Niklas Grip
Hilbert space frames and bases,
a comparison of Gabor and wavelet frames and applications to multicarrier digital
communications
LICENTIATE THESIS
Licentiate thesis Institutionen för Matematik
Avdelningen för -
2000:12 • ISSN: 1402-1757 • ISRN: LTU-LIC--00/12--SE
Hilbert space frames and bases,
a comparison of Gabor and wavelet frames and applications to multicarrier digital
communications
Niklas Grip
Department of Mathematics Lule˚ a University of Technology
SE-971 87 Lule˚ a, Sweden.
tel: 0920-91 974, fax: 0920-91 073, e-mail: grip@sm.luth.se
Both this licentiate thesis and the original papers are available at http://www.sm.luth.se/˜grip/Research/publications.html
March 20, 2000
Technical note: Typeset with LaTEX2ε, BibTEX, MakeIndex, AMS-LaTEX version 1.2 andAMSFonts), running on a Sun Sparcstation 4.
Abstract
Several signal processing applications today are based on the use of different transforms.
The signals under consideration are written as a linear combination (or series) of some predefined set of functions. Traditionally, orthogonal bases have been used for this purpose, for example, in the discrete Fourier transform. The theory for orthogonal bases for Hilbert spaces can, however, be generalized to other sequences of functions.
This is the topic of the first part of this thesis. It begins with an application-oriented introduction to the theory of frames and bases for separable Hilbert spaces. We explain similarities with and differences from the theory of orthogonal bases. Special attention is given to the relatively new theory of Gabor and Wavelet frames for L2(R). We explain how they can be used for so-called time-frequency analysis. The main emphasis is focused on explaining fundamental similarities and differences between Gabor and wavelet frames. We also give an example of an application (OFDM) related to the second part of the thesis, for which nonorthogonal Gabor frames are superior to any orthogonal basis.
The second part of this thesis concerns the current development of a standard for very high speed digital communication in ordinary telephone copper wires. It is the result of a cooperation with the Division of Signal Processing and Telia Research. We present a novel duplex method for Very high bit rate Digital Subscriber Lines (VDSL), called Zipper. It is intended to provide bit rates up to 52 Mbit per second, about 1000 times faster than the most common modems today. Zipper is based on Discrete Multi Tone (DMT) modulation. It uses an orthogonal basis of Gabor type for the signal transmission. Certain cyclical extensions are used to ensure the orthogonality between the basis functions. Zipper is proposed as a standard for VDSL to both the American National Standards Institute (ANSI) T1E1.4 group and in the European Telecommunication Standards Institute (ETSI) TM6 group. It will also be presented for the International Telecommunication Union (ITU). Telia Research is currently building a prototype together with ST Microelectronics (former SGS-Thomson), France. The first Zipper-VDSL modems are expected to be available on the mass market in the year 2001. The second part of this text consists of a brief introduction to Zipper, an ANSI standard contribution and three conference papers. The standard contribution compares Zipper performance with competing standard proposals at that time. In the first conference paper we present a new and patented method for reducing the interference that the unshielded copper wires experience from radio transmissions. The two last conference papers present a low complexity method for reducing the so-called peak to average power ratio (PAR) of the transmitted signal. PAR is a measure for the amount of rare but very high peaks in the signal. A reduced PAR allows for using a cheaper digital-to-analog converter and amplifier in the transmitter.
Abstract iii
Acknowledgements xi
Introduction 1
Outline of the thesis . . . 1
Notation . . . 3
Some recommendations for further reading . . . 4
I Theory 5
1 Hilbert space frames and bases 7 1.1 Hilbert and Banach space bases . . . 81.2 Signal processing basics . . . 11
1.3 Frames . . . 14
1.4 Riesz bases . . . 17
1.5 Non-orthogonal series expansions . . . 18
1.6 Sorting out overlapping definitions . . . 19
1.7 Snug frames . . . 22
1.8 The S−12 trick for making a frame tight . . . 24
1.9 Finite dimensional frames and bases . . . 25
1.10 In between the frames (fk) and (S−12fk) . . . 25
2 Gabor and Wavelet frames 28 2.1 Time-frequency analysis . . . 29
2.2 Uncertainty principles . . . 31
2.3 Different operator group structures . . . 33
2.4 Gabor and wavelet time-frequency sampling . . . 34
2.5 Wexler-Raz duality for Gabor frames . . . 41
2.6 Discrete time Gabor analysis . . . 44
2.7 Applications to image compression . . . 46
2.8 Conclusions . . . 46
iv
CONTENTS v
II Applications to multicarrier digital communications 49
3 DMT and OFDM signal transmission 51
3.1 Multicarrier signal transmission . . . 51
3.2 Choosing basis functions . . . 52
3.3 Digital subscriber lines . . . 54
3.4 Zipper — A flexible duplex scheme for VDSL . . . 55
4 Zipper performance when mixing ADSL and VDSL in terms of reach and capacity 58 Daniel Bengtsson, Petra Deutgen, Niklas Grip, Mikael Isaksson, Lennart Ols- son, Frank Sj¨oberg and Hans ¨Ohman, “Zipper performance when mixing ADSL and VDSL in terms of reach and capacity”, ANSI Technical Report T1E1.4/97-138, Clearwater Beach, Florida, May 12–17 1997. 4.1 Introduction . . . 58
4.2 Models for ADSL and VDSL signals and crosstalk . . . 59
4.2.1 ADSL-crosstalk into VDSL . . . 60
4.2.2 VDSL-crosstalk into ADSL . . . 60
4.3 Performance Evaluation and Comparison of Zipper, TDD and FDD . . 61
4.3.1 System parameters . . . 62
4.3.2 Asymmetrical rates (8:1) . . . 62
4.3.3 Symmetrical rates . . . 63
4.4 Results from different combinations of ADSL/VDSL . . . 64
4.4.1 Environment with no ADSL Disturber . . . 64
4.4.2 Environment with 5 ADSL Disturbers . . . 65
4.4.3 Environment with 25 ADSL Disturbers . . . 65
4.4.4 Environment with 44 ADSL Disturbers . . . 65
4.4.5 Performance of Zipper . . . 66
4.5 Conclusions . . . 67
5 Digital RFI suppression in DMT-based VDSL systems 68 Frank Sj¨oberg, Rickard Nilsson, Niklas Grip, Per Ola B¨orjesson, Sarah Kate Wilson and Per ¨Odling, “Digital RFI suppression in DMT-based VDSL sys- tems”, International Conference on Telecommunications ’98 (ICT), volume 2, pages 189–193, Chalkidiki, Greece, June 1998. 5.1 Introduction . . . 68
5.2 Radio Frequency Interference . . . 69
5.3 Suppression of RF-ingress . . . 70
5.3.1 Signal Model . . . 70
5.3.2 Parametrization of the disturbance . . . 71
5.3.3 Model parameter estimation . . . 72
5.3.4 Disturbance estimation and cancellation . . . 73
5.4 Simulations . . . 73
5.5 Summary . . . 75
A Linearization of narrowband RFI . . . 75
6 A low-complexity PAR-reduction method for DMT-VDSL 77 Per Ola B¨orjesson, Hans G. Feichtinger, Niklas Grip, Mikael Isaksson, Nor- bert Kaiblinger, Per ¨Odling and Lars-Erik Persson, “A low-complexity PAR- reduction method for DMT-VDSL”, Proceedings of the 5th International Sym- posium on DSP for Communication Systems (DSPCS’99), pages 164–169, Perth, Australia, February 1999.
6.1 Introduction . . . 77
6.2 The intended target system . . . 78
6.3 A non-iterative PAR-reduction method . . . 79
6.4 Computer simulation results . . . 81
6.4.1 Choosing the peak reduction factor . . . 81
6.4.2 Performance . . . 82
6.4.3 Reducing complexity . . . 82
6.5 Summary and conclusions . . . 83
7 DMT PAR-reduction by weighted cancellation waveforms 84 Per Ola B¨orjesson, Hans G. Feichtinger, Niklas Grip, Mikael Isaksson, Nor- bert Kaiblinger, Per ¨Odling and Lars-Erik Persson, “DMT PAR-reduction by weighted cancellation waveforms”, Proceedings of RadioVetenskap och Kom- munikation 99 (RVK 99), pages 303–307, Karlskrona, Sweden June 1999. 7.1 Introduction . . . 84
7.2 Weighted peak reduction . . . 86
7.3 Computer simulation results . . . 88
7.3.1 Clip rate . . . 88
7.3.2 Clip power . . . 89
7.3.3 Complexity . . . 89
7.4 Summary and conclusions . . . 89
8 On clip reduction in the receiver 91 8.1 The basic idea . . . 91
8.2 Signal Model . . . 92
8.3 Complexity . . . 93
References 93
Index 103
List of Figures
0.1 Main structure of this thesis . . . 1
1.1 DMT signal transmission synthesis and analysis mappings . . . 13
1.2 Compression and noise reduction analysis and synthesis mappings . . . 13
1.3 Venn diagram of Hilbert space frames and bases defined in this text . . 20
2.1 Time-frequency analysis in the inner ear. . . 28
2.2 Time-frequency analysis with the STFT and continuous wavelet transform. 29 2.3 Dilation and translation of a mother wavelet . . . 31
2.4 Heisenberg boxes . . . 32
2.5 Wavelet subspaces . . . 39
2.6 Walnut representation of the Gabor frame operator S . . . 45
3.1 Basic multicarrier transmission . . . 52
3.2 Signal constellations . . . 52
3.3 The xDSL network . . . 54
3.4 Coexistense of Zipper and other transmission methods . . . 55
3.5 Zipper and ADSL spectral compatibility . . . 55
3.6 Near and far end crosstalk . . . 56
3.7 VDSL channel impulse response . . . 56
3.8 Cyclic prefix and suffix. . . 57
3.9 Basic DMT system . . . 57
4.1 Zipper coexisting with ADSL . . . 59
4.2 ADSL and VDSL scenario . . . 59
4.3 PSD of VDSL signal with disturbing NEXT and FEXT from 5 ADSL- and 44 VDSL-disturbers . . . 60
4.4 SNR of an upstream and a downstream VDSL signal . . . 61
4.5 Power spectral density of an ADSL signal received at the NT . . . 61
4.6 Downstream and 8*upstream rate with 44 VDSL- and 5 ADSL- disturbers 63 4.7 Achievable (8:1) asymmetrical rates with Zipper, TDD and FDD on TP1 64 4.8 Up- and down-stream bit rates in symmetrical case . . . 64
4.9 Achievable symmetrical rates with Zipper and TDD on TP1 . . . 65
5.1 The DFT coefficients Sk of a disturber s(t) = A cos(2πfct + θ). . . 71
5.2 Average SNR-loss due to RFI at 3.65 MHz, with/without RFI-cancellation. 73 5.3 Average signal power before/after RFI-cancelling. . . 74
5.4 SNR-loss after RFI-cancellation for 0≤ | bfc− fc| ≤ 20 kHz. . . 74
vii
6.1 Peak reduction in a DMT transmitter . . . 79
6.2 The Cancellation-peak generator . . . 80
6.3 Frame clip rates with µpeakred= 0.98µclip . . . 82
7.1 Peak reduction in a DMT transmitter . . . 85
7.2 Weighted peak reduction . . . 86
7.3 Frame clip rate with weighting . . . 88
7.4 SCR vs frame clip rate with weighting . . . 90
8.1 A DMT receiver with clip noise reduction . . . 91
List of Tables
0.1 Some frequently used symbols . . . 3 4.1 Performance in an environment with 49 VDSL and no ADSL disturbers 65 4.2 Performance in an environment with 44 VDSL and 5 ADSL disturbers. 66 4.3 Performance in an environment with 24 VDSL and 25 ADSL disturbers. 66 4.4 Performance in an environment with 5 VDSL and 44 ADSL disturbers 66 4.5 Performance of Zipper at different bit rates and with different number
of ADSL disturbers . . . 67 5.1 Four UK amateur radio bands overlapping the VDSL spectrum. . . 69 5.2 Average SNR-loss with and without RFI-cancellation for a 6kHz wide
RFI-disturber in different HAM-bands. . . 75
7.1 Clip level reduction . . . 88
ix
Acknowledgements
First of all, I want to express my deepest gratitude to my main supervisor Professor Lars-Erik Persson for believing in me and giving me this opportunity in life. My research field is fairly new at our department and I would never have come this far if he had not been so good at establishing connections with all the talented and wonderful people I have had the pleasure to work together with. I have been very lucky to have Dr. Stefan Eriksson and Professor Jan-Olov Str¨omberg as assisting supervisors. It is always very rewarding to discuss problems with any of them. I also want to express my deepest gratitude to Professor Dr. Hans Georg Feichtinger and Dr. Norbert Kaiblinger at the Numerical Harmonic Analysis Group (NuHAG) at University of Vienna, Austria.
They have taught me practically everything I know about Gabor frames. I highly appreciate all enlightening discussions about both work and other matters, and I hope that I will be able to visit Vienna again in the future. I do also want to thank Professor Karlheinz Gr¨ochenig for the preprint of his book [Gr¨o00]. It has been an invaluable support when I was writing this thesis. I also owe Professors J¨oran Bergh and Natan Krougljak many thanks for interesting discussions and good advice during their visits at our department.
A substantial part of this thesis concerns applications to multicarrier digital com- munications. I am very grateful to Professor Per Ola B¨orjesson, who made it possible for me to work together with the Division of Signal Processing and the staff at Telia Research. It was a wonderful environment to work in and it gave me invaluable expe- riences from working in an industrial environment, which most Ph. D. students never get. I owe Assistant Professors Sarah Kate Wilson and Per ¨Odling many thanks for their guidance during this time. Katie has given me great support without being afraid to immediately tell when she thinks something is wrong or should be done in a different way. Per has generously shared his great technical knowledge and been a never-ending source of inspiration and good advice about technical writing and oral presentations. I do also want to thank Mikael Isaksson and his staff at Telia Research, as well as Rickard Nilsson, Frank Sj¨oberg and the remaining Signal Processing staff for a wonderful time.
I also want to show my appreciation towards TFR (Swedish Research Council for Engineering Sciences) and Telia Research for financial support.
Last, and most importantly, I would like to thank my family and friends for every- thing.
xi
Introduction
Outline of the thesis
This thesis consists of two main parts. The first part is an introduction to the theory of Hilbert space frames and bases in general and Gabor and Wavelet frames in particular.
The second part is about applications of this theory to multicarrier communications.
The main emphasis is focused on very high speed communication in ordinary telephone copper wires. Figure 0.1 shows the main structure of the text.
In Chapter 1 we show how the theory of orthonormal bases for Hilbert spaces can be generalised to so-called frames. We show that a sequence of elements in a Hilbert space has to be a frame in order to provide numerically stable series expansions of all elements in the space. We point out important similarities and differences between frames and bases and give some examples of signal processing applications. Most results in this
Figure 0.1: The main structure of this thesis.
1
introduction are essentially known in the literature. However, the presentation here is partly new and particularly fitted to the other chapters of this thesis. Moreover, some of the results (such as Proposition 1.25, Proposition 1.45 and Figure 1.3) are presented and proved in a more general form than previously seen by the author.
Chapter 2 is an overview and comparison of wavelet and Gabor frames. The main emphasis is on similarities, differences and properties that make either wavelet of Gabor frames particularly useful in different types of applications, such as those described in the following chapters. Like in the previous chapter, most of the individual results are known since before, except for Proposition 1.55, which is a preliminary result from an ongoing co-operation with the Numerical Harmonic Analysis Group (NuHAG) at University of Vienna, Austria.
In Chapter 3 we explain how wavelet and Gabor frames can be used for multicarrier transmission. We shortly discuss how to choose optimal basis functions for mobile radio channels, and argue that certain non-orthogonal frames are optimal for this particular application. The main emphasis is however on wireline systems. We present some results of a co-operation with the Division of Signal Processing and Telia Research in Lule˚a. We present a novel duplex method for Very high bit rate Digital Subscriber Lines (VDSL), called Zipper. It is intended to provide bit rates up to 52 Mbit per second in ordinary telephone copper wires. This is about 1000 times faster than the most common modems today. Zipper is based on Discrete Multi Tone (DMT) modulation. It uses an orthogonal basis of Gabor type for the signal transmission. Certain cyclical extensions are used to ensure the orthogonality between the basis functions. Zipper is proposed as a standard for VDSL to both the American National Standards Institute (ANSI) T1E1.4 group and in the European Telecommunication Standards Institute (ETSI) TM6 group. It will also be presented for the International Telecommunication Union (ITU).
Telia Research is currently building a prototype together with ST Microelectronics (former SGS-Thomson), France. The first Zipper-VDSL modems are expected to be available on the mass market in the year 2001. Chapters 4–7 consists of an ANSI contribution and three conference papers.
Chapter 4 is a technical report [BDG+97] submitted to the ANSI Standards Sub- committee T1E1.4. It consists of a comparison of Zipper with two competing proposals at that time, namely TDD and FDD. In all studied cases, Zipper had best performance in terms of reach and capacity.
Chapter 5 is a conference paper [SNG+98], in which we propose a new frequency domain method for reducing the radio frequency interference that mainly radio ama- teurs causes to the unshielded copper wires of a DMT-system. This method is now patented [BBI+97, BBI+98] by Telia Research and the authors.
Chapters 6 and 7 contains two related conference papers [BFG+99a, BFG+99b]. In the first paper we propose a low-complexity method for reducing the peak to average power ratio (PAR) of a DMT signal. The PAR measures the amount of rare but high peaks in the transmitted signal. The PAR reduction makes it possible to use simpler and cheaper amplifier and digital-to-analog converter in the DMT transmitter. In the second paper we present an improvement of our PAR-reduction method. We show how to improve the performance with almost no increase in complexity.
Chapter 8, finally, is a short presentation of a proposed method for reducing the effects of high PAR in a DMT receiver.
Introduction 3
Notation
Most symbols and abbreviations are defined as they appear in the text and a reference to this definition is included in the index (starting at page 103). Some frequently used symbols are also defined in table 0.1. In the index, boldface references indicate references to definitions or detailed explanations.
For sequences as well as sums and series over a finite or countable index set, we usually use shorthand notation like (ek) andP
kckfk. It is then understood that the choice of index set is of no importance for the current discussion. Moreover the notation P
kckfk indicates unconditional convergence of a series, that is, the convergence is independent of the order of summation (see, for instance, [HW96, Section 5.2]).
Equations and different results are numbered consecutively. The reason for this is to make it easier to follow references to earlier results. For example, if one page contains Proposition 3.3, equation (3.37) Lemma 3.1 and Example 3.12, then that gives no information about where to find Theorem 3.7.
symbol meaning
δk[n] The natural basis for lpδk[n] =
(1 if n = k, 0 otherwise.
χI The characteristic function χI(x) =
(1 if x∈ I, 0 otherwise.
i The complex number i, i2= 1.
infkrk The largest r∈ R such that r ≤ rkfor all k.
supkrk The smallest r∈ R such that r ≥ rkfor all k.
lp, 1≤ p < ∞ The vector space of complex-valued (unless otherwise stated) sequences (ek) with finite normk(ek)k = (P
|ek|pdx)p1 [Kre78].
Lp(I), 1 ≤ p < ∞ The vector space of Lebesgue measurable functions defined onI and having finite normkf k = R
I|f (x)|pdx1p
[Rud87].
L∞(I), 1 ≤ p < ∞ The vector space of essentially bounded Lebesgue measurable functions de- fined onI [Rud87].
A 1: The closure [Rud87] of a set A. 2: Complex conjugate of A∈ C.
A∗ 1: Hilbert-adjoint operator of A. 2: Conjugate transpose of matrix A. 3:
complex conjugate of A∈ C.
Adef= B A is defined to be equal to B
B A Banach space
C The complex numbers.
H A Hilbert space
N The nonnegative integers or natural numbers.
R The real numbers.
R+ The positive real numbers.
< The real part of a complex number.
Z The integers.
Z+ The positive integers.
a The complex conjugate of a.
Table 0.1: Some frequently used symbols. When a symbol is defined by an example, the defined symbol is indicated with bold face printing (for example, Adef= B).
Some recommendations for further reading
There are plenty of references it the text. Here we have collected some special recom- mendations for a quicker reference.
Bases for Banach spaces: Singer [Sin70, Sin81] gives an encyclopaedic treatment of bases in Banach spaces. [LT96] is a classical concise treatment of classical Banach spaces and [Woj91] gives an introduction to modern Banach space theory.
Frame theory: See, for example, [FZ97, You80, Teo98, Gr¨o00, Dau92] and the refer- ences given in the text.
Wavelets: There are many textbooks about wavelets. The following are some books that the author has found particularly useful. The books [BEL99, BGG98] give an application oriented introduction to multiresolution analysis (MRA) wavelets without too many technical details. A more mathematically complete descrip- tion is given in [HW96] and the classical [Dau92]. A more advanced classical book is [Mey92]. The continuous wavelet transform and its use in mathematical analysis is described in [Hol95]. This book also covers wavelet frames and MRA.
Another book that includes a treatment of also introduces inexact wavelet frames is [Teo98] .
Gabor analysis: There are less textbook about Gabor analysis than about wavelets.
A good introduction to the subject is however given in [Gr¨o00]. More specialised results are covered in [FS98].
Telecommunication systems: The books [ET96, ET98a] (with English translations [ET97, ET98b]) give an elementary introduction to telecommunications in gen- eral. See [SCS99] for a more detailed description of digital subscriber line technol- ogy, from ISDN to VDSL. This book also includes a CD-ROM containing drafts of the xDSL standards as well as selected papers presented at the T1E1.4 Stan- dards Committee meetings between 1990 and 1998 (including the one reprinted as Chapter 4 in this text).
Part I
Theory
5
Chapter 1
Hilbert space frames and bases
This is basically a chapter about series expansions in separable Hilbert spaces. It also serves as a theoretical framework for the remaining thesis. We begin with a summary of the topics that are treated in this chapter:
It is a standard trick in signal processing to write an arbitrary element h in a (separable) Hilbert space H as a series expansionP
kckek. The mapping T of an l2- sequence c = (ck) of coefficients to T c def= P
ckek is called a synthesis mapping. We show that the Hilbert-adjoint operator T∗ of a synthesis mapping T is an analysis mapping, which maps any h ∈ H to the sequence T∗h = (hh, ekiH). We give some examples of how T∗ and T appear in typical applications in the form of, for example, a Fourier transform and its inverse transform. We also show that T∗ is numerically stable and has a numerically stable left inverse if and only if (ek) is a so-called frame (with frame bounds “sufficiently close” to each other). The same holds for T if and only if (ek) is a special kind of frame called Riesz basis (with bounds “sufficiently close”
to each other).
Every frame (ek) has at least one dual frame (eek), which provides a series expansion h =P
khh, eekiHek. Contrary to orthonormal bases, this expansion may be non-unique, but the canonical dual frame eek def= S−1ek always gives minimal l2-norm coefficients.
Here Sdef= T T∗ is the very important frame operator.
We give a rather detailed introduction to this subject and discuss important similar- ities with and differences from the theory of orthonormal bases. For example, conver- gence issues in infinite dimensional spaces are more complicated for frames. Moreover, a dual frame has to be computed efficiently. Thus orthogonal bases are in one sense simpler, but we will see in later chapters, that in some applications, frames outper- forms orthonormal bases. We do also explain how different types of frames and bases are related to each other and summarize those results in a Venn diagram (Figure 1.3, page 20).
7
1.1 Hilbert and Banach space bases
Much of the theory in this chapter can be generalized to Banach spaces (see [Gr¨o91, CH97, FZ98]), but generally, things get more complicated (we give some example be- low). For simplicity, we will usually restrict ourselves to Hilbert spaces. We begin with some fundamental definitions and properties of Banach and Hilbert spaces. Readers not acquainted with basic functional analysis terminology (such as vector spaces) can consult, for instance, the first chapters of [Kre78].
Definition 1.1 ((Separable) Banach and Hilbert spaces). A Banach spaceB is a normed vector space which is complete in the metric defined by the norm. B is separable if it contains a countable dense subset. A Hilbert spaceH is a Banach space for which the norm is obtained from an inner product h·, ·i.
The existence of an inner product (providing geometry) and tricks like Gram- Schmidt orthonormalization give Hilbert spaces a simpler structure. Therefore much of the theory in this thesis is developed to a much greater depth for Hilbert spaces than for Banach spaces in general. Moreover, in many practical applications different kinds of energy are the natural quantities of importance, leading to computations in the Hilbert spaces l2or L2. This does not mean that only Hilbert spaces are important for applications, however. For example, DeVore, Jawerth and Lucier [DJL92] argue that from all Lp-norms, L1is best suited for image compression. They conclude from exper- imental research that L1most closely resembles the human eye’s way of measuring the error in the compressed image. Computer-aided design is another application where a non-Hilbert space norm is used [DJL92]: A function that describes the boundaries of some object is restricted to some maximum amplitude, that is, an L∞norm is used.
Schauder bases are useful tools in all Banach spaces.
Definition 1.2 ((Schauder) basis). A sequence (ek) in a Banach space B is called a (Schauder) basis forB if, for every h ∈ B, there is a unique sequence of scalarsa (ck) such that h =Pckek with convergence in the norm ofB.
We will often omit the word Schauder (but Riesz bases will always be called exactly that). A Banach spaceB with a basis (ek) is separable (it is not difficult to show that the set of linear combinationsPn
k=1(αk+iβk)ekwith rational αk, βkis dense inB). Banach himself posed the question in 1932 whether every separable Banach space has a basis.
This remained one of the outstanding unsolved problems of functional analysis [You80]
until 1973, when Per Enflo [Enf73] found a separable Banach space which does not have a Schauder basis. Nonetheless Schauder bases have been found for almost all known infinite-dimensional separable Banach spaces [You80]. Separable Hilbert spaces, however, always have a basis (Proposition 1.4). We refer to, for instance, [LT96, Sin70, Sin81, Woj91] for a more complete survey of Banach space theory.
Every basis for B is total in B:
Definition 1.3 (Total sequence). A countable set {ek} (or sequence) in B is said to be totalbif its linear span is dense inB, that is, for each h ∈ H and each ε > 0 there is a linear combinationPn
k=k0ckek such that
h −Pn
k=k0ckek < ε.
aIn functional analysis, the scalar field is eitherC or R.
bWe choose to use the same notation as in [Kre78]. Some equivalent names for total sequences are “complete”, “closed” and “fundamental”. Note, however, that the name “fundamental sequence”
sometimes is used for Cauchy sequences.
1.1. HILBERT AND BANACH SPACE BASES 9
Remark. From the definition and experience from finite dimensional spaces, it is tempt- ing to assume that a total and linearly independentcsequence is a basis, or can be made into a basis by removing elements. This is not true, however. Not even in Hilbert spaces, as we will see in Examples 1.12–1.14. If the total sequence is orthonormal, however, then it is a basis. This implies the following important property of separable Hilbert spaces.
Proposition 1.4. Every separabled Hilbert space H contains an orthonormal basis.
Proof. We can construct a total orthonormal sequence (ek) by applying the Gram- Schmidt orthonormalizing process [Kre78] and induction to a countable dense subset ofH. That (ek) is a basis forH follows from next proposition.
Proposition 1.5. Every total orthonormal sequence (ek) in a Hilbert space H is also a basis forH.
Proof. The analysis mapping T∗h = (hh, eki) maps each ek to δk ∈ l2 given by
δk[n] =
(1 if n = k,
0 otherwise. (1.6)
T∗is onto, because it is bounded (by next lemma) and (δk) is an basis for l2(with the same index set and scalars). We claim that T∗ also is one-to-one and preserves the inner product, that is, it is an isomorphism [Kre78]. Thus (ek) is an orthonormal basis forH because it is the inverse mapping of the orthonormal l2-basis (δk).
It remains to prove that T∗ is one-to-one and preserves the inner product. Next lemma shows thatkT∗hk = khk. By the polarization identity [Kre78], T∗also preserves the inner product: hT∗g, T∗hi = hg, hi. The injectivity immediately follows from kT∗g− T∗hk = kT∗(g− h)k = kg − hk.
We refer to [Kre78, Theorem 3.6-3] for a proof of the following
Lemma 1.7 (Total orthonormal sets and the Parseval relation). An ortho- normal sequence (ek) in a Hilbert spaceH is total if and only if the Parseval relation holds, that is if and only if
X
k
|hh, eki|2=khk2 for all h∈ H. (1.8)
A basis has the property that if one element is removed, it is no longer a basis. It is not even total. To show this, we need the following proposition.
Proposition 1.9. A sequence (ek) in a separable Hilbert space H is total if and only if no nonzero h∈ H is orthogonal to all ek.
Proof. See, for instance, [Kre78, Lemma 3.3-7].
We can now show that no subsequence of a basis (bk) is total. (Examples 1.12 and 1.13 show that this is a stronger result than the more obvious fact that there is no series expansionP
n6=kckbk= bn.)
cA sequence (ek) is linearly independent ifPn
k=k0ckek= 0 imply that ck0=· · · = cn= 0.
dActually, the same result holds for all Hilbert spaces. Then (ek) may be uncountable, but every x has a series expansionP
kckekwith at most countably many nonzero ck[Kre78, page 165].
Proposition 1.10. Suppose that (bk) is a basis for a separable Hilbert space H. Re- moving an element bnfrom (bk) leaves a sequence which is not total in H.
Proof. Let C be the closure of the span of (bk)k6=n. Since C is a closed subspace of H, there is an orthogonal decomposition [Kre78, Theorem 3.3-4] bn= c + o such that c∈ C and o is orthogonal to every bk, k6= n. Moreover, bn6∈ C (or (bk) would not be a basis), that is, o is a nonzero element orthogonal to every bk, k6= n. Therefore (by Proposition 1.9), (bk)k6=nis not total.
In finite dimensional Hilbert spaces, a linearly independent sequence that spans the space is a basis. We have seen that for infinite dimensional separable Hilbert spaces, an orthogonal total sequence is a basis. The same does not hold for linearly independent total sequences, however. Examples 1.12 and 1.13 show that some h∈ H might not have a series expansion. Example 1.14 show that there may be non-unique series expansions. Examples 1.12 and 1.13 make use of the following lemma:
Lemma 1.11 (Biorthogonal bounded coefficient functionals). Suppose that a sequence (ek)∞k=k0 provides at least one series expansion of every b in a Banach space B. If there exists a sequence of linear bounded functionals αk: B → C, such that αk(en) = δk[n], then (ek) is a basis forB and b =P∞
k=k0αk(b)ek. Proof. LetP∞
n=k0cnenbe a series expansion of some b∈ B. Since αk is bounded, and thus continuous, αk(b) = αk P∞
n=k0cnen
= P∞
n=k0cnαk(en) = ck. Consequently b =P∞
k=k0αk(b)ek.
Example 1.12 (No series expansion). ([Sin70, Example 4.1]) Let C2πbe the space of all real-valued 2π-periodic continuous functions defined on R, with norm kxk∞ = maxt∈R|x(t)|. For n ∈ Z+, let x0(t) = 12, x2n−1(t) = sin(nt) and x2n(t) = cos(nt).
Then (xn) is dense in C2πby the Weierstrass approximation theorem for trigonometric polynomials [Rud87, Theorem 4.25]. The Fourier coefficient functionals (αk) are given by α2(n−1)(x) = 1πRπ
−πx(t) cos((n− 1)t)dt and α2n−1(x) = π1Rπ
−πx(t) sin(nt)dt. Since αk(xn) = δk[n] and|αk(x)| ≤ 2π kxk∞, Lemma 1.11 shows that no other choice of co- efficients can give a series expansion in C2π. However, there are real-valued continuous functions whose Fourier series does not converge uniformly [Kre78, page 252]. Thus these functions have no C2π series expansionP∞
k=1ckxk. This was not a Hilbert space example, but the next example is.
Example 1.13 (No series expansione). The sequence of functions
γk[n] =
(1 if n≤ k,
0 otherwise, k∈ Z+
is total in the Hilbert space l2 with index setZ+, because δ1= γ1 and δk = γk− γk−1
for k > 1, where (δk) is the natural basis defined by (1.6). The linear functionals αk(h) = h[k]− h[k + 1] satisfy αk(γn) = δk[n]. They are bounded since |αk(h)| ≤
|h[k]|+|h[k + 1]| ≤ 2 khk. Thus Lemma 1.11 shows that if h ∈ l2has a series expansion
1.2. SIGNAL PROCESSING BASICS 11 P∞
k=1ckγk, then it has partial sums Xn
k=1
αk(h) γk = Xn k=1
(h[k]− h[k + 1]) γk= Xn k=1
h[k]γk−
n+1X
k=2
h[k]γk−1
= h[1]γ1+ Xn k=2
h[k] (γk− γk−1)− h[n + 1]γn
= Xn k=1
h[k]δk− h[n + 1]γn,
where the sum Pn
k=1h[k]δk → h. Consequently, h has a series expansionP∞ k=1ckγk
only if kh[n + 1]γnk → 0. But this is not the case for h = P
k1
kδk4+1, since then h[k4+ 1]γk4
= 1kkγk4k = 1kk2→ ∞ as k → ∞. Thus h has no l2 series expansion P∞
k=1ckγk.
Example 1.14 (Non-unique series expansion). Let f ∈ l2 (with index set N) be given by f[k] = e−k. Then the sequence (ek)def= (f, δ1, δ2, . . . ) is linearly independent and total in l2. Since 0 = f −P
kf[k]δk, every x ∈ l2 has infinitely many different series expansions of the type x = αf +P(x[k]− αf[k]) δk, where α∈ C.
Remark. In Example 1.14P
k|hh, eki|2 =|hh, fi|2+P
k|h[k]|2 =|hh, fi|2+khk2 for every h ∈ l2. Thus khk2 ≤ P
k|hh, eki|2 ≤
1 +kfk2
khk2 (by Schwarz inequal- ity [Kre78]). This implies that (ek) is a frame (Definition 1.23). We will see in Propo- sition 1.43 that non-unique series expansions are possible only for frames that are inexact, that is, frames for which we can remove one element (here f) and still have a frame. In this simple example there is no reason not to remove f, but we will show later on that inexact frames may be preferable because of properties such as better time-frequency localization (see Theorem 2.20). Moreover, in Section 3.2 we give a concrete example of an application for which inexact frames have better performance than exact frames or orthonormal bases.
Remark. Convergence issuses often makes infinite dimensional spaces more compli- cated, but not always. For example, sampling and discretization makes it difficult to explain properties connected to regularity and dilation. Then results in infinite di- mensional spaces can give information about asymptotic results when the sampling intervals decrease towards zero [Mal98, Section 1.3.1]. Another example is the multi- carrier transmission systems in chapters 3–8. The transmitted signal is built up from a very large but finite number of basis functions. The number of basis functions is, however, so large that any convergence issues that are important for an infinite-length signal is important for the system. For instance, we show in Propositions 1.25 and 1.35 that for numerical stability, the carrier functions must be a so-called Riesz basis. (We mention other requirements, such as time localization, in Chapter 3.)
1.2 Signal processing basics
The analysis and synthesis mappings play a central role in many different signal pro- cessing applications. In this section we define these mappings and give some examples of their use in signal processing applications.
Definition 1.15 (Analysis and synthesis mapping). Suppose that a sequence (ek) in a Hilbert space H is such that the equation T c = P
kckek defines a linear bounded mapping T : l2 → H. Then T is called the synthesis mapping of (ek). The Hilbert-adjoint [Kre78] operator T∗: H → l2 of T is called the analysis mapping of (ek).
In next section we define a class of sequences for which T and T∗ are well defined and have certain nice properties. Next proposition shows that T∗h = (hh, eki) and that T exists if and only if the mapping h7→ (hh, ekiH) is a linear and bounded mapping into l2.
Proposition 1.16. Let (ek) be a sequence in a Hilbert space H. If Dh def= (hh, eki) defines a linear bounded mapping D :H → l2, then (ek) has a synthesis mapping T and D = T∗. Conversely, if T is the synthesis mapping of a Hilbert space sequence, then T∗h = (hh, eki).
Proof. (Based on [Dau92, page 101]) Let (ekn)n∈Z+ be any permutation of (ek). We show in two steps that if D is linear and bounded, then T c = P
kckek defines a linear bounded mapping T : l2 → H, that is, the synthesis mapping. First, Cauchy convergence ofP∞
n=1cnkenk follows from a standard representation of the norm (given by Riesz representation theorem), continuity of the inner product and the Cauchy- Schwarz inequality [Kre78]:
n1
X
n=n0
cknekn
= sup
khkH=1
* n1 X
n=n0
cknekn, h+
= sup
khkH=1
n1
X
n=n0
cknhekn, hi
≤ sup
khkH=1
vu ut
n1
X
n=n0
|ckn|2 vu utX∞
n=1
|hekn, hi|2
= vu ut
n1
X
n=n0
|ckn|2 sup
khkH=1kDhkl2 → 0 as n0, n1→ ∞, since supkhk
H=1kDhkl2 =kDk and c ∈ l2. This shows thatP∞
n=1cknekn converges in norm and (by the continuity of the inner product)
* ∞ X
n=1
cknekn, h +
H
= X∞ n=1
hcknekn, hiH=X
k
ckhh, ekiH=hc, Dhil2, (1.17)
where the second last equalityf follows from absolute convergence (by the Cauchy- Schwarz inequality). This shows that D∗h = P∞
n=1cknekn for every rearrangement (ekn) of (ek). Thus the convergence is unconditional and D∗ = T is the synthesis mapping of (ek), that is, T∗= D.
Conversely, if T is the synthesis mapping of an Hilbert space sequence (ek), then (1.17) shows that hT c, hiH = hc, (hh, eki) il2 for all h ∈ H and c ∈ l2. Thus T∗h = (hh, eki).
fAs explained on page 3, the notationP
kis used for unconditionally convergent series.
1.2. SIGNAL PROCESSING BASICS 13
Figure 1.1: Synthesis T∗ and analysis T in a DMT signal transmission system.
Figure 1.2: Synthesis T∗ and analysis T in compression or noise reduction of a signal.
Consequently, it is not necessary to require T c to be unconditionally convergent in Definition 1.15:
Corollary 1.18. Suppose that (ek) is a sequence in a Hilbert space H. If (ekn) is a permutation of (ek) and T c =P∞
n=1cknekn defines a linear bounded mapping T : l2→ H, then T is the synthesis mapping of (ek).
Proof. It follows from the second half of Propostion 1.16 that T∗h = (hh, ekni). But then the first part of the proposition tells that T c =P
kckek.
We now give some examples of where T and T∗ appear in some typical signal processing applications. The introduction of [FS98] contains many more examples. In a typical applicationH is finite-dimensional. The actual number of dimensions and the number of elements in the sequence (ek) may however be very large. If this is the case, then it may be important to choose the elements cleverly, as in the following example.
Example 1.19. Figure 1.1 shows a simplified DMT transmission system for high speed signal transmission in ordinary telephone copper wires. (The subject of Part II, pages 51–93.) The transmitted signal is built up from a very large number of functions fk. For efficient computation, these are chosen to be a so-called Gabor Riesz basis.
We will explain this in more detail later on. For now, it is sufficient to know that this means that the basis functions are collected into relatively short subsequences of functions that are well localized in different short intervals of time. In the DMT system that we describe later, a basis consisting of 4096 functions is used in each subinterval.
They are chosen so that the synthesis mapping is efficiently computed, for one inter- val at a time, by an inverse fast Fourier transform (IFFT) and an digital-to-analog converter. The signal is sent through a channel (the copper wires), which distorts h and adds different types of noise. An equalizer neutralizes the distortion but not the noise. The analysis mapping T∗ (here an analog-to digital converter combined with a fast Fourier transform, FFT) maps the received signal eh onto its Fourier coefficients eck =hh, fki. The received coefficients ec are then used to estimate the transmitted data c.
Example 1.20. Two other examples are compression and noise reduction (Figure 1.2).
Here T is used before T∗ and h is now the input signal. The coefficients c are modified in some way to reduce noise or to reduce the number of nonzero coefficients. Then the new coefficients are used to construct the modified signal eh.
1.3 Frames
If a sequence (ek) is an orthonormal basis for a Hilbert spaceH, then there is a unique series expansion h = T T∗h =P
khh, eki ekof every h∈ H. We will see in Theorem 1.37 that similar series expansions with coefficients c = T∗h exist for a much larger class of sequences (ek), called frames. If the coefficients shall be of any practical use, we do not want them to be sensitive to small changes in h. Thus if eh is an approximation of h, then it is reasonable to require the relative error in the coefficients T∗eh to be small whenever the relative error in eh is small. That is, for some positive constant r,
h − eh
khk ≤ r T∗
h− eh kT∗hk and
T∗
h− eh kT∗hk ≤ r
h − eh
khk
for all nonzero h and h− eh in H. The second inequality tells that T∗ has condi- tion number r. The condition number is a popular measure for numerical stability in numerical analysis and linear algebra (see [Str88, IK94]). We will see below that r≥ 1. For numerical stability, r should be “sufficiently” close to 1 for the application at hand. The first inequality implies that T∗ is one-to-one and thus has a left inverseg L, which also has condition number r. Note also that the first inequality can be rewrit- ten kT∗hk
kT∗(h−eh)k ≤r khk
kh−ehk. Hence it is equivalent to the second equality. (Exchanging the roles of h and h− eh makes no difference since both inequalities are to hold for all nonzero h and h− eh in H.) Both inequalities are therefore equivalent to
T∗
h− eh
h − eh · khk
kT∗hk≤ r for all nonzero h, eh ∈ H, (1.21) where we assume the constant r to be optimal in the sense that it is the smallest constant for which T∗ has this property. In other words, for every ε∈ [0, 1] there exist h and h− eh such that kT∗(h−eh)k
kh−ehk ·kTkhk∗hk > (1− ε)r. By taking the supremum first over all h and then over all h− eh, we see that the left inverse L is bounded, kT∗k · kLk ≤ r andkT∗k · kLk > (1 − ε)r for all ε ∈ [0, 1], that is, kT∗k · kLk = r. Thus we have shown that (1.21) implies that
kT∗k = r√
A and kLk = 1
√A, (1.22)
where we have introduced the constant A = kLk−2. Conversely, (1.21) follows from (1.22). To see this, note that (1.21) can be seen as the multiplication of two indepen- dent inequalities kT∗(h−eh)k
kh−ehk ≤ r√
A and kTkhk∗hk ≤ √1A, which both follow from (1.22).
Moreover, it also follows that that r≥ 1, since 1 = kIk = kLT∗k ≤ kLk · kT∗k = r.
We have thus shown that the analysis mapping T∗of a sequence is numerically sta- ble if and only if (1.22) holds (for all “sufficiently small” r). Such a sequence is called a frame. Frames are usually defined by equation (1.22) rewritten in the equivalent form Akhk2 ≤ kT∗hk2 ≤ B khk2, where p
B/A = r. Note that this definition im- plies a bounded and thus well defined analysis mapping T∗and (by Propostition 1.16)
gAn operator L is called a left inverse of T∗if LT∗is the identity operator I : T∗(H) → H.
1.3. FRAMES 15
synthesis mapping T . Frames were first introduced in 1952 by Duffin and Schaef- fer [DS52], but occurred implicitly at several places in the mathematical literature before that [Gr¨o00]. We will often use the notation (fk) to indicate that a sequence is a frame.
Definition 1.23 (Hilbert frames and Bessel sequences). Suppose that (fk) be a sequence in a Hilbert spaceH. We say that (fk) is a (Hilbert) frame forH with frame bounds A, B > 0 if it has an analysis mapping T∗ satisfying
Akhk2H≤ kT∗hk2l2 ≤ B khk2H for all h∈ H. (1.24) The frame is called tight if A = B. It is exact if it is no longer a frame when any one of its elements is removed, otherwise it is said to be inexact. Each frame bound is called sharp if the corresponding inequality is sharp, that is, if there is no larger A or smaller B such that equation (1.24) is satisfied. Unless otherwise stated, we will always assume the frame bounds to be sharp. If there only is an upper bound (that is, A = 0), then we call the sequence a Bessel sequence, and we usually denote it (bk).
We have now defined quite many different types of frames and bases. It is in no way clear how they are related to each other or if there really exist examples of all defined frames and bases. Other texts known by the author try to reduce the confusion by giving some (but not all) examples of how some of these definitions are partially overlapping or equivalent. One of the aims of this text has been to, once and for all, give simple examples of all existing combinations of frames and bases defined in this text and to show in a Venn diagram how they are related to each other (and to Riesz bases, which we soon will define). This will require some more work, but the reader can already now take a look at the resulting Venn diagram in Figure 1.3 (page 20).
We do now summarize the discussion that lead to the definition of a frameh: Proposition 1.25. For every Hilbert space sequence (ek) the following statements are equivalent:
i. For some A, r > 0, (ek) is a frame with bounds A and B = r2A.
ii. (ek) has a one-to-one analysis mapping T∗. Moreover, there is a minimal con- stant r≥ 1 such that both T∗ and its left inverse have condition number r.
Some other equivalent definitions of a frame follows in Proposition 1.29. To get there, we first need to define the very important frame operator:
Definition 1.26 (Frame operator). Let T and T∗ be the synthesis and analysis mapping of a Hilbert space sequence (ek). With (ek) we associate its frame operator S :H → H,
S def= T T∗, that is,
Sh = T T∗h =X
k
hh, eki h for all h ∈ H.
hThe author has not previously seen Proposition 1.25 or 1.35 in print, but Proposition 1.25 is a generalization of the fact that i⇒ ii when r = 1, which was pointed out in [BEL99, page 53].
For later use in Propostition 1.55, Proposition 1.29iii express the frame bounds in terms of the frame operator spectrum σ(S), which is a generalization of the eigenvalues of operators in finite-dimensional spaces. For completeness, we include the definition of σ(S) and refer to, for instance, [Kre78] for more about spectral theory in normed spaces.
Definition 1.27 (Spectrum). Let X be a complex normed space and P :D (P ) → X a linear operator with domain D (P ) ⊆ X 6= {0}. With P we associate the linear operator Pλ= P− λI where λ is a complex number and I is the identity operator on D (P ). If Pλ has an inverse, we denote it by Rλ(P ), that is,
Rλ(P ) = Pλ−1= (P − λI)−1
and call it the resolvent operator of P or, simply, the resolvent of P .
A regular value λ of P is a complex number such that Rλ(P ) exists, is bounded and is defined on a set which is dense in X. The resolvent set ρ(P ) is the set of all regular values λ of P . Its complement σ(P ) = C − ρ(P ) is called the spectrum of P and a λ∈ σ(P ) is called a spectral value of P .
Proposition 1.29 also express the frame bounds in terms of the following partial ordering:
Definition 1.28. Given a complexi Hilbert space H and bounded self-adjoint linear operators T1, T2: H → H, we write T1≤ T2or T2≥ T1if and only ifhT1h, hi ≤ hT2h, hi for all h∈ H. T1 is said to be positivej if and only if T1≥ 0. (In some texts, positive operators are called positive semidefinite or non-negative (definite).)
It is shown in [Kre78, pages 474 and 661] that this makes≤ a partial ordering. We are now ready to introduce two equivalent ways to write the frame definition inequalities (1.24).
Proposition 1.29 (Frame and Bessel bound defintions). Let T be the synthesis mapping of a sequence (fk) in a Hilbert spaceH 6= {0}. If S = T T∗, then the following statements are equivalent:
i. Akhk2≤P
k|hh, eki|2≤ B khk2
ii. AI≤ S ≤ BI, where I is the identity operator.
iii. A = min σ(S) and B = max σ(S)
Proof. The equivalence i⇔ ii follows immediately from X
k
|hh, eki|2=kT∗hk2=hT∗h, T∗hi = hT T∗h, hi = hSh, hi (1.30)
and the fact that S = T T∗ is self-adjoint.
ii ⇔ iii: S is linear, bounded, self-adjoint and (by (1.30)) positive. Hence, it follows from [Kre78, Theorem 9.2-1] that the spectrum of S lies in the closed interval [m, M ], where mdef= infkhk=1hSh, hi = A and M def= supkhk=1hSh, hi = B. Moreover, ifH 6= {0}, then m and M are spectral values (see [Kre78, Theorem 9.2-3]).
iA complex Hilbert space is one that has complex-valued vector space scalars.
jAlthough, “non-negative” would have been a better name.
1.4. RIESZ BASES 17
Later on, we will also need the following observation.
Proposition 1.31 (Operator norms). If B is the upper frame bound corresponding to the operators in Definition 1.26, then
kT k = kT∗k =√
B (1.32a)
and
kSk = B. (1.32b)
Proof. (1.32a) follows immediately from the definition of the (sharp!) upper frame bound and the fact [Kre78] thatkT∗k = kT k. Now (1.32b) follows immediately from the formulakSk = kT∗Tk = kT T∗k = kT k2, which holds for every product of an linear bounded operator and its Hilbert adjoint operator [Kre78, Theorem 3.9-4].
1.4 Riesz bases
We showed in Section 1.3 that the demand for a numerically stable analysis mapping leads to the definition of a frame. Similarly, we will now show that a basis (rk) for a Hilbert spaceH has a numerically stable synthesis mapping if and only if (bk) is a so-called Riesz basis with bound ratio B/A “sufficiently” close to 1. More precisely, let ec be an approximation of some c ∈ l2. In any application where the synthesis mapping T of (rk) has to be computed, we want Tec to be a “good” approximation of c whenever ec is a “good” approximation of c. In other words, we require that, for some r ≥ 1,
kT (c − ec)k
kT ck ≤ rkc − eck
kck and kc − eck
kck ≤ rkT (c − ec)k
kT ck for all nonzero c,ec ∈ l2.
The second inequality tells that T is one-to-one, that is, it has a left inverse. Note that this means that (rk) is a basis for the image T l2of T . Moreover, by repeating the argument in the beginning of Section 1.3, we see that these inequalities are satisfied if and only if there is an A > 0 and B = r2A such that Akck2≤ kT ck2 ≤ B kck2 for all c ∈ l2. If (ek) have this property and also is a basis for entire H, then it is called a Riesz basis forH:
Definition 1.33 (Riesz basis). A basis (rk) for a Hilbert space H is said to be a Riesz basis if there are positive Riesz bounds A and B such that
Akck2l2 ≤ kT ck2H≤ B kck2l2 for all c∈ l2. (1.34) A Riesz basis for a subspace ofH is sometimes called a Riesz projection basis [FS98].
Remark. This is the same definition as in, for example, [FS98, HW96, Mal98, Mey92, Woj97]. Thus we do not require that A≤ 1 ≤ B, as in [BEL99]. Note, however, that if (rk) is a Riesz basis, then B1rk
is a Riesz basis that satisfies this extra condition.
We now summarize our initial argument:
Proposition 1.35. For every Hilbert space sequence (rk) the following statements are equivalent:
i. For some A, r > 0, (rk) is a Riesz basis with bounds A and B = r2A.
ii. (rk) has an invertible synthesis mapping T : l2→ H. Moreover, there is a minimal constant r > 0 such that both T and T−1 have condition number r.
