Table Of ContentIndustrial and Applied Mathematics
Yu. A. Farkov
Pammy Manchanda
Abul Hasan Siddiqi
Construction
of Wavelets
Through Walsh
Functions
Industrial and Applied Mathematics
Editor-in-chief
Abul Hasan Siddiqi, Sharda University, Greater Noida, India
Editorial Board
Zafer Aslan, Istanbul Aydin University, Istanbul, Turkey
Martin Brokate, Technical University, Munich, Germany
N.K. Gupta, Indian Institute of Technology Delhi, New Delhi, India
Akhtar A. Khan, Rochester Institute of Technology, Rochester, USA
René Pierre Lozi, University of Nice Sophia-Antipolis, Nice, France
Pammy Manchanda, Guru Nanak Dev University, Amritsar, India
Zuhair Nashed, University of Central Florida, Orlando, USA
Govindan Rangarajan, Indian Institute of Science, Bengaluru, India
Katepalli R. Sreenivasan, NYU Tandon School of Engineering, Brooklyn, USA
TheIndustrialandAppliedMathematicsseriespublisheshigh-qualityresearch-level
monographs, lecture notes and contributed volumes focusing on areas where
mathematics is used in a fundamental way, such as industrial mathematics,
bio-mathematics, financial mathematics, applied statistics, operations research and
computer science.
More information about this series at http://www.springer.com/series/13577
Yu. A. Farkov Pammy Manchanda
(cid:129) (cid:129)
Abul Hasan Siddiqi
Construction of Wavelets
Through Walsh Functions
123
Yu.A.Farkov Pammy Manchanda
Department ofApplied Information Department ofMathematics
Technology,School ofPublic Policy Guru NanakDev University
Russian Presidential Academy of National Amritsar, Punjab,India
Economy andPublic Administration
(RANEPA)
Moscow,Russia
AbulHasan Siddiqi
Schoolof BasicSciences andResearch
andCentrefor Advanced Researchin
AppliedMathematics andPhysics
Sharda University
Greater Noida, Uttar Pradesh, India
ISSN 2364-6837 ISSN 2364-6845 (electronic)
Industrial andAppliedMathematics
ISBN978-981-13-6369-6 ISBN978-981-13-6370-2 (eBook)
https://doi.org/10.1007/978-981-13-6370-2
LibraryofCongressControlNumber:2019930359
©SpringerNatureSingaporePteLtd.2019
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore
Preface
ThebookisessentiallydevotedtotheconstructionofwaveletsbyWalshfunctions
andtheirtransforms.Thisembodiesthecontemporaryresearchonwaveletsdefined
onpositivehalfline,waveletsonVilenkingroups,Haar–Vilenkinwavelets,studied
bytheauthorsofthisbookandtheircollaborators.Moreprecisely,thisbookdeals
with wavelet theory on Vilenkin groups, particularly on the Cantor group, which
canalsobeinterpretedaswavelettheoryonthepositivehalfline.Thistheoryisin
theframeworkofWalshanalysis(dyadicanalysisinthecaseoftheCantorgroup),
which has been actively studied for the last four decades. The foundation of these
developments is discussed in books authored by Schipp, Wade, and Simon;
Golubov, Efimov, and Skvortsov; Maqusi; Siddiqi; and Beauchamp. Recent
research work not included in these books constitute important ingredients of the
present volume. These themes are an approximation of special cases of matrix
transform of Walsh–Fourier series in dyadic homogeneous Banach spaces and
Hardy spaces, Haar–Vilenkin wavelets and their properties, orthogonal and
biorthogonal wavelets and frames, nonuniform multiresolution analysis on the
positive half line, and vector-valued wavelets on the positive half line.
The book contains nine chapters and one appendix. Chapter 1 is devoted to the
basic results of Walsh analysis and wavelets on the real line. An exhaustive
background for Chaps. 1–3 can be found in Schipp et al. [1]. It has been observed
that Walsh functions were used for the transposition of conductors in open wire
linesasearlyasthelate1800s,andacompletesystemofWalshfunctionsseemsto
have been found around 1900. Rademacher functions were introduced to the
mathematical world in 1922 by Rademacher. His widow told an audience of
mathematicians that Rademacher knew the concept of Walsh functions, but he
abandoned the idea as advised by his academic mentors. The concept of wavelets
was introduced by the French Geophysicist, Jean Morlet, in 1982. Its literal
meaning is “small wave”. Alex Grossmann, French Theoretical Physicist, studied
inverseformulaforthewavelettransforms.In1984,thecollaborationofMorletand
Grossmann yielded a detailed mathematical study of the continuous wavelet
transforms and their various applications, of course, without realizing that similar
results had already been obtained 20–50 years earlier by Calderon, Littlewood,
v
vi Preface
Paley, and Franklin. In 1985, Yves Meyer, recipient of the Abel Prize of 2017,
accidentally found the then existing literature of wavelets. With the knowledge
oftheCalderon–ZygmundoperatorsandtheLittlewood–Paleytheory,hewasable
to lay the mathematical foundation of wavelet theory. The first major achievement
is due to Daubechies, Grossman, and Meyer for constructing a painless
non-orthogonal expansion in 1986.
Chapter2dealswiththepropertiesofWalsh–Fouriercoefficients,theconvergence
of Walsh–Fourier series, and an approximation of functions belonging to different
spacesbyspecialclassesofmatrixtransformofWalsh–Fourierseries.Thehistorical
note onmostofthe themesofthischapter iswelldocumentedinthe monograph of
Schipp, Wade, and Simon. Here, we discuss approximation by matrix transform of
Walsh–Fourierseries.AftertheresultofYanoforarithmeticandCesáromeans,there
were no significant results on this theme till 1992. During the visit of Schipp in
January1992andMoriczin1994toAligarhMuslimUniversity,India,theproblem
of generalizing the results of Yano, Jastrebova, and Skvorcov on the rate of
approximationbyCesáromeanswasgiventoppriority.MoriczandSiddiqipublished
an interesting paper generalizing all earlier results (seeTheorems 2.24–2.26).Fridli,
Manchanda, and Siddiqi studied approximation by Norlund means in dyadic
homogeneous Banach spaces and Hardy spaces and extended the earlier results of
Moricz and Siddiqi (see Theorems 2.27–2.29).
Chapter3presentsHaarsystemsanditsgeneralization,Haar–Fourierseries,the
relation between Haar and Walsh functions, and applications of Haar wavelets to
initial and boundary value problems. Alfred Haar was a Ph.D. student of David
Hilbert at Göttingen University. Haar function is the simplest example of mother
wavelet. Haar wavelets and its generalizations were studied in detail by Devore
et al. [2] and Dubeau et al. [3]. In a recent monograph, Lepik and Hein [4] have
studied the applications of Haar wavelets to initial and boundary problems.
Chapter 4 is devoted to the construction of dyadic wavelets and frames through
Walsh functions. This chapter is based on the research work of Farkov [5–12].
In Sect. 4.1 of this chapter, preliminary results needed for the subsequent chapters
are explained. Section 4.2 of this chapter presents the orthogonal wavelets and
multiresolution(MRA).Section4.3ofthischapterisdevotedtoorthogonalwavelets
with compact support on the positive half line. Section 4.4 introduces the estimates
of the smoothness of the scaling functions. Section 4.5 presents the approximation
properties of dyadic wavelets.
Chapter 5 is devoted to orthogonal and periodic wavelets on Vilenkin groups.
ThischapterisessentiallybasedontheresearchworkofFarkovandhiscollaborators.
As we know, Walsh functions are identified with characters of the Cantor dyadic
group.ThisfactwasrecognizedbyGelfandasbackasin1940.Vilenkingroupwas
introduced in 1947. In a series of papers published after 2005, Farkov and his
research collaborators have studied orthogonal and periodic wavelets [6, 13–19].
The results of Chap. 4 and 5 may be extended for results given in [20]. Section5.1
deals with multiresolution analysis on Vilenkin groups. Section 5.2 presents com-
pactlysupportedorthogonalp-wavelets.Section5.3introducestheperiodicwavelets
Preface vii
onVilenkingroups.Section5.4introducesperiodicwaveletsrelatedtotheVilenkin–
Christenson transform. Section 5.5 presents applications to the coding of fractal
functions.
Chapter6isdevotedtoHaar–Vilenkinwaveletsandtheirproperties.Haar–Vilenkin
system was introduced around 1947. The concept of Haar–Vilenkin wavelets was
introduced and studied by Manchanda, Meenakshi, and Siddiqi [21, 22].
Chapter 7 is devoted to biorthogonal wavelets and frames on Vilenkin groups.
This chapter is based mainly on the research papers of Farkov published between
2008 and 2012; [14, 23, 24]. Section 7.1 introduces biorthogonal wavelets on the
positive half line. Section 7.2 presents biorthogonal wavelets on Vilenkin groups.
ParsevalframesonVilenkingroupsarealsopresentedinthischapter.Thischapter
is concluded by applications of biorthogonal dyadic wavelets.
Chapter8dealswithwaveletsassociatedwithnonuniformmultiresolutionanalysis
on the positive half line. This chapter is based on the research of Manchanda,
Meenakshi, Siddiqi, and Vikram published during 2012 and 2017 [25–28].
Chapter 9 presents the concept of orthogonal vector-valued wavelets on the
positive half line.Section 9.4 presentsvector-valued wavelets and wavelet packets
associated with the nonuniform multiresolution analysis. The study of the concept
oforthogonalvector-valuedwaveletsonthereallinewasinitiatedbyXiaandSuter
(1996).ThisconceptonthepositivehalflinewasinitiatedbyFarkovin2005[16].
The chapter contains the results obtained by Manchanda, Meenakshi, Siddiqi, and
Vikram in the recent past [29–32].
Theappendixcontainstopicsonthebasicresultsoffunctionalanalysis,Vilenkin
systems, Vilenkin–Pontryagin class of functions, and real and Fourier analysis
including pointwise, uniform, and absolute convergence.
The book will serve as a reference material for research in Fourier analysis.
The book, however, could also be used for a special course for graduate and
advanced undergraduate students of mathematics and engineering. We take this
opportunitytoexpressourgratitudetoourresearchgroups.Wealsoliketoexpress
our sincere gratitude to the chancellor of Sharda University, P. K. Gupta, for
extending full support during the preparation of this book. We also take this
opportunity to thank Pooja and Mamta for meticulously preparing the final
manuscript. We take this opportunity to thank the World Scientific Publishing
Company for using results published in International Journal of Wavelets,
Multiresolution and Information Processing.
Moscow, Russia Yu. A. Farkov
Amritsar, India Pammy Manchanda
Greater Noida, India Abul Hasan Siddiqi
viii Preface
References
1. Schipp,F.,Wade,W.R.,&Simon,P.(1990).Walshseries.AdamHilger.
2. Pfander, G. E., & Benedetto J. J. (2000). Periodic wavelet transforms and periodicity
detection.SIAMJournalofAppliedMathematics,62(4),1329–1368.
3. Dubeau, F., Elmejdani, S., & Krantini, R. (2004). Nonuniform haar wavelets. Applied
MathematicsandComputation,159,675–693.
4. Lepik,U.,&Hein,H.(2014).Haarwaveletswithapplications.Berlin:Springer.
5. Farkov,Yu.A.,&ProtasovV.Yu.(2006).Dyadicwaveletsandrefinablefunctionsonahalf
line.MatematicheskiiSbornik,197(10),129–160.Englishtranslation,Sbornik:Mathematics,
197,1529–1558.
6. Farkov,Yu.A.(2005).Orthogonalwaveletswithcompactsupportonlocallycompactabelian
groups. Izv. Ross. Akad. Nauk Ser. Mat. 69(3), 193–220. English translation, Izvestia:
Mathematics,69(3),623–650.
7. Farkov, Yu. A. (2009). On wavelets related to Walsh series. Journal of Approximation
Theory,161,259–279.
8. Farkov, Yu. A. (2010). Wavelets and frames based on Walsh-Dirichlet type kernels.
CommunicationsinMathematicsandApplications,1,27–46.
9. Farkov, Yu. A., & Stroganov, S. A. (2011). The use of discrete dyadic wavelets in image
processing.RussianMathematics(Iz.Yuz),55(7),47–55.OriginalRussiantextpublishedin
Izvestiya.UchebnykhZavednic.Mathematica,7,57–66.
10. Farkov,Yu.A.(2012).PeriodicwaveletsinWalshanalysis.CommunicationinMathematics
andApplications,3(3),223–242.
11. Farkov, Yu. A., & Borisov, M. E. (2012). Periodic dyadic wavelets and coding offractal
functions.RussianMathematics(IzvestiyaVuz.Matematika),56(9),46–56.
12. Farkov,Yu.A.(2014).WaveletexpansionsontheCantorgroup.MathematicalNotes,96(6),
996–1007.
13. Farkov, Yu. A. (2007). Orthogonal wavelets on direct products of cyclic groups.
Matematicheskie Zametki, 82(6), 934–952. English translation, Mathematical Notes, 82(6),
843–859.
14. Farkov, Yu. A. (2008). Multiresolution analysis and wavelets on Vilenkin groups. Facta
Univers.(Nis)ser.:Elec.Engineering,21(3),309–325.
15. Farkov, Yu. A. (2009). Biorthogonal wavelets on Vilenkin groups. Tr. Mat. Inst. Steklova,
265(1), 110–124. English translation, Proceedings of the Steklov Institute of Mathematics,
265(1),101–114.
16. Farkov, Yu. A. (2005). Orthogonal p-wavelets on R+. In Proceedings of International
ConferenceonWaveletsandSpilnes,St.Petersburg,Russia,July3–8(p.426).St.Petersberg:
St.PetersbergUniversityPress.
17. Farkov, Yu. A. (2011). Discrete wavelets and the Vilenkin Chrestensen transform.
MathematicalNotes,89(6),871–884.
18. Farkov, Yu. A. (2011). Periodic wavelets on the p-adic Vilenkin group. P-Adic Numbers,
UltrametricAnalysis,andApplications,3(4),281–287.
19. Farkov,Yu.A.,&Rodionov,E.A.(2011).AlgorithmsforwaveletconstructiononVilenkin
groups.P-AdicNumbers,UltrametricAnalysis,andApplications,3(3),181–195.
20. Krivoshein,A.V.,Protasov,V.Yu.,&Skopina,M.A.(2016).Multivariatewaveletframes.
Berlin:Springer.
21. Manchanda, P., Meenakshi, & Siddiqi, A. H. (2008). Haar-Vilenkin wavelet. The Aligarh
BulletinofMathematics,27(1),59–73.
22. Manchanda, P., & Meenakshi. (2009). New classes of wavelets. In A. H. Siddiqi,
A.K.Gupta,&M.Brokate(Eds.),ProceedingsofConferenceonModelingofEngineering
andTechnologicalProblems,NewYork(vol.1146,pp.253–271).
Preface ix
23. Farkov, Yu. A., Maksimov A. Yu., & Stroganov, S. A. (2011). On biorthogonal wavelets
related to the Walsh functions. International Journal of Wavelets, Multiresolution and
InformationProcessing,9(3),485–499.
24. Farkov, Yu. A. (2012). Examples of frames on the Cantor dyadic group. Journal of
MathematicalSciences,187(1),22–34.
25. Manchanda, P., & Sharma, V. (2013). Wavelet packets with nonuniform multiresolution
analysisonpositivehalf-line.Asian-EuropeanJournalofMathematics,6(1),1350007.
26. Meenakshi,Manchanda, P.,& Siddiqi, A.H. (2012).Wavelets associatedwith nonuniform
multiresolution analysis on positive half line. International Journal of Wavelets,
MultiresolutionandInformationProcessing,10(2),1250018,1–27.
27. Sharma, V., & Manchanda, P. (2015a). Nonuniform wave packet frames in L2(R). Indian
JournalofIndustrialandAppliedMathematics,6(2),139–152.
28. Sharma,V.,&Manchanda,P.(2015b).NonuniformwaveletframesinL2(R).AsianEuropean
JournalofMathematics,WorldScientific,8(2),1550034(1–15).
29. Manchanda, P., & Sharma, V. (2012). Orthogonal vector valued wavelets on R+.
InternationalJournalofPureandAppliedMathematics,75(4),493–510.
30. Manchanda, P., Meenakshi, & Siddiqi, A. H. (2014). Construction of vector-valued
nonuniform wavelets and wavelet packets. Special Issue of Indian Journal of Industrial
andAppliedMathematics.
31. Meenakshi,Manchanda,P.,&Siddiqi,A.H.(2014).Waveletsassociatedwithvectorvalued
non-uniformmultiresolution analysis. ApplicableAnalysis:AnInternationalJournal,93(1),
84–104.
32. Meenakshi & Manchanda, P. (2017). Vector-valued nonuniform wavelet packets.
Numerical Functional Analysis and Optimization. Published online: https://doi.org/
10.1080/01630563.2017.1355814(TaylorandFrancis).
