 |
Orthogonal transformations with applications
|
Help 
Print 
|
Here you can view and search the projects funded by NKFI since 2004
Back »
|
| |
Details of project |
|
|
| Identifier |
115804 |
| Type |
K |
| Principal investigator |
Weisz, Ferenc |
| Title in Hungarian |
Ortogonális transzformációk és alkalmazásaik |
| Title in English |
Orthogonal transformations with applications |
| Keywords in Hungarian |
Gábor-tarnszformált, wavelet-transzformált, Fourier-transzformált, Fourier-sorok, Walsh-sorok, ortogonális polinomok, jelfeldolgozás, EKG |
| Keywords in English |
Gábor tarnsform, wavelet transform, Fourier transform, Fourier series, Walsh series, orthogonal polynomilas, signal processing, ECG |
| Discipline |
| Mathematics (Council of Physical Sciences) | 100 % | | Ortelius classification: Fourier analysis |
|
| Panel |
Mathematics and Computing Science |
| Department or equivalent |
Department of Numerical Analysis (Eötvös Loránd University) |
| Participants |
Fridli, Sándor
Kovács, Péter
Lócsi, Levente
Schipp, Ferenc
Szarvas, Kristóf
Szili, László
Vértesi, Péter
|
| Starting date |
2015-09-01 |
| Closing date |
2020-08-31 |
| Funding (in million HUF) |
21.040 |
| FTE (full time equivalent) |
16.10 |
| state |
closed project |
Summary in Hungarian A kutatás összefoglalója, célkitűzései szakemberek számára
Itt írja le a kutatás fő célkitűzéseit a témában jártas szakember számára.
A konvergencia-problémák és az összegzési eljárások a Fourier-analízis egy fontos ágát alkotják. Ismeretes, hogy ha egy Fourier-sor részletösszegei divergensek, egy alkalmas összegzési eljárással kaphatunk konvergenciát. Hasonló igaz az inverz Fourier-transzformáltra is.
A Gábor-transzformált az elmúlt 20 évben került az érdeklődés középpontjába. A magyar Gábor Dénes ötletén alapulva Janssen, Daubechies, Feichtinger, Benedetto és Gröchenig munkássága nyomán vált az idő-frekvencia analízis a harmonikus analízis önálló, gyorsan fejlődő ágává. Hasonlóan a wavelet-analízissel is 20-30 évvel ezelőtt kezdtek el foglalkozni. A Gábor- és wavelet-analízisnek számos gyakorlati alkalmazása van, pl. a kép- és jelfeldolgozásban, illetve a kommunikációelméletben (mobil telefonok).
A magyar approximációs iskola számos képviselője foglalkozott klasszikus interpolációs problémákkal és Fourier-sorokkal kapcsolatos kérdésekkel, és ért el alapvető tételeket.
Az ortogonális racionális rendszereket kiválóan lehet alkalmazni a biológiai jelek feldolgozása területén, az irányításelméletben és az automatizálásban.
Ezen pályázat keretén belül különböző Fourier-sorok és Fourier-transzformáltak konvergencia tulajdonságait és összegzéseit szeretnénk vizsgálni, valamint hasonló problémákat a Gábor- és wavelet-transzformáltra is. Ezzel kapcsolatban vizsgálnánk egy új Hardy-teret, az ún. lokális Hardy-teret. Továbbá súlyozott approximációs és interpolációs problémákkal is foglalkoznánk, illetve az alkalmazások területén is érnénk el új eredményeket.
Több dolgozat publikálását tervezzük referált nemzetközi szakfolyóiratokban.
Mi a kutatás alapkérdése?
Ebben a részben írja le röviden, hogy mi a kutatás segítségével megválaszolni kívánt probléma, mi a kutatás kiinduló hipotézise, milyen kérdéseket válaszolnak meg a kísérletek.
Különböző ortogonális, vagy ortogonális jellegű transzformációt és Fourier-sort, valamint ezek alkalmazásait szeretnénk vizsgálni. Többek között Fourier- és Walsh-sorokkal, wavelet- és Gábor-sorokkal, ortogonális polinomok és racionális törtfüggvények szerinti sorfejtéssel, súlyozott approximációs és interpolációs problémákkal, Fourier-, wavelet- és Gábor-transzformálttal, valamint az alkalmazáson belül jel- és képfeldolgozással illetve EKG jelek feldolgozásával foglalkoznánk.
Mi a kutatás jelentősége?
Röviden írja le, milyen új perspektívát nyitnak az alapkutatásban az elért eredmények, milyen társadalmi hasznosíthatóságnak teremtik meg a tudományos alapját. Mutassa be, hogy a megpályázott kutatási területen lévő hazai és a nemzetközi versenytársaihoz képest melyek az egyediségei és erősségei a pályázatának!
Ezek a témakörök érdekesek, modernek, számos világhírű kutató foglalkozik velük. Több nyitott probléma megoldását tervezzük, amelyek várhatóan nemzetközi szinten is figyelemre méltóak lesznek.
A kutatás összefoglalója, célkitűzései laikusok számára
Ebben a fejezetben írja le a kutatás fő célkitűzéseit alapműveltséggel rendelkező laikusok számára. Ez az összefoglaló a döntéshozók, a média, illetve az érdeklődők tájékoztatása szempontjából különösen fontos az NKFI Hivatal számára.
Magyar és külföldi matematikusok munkásságát folytatva foglalkoznánk Fourier-sorokkal, ortogonális polinomok és racionális törtfüggvények szerinti sorfejtéssel, súlyozott approximációs és interpolációs problémákkal, Fourier-, wavelet- és Gábor-transzformálttal. Továbbá ezek alkalmazásain, pl. jel- és képfeldolgozás, illetve az EKG jelek feldolgozása, is dolgoznánk és érnénk el új eredményeket. Az előző témakörökben több nyitott probléma megoldását tervezzük.
| Summary Summary of the research and its aims for experts
Describe the major aims of the research for experts.
Convergence problems and summability methods constitute a central part of classical Fourier analysis. They have arisen from the observation that the partial sums of the Fourier series of a function may show a bad behaviour, which may be very much ameliorated by the use of summability methods. Fourier transforms have similar properties.
Gábor transforms has got into the focus of interest in the last 20 years. Based on the idea of the Hungarian Dénes Gábor, in the wake of the works of Janssen, Daubechies, Feichtinger, Benedetto and Gröchenig, time-frequency analysis has been established itself as a separate, fertile branch of harmonic analysis. Similarly, wavelet analysis has been investigated since 20-30 years. Gábor and wavelet analysis have a lot of links to real world applications, e.g. in image- and signal processing, in communication theory (cell phones).
Several representatives of the Hungarian approximation school have investigated classical interpolation problems and have reached basic theorems.
The orthogonal rational systems can perfectly be used in the field of control theory, automation and biological signal processing.
We would like to investigate convergence problems and summations of different Fourier series and Fourier transforms and similar problems for Gábor and wavelet transforms. Moreover, in connection with this, we would like to investigate a new Hardy space, the so called local Hardy space. Moreover, we consider weighted approximation and interpolation problems, and we reach new results in the fields of applications.
We plan to publish several papers in international journals.
What is the major research question?
Describe here briefly the problem to be solved by the research, the starting hypothesis, and the questions addressed by the experiments.
We would like to investigate orthogonal or orthogonal type transformations and Fourier series and their applications. Amongst others, we consider Fourier and Walsh series, wavelet and Gábor series, series with respect to orthogonal polynomials and rational systems, weighted approximation and interpolation problems, Fourier, wavelet and Gábor transforms, and signal and image processing and processing of EKG signals.
What is the significance of the research?
Describe the new perspectives opened by the results achieved, including the scientific basics of potential societal applications. Please describe the unique strengths of your proposal in comparison to your domestic and international competitors in the given field.
These topics are interesting and current, many famous researchers deal with them. We plan to solve several open problems in these topics, which are expected to be remarkable internationally.
Summary and aims of the research for the public
Describe here the major aims of the research for an audience with average background information. This summary is especially important for NRDI Office in order to inform decision-makers, media, and others.
Following the work of Hungarian and international mathematicians we investigate Fourier series, series with respect to orthogonal polynomials and rational systems, weighted approximation and interpolation problems, Fourier, wavelet and Gábor transforms. Moreover, in the field of applications, we work on signal and image processing and processing of ECG signals. We plan to solve several open problems in these areas.
|
|
|
|
|
|
|
|
| |
List of publications |
|
|
| L. Szili and P. Vértesi: On barycentric interpolation. III. (On convergent type processes), Annales Univ. Sci. Budapest., Sect. Comp. 51, 2020 | | Paul Y., Fridli S.: Epileptic Seizure Detection Using Piecewise Linear Reduction, Computer Aided Systems Theory – EUROCAST 2019. EUROCAST 2019. Lecture Notes in Computer Science, vol 12014. Springer, 2020 | | Weisz Ferenc: $\ell_1$-summability and Lebesgue points of $d$-dimensional Fourier transforms, Adv. Oper. Theory (to appear), 2018 | | Weisz Ferenc: $\ell_1$-summability and Lebesgue points of $d$-dimensional Fourier transforms, Adv. Oper. Theory 4, 284-304, 2019 | | Weisz Ferenc: Variable Hardy and Hardy-Lorentz spaces and applications in Fourier analysis, Stud. Univ. Babes-Bolyai Math. 63, 381-393, 2018 | | J. Liu and F. Weisz and D. Yang and W. Yuan: Variable anisotropic Hardy spaces and their applications, Taiwanese J. Math. 22, 1173-1216, 2018 | | Weisz Ferenc: Summability of Fourier transforms in variable Hardy and Hardy- Lorentz spaces, Jaen J. Approx. 10, 101-131, 2018 | | G. Xie and F. Weisz and D. Yang and Y. Jiao: New martingale inequalities and applications to Fourier analysis, Nonlinear Analysis 182, 143-192, 2019 | | J. Liu and F. Weisz and D. Yang and W. Yuan: Littlewood-Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications, J. Fourier Anal. Appl. 25, 874-922, 2019 | | K. Szarvas and F. Weisz: The boundedness of the Cesaro- and Riesz means in variable dyadic Hardy spaces, Banach J. Math. Anal. 13, 675-696, 2019 | | Zoltán Fazekas, Levente Lócsi, Alexandros Soumelidis, Ferenc Schipp, Zsolt Németh: Rational Zernike functions capture the rotations of the eye-ball, Proceedings of the 20th European Conference on Mathematics for Industry (ECMI 2018), to appear., 2019 | | Levente Lócsi: Introducing p-eigenvectors, exact solutions fot some simple matrices, Annales Univ. Sci. Budapest., Sect. Comp. 49, to appear., 2019 | | Kovács, P., Fekete, A. M.: Nonlinear least squares spline fitting with variable knots, Applied Mathematics and Computation. 354, 490–501, 2019 | | Kovács, P., Fridli, S., Schipp, F: Generalized Rational Variable Projection with Application in ECG Compression, IEEE Transactions on Signal Processing. to appear, 2019 | | Gergő Bognár, Sándor Fridli, Péter Kovács, and Ferenc Schipp: Adaptive Rational Transformations in Biomedical Signal Processing, ECMI2018_Proceeding. 1-8, 2018 | | Gergő Bognár, Sándor Fridli: On the Pole Stability of Rational Approximation, Annales Univ. Sci. Budapest., Sect. Comp. 49, 1-17., 2019 | | Yash Paul, Sándor Fridli: Epileptic seizure detection using piecewise linear reduction, Lecture Notes in Computer Science, Proc. Eurocast, to appear, 2019 | | Levente Lócsi: Introducing p-eigenvectors, exact solutions for some simple matrices, Annales Univ. Sci. Budapest., Sect. Comp. 49, 325–345., 2019 | | Levente Lócsi, Zsolt Németh: On the construction of p-eigenvectors, Annales Univ. Sci. Budapest., Sect. Comp. 50, 2020 | | Jiao Yong, Weisz Ferenc, Wu Lian, Zhou Dejian: Variable martingale Hardy spaces and their applications in Fourier analysis, DISSERTATIONES MATHEMATICAE 550: pp. 1-67., 2020 | | Németh Zsolt, Schipp Ferenc, Weisz Ferenc: Hyperbolic Transformations of Zernike Functions and Coefficients, In: Quesada-Arencibia, Alexis; Pichler, Franz; Moreno-Díaz, Roberto (szerk.) Computer Aided Systems Theory – EUROCAST 2019, Springer International Publishing (2020) pp. 380-387., 2020 | | Szarvas K, Weisz F: Applications of mixed martingale Hardy spaces in Fourier analysis, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 492: (1) 124403, 2020 | | Weisz F: Boundedness of dyadic maximal operators on variable Lebesgue spaces, ADVANCES IN OPERATOR THEORY 5: (4) pp. 1588-1598., 2020 | | Weisz F.: Cesàro and Riesz summability with varying parameters of multi-dimensional Walsh–Fourier series, ACTA MATHEMATICA HUNGARICA 161: (1) pp. 292-312., 2020 | | Weisz F.: Summability of Fourier series in periodic Hardy spaces with variable exponent, ACTA MATHEMATICA HUNGARICA, 2020 | | Weisz F: Weighted Hardy spaces and the Fejér means of Walsh-Fourier series, ANNALES UNIVERSITATIS SCIENTIARUM BUDAPESTINENSIS DE ROLANDO EOTVOS NOMINATAE SECTIO COMPUTATORICA 49: pp. 411-424., 2019 | | Weisz f: Continuation of the laudation to Professor Ferenc Schipp on his 80-th birthday, ANNALES UNIVERSITATIS SCIENTIARUM BUDAPESTINENSIS DE ROLANDO EOTVOS NOMINATAE SECTIO COMPUTATORICA 49: pp. 9-10., 2019 | | Weisz Ferenc: Hardy spaces with variable exponents and applications in Fourier analysis, NONLINEAR STUDIES 26: (4) pp. 1015-1026., 2019 | | Samiee, K., Kovács, P., Gabbouj, M.: Epileptic seizure detection in long-term EEG records using sparse rational decomposition and local Gabor binary patterns feature extraction, KnowledgeBased Systems, 118: 228–240,, 2017 | | Kovács, P.: Rational Variable Projection Methods in Signal Processing,, Lecture at the 16th International Conference on Computer Aided Systems Theory (EUROCAST), Las Palmas de Gran Canaria, Spain,, 2017 | | F. Weisz: Multi-parameter martingales., Lecture at the Central South University, Changsha, China., 2016 | | F. Weisz: Multi-parameter martingales and applications., Lecture at the Wuhan University, Wuhan, China., 2016 | | F. Weisz: Multi-parameter martingales and some applications in Fourier analysis, Lecture at the Multi-parameter martingales and some applications in Fourier analysis. Normal University, Beijing, China, 2016 | | F. Weisz: Kétdimenziós Fourier-transzformáltak Lebesgue-pontjai., Lecture at the Mathematical Institute, University of Pécs, Hungary, 2016 | | F. Weisz: The inverse of the continuous wavelet transform., Lecture at the 16th International Conference on Computer Aided Systems Theory (EUROCAST), Las Palmas de Gran Canaria, Spain,, 2017 | | F. Weisz: Fourier-transzformáltak és Lebesgue-pontok., Lecture at the Rényi Institute of Mathematics, Budapest, Hungary, 2017 | | F. Weisz: Rectangular summability and Lebesgue points of higher dimensional Fourier transforms., Lecture at the VIII Jaen Conference on Approximation Theory, July 2 - 7, Ubeda, Jaen, Spain, 2017 | | F. Weisz: Convergence of rectangular summability and Lebesgue points of higher dimen- sional Fourier transforms., Lecture at the 6th Workshop on Fourier Analysis and Related Fields, Pécs, August 24-31, 2017 | | Bognár Gergő, Fridli Sándor,: Heartbeat Classication of ECG Signals Using Rational Function Systems, Lecture at the 16th International Conference on Computer Aided Systems Theory (EUROCAST), Las Palmas de Gran Canaria, Spain,, 2017 | | Fridli Sándor: Approximation problems in ECG signal processing, Lecture at the VIII Jaen Conference on Approximation Theory, July 2 - 7, Ubeda, Jaen, Spain,, 2017 | | Levente Lócsi, Ferenc Schipp: Rational Zernike functions, Annales Univ. Sci. Budapest., Sect. Comp. 46, 177–190, 2017 | | Zsolt Németh, Ferenc Schipp: Dicrete orthogonality of Zernike-Blaschke functions, SIAM Journal on Numerical Analysis (to appear), 2018 | | G. Mastroianni, I. Notarangelo, L. Szili and P. Vértesi: Some new results on orthogonal polynomials for Laguerre type exponential weights, Acta Math. Hungar., 155, 466–478, 2018 | | G. Mastroianni, I. Notarangelo, L. Szili and P. Vértesi: A note on Hermite–Fejér interpolation at Laguerre zeros, Calcolo 55:39, 2018 | | Szarvas Kristóf: Változó indexű Lebesgue-terek és alkalmazásuk a Fourier-analízisben, ELTE, PhD disszertáció, 2017 | | G. Bognár, S. Fridli: Heartbeat Classification of ECG Signals Using Rational Function Systems, R. Moreno Díaz et al. (eds.) Proc. 16th EUROCAST 2017, Part II, Lecture Notes in Computer Science 10672, Springer, 2018, 187-195, 2018 | | G. Bognár, F. Schipp: Geometric interpretation of QRS complexes in ECG signals by rational functions, Ann. Univ. Sci. Budapest., Sect. Comp., 47 (2018), 155-166, 2018 | | Weisz Ferenc: The inverse of the continuous wavelet transform, In: Roberto Moreno-Díaz, Franz Pichler, Alexis Quesada-Arencibia (szerk.) (szerk.) Computer Aided Systems Theory - EUROCAST 2017. Berlin: Springer, 2018. pp. 246-253. (Lecture Notes in Computer Science; 10672.), 2018 | | Weisz Ferenc: Summability in mixed-norm Hardy spaces, ANN UNIV SCI BP R EÖTVÖS NOM SECT COMPUT 48: pp. 233-246., 2018 | | Weisz Ferenc: Marcinkiewicz summability of Fourier series, Lebesgue points and strong summability, ACTA MATH HUNG 153: pp. 356-381., 2017 | | Weisz Ferenc: Lebesgue points and convergence over cone-like sets, JAEN J APPROX 9: pp. 65-83., 2017 | | Weisz Ferenc: Convergence and Summability of Fourier Transforms and Hardy Spaces, Basel; Boston; Berlin: Springer (Basel), 446 p. (Applied and Numerical Harmonic Analysis), 2017 | | Weisz Ferenc: Dual spaces of multi-parameter martingale Hardy spaces, Q J MATH 67: pp. 137-145., 2016 | | Weisz Ferenc: Multi-dimensional summability theory and continuous wavelet transform, In: H Dutta, B E Rhoades (szerk.) (szerk.) Current Topics in Summability Theory and Applications. Singapore: Springer, 2016. pp. 241-311., 2016 | | Weisz F: Convergence of the inverse continuous wavelet transform in Wiener amalgam spaces, ANALYSIS (MUNICH) 35: (1) pp. 33-46., 2015 | | Weisz Ferenc: Lebesgue points of two-dimensional Fourier transforms and strong summability, J FOURIER ANAL APPL 21: pp. 885-914., 2015 | | Weisz Ferenc: Strong summability of Fourier transforms at Lebesgue points and Wiener amalgam spaces, J FUNCT SPACE APPL 2015: Paper 420750. 10 p. , 2015 | | Weisz Ferenc: Lebesgue points of double Fourier series and strong summability, J MATH ANAL APPL 432: pp. 441-462., 2015 | | Weisz Ferenc: Restricted convergence of the inverse continuous wavelet transform, ACTA SCI MATH (SZEGED) 81: pp. 535-547., 2015 | | Weisz Ferenc: Inverse continuous wavelet transform in Pringsheim’s sense on Wiener amalgam spaces, ACTA MATH HUNG 145: pp. 392-415., 2015 | | Szarvas Kristóf, Weisz Ferenc: Almost everywhere and norm convergence of the inverse continuous wavelet transform in Pringsheim's sense, ACTA SCI MATH (SZEGED) 82: 125-146, 2016 | | Weisz Ferenc: Dual spaces of multi-parameter martingale Hardy spaces, Q J MATH 67: 137-145, 2016 | | Weisz Ferenc: Multi-dimensional summability theory and continuous wavelet transform, In: H Dutta, B E Rhoades (szerk.) (szerk.) Current Topics in Summability Theory and Applications. Singapore: Springer, 2016. pp. 241-311., 2016 | | Weisz Ferenc: Strong summability of Fourier transforms at Lebesgue points and Wiener amalgam spaces, J FUNCT SPACE APPL 2015: , 2015 | | Weisz Ferenc: Restricted convergence of the inverse continuous wavelet transform, ACTA SCI MATH (SZEGED) 81: 535-547, 2015 | | Szili László: Egyenletesen konvergens polinomapproximációs eljárások, ELTE, habilitáció, 2016 | | Weisz Ferenc: Two-dimensional Fourier transforms and Lebesgue points, Lecture at the 11th Joint Conference on Mathematics and Computer Science, Eger, Hungary, May 20–22, 2016, 2016 | | Weisz Ferenc: Lebesgue points of two-dimensional Fourier transforms, Lecture at the Time-Frequency Analysis and Related Topics, June 6 - 10, Strobl, Austria, 2016, 2016 | | Weisz Ferenc: Generalizations of Lebesgue points for two-dimensional functions, Lecture at the VII Jaen Con- ference on Approximation Theory, July 3 - 8, ´ Ubeda, Ja´ en, Spain, 2016, 2016 | | Fridli Sándor: On the integrability of dyadic maximal Walsh series, Acta Sci. Math. (Szeged) 81, 561–574, 2015 | | Fridli Sándor: Sufficient conditions for the integrability of dyadic maximal Walsh series, Lecture at the 11th Joint Conf. on Math. and Comp. Sci., May 20-22, 2016, 2016 | | Bognár,Gergő, Gilián Zoltán, Fridli Sándor: Orthogonal transformations in ECG processing, Lecture at the BJMT Applied Mathematical Conference, Győr, June 1-3, 2016, 2016 | | Vértesi Péter: The Bernstein Erdos Conjecture for certain Haar (Tchebycheff) Systems, Lecture at the VII Jaen Con- ference on Approximation Theory, July 3 - 8, ´ Ubeda, Ja´ en, Spain, 2016, 2016 | | Szarvas Kristóf: Convergence of integral operators in variable Lebesgue spaces, Lecture at the 11th Joint Conference on Mathematics and Computer Science, Eger, Hungary, May 20–22, 2016, 2016 | | Szarvas Kristóf: Variable Lebesgue spaces and integral operators, Conference poster at the Time-Frequency Analy- sis and Related Topics, June 6 - 10, Strobl, Austria, 2016, 2016 | | Kovács Péter: Optimization problems in signal compression, Lecture at the 11th Joint Conference on Mathematics and Computer Science, Eger, Hungary, May 20–22, 2016, 2016 | | Kovács Péter, Schipp Ferenc: Model reduction and its applications, Lecture at the BJMT Applied Mathematical Conference, Győr, June 1-3, 2016, 2016 | | Szarvas Kristóf: Variable Lebesgue spaces and continuous wavelet transforms, Acta Mathematica Academiae Paedagogicae Ny ́ıregyh ́aziensis, 32:313–325, 2016 | | F Weisz: Lebesgue points and restricted convergence of Fourier transforms and Fourier series, ANAL APPL 15: 107-121, 2017 | | K Szarvas, F Weisz: Convergence of multi-dimensional integral operators and applications, PERIOD MATH HUNG 74: 40-66, 2017 | | Weisz F: Triangular summability and lebesgue points of 2-dimensional fourier transforms, BANACH J MATH ANAL 11: (1) 223-238, 2017 | | Weisz Ferenc: Walsh-Lebesgue points and restricted convergence of multi-dimensional Walsh-Fourier series, STUD SCI MATH HUNG 54: (1) 97-118, 2017 | | Y Jiao, D Zhou, F Weisz, Z Hao: Corrigendum: Fractional integral on martingale Hardy spaces with variable exponents, FRACT CALC APPL ANAL 20: 1051-1052, 2017 | | F Weisz: Some generalizations of Lebesgue's theorem for two-dimensional functions, ANN UNIV SCI BP R EÖTVÖS NOM SECT COMPUT 45: 277-290, 2016 | | K Szarvas, F Weisz: Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces, CZECH MATH J 66: 1079-1101, 2016 | | Weisz F: Multi-dimensional Fourier Transforms, Lebesgue Points and Strong Summability, MEDITERR J MATH 13: 3557-3587, 2016 | | Weisz Ferenc: Convergence of trigonometric and Walsh-Fourier series, ACTA MATH ACAD PAEDAG NYÍREGYH 32: 277-301, 2016 | | Z Hao, Y Jiao, F Weisz, D Zhou: Atomic subspaces of $L_1$-martingale spaces, ACTA MATH HUNG 150: 423-440, 2016 |
|
|
|
|
|
|
|
|
|
