERS ASSOCIATED WITH THE TWISTED LAPLACIAN [J]. Acta mathematica scientia,Series B, 2017, 37(1): 280-292.
[12]	Ru PENG, You WU, Fangwen DENG. CARLESON MEASURES, RIEMANN-STIELTJES OPERATORS AND MULTIPLIERS ON F(p, q, s) SPACES IN THE UNIT BALL OF Cⁿ [J]. Acta mathematica scientia,Series B, 2016, 36(3): 635-654.
[13]	LIN Hai-Bo, YANG Da-Chun. POINTWISE MULTIPLIERS FOR LOCALIZED MORREY-CAMPANATO SPACES ON RD-SPACES [J]. Acta mathematica scientia,Series B, 2014, 34(6): 1677-1694.
[14]	ZHAO Ji-Hong, LIU Qiao. LOGARITHMICALLY IMPROVED REGULARITY CRITERION FOR THE 3D GENERALIZED MAGNETO-HYDRODYNAMIC EQUATIONS [J]. Acta mathematica scientia,Series B, 2014, 34(2): 568-574.
[15]	R. M. EL-ASHWAH. SOME CONVOLUTION AND INCLUSION PROPERTIES FOR SUBCLASSES OF MEROMORPHIC p-VALENT FUNCTIONS INVOLVING INTEGRAL OPERATOR [J]. Acta mathematica scientia,Series B, 2013, 33(6): 1749-1758.