Cosine Manifestations of the Gelfand Transform

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1.1 What is the Gelfand transform? Without further ado we begin with the basic principles of the Banach algebra theory. In his tour de force “A Course in Commutative Banach Algebras”1 Eberhard Kaniuth deﬁnes a normed algebra as a normed linear space A, over the ﬁeld of complex numbers C, whose norm is submultiplicative, i.e., p }¨}q

f,g A f g f g . @ P } ¨ }ď} }¨} } If A, is complete (i.e., it is a Banach space), we say that it is a Banach algebra (a multitude of p }¨}q examples of Banach algebras is provided in Chapter 6 in [8]). Given two Banach algebras A1 and A2 we say that a map Ψ : A1 A2 is a Banach algebra homomorphism if it is ÝÑ • continuous, • C linear, and ´ • multiplicative, i.e.,

f,g A1 Ψ f g Ψ f Ψ g . @ P p ¨ q“ p q ¨ p q A structure space2 ∆ A of a commutative Banach algebra A is the set of all nonzero Banach algebra homomorphisms m : A p q C. The set ∆ A becomes a locally compact space3 when endowed with the weak* topology4 andÝÑ its raison d’être liesp q in the fact that there exists a norm-decreasing, algebra homomorphism Γ : A C0 ∆ A given by the formula ÝÑ p p qq

A Γ f m : m f . @mP∆p q p qp q “ p q

The function Γ f C0 ∆ A is commonly referred to as the Gelfand transform of the element f A p q P p p qq P and the notation “Γ f ” is usually reduced to f. In order to reinforcep q our intuition regarding the abstract theory above let us focus on a particular p instance of a commutative Banach algebra, which will permeate the rest of the paper. Let G be a locally compact abelian group and let µ be its Haar measure,5 i.e., a nonzero, Borel measure which is • ﬁnite on compact sets, • inner regular, i.e., for every open set U in G we have

µ U sup µ K : K U, K compact , p q“ t p q Ă ´ u 1See [26], p. 1. 2For a thorough exposition of this concept see [13], p. 43 or [17], p. 5 or [26], p. 46. 3Every topological space that appears in the paper is assumed to be Hausdorﬀ so we refrain from writing that explicitly. 4For a detailed exposition of the weak* topology τ ˚ see Chapter 3.4 in [9], p. 62 or Chapter 2.4 in [36], p. 62. The fact that ∆pAq is a locally compact space (under τ ˚) can be found as Theorem 2.4.5 in [13], p. 46 or Theorem 2.2.3 in [26], p. 52. The former theorem also establishes the existence of the Gelfand transform – for another point of view see Theorem 1.13 in [17], p. 7. 5See [21] for the original paper by Haar as well as [10, 14] or [41] (chapter 2) for a further study of the subject.

2 • outer regular, i.e., for every Borel set A in G we have

µ A inf µ U : A U, U open , p q“ t p q Ă ´ u • translation-invariant, i.e., for every x G and every Borel set A we have µ x A µ A . P p ` q“ p q For every two functions f,g L1 G we deﬁne their convolution f g with the formula P p q ‹

xPG f g x : f y g x y dy, (1) @ ‹ p q “ żG p q p ´ q where the integration is with respect to the Haar measure µ.6 This operation turns L1 G into a commutative Banach algebra (obviously L1 G is already a Banach space, so deﬁnes the “mutliplication”p q of the elements). p q ‹ It is one of the gems of abstract harmonic analysis that the structure space ∆ L1 G , is homeomor- p p q ‹q phic7 to the dual group G, i.e., the group of all nonzero, multiplicative homomorphisms (called characters) χ : G S1. The homeomorphism H : ∆ L1 G , G in question assigns a unique χ G to every p multiplicativeÝÑ linear functional m ∆ L1 Gp , p inq ‹q such ÝÑ a way that P P p p q ‹q p p

fPL1pGq m f f x χ x dx. @ p q“ żG p q p q

1 1 As a consequence of ∆ L G , and G being homeomorphic, the Banach algebra C0 ∆ L G , is iso- p p q ‹q p p p q ‹qq metrically isomorphic with C G .8 It follows that we may treat the Gelfand transform f C ∆ L1 G , 0 p 0 1 p q P p p p q ‹qq of f L G as an element of C0 G given by the formula P p q p p q p p fPL1pGq f χ f x χ x dx. @ p q“ żG p q p q p 1 To put it in other words, the Gelfand transform of f L G is the Fourier transform f C0 G . P p q P p q p p 1.2 Context The aim of the current subsection is to provide the context for the upcoming sections by giving a brief overview of the present state of the literature. We strive to substantiate the claim that the cosine transform is not only an invaluable tool in the mathematical toolbox (especially for those working in the ﬁeld of harmonic analysis), but its usage stretches outside of the mathematical realm, to ﬁelds such as engineering and technology. Although the concept of the cosine transform goes as far back as the monumental work “The analytical theory of heat” by Jean Baptiste Fourier,9 it is widely agreed that the “modern history” of the operator starts roughly 50 years ago. As Nasir Ahmed recollects,10 in late ’60s and early ’70s “there was a great

6Formally, we probably should write “dµpyq” when integrating with respect to the Haar measure, yet we feel that this is an unnecessary complication of the nomenclature. Hence, we stick to an abbreviated form “dy” for the sake of simplicity. 7See Theorem 3.2.1 in [13], p. 67 or Theorem 2.7.2 in [26], p. 89. 8 See Corollary 2.2.13 in [26], p. 57 for a more general result stating that Banach algebras C0pXq and C0pY q are isomet- rically isomorphic if and only if locally compact Hausdorﬀ spaces X and Y are homeomorphic. 9For an english translation of Fourier’s book with commentary by Alexander Freeman see [18]. 10See [2].

3 deal of research activity related to digital orthogonal transforms” and a large number of transforms were introduced “with claims of better performance relative to others transforms”. Inspired by the Karhunen- Loeve transform, Ahmed came up with the idea of the cosine transform and issued a proposal to the National Science Foundation to study the newly-invented operator. His proposal was dismissed due to the idea being “too simple”, but Ahmed continued to work on the concept with his Ph.D. student T. Natarajan and a colleague at the University of Texas, Dr. K. R. Rao. The team was quickly surprised how well the cosine transform performed relative to other transforms and published their results in 1974.11 17 years later, Ahmed wrote “Little did we realize at that time that the resulting DCT12 would be widely used in the future! It is indeed gratifying to see that the DCT is now essentially a standard in the area of image data compression via transform coding techniques”. A ﬂeeting glimpse at the literature reveals that Ahmed’s claim is not unsubstantiated. The cosine transform has found applications in: • algorithmics and computational complexity theory ([5, 7, 11, 20, 23, 30, 31, 32, 35, 37, 40]), • data compression (see [22]), • image compression and JPEG format (see [1, 3, 39]), • signal processing (see [25]). We believe that such broad interest in the subject of cosine transform is not coincidental and reﬂects the fact that the topic is still within the scope of active mathematical research.

1.3 Layout of the paper In this ﬁnal part of the introduction we summarize the structure of the paper. The goal of this quick outline is to facilitate the comprehension of the “big picture” before delving into technical subtleties. Section 2 begins with the deﬁnition of the cosine convolution and what follows is an investigation of its basic features like the d’Alambert property (see Theorem 2). We proceed with establishing a bijection 1 βG between the structure space ∆ L G , c and the cosine class COS G (see Theorem 4). Section 3 adds a topological layerp top q our‹ q analysis – by topologizing thep q set COS G with the topology p q of uniform convergence on compacta we discover that βG is in fact an open map. The section goes on to prove that COS G is homeomorphic to p q • R 0 if G R, ` Yt u “ • S1 1 if G Z, where ` Yt u “ S1 : z S1 : Im z 0 , ` “ " P p qą *

1 • N0 if G S , “ • Z n`1 if G Zn (x rxs is the ceiling function). r 2 s “ ÞÑ 11See [4]. 12DCT stands for “discrete cosine transform”.

4 The focal point of Section 4 is the computation of particular instances of (what we call) cosine structure 1 1 spaces. It turns out (see Theorems 11, 12, 13 and 14) that if G R, Z,S or Zn then βG : ∆ L G , c COS G is continuous (and thus a homeomorphism in view of“ what we said earlier). The sectionp p concludesq ‹ q ÝÑ withp bringingq all the pieces of the puzzle as we witness the Gelfand transform manifest itself in the form of the cosine transforms. Our ﬁnal remarks take the form of an Epilogue, in which we indicate a possible direction of future research. Naturally, the paper concludes with a bibliography.

2 Cosine convolution

1 1 We begin our story with the definition of the cosine convolution operator c : L G L G C with the formula: ‹ p qˆ p q ÝÑ g x y g x y xPG f c g x : f y p ` q` p ´ q dy. (2) @ ‹ p q “ żG p q ¨ 2 The question that immediately springs to mind when looking at formula (2) is whether the cosine convo- 1 lution is well-defined? In other words, given two functions f,g L G does f c g x make sense for at P p q ‹ p q least some values x G? The answer is (luckily) affirmative and in fact, f c g defines a multiplication on L1 G turning it intoP a Banach algebra: ‹ p q 1 Theorem 1. If f,g L G then f c g x is well-defined (i.e., finite for almost every x G) and we have P p q ‹ p q P f c g 1 f 1 g 1. } ‹ } ď} } } } 1 Consequently, L G , c is a Banach algebra. p p q ‹ q We have taken the liberty of omitting the proof of this result as it is almost a verbatim rewrite of the proof of Theorem 1.6.2 in [12], p. 26 where Deitmar shows (with painstaking precision) that L1 G is a Banach algebra under the standard convolution defined by (1). Whoever shall read Deitmar’s reasoningp q will surely discover that his proof works equally‹ well for the anticonvolution:

xPG f a g x : f y g x y dy. @ ‹ p q “ żG p q p ` q

‹`‹a Thus, if we view the cosine convolution c as an “arithmetic mean of convolutions”, i.e., 2 then Theorem 1 becomes trivial and we shall waste no‹ more time discussing it. Amongst all properties of the cosine convolution we focus on the one bearing a close resemblance with the classical d’Alambert functional equation φ x y φ x y x,y R φ x φ y p ` q` p ´ q, @ P p q p q“ 2 whose continuous and bounded (and nonzero) solutions are the functions x cos ωx , ω R.13 After all, it is hardly surprising that the cosine convolution should have so muchÞÑ in commonp q withP the cosine function itself! Without further ado, here is the d’Alambert property of the cosine convolution:

13For completeness we should mention that if we allow for unbounded (yet still continuous) solutions then the family of functions x ÞÑ coshpωxq, ω P R also satisﬁes the classical d’Alambert functional equation. What is more, apart from the zero function and the two families x ÞÑ cospωxq, x ÞÑ coshpωxq, ω P R there are no other solutions.

5 Theorem 2. Let x, y G. If g L1 G is an even function, then P P p q Lx`yg Lx´yg Lyg c Lxg g c ` , (3) ‹ “ ‹ 2

1 1 where for every z G the operator Lz : L G L G is given by P p q ÝÑ p q

u G Lzf u : f u z . @ P p q “ p ´ q

Proof. First we note that for every u G we have P vÞÑv`y Lyg v Lxg u v dv g v Lxýg u v dv, ż p q p ` q “ ż p q p ` q G G (4) vÞÑ´v`y Lyg v Lxg u v dv g v Lxýg u v dv g v Lxýg u v dv, żG p q p ` q “ żG p´ q p ´ q “ żG p q p ´ q and analogously

vÞÑv`y Lyg v Lxg u v dv g v Lx`yg u v dv, ż p q p ´ q “ ż p q p ´ q G G (5) vÞÑ´v`y Lyg v Lxg u v dv g v Lx`yg u v dv g v Lx`yg u v dv. żG p q p ´ q “ żG p´ q p ` q “ żG p q p ` q Summation of equations (4) (and division by 2) yields

Lxýg u v Lxýg u v uPG Lyg v Lxg u v dv g v p ` q` p ´ q dv g c Lxýg u , (6) @ żG p q p ` q “ żG p q ¨ 2 “ ‹ p q while the summation of equations (5) (and again division by 2) yields

Lx`yg u v Lx`yg u v uPG Lyg v Lxg u v dv g v p ` q` p ´ q dv g c Lx`yg u . (7) @ żG p q p ´ q “ żG p q ¨ 2 “ ‹ p q Finally, summation of equations (6) and (7) (and division by 2 for the last time) produces the desired d’Alambert property (3). With the d’Alambert property of the cosine convolution in our toolbox we begin to investigate the 1 1 structure space ∆ L G , c . Our ﬁrst goal is to prove that ∆ L G , c is in bijective correspondence with the cosine classp p q ‹ q p p q ‹ q

b φ x y φ x y COS G : φ C G : φ 0, x,y G φ x φ y p ` q` p ´ q . p q “ " P p q ‰ @ P p q p q“ 2 *

In order to demonstrate this bijection, we will use the following lemma:

1 1 Lemma 3. For every m ∆ L G , c there exists an even function g L G such that m g 0. P p p q ‹ q ˚ P p q p ˚q‰

6 Proof. Let ι : G G be the inverse function ι x : x. Then ÝÑ p q “´ f x y f x y fPL1pGq f ι c f x f ι y p ` q` p ´ q dy @ p ˝ q‹ p q“ żG ˝ p q ¨ 2 yÞÑ´y f x y f x y f y p ´ q` p ` q dy f c f, “ żG p q ¨ 2 “ ‹ which leads to 2 1 m f ι m f m f ι c f m f c f m f . (8) @fPL pGq p ˝ q p q“ pp ˝ q‹ q“ p ‹ q“ p q 1 1 Since m is nonzero by the deﬁnition of ∆ L G , c , then there must exist a function f L G such p p q ‹ q ˚ P p q that m f˚ 0. Consequently, equations (8) yield m f˚ ι m f˚ . Finally, we put g˚ : f f ι and observep thatq‰ p ˝ q“ p q “ ` ˝ m g m f f ι m f m f ι 2m f 0, p ˚q“ p ˚ ` ˚ ˝ q“ p ˚q` p ˚ ˝ q“ p ˚q‰ which concludes the proof.

1 We are now in position to prove a bijective correspondence between ∆ L G , c and COS G : p p q ‹ q p q 1 Theorem 4. If mφ : L G C is given by p q ÝÑ

mφ f : f x φ x dx (9) p q “ żG p q p q 1 1 for some φ COS G , then mφ ∆ L G , c . Furthermore, for every m ∆ L G , c there exists a P p q P p p q ‹ q P p p q ‹ q unique φ COS G such that m mφ. P p q “ Proof. Obviously, if mφ is given by formula (9) then it is a linear functional, which satisﬁes

1 mφ f φ f 1, @fPL pGq | p q|ď} }8} } where 8 stands for the supremum norm. The inequality above means that mφ is a bounded (and thus }¨} 1 1 continuous) functional on L G . Thus, in order to conclude that mφ ∆ L G , c we simply need to p q P p p q ‹ q demonstrate the muliplicativity of mφ. We have

f,gPL1pGq mφ f c g f c g x φ x dx @ p ‹ q“ żG ‹ p q p q g x y g x y f y p ` q` p ´ q dy φ x dx “ żG ˆ żG p q ¨ 2 ˙ p q g x y g x y f y p ` q` p ´ q φ x dxdy “ żG żG p q ¨ 2 ¨ p q 1 1 f y g x y φ x dxdy f y g x y φ x dxdy. “ 2 żG żG p q p ` q p q ` 2 żG żG p q p ´ q p q Applying the substition x x y to the ﬁrst double integral and the substitution x x y to the second double integral in the lastÞÑ line´ we obtain ÞÑ ` φ x y φ x y f,gPL1pGq mφ f c g f y g x p ´ q` p ` q dxdy @ p ‹ q“ żG żG p q p q ¨ 2 φPCOSpGq f y g x φ x φ y dxdy mφ f mφ g . “ żG żG p q p q p q p q “ p q p q

7 1 This implies that mφ ∆ L G , c and concludes the first part of the proof. P p p q ‹ q 1 For the second part of the proof, we fix an element m ∆ L G , c and seek to show that there P p p q ‹ q 1 exists a unique φ COS G such that m mφ. By Lemma 3 there exists an even function g˚ L G such that m g P0. Moreover,p q by Theorem“ 2 we know that P p q p ˚q‰ Lx`yg˚ Lxýg˚ x,y G Lyg c Lxg g c ` . @ P ˚ ‹ ˚ “ ˚ ‹ 2 Applying the functional m to this equation and using its linearity and multiplicativity we obtain

m Lx`yg˚ m Lx´yg˚ x,y G m Lyg m Lxg m g p q` p q. (10) @ P p ˚q p ˚q“ p ˚q ¨ 2 Next, we deﬁne the function φ : G C by the formula ÝÑ m Lxg˚ x G φ x : p q, (11) @ P p q “ m g p ˚q which is • nonzero, because φ 0 1, p q“ • continuous by Lemma 1.4.2 in [12], p. 18, and

• bounded, because Lxg 1 g 1 for every x G. } ˚} “} ˚} P Dividing equation (10) by m g 2 we may rewrite it in the form p ˚q φ x y φ x y x,y G φ y φ x p ` q` p ´ q, @ P p q p q“ 2 which means that φ COS G . We have P p q φPCOSpGq φ x φ x fPL1pGq f x φ x dx f x p q` p´ q dx @ żG p q p q “ żG p q ¨ 2 (11) m Lxg m L xg f x p ˚q` p ´ ˚q dx “ ż p q ¨ 2m g˚ G p q 1 Lxg L xg m f x ˚ ` ´ ˚ “ m g˚ ¨ ˆż p q ¨ 2 ˙ p q G 1 1 m f c g m f m g m f , “ m g ¨ p ‹ ˚q“ m g ¨ p q p ˚q“ p q p ˚q p ˚q where the third equality holds true due to Lemma 11.45 in [6], p. 427 (or Proposition 7 in [15], p. 123). We have thus proved that m mφ. “ Finally, the fact that (11) is a unique φ such that m mφ follows from a technique, which is well-known “ in the ﬁeld of variational calculus: suppose that there exist φ1, φ2 COS G such that m mφ1 mφ2 . Consequently, we have P p q “ “

fPL1pGq f x φ1 x φ2 x dx 0. (12) @ żG p qˆ p q´ p q˙ “

Assuming the existence of an element x G such that φ1 x φ2 x , we can choose an open neigh- ˚ P p ˚q ‰ p ˚q bourhood U˚ of x˚ such that

8 • φ1 φ2 is of constant sign on U , and ´ ˚ • U˚ is compact (because we constantly work under the assumption that the group G is locally compact).

It remains to observe that for f˚ : φ1 φ2 1 we have “ p ´ q U˚ 2 (12) 0 f˚ x φ1 x φ2 x dx φ1 x φ2 x dx 0, “ żG p qˆ p q´ p q˙ “ żU˚ ˆ p q´ p q˙ ą

which is a contradiction. This means that φ1 φ2 and concludes the proof. “ 3 Structure spaces and cosine classes

To summarize the climactic point of the previous section, Theorem 4 establishes that the function 1 βG : ∆ L G , c COS G given by βG m : φ (where φ is a unique element of COS G such that p p q ‹ q ÝÑ p q p q “ p q m mφ) is a bijection. Our next task is to prove that βG is in fact more than just a bijection of two sets. Before“ we discuss the topology that we impose on the cosine class COS G let us examine its elements a little closer: p q

Theorem 5. For every φ COS G there exists a character χφ G such that P p q P

χφ x χφ x p x G φ x p q` p q . (13) @ P p q“ 2 In particular, φ 1. } }8 ď ˚ Proof. By Corollary 1 in [27] there exists a continuous homomorphism χφ : G C such that ÝÑ ´1 χφ x χφ x x G φ x p q` p q . (14) @ P p q“ 2

Furthermore, since φ is bounded, then so is χφ. We ﬁx y G and observe that the muliplicative property ˚ P x G χφ x χφ y χφ x y @ P p q p ˚q“ p ` ˚q implies sup χφ x χφ y˚ sup χφ x y˚ sup χφ z . xPG | p q|| p q| “ xPG | p ` q| “ zPG | p q|

By the ﬁniteness of sup χφ x sup χφ z we obtain χφ y 1. Since the element y was xPG | p q| “ zPG | p q| | p ˚q| “ ˚ chosen arbitrarily, then we have established that χφ is in fact a character, i.e., χφ G. Formula (13) ´1 P follows from (14) and the fact that χφ x χφ x for every x G. p q “ p q P p Knowing what the elements of COS G look like, we intend to show that βG is an open map if the cosine class is endowed with the properp topology.q 14 What do we mean by “proper”? Well, COS G is a subspace of Cb G so it is natural to endow it with the topology of uniform convergence on compactap q 15 p q τucc. Our next result conﬁrms that this is the right choice: 14 1 ˚ As we have already mentioned in the introductory section, ∆pL pGq, ‹cq is endowed with the weak* topology τ , which makes it a locally compact space. 15 For a detailed discussion on the properties of τucc see Chapter 7 in [28], Chapter 46 in [33] or Chapter 43 in [42].

9 1 ˚ Theorem 6. βG : ∆ L G , c , τ COS G , τucc is an open map. p p p q ‹ q q ÝÑ p p q q Proof. Our task is to prove that the image (under βG) of an arbitrary set

1 U : m ∆ L G , c : n 1,...,N m fn mφ˚ fn ε , “ " P p p q ‹ q @ “ | p q´ p q| ă *

1 N 1 where ε 0, mφ˚ ∆ L G , c and fn n“1 L G are ﬁxed, is τucc open. We ﬁx φ˚˚ βG U , which meansą that thereP p existsp qδ‹ q0, 1 suchp q thatĂ p q ´ P p q P p q

n 1,...,N mφ˚˚ fn mφ˚ fn δε. (15) @ “ | p q´ p q| ă Furthermore, we pick K to be a compact subset of G such that 1 δ ε n“1,...,N fn x dx p ´ q . (16) @ żGzK | p q| ď 4 Last but not least, we put 1 δ ε V : φ COS G : sup φ x φ˚˚ x p ´ q , “ " P p q x K | p q´ p q| ă 2maxn 1,...,N fn 1 * P “ } } which is a τucc open neighbourhood of φ . Finally, we calculate that ´ ˚˚

fn x φ x φ˚˚ x dx fn x φ x φ˚˚ x dx δε ď żK | p q|| p q´ p q| ` żGzK | p q|| p q´ p q| ` Lemma 5 fn x φ x φ˚˚ x dx 2 fn x dx δε ď żK | p q|| p q´ p q| ` żGzK | p q| ` (16) 1 δ ε fn x φ x φ˚˚ x dx p ` q ď żK | p q|| p q´ p q| ` 2 φPV 1 δ ε 1 δ ε p ´ q fn x dx p ` q ε. ă 2maxn“1,...,N fn 1 ¨ ż | p q| ` 2 ď } } K We conclude that V is a τucc open neighbourhood of (an arbitrarily chosen) φ and V βG U . Thus ´ ˚˚ Ă p q βG is an open map.

The question of continuity of βG is more subtle. In fact, we do not know whether this function is continuous for an arbitrary locally compact abelian group G, but it turns out to be true for very important particular cases. We will come back to this issue in the next section. However, before we do that we wish to investigate the cosine classes a little further. 16 1 Our goal is to “compute” the cosine classes COS G if G R, Z,S and Zn. We will refer to these p q “ 1 families as the canonical cosine classes since the four groups R, Z,S and Zn play a fundamental role in commutative harmonic analysis.

16By “computing” the cosine class COSpGq we mean “ﬁnding a (relatively simple) topological space T which is homeomorphic to COSpGq”.

10 It is well-known in the literature17 that

R x e2πiyx : y R , “ " ÞÑ P * p Z k zk : z S1 , “ " ÞÑ P * p S1 x e2πikx : k Z , “ " ÞÑ P *

x 2πilk Zn k e n : l Zn , “ " ÞÑ P * x so by Theorem 5 we have e2πiyx e´2πiyx COS R x ` cos 2πyx : y R 0 , p q“ " ÞÑ 2 “ p q P ` Yt u* zk z´k COS Z k ` : z S1 1 , where S1 : z S1 : Im z 0 , p q“ " ÞÑ 2 P ` Yt u* ` “ " P p qą * 2πikx ´2πikx 1 e e COS S x ` cos 2πkx : k N0 , p q“ " ÞÑ 2 “ p q P * 2πilk ´ 2πilk e n e n 2πlk COS Zn k ` cos : l Z n`1 . p q“ " ÞÑ 2 “ ˆ n ˙ P r 2 s*

We go on to prove that COS R , τucc is homeomorphic to R` 0 but ﬁrst we need the following technical lemma: p p q q Yt u

Lemma 7. If y˚ R` 0 and yn R` 0 is an unbounded sequence, then there exists a compact set K R such thatP theYt sequenceu p qĂ Yt u Ă n sup cos 2πynx cos 2πy˚x ÞÑ xPK | p q´ p q| does not converge to zero.

Proof. If y 0 then for any yn 1 the function x cos 2πynx has period Tn 1 so ˚ “ ě ÞÑ p q ď

yně1 sup cos 2πynx 1 2. @ xPr0,1s | p q´ |ě

1 Suppose therefore that y˚ 0 and put K : 0, . Then ‰ “r 8y˚ s ?2 x K cos 2πy x @ P p ˚ qě 2 1 whereas for any yn 4y˚ the function x cos 2πynx has period Tn and thus attains the value ě ÞÑ p q ď 4y˚ 1 on K. Consequently, we have ´ ?2 yně4y˚ sup cos 2πynx cos 2πy˚x 1 , @ xPK | p q´ p q| ě ` 2 which concludes the proof. 17See Proposition 7.1.6 in [12], p. 106 or Theorem 4.5 in [17], p. 89.

11 Theorem 8. The function HR : COS R , τucc R 0 given by the formula p p q q ÝÑ ` Yt u HR x cos 2πyx : y p ÞÑ p qq “ is a homeomorphism.

Proof. It is easy to check that HR is a bijection, so we focus on the topological properties of this function. By Proposition 1.2 in [24], p. 152 the space COS R , τucc is second-countable and thus, by Theorem p p q q 1.6.14 in [16], p. 53 it is sequential. This means that for HR to be a homeomorphism it is necessary and ´1 suﬃcient that HR satisﬁes

´1 ´1 yn y HR yn τ HR y . ÝÑ ˚ ðñ p q ÝÑ ucc p ˚q

First, suppose that yn R 0 is a sequence convergent to y . Then p qĂ ` Yt u ˚ yn xPR cos 2πynx cos 2πy˚x 2π x sin 2πyx dy 2π x yn y˚ , @ | p q´ p q| “ | | ˇż p q ˇ ď | || ´ | ˇ y˚ ˇ ˇ ˇ so for every compact K R we have ˇ ˇ Ă

sup cos 2πynx cos 2πy˚x 0 xPK | p q´ p q| ÝÑ

´1 ´1 as n . Consequently, HR yn τ HR y as desired. Ñ8 p q ÝÑ ucc p ˚q For the reverse implication (i.e., “ ”) we suppose that yn R` 0 is a sequence such that ´1 ´1 ðù p q Ă Yt u HR yn τ HR y for some y R 0 . By Lemma 7 the sequence yn is necessarily bounded, p q ÝÑ ucc p ˚q ˚ P ` Yt u p q so using the Bolzano-Weierstrass theorem there exists a convergent subsequence ynk . If y˚˚ denotes the ´1 p q ´1 limit of this subsequence, then by the ﬁrst part of the reasoning we have HR ynk τucc HR y˚˚ . Since ´1 ´1 p q ÝÑ p q τucc is a Hausdorﬀ topology then it follows that HR y HR y , which in turn implies the equality p ˚˚q“ p ˚q y˚ y˚˚. Since the reasoning works for an arbitary choice of the subsequence we have yn y˚, which concludes“ the proof. ÝÑ Let us prove a corresponding result for the cosine class COS Z : p q 1 Theorem 9. The function HZ : COS Z , τucc S 1 given by the formula p p q q ÝÑ ` Yt u zk z´k HZ k ` : z ˆ ÞÑ 2 ˙ “ is a homeomorphism. Proof. Arguing as in Theorem 8 it is enough to prove that

´1 ´1 zn z HZ zn τ HZ z . ÝÑ ˚ ðñ p q ÝÑ ucc p ˚q 1 First, suppose that zn S 1 is a sequence convergent to z . Then p qĂ ` Yt u ˚ k ´k k ´k k k ´k ´k ´1 ´1 zn zn z˚ z˚ zn z˚ zn z˚ zn z˚ zn z˚ |zn|“|z˚|“1 kPZ ˇ ` ` ˇ | ´ | ` | ´ | k | ´ | ` | ´ | k zn z˚ , @ ˇ 2 ´ 2 ˇ ď 2 ď ¨ 2 “ | ´ | ˇ ˇ ˇ ˇ ˇ ˇ 12 so for every compact (i.e., ﬁnite) K Z we have Ă k ´k k ´k zn zn z˚ z˚ sup ˇ ` ` ˇ 0 kPK ˇ 2 ´ 2 ˇ ÝÑ ˇ ˇ ˇ ˇ ´1 ˇ ´1 ˇ as n . Consequently, HZ zn τucc HZ z˚ as desired. Ñ8 p q ÝÑ p q 1 For the reverse implication (i.e., “ ”) we suppose that zn S` 1 is a sequence such that H´1 H´1 ðù 2πiα˚ 1 p q Ă Yt u 1 Z zn τucc Z z˚ for some z˚ e S` 1 . Moreover, let αn 0, 2 be such that p 2qπiα ÝÑ p q “ P Yt u p qĂr q zn e n . Since the sequence αn is bounded, then using the Bolzano-Weierstrass theorem there exists “ p q 2πiαnk a convergent subsequence αnk . If α˚˚ denotes the limit of this subsequence, then znk e 2πiα˚˚ p q ´1 ´1 “ ÝÑ z˚˚ e and by the ﬁrst part of the reasoning we have HZ znk τucc HZ z˚˚ . We conclude the proof“ just as in Theorem 8. p q ÝÑ p q In a similar vein (with even simpler proofs) one can prove analogous results for the remaining two cosine classes: Theorem 10. The functions

1 HS1 : COS S , τucc N0, HS1 x cos 2πkx : k, p p q q ÝÑ p ÞÑ p qq “ 2πlk HZ : COS Zn , τucc Z n`1 , HZ k cos : l, n p p q q ÝÑ r 2 s n ˆ ÞÑ ˆ n ˙˙ “ are homeomorphisms.

4 Canonical cosine structure spaces and transforms

In the previous section we have investigated the canonical cosine classes and found relatively simple 1 ˚ spaces to which they are homeomorphic. We have also shown (see Theorem 6) that βG : ∆ L G , τ p p q q ÝÑ COS G , τucc is always an open map. This raises a natural question: can we compute the canonical p p q q 1 1 1 1 1 18 cosine structure spaces ∆ L R , c , ∆ L Z , c , ∆ L S , c and ∆ L Zn , c ? A major part of the present section is devotedp p toq answering‹ q p p thisq ‹ questionq p p aﬃqrmatively.‹ q p p q ‹ q

1 ˚ Theorem 11. ∆ L R , c , τ is homeomorphic to R 0 . p p p q ‹ q q ` Yt u 1 ˚ Proof. By Theorem 6 we know that βR : ∆ L R , c , τ COS R , τucc is an open map and by p p p q ‹ q q ÝÑ p p q q Theorem 8 the function HR : COS R , τucc R 0 is a homeomorphism. Consequently, it suﬃces p p q q ÝÑ ` Yt u to prove that HR βR is continuous. To this end we ﬁx y˚ R` 0 as well as its arbitrary open neighbourhood ˝ P Yt u

Uε : y R 0 : y y ε “ " P ` Yt u | ´ ˚|ă * where ε 0. Our task is to prove that ą

´1 1 HR βR Uε m ∆ L R , c : HR βR m y ε p ˝ q p q“ " P p p q ‹ q | ˝ p q´ ˚|ă *

18 1 Again, by “compute” we mean “ﬁnd a topological space T , which is homeomorphic to ∆pL pGq, ‹cq.”

13 ´1 is weak* open and we do it by fixing an arbitrary element m HR βR Uε and constructing a ˚˚ P p ˝ q p q weak* open set W˚˚ such that ´1 m W HR βR Uε . ˚˚ P ˚˚ Ă p ˝ q p q ´1 To begin with, since m˚˚ HR βR Uε then y˚˚ : HR βR m˚˚ satisfies y˚˚ y˚ δε for some δ 0, 1 . Further reasoningP p depends˝ q onp whetherq y is“ zero˝ or not:p q | ´ | ă Pr q ˚˚ • If y 0 then we define a function g : R 0 R with the formula ˚˚ ‰ ` Yt u ÝÑ 1 π π g z : sinc z 1 . p q “ 2πy˚˚ ¨ ´ 2 ¨ ´ 2 ¨ ¯ ´ ¯

Its crucial property is that there exists η 0 such that ą 1 δ ε zPR`Yt0u g z η z 1 p ´ q . (17) @ | p q| ă ùñ | ´ |ă y˚˚

We claim that

1 W m L R , m 1 1 m 1 1 η ˚˚ : ∆ c : r0, 4 s ˚˚ r0, 4 s “ " P p p q ‹ q ˇ ´ y˚˚ ¯ ´ ´ y˚˚ ¯ˇ ă * ˇ ˇ ˇ ˇ is the desired weak* open neighbourhood of m˚˚. Indeed, we have

1 4y˚˚ 1 R m 1 1 m 1 1 cos 2πyx cos 2πy x dx mP∆pL p q,‹cq r0, 4y s ˚˚ r0, 4y s ˚˚ @ ´ ˚˚ ¯ ´ ´ ˚˚ ¯ “ ż0 p q´ p q π xÞÑ x 2 2πy˚˚ 1 y cos x cos x dx “ 2πy˚˚ ¨ ż0 ˆy˚˚ ¨ ˙ ´ p q 1 π π y y sinc 1 g , “ 2πy˚˚ ¨ ˆ 2 ¨ ˆ 2 ¨ y˚˚ ˙ ´ ˙ “ ˆy˚˚ ˙

H y where y R βR m . If m W˚˚ then g y η, so by (17) we have y y˚˚ 1 δ ε. “ ˝ p q P ˇ ´ ˚˚ ¯ˇ ă | ´ | ă p ´ q Finally, we have ˇ ˇ ˇ ˇ

m W y y y y y y δε 1 δ ε ε, @ P ˚˚ | ´ ˚| ď | ´ ˚˚| ` | ˚˚ ´ ˚|ă ` p ´ q “

´1 which proves that m W HR βR Uε . ˚˚ P ˚˚ Ă p ˝ q p q • If y˚˚ 0 then we deﬁne a function g : R` 0 R with the formula g z : sinc 2πz 1. Its crucial“ property is that there exists η 0 suchYt thatu ÝÑ p q “ p q´ ą R g z η z 1 δ ε. (18) @zP `Yt0u | p q| ă ùñ | | ă p ´ q We claim that

1 W : m ∆ L R , c : m 1 m 1 η ˚˚ “ " P p p q ‹ q r0,1s ´ ˚˚ r0,1s ă * ˇ ` ˘ ` ˘ˇ ˇ ˇ 14 is the desired weak* open neighbourhood of m˚˚. Indeed, we have 1 1 1 1 mP∆pL pRq,‹ q m r0,1s m˚˚ r0,1s cos 2πyx 1 dx sinc 2πy 1 g y . @ c ´ “ ż p q´ “ p q´ “ p q ` ˘ ` ˘ 0

If m W˚˚ then g y η, so by (18) we have y 1 δ ε. We conclude the reasoning as in the previousP case. | p q| ă | | ă p ´ q

1 ˚ 1 Theorem 12. ∆ ℓ Z , c , τ is homeomorphic to S 1 . p p p q ‹ q q ` Yt u Proof. As in Theorem 11 we argue that it is suﬃcient to prove that HZ βZ is continuous so we choose an arbitrary open neighbourhood ˝

1 Uε : z S 1 : z z ε “ " P ` Yt u | ´ ˚|ă * of a ﬁxed element z S1 1 . Our task is to prove that ˚ P ` Yt u

´1 1 HZ βZ Uε m ∆ ℓ Z , c : HZ βZ m z ε p ˝ q p q“ " P p p q ‹ q | ˝ p q´ ˚|ă *

´1 is weak* open and we do it by fixing an arbitrary element m HZ βZ Uε and constructing a weak* ˚˚ P p ˝ q p q open set W˚˚ such that ´1 m W HZ βZ Uε . ˚˚ P ˚˚ Ă p ˝ q p q ´1 To begin with, since m˚˚ HZ βZ Uε then z˚˚ : HZ βZ m˚˚ satisfies z˚˚ z˚ δε for some δ 0, 1 . We define a functionP pg : S˝1 q 1p q R with the“ formula˝ p q | ´ |ă P p q ` Yt u ÝÑ z z´1 z z´1 g z : ` ˚˚ ` ˚˚ . p q “ 2 ´ 2 Its crucial property is that there exists η 0 such that ą z S1 1 g z η z z˚˚ 1 δ ε. (19) @ P `Yt u | p q| ă ùñ | ´ | ă p ´ q We claim that 1 W : m ∆ ℓ Z , c : m 1 m 1 η ˚˚ “ " P p p q ‹ q t1u ´ ˚˚ t1u ă * ˇ ` ˘ ` ˘ˇ ˇ ˇ is the desired weak* open neighbourhood of m˚˚. Indeed, we have

k ´k k ´k z z z˚˚ z˚˚ 1 Z m 1 m 1 1 k ` ` @mP∆pℓ p q,‹cq t1u ´ ˚˚ t1u “ t1up q ¨ ˆ 2 ´ 2 ˙ ` ˘ ` ˘ kÿPZ z z´1 z z´1 ` ˚˚ ` ˚˚ g z , “ 2 ´ 2 “ p q

where z HZ βZ m . If m W˚˚ then g z η, so by (19) we have z z˚˚ 1 δ ε. Finally, we have “ ˝ p q P | p q| ă | ´ | ă p ´ q m W z z z z z z δε 1 δ ε ε, @ P ˚˚ | ´ ˚| ď | ´ ˚˚| ` | ˚˚ ´ ˚|ă ` p ´ q “ ´1 which proves that m W HZ βZ Uε . We conclude the reasoning as in Theorem 11. ˚˚ P ˚˚ Ă p ˝ q p q

15 1 1 ˚ Theorem 13. ∆ L S , c , τ is homeomorphic to N0. p p p q ‹ q q Proof. As in Theorem 11 we argue that it is suﬃcient to prove that HS1 βS1 is continuous. Since the ˝ topology on N0 is discrete, then we have to show that

´1 m : HS1 βS1 k t ˚u “ p ˝ q pt ˚uq is weak* open for every k N0. Further reasoning depends on whether k is zero or not: ˚ P ˚ • If k 0 then we deﬁne a function g : N0 R with the formula ˚ ‰ ÝÑ 1 π π k g k : sinc 1 . p q “ 2πk˚ ¨ ˆ 2 ¨ ˆ 2 ¨ k˚ ˙ ´ ˙

Its crucial property is that there exists η 0 such that ą

k N0 g k η k k . (20) @ P | p q| ă ùñ “ ˚ We claim that

1 1 W m L S , m 1 1 m 1 1 η ˚ : ∆ c : r0, 4 s ˚ r0, 4 s “ " P p p q ‹ q ˇ ´ k˚ ¯ ´ ´ k˚ ¯ˇ ă * ˇ ˇ ˇ ˇ is the desired weak* open neighbourhood of m˚. Indeed, we have

1 4k˚ 1 1 m 1 1 m 1 1 cos 2πkx cos 2πk x dx mP∆pL pS q,‹cq r0, 4k s ˚ r0, 4k s ˚ @ ´ ˚ ¯ ´ ´ ˚ ¯ “ ż0 p q´ p q π xÞÑ x 2 2πk˚ 1 k cos x cos x dx “ 2πk˚ ¨ ż0 ˆk˚˚ ¨ ˙ ´ p q 1 π π k sinc 1 g k , “ 2πk˚ ¨ ˆ 2 ¨ ˆ 2 ¨ k˚ ˙ ´ ˙ “ p q

where k HS1 βS1 m . If m W then g k η, so by (20) we have k k . This proves that “ ˝ p q P ˚ | p q| ă “ ˚ ´1 m W HS1 βS1 k . t ˚u“ ˚ “ p ˝ q pt ˚uq

• If k˚ 0 then we deﬁne a function g : N0 R with the formula g k : sinc 2πk 1. Its crucial property“ is that there exists η 0 such thatÝÑ p q “ p q´ ą

k N0 g k η k 0. (21) @ P | p q| ă ùñ “ We claim that 1 1 W : m ∆ L S , c : m 1S1 m 1S1 η ˚ “ " P p p q ‹ q | p q´ ˚ p q| ă *

is the desired weak* open neighbourhood of m˚. Indeed, we have

1 1 1 1 1 1 1 mP∆pL pS q,‹cq m S m˚ S cos 2πkx 1 dx sinc 2πk 1 g k . @ p q´ p q“ ż0 p q´ “ p q´ “ p q

16 If m W then g k η, so by (21) we have k 0. This proves that P ˚ | p q| ă “ ´1 m W HS1 βS1 0 , t ˚u“ ˚ “ p ˝ q pt uq which concludes the proof.

1 ˚ Theorem 14. ∆ ℓ Zn , c , τ is homeomorphic to Z n`1 , where x rxs is the ceiling function. p p p q ‹ q q r 2 s ÞÑ Proof. As in Theorem 13 our task is to prove that

´1 m : HZ βZ k t ˚u “ p n ˝ n q pt ˚uq

is a weak* open neighbourhood of m˚ for every k˚ Zn. We deﬁne a function g : Z n`1 R with the P r 2 s ÝÑ formula 2πk 2πk g k : cos cos ˚ . p q “ ˆ n ˙ ´ ˆ n ˙ Its crucial property is that there exists η 0 such that ą

k Z g k η k k . (22) @ P n | p q| ă ùñ “ ˚ We claim that 1 W : m ∆ ℓ Zn , c : m 1 m 1 η ˚ “ " P p p q ‹ q t1u ´ ˚ t1u ă * ˇ ` ˘ ` ˘ˇ ˇ ˇ is the desired weak* open neighbourhood of m˚. Indeed, we have

2πk 2πk˚ 1 Z m 1 m 1 cos cos g k , @mP∆pℓ p nq,‹cq t1u ´ ˚ t1u “ ˆ n ˙ ´ ˆ n ˙ “ p q ` ˘ ` ˘ where k HZ βZ m . If m W then g k η, so by (22) we have k k . This proves that “ n ˝ n p q P ˚ | p q| ă “ ˚ ´1 m W HZ βZ k , t ˚u“ ˚ “ p n ˝ n q pt ˚uq which concludes the proof. The last four theorems can be summarized as follows:

1 Theorem 15. The cosine structure space ∆ L G , c is homeomorphic to: p p q ‹ q • R 0 if G R, ` Yt u “ • S1 1 if G Z, ` Yt u “ 1 • N0 if G S , “ • Z n`1 if G Zn. r 2 s “

It is high time we reaped what we have sown and enjoyed the fruits of our labour. Due to Theorem 15 we know that

17 1 • C0 ∆ L R , c is homeomorphic to C0 R 0 , p p p q ‹ qq p ` Yt uq 1 1 • C0 ∆ ℓ Z , c is homeomorphic to C0 S 1 , p p p q ‹ qq p ` Yt uq 1 1 • C0 ∆ L S , c is homeomorphic to C0 N0 , p p p q ‹ qq p q 1 1 r n` s • C0 ∆ ℓ Zn , c is homeomorphic to C0 Z n`1 C Z n`1 C 2 . p p p q ‹ qq p r 2 sq“ p r 2 sq“ 1 1 Hence, the Gelfand transform f C0 ∆ L G , c of a function f L G manifests itself as P p p p q ‹ qq P p q • the classical cosine transformp

yPR`Yt0u f y f x cos 2πyx dx, @ p q“ żR p q p q p if G R, “ • the discrete-time cosine transform zk z´k z S1 1 f z f k ` , @ P `Yt u p q“ p q ¨ 2 kÿPZ p if G Z, “ • the k th cosine coeﬃcient in the Fourier series ´ 1

kPN0 f k f x cos 2πkx dx, @ p q“ ż0 p q p q p if G S1, “ • the discrete cosine transform n 2πlk lPZ n`1 f l f k cos , @ r 2 s p q“ p q ˆ n ˙ kÿ“1 p if G Zn. “ Epilogue

Our journey has come to an end and it is instructive to pause one last time and, with the benefit of hindsight, reflect on how far we have travelled and what lies ahead. Last section taught us that 1 1 ∆ L G , c is (homeomorphic to) a relatively simple topological space if G R, Z,S or Zn. Asa result,p p weq ‹ rediscoveredq the cosine transforms as special manifestations of the Gelfand“ transform. However, one would be wrong thinking that the topic has been exhausted. Four homeomorphisms, which appear 1 in Theorem 15, force all four functions βR,βZ,βS and βZn to be homeomorphisms as well. This raises 1 the very natural question: is it true that βG : ∆ L G , c COS G is a homeomorphism for every locally compact abelian group G? Unfortunately,p despitep q ‹ ourq ÝÑ best effortsp q we were not able to answer that question. Thus, we leave it as an open problem with the intention of stimulating future research in the fascinating field of cosine transforms.

18 References

[1] Agui T., Arai Y., Nakajima M. : A fast DCT-SQ scheme for images, Trans. IEICE, Vol. 71 (11), p. 1095-1097 (1988) [2] Ahmed N. : How I Came Up with the Discrete Cosine Transform, Digit. Signal Process., Vol. 1, p. 4-5 (1991) [3] Ahmed N., Magotra N., Mandyam G. : Lossless Image Compression Using the Discrete Cosine Trans- form, J. Vis. Commun. Image R., Vol. 8 (1), p. 21-26 (1997) [4] Ahmed N., Natarajan T., Rao R. K. : Discrete cosine transform, IEEE T. Comput., Vol. C-23 (1), p. 90-93 (1974) [5] Akopian D., Astola J., Saarinen J., Takala J. : Constant geometry algorithm for discrete cosine transform, IEEE Trans. Signal Process., Vol. 48 (6), p. 1840-1843 (2000) [6] Aliprantis C. D., Border K. C. : Infinite Dimensional Analysis. A Hitchhiker’s Guide, Springer-Verlag, Berlin, 2006 [7] Arguello F., Zapata E. L. : Fast cosine transform based on the successive doubling method, Electronic Lett., Vol. 26 (19), p. 1616-1618 (1990) [8] Bobrowski A. : Functional Analysis for Probability and Stochastic Processes, Cambridge University Press, Cambridge, 2005 [9] Brezis H. : Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2010 [10] Cartan, H. : Sur la mesure de Haar, C. R. Math. Acad. Sci. Paris, Vol. 211, p. 759-762 (1940) [11] Chen W. H., Fralick S. C., Smith C. H. : A fast computational algorithm for the discrete cosine transform, IEEE Trans. Comm., Vol. 25 (9), p. 1004-1009 (1977) [12] Deitmar A. : A First Course in Harmonic Analysis, Springer-Verlag, New York, 2005 [13] Deitmar A., Echterhoff S. : Principles of Harmonic Analysis, Springer, New York, 2009 [14] Diestel J., Spalsbury A. : The Joys of Haar measure, American Mathematical Society, Providence, 2014 [15] Dinculeanu N. : Vector measures, Pergamon Press, Berlin, 1967 [16] Engelking R. : General Topology, Heldermann Verlag, Berlin, 1989 [17] Folland G. B. : A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, 1995 [18] Fourier J. B. : The analytical theory of heat, Cambridge University Press, Cambridge, 2009 [19] Gohberg I., Goldberg S., Kaashoek M. A. : Classes of Linear Operators, Vol. 1, Springer, Basel, 1990 [20] Guo Z., Shi B., Wang N. : Two new algorithms based on product system for discrete cosine transform, Signal Process., Vol. 81, p. 1899-1908 (2001)

19 [21] Haar, A. : Der Massbegriﬀ in der Theorie der kontinuierlichen Gruppen, Ann. Math., Vol. 34 (1), p. 147-169 (1933) [22] Hoﬀman R. : Data Compression in Digital Systems, Springer Science+Business Media Dordrecht, 1997 [23] Hou H. S. : A fast algorithm for computing the discrete cosine transform, IEEE Trans. Acoust. Speech Signal Process., Vol. 35 (10), p. 1455-1461 (1987) [24] Hu Sz.-T. : Elements of general topology, Holden-Day, San Francisco, 1969 [25] Johnson S. G., Shao X. : Type-II/III DCT/DST algorithms with reduced number of arithmetic operations, Signal Process., Vol. 88 (6), p. 1553-1564 (2008) [26] Kaniuth E. : A Course in Commutative Banach Algebras, Springer-Verlag, New York, 2009 [27] Kannappan P. : The functional equation f xy f xy´1 2f x f y for groups, Proc. Am. Math. Soc., Vol. 19, p. 69-74 (1968) p q` p q“ p q p q [28] Kelley J. L. : General Topology, Springer-Verlag, New York, 1975 [29] Kok C. W. : Fast algorithm for computing discrete cosine transform, IEEE Trans. Signal Process., Vol. 45 (3), p. 757-760 (1997) [30] Lee B. G. : A new algorithm to compute the discrete cosine transform, IEEE Trans. Acoust. Speech Signal Process., Vol. 32 (6), p. 1243-1245 (1984) [31] Li W. : A new algorithm to compute the DCT and its inverse, IEEE Trans. Signal Process., Vol. 39 (6), p. 1305-1313 (1991) [32] Miller W. C., Tseng B. D. : On computing the discrete cosine transform, IEEE Trans. Comput., Vol. 27 (10), p. 966-968 (1978) [33] Munkres J. R. : Topology, Prentice Hall, Upper Saddle River, 2000 [34] Narasimha M. J., Peterson A. M. : On the computation of the discrete cosine transform, IEEE Trans. Comm., Vol. 26 (6), p. 934-936 (1978) [35] Nussbaumer H. J., Vetterli M. : Simple FFT and DCT algorithms with reduced number of operations, Signal Process., Vol. 6 (4), p. 267-278 (1984) [36] Pedersen G. K. : Analysis Now, Springer-Verlag, New York, 1989 [37] Plonka G., Tasche M. : Fast and numerically stable algorithms for discrete cosine transforms, Linear Algebra Appl., Vol. 394, p. 309-345 (2005) [38] Terras A. : Fourier Analysis on Finite Groups and Applications, Cambridge University Press, Cam- bridge, 1999 [39] Wallace G. K. : The JPEG still picture compression standard, Commun. ACM, Vol. 34 (4), p. 30-44 (1991)

20 [40] Wang Z. : A fast algorithm for the discrete sine transform implemented by the fast cosine transform, IEEE Trans. Acoust. Speech Signal Process., Vol. 30 (5), p. 813-815 (1982) [41] Weil A. : L’intégration dans les groupes topologiques et ses applications, Hermann, Paris, 1965 [42] Willard S. : General topology, Addison-Wesley Publishing Company, Reading, 1970