scispace - formally typeset
Search or ask a question

Showing papers on "Space (mathematics) published in 2003"


Book
01 Jan 2003
TL;DR: As one of the part of book categories, introduction to operator space theory always becomes the most wanted book.
Abstract: The theory of operator spaces is very recent and can be described as a non-commutative Banach space theory. An 'operator space' is simply a Banach space with an embedding into the space B(H) of all bounded operators on a Hilbert space H. The first part of this book is an introduction with emphasis on examples that illustrate various aspects of the theory. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C*-algebras. The third (and shorter) part of the book describes applications to non self-adjoint operator algebras, and similarity problems. In particular the author's counterexample to the 'Halmos problem' is presented, as well as work on the new concept of 'length' of an operator algebra. Graduate students and professional mathematicians interested in functional analysis, operator algebras and theoretical physics will find that this book has much to offer.

817 citations


Journal ArticleDOI
TL;DR: In this article, the moduli space of pairs (C,ω) is considered, where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeros.
Abstract: Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe connected components of this space. This classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.

502 citations


Journal ArticleDOI
TL;DR: In this article, a central limit theorem for Frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds.
Abstract: This article develops nonparametric inference procedures for estimation and testing problems for means on manifolds. A central limit theorem for Frechet sample means is derived leading to an asymptotic distribution theory of intrinsic sample means on Riemannian manifolds. Central limit theorems are also obtained for extrinsic sample means w.r.t. an arbitrary embedding of a differentiable manifold in a Euclidean space. Bootstrap methods particularly suitable for these problems are presented. Applications are given to distributions on the sphere Sd (directional spaces), real projective space ℝPN−1 (axial spaces), complex projective space ℂPk−2 (planar shape spaces) w.r.t. Veronese–Whitney embeddings and a three-dimensional shape space Σ34.

434 citations


Journal ArticleDOI
TL;DR: In this article, a supersymmetric string compactification to 4D Minkowski space is proposed, which involves type II strings propagating on (orientifolds of) non Calabi-Yau spaces in the presence of background NS and RR fluxes.
Abstract: We describe a new class of supersymmetric string compactifications to 4d Minkowski space. These solutions involve type II strings propagating on (orientifolds of) non Calabi-Yau spaces in the presence of background NS and RR fluxes. The simplest examples have descriptions as cosets, generalizing the three-dimensional nilmanifold. They can also be thought of as twisted tori. We derive a formula for the (super) potential governing the light fields, which is generated by the fluxes and certain "twists" in the geometry. Detailed consideration of an example also gives strong evidence that in some cases, these exotic geometries are related by smooth transitions to standard Calabi-Yau or G(2) compactifications of M-theory.

408 citations


Proceedings ArticleDOI
03 Dec 2003
TL;DR: An overview of colour spaces used in electrical engineering and image processing can be found in this paper, where the authors stress the importance of the perceptual, historical and applicational background that led to a colour space.
Abstract: In this paper, we present and overview of colour spaces used in electrical engineering and image processing. We stress the importance of the perceptual, historical and applicational background that led to a colour space. The colour spaces presented are: RGB, opponent-colour spaces, phenomenal colour spaces, CMY, CMYK, TV colour spaces (YUV and YIQ), PhotoYCC, CIE XYZ, Lab and Luv colour spaces.

370 citations


Book
04 Dec 2003
TL;DR: In this paper, the theory of classical electrodynamics is used to unify space observations of large-scale electrodynamic phenomena observed in space; formal equations and definitions; basic properties of stationary electromagnetic fields in the absence of local currents; motions of single particles; and the physics of a collection of particles treated as a fluid.
Abstract: Electrodynamic phenomena in space are reviewed in a systematic manner using the theory of classical electrodynamics to unify space observations Topics discussed include large-scale electrodynamic phenomena observed in space; formal equations and definitions; basic properties of stationary electromagnetic fields in the absence of local currents; motions of single particles; and the physics of a collection of particles treated as a fluid Further, the discussion focuses on plasma convective phenomena in the magnetosphere; interaction of conducting fluids and magnetic fields; the physics of boundaries; wave generation in MHD fluids; the physics of shocks; and instabilities in space

330 citations


Journal ArticleDOI
TL;DR: In this paper, the maximally supersymmetric solutions of the eleven and ten-dimensional supergravity theories were classified up to local isometry, and it was shown that the AdS solutions, the Hpp-waves and the flat space solutions exhaust them.
Abstract: We classify (up to local isometry) the maximally supersymmetric solutions of the eleven- and ten-dimensional supergravity theories. We find that the AdS solutions, the Hpp-waves and the flat space solutions exhaust them.

251 citations


Journal ArticleDOI
TL;DR: In this article, mental representations of space are constructions based on elements, the things in space, and the spatial relations among them relative to a reference frame, and different elements and spatial relations are central for functioning in different spaces, yielding different mental representations.
Abstract: Human activity takes place in space. To act effectively, people need mental representations of space. People’s mental representations of space differ from space as conceived of by physicists, geometers, and cartographers. Mental representations of space are constructions based on elements, the things in space, and the spatial relations among them relative to a reference frame. People act in different spaces depending on the task at hand. The spaces considered here are the space of the body, the space around the body, the space of navigation, and the space of graphics. Different elements and spatial relations are central for functioning in the different spaces, yielding different mental representations.

251 citations


01 Jan 2003
TL;DR: The U-Matrix, as defined here, is a display of the density relationships in the data space using Pareto Density Estimation and can be used for a non-ambiguous display of a non planar neuron space.
Abstract: -The U-Matrix is a canonical tool for the display of distance structures in data space using emergent SOM (ESOM). The U-Matrix defined originally for planar map spaces is extended in this work to toroid neuron spaces. Embedding the neuron space in a finite but borderless space, such as a torus, avoids border effects of planar spaces. A planar display of a toroid map space disrupts, however, coherent U-Matrix structures. Tiling multiple instances of the U-Matrix solves this problem at the cost of multiple images of data points. The P-Matrix, as defined here, is a display of the density relationships in the data space using Pareto Density Estimation. While the P-Matrix is useful for clustering, it can also be used for a non-ambiguous display of a non planar neuron space. Centering the display for high density regions and removing ambiguous images of data points leads to U-Maps and P-Maps. U-Maps depict the distance structure of a data space as a borderless three dimensional landscape whose floor space is ordered according to the topology preserving features of ESOM. P-Maps display the density structures. Both maps are specially suited for data mining and knowledge discovery.

218 citations


Journal ArticleDOI
TL;DR: In this article, Calderon-Zygmund singular integral operators have been studied in the context of discrete groups of dilations, and they have been shown to be an unconditional basis for the anisotropic Hardy space H A.
Abstract: In this paper, motivated in part by the role of discrete groups of dilations in wavelet theory, we introduce and investigate the anisotropic Hardy spaces associated with very general discrete groups of dilations. This formulation includes the classical isotropic Hardy space theory of Fefferman and Stein and parabolic Hardy space theory of Calderon and Torchinsky. Given a dilation A, that is an n × n matrix all of whose eigenvalues λ satisfy |λ| > 1, define the radial maximal function M φf(x) := sup k∈Z |(f ∗ φk)(x)|, where φk(x) = | detA|φ(Ax). Here φ is any test function in the Schwartz class with ∫ φ 6= 0. For 0 < p < ∞ we introduce the corresponding anisotropic Hardy space H A as a space of tempered distributions f such that M φf belongs to L (R). Anisotropic Hardy spaces enjoy the basic properties of the classical Hardy spaces. For example, it turns out that this definition does not depend on the choice of the test function φ as long as ∫ φ 6= 0. These spaces can be equivalently introduced in terms of grand, tangential, or nontangential maximal functions. We prove the Calderon-Zygmund decomposition which enables us to show the atomic decomposition of H A. As a consequence of atomic decomposition we obtain the description of the dual to H A in terms of Campanato spaces. We provide a description of the natural class of operators acting on H A, i.e., Calderon-Zygmund singular integral operators. We also give a full classification of dilations generating the same space H A in terms of spectral properties of A. In the second part of this paper we show that for every dilation A preserving some lattice and satisfying a particular expansiveness property there is a multiwavelet in the Schwartz class. We also show that for a large class of dilations (lacking this property) all multiwavelets must be combined minimally supported in frequency, and thus far from being regular. We show that r-regular (tight frame) multiwavelets form an unconditional basis (tight frame) for the anisotropic Hardy space H A. We also describe the sequence space characterizing wavelet coefficients of elements of the anisotropic Hardy space. 2000 Mathematics Subject Classification. Primary 42B30, 42C40; Secondary 42B20, 42B25.

213 citations


Journal ArticleDOI
TL;DR: 3. S. Thomson, Real Analysis, Prentice Hall, Upper Saddle River, NJ, 1997.
Abstract: (2003). When Does the Position Vector of a Space Curve Always Lie in Its Rectifying Plane? The American Mathematical Monthly: Vol. 110, No. 2, pp. 147-152.

Journal ArticleDOI
TL;DR: In this paper, it was shown that if a function f is in BV, its coefficient sequence in a normalized wavelet basis satisfies a class of weak-� 1 type estimates.
Abstract: We establish new results on the space BV of functions with bounded variation. While it is well known that this space admits no unconditional basis, we show that it is “almost” characterized by wavelet expansions in the following sense: if a function f is in BV, its coefficient sequence in a BV normalized wavelet basis satisfies a class of weak-� 1 type estimates. These weak estimates can be employed to prove many interesting results. We use them to identify the interpolation spaces between BV and Sobolev or Besov spaces, and to derive new Gagliardo-Nirenberg-type inequalities. 1. Background and main results

Journal ArticleDOI
TL;DR: In this article, the authors consider a metric measure space (M, d, p) and a heat kernel p t (x,v) on M satisfying certain upper and lower estimates, which depend on two parameters a and β.
Abstract: We consider a metric measure space (M, d, p) and a heat kernel p t (x,v) on M satisfying certain upper and lower estimates, which depend on two parameters a and β We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M,d,μ) Namely, a is the Hausdorff dimension of this space, whereas β, called the walk dimension, is determined via the properties of the family of Besov spaces W σ,2 on M Moreover, the parameters a and β are related by the inequalities 2 < β < a + 1 We prove also the embedding theorems for the space W β/2,2 , and use them to obtain the existence results for weak solutions to semilinear elliptic equations on M of the form -?u + f(x,u) = g(x), where? is the generator of the semigroup associated with p t The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpinski carpet in R n

Journal ArticleDOI
TL;DR: In this article, the authors generalize the results due to Eisenbud, Harris, and Mumford on moduli spaces of curves of genus g to the moduli space of curves with n marked points.
Abstract: The purpose of this paper is to generalize the methods and results due to Eisenbud, Harris, and Mumford on the moduli space of curves of genus g to the moduli space of curves of genus g with n marked points. The main result is the determination, for each g with 4 ≤ g ≤ 23, of an n such that the moduli space is of general type. 1. Introduction. In the early 1980s, Eisenbud, Harris, and Mumford stud- ied Mg, the moduli space of curves of a given genus, and succeeded in proving that if the genus is at least 24, the moduli space is of general type. This was contrary to many people's expectations, as it had been known for some time that moduli spaces of curves of small genus were rational or unirational. Briefly, what is involved in their proof is the computation of the canonical divisor of Mg, the exhibition of certain effective divisors and the determination of their classes, and the verification that the canonical divisor is the sum of an ample and an effective divisor. In the other direction, it is known that Mg has Kodaira dimension −∞ when g ≤ 13 or g = 15. One can generalize the definition of Mg by asking for a parametrization of the curves of genus g together with n distinct nonsingular points on them. This amounts to studying, instead of families of curves, families of curves with n disjoint sections. As before, it turns out that one obtains a noncompact moduli space, which is compactified by replacing "nonsingular curve" by "stable curve" in the definitions. In order to be considered stable, an n-pointed curve is required only to have finitely many automorphisms that fix all the marked points. The space can still be compact even when the marked points are required to be distinct and nonsingular, because coincident or singular marked points can be blown up on the total space of the family.

Journal ArticleDOI
TL;DR: A generalization of the quantum Hall effect where particles move in an eight-dimensional space under an SO(8) gauge field is constructed, where two fundamentally different liquids with distinct configuration spaces can be constructed, depending on whether the particles carry spinor or vector SO( 8) quantum numbers.
Abstract: We construct a generalization of the quantum Hall effect where particles move in an eight-dimensional space under an SO(8) gauge field. The underlying mathematics of this particle liquid is that of the last normed division algebra, the octonions. Two fundamentally different liquids with distinct configuration spaces can be constructed, depending on whether the particles carry spinor or vector SO(8) quantum numbers. One of the liquids lives on a 20-dimensional manifold with an internal component of SO(7) holonomy, whereas the second liquid lives on a 14-dimensional manifold with an internal component of G2 holonomy.

Journal ArticleDOI
01 Dec 2003-Calcolo
TL;DR: For these discretizations of the heat equation by A-stable θ-schemes in time and conforming finite elements in space, residual a posteriori error indicators are derived that yield upper bounds on the error and lower bounds that are global in space and time and local in time.
Abstract: We consider discretizations of the heat equation by A-stable θ-schemes in time and conforming finite elements in space. For these discretizations we derive residual a posteriori error indicators. The indicators yield upper bounds on the error which are global in space and time and yield lower bounds that are global in space and local in time. The ratio between upper and lower bounds is uniformly bounded in time and does not depend on any step-size in space or time. Moreover, there is no restriction on the relation between the step-sizes in space and time.

Journal ArticleDOI
TL;DR: In this article, the Sierpinski gasket and effective resistance metric of Kigami were used to construct function spaces for post-critically finite self-similar fractals.

Journal ArticleDOI
TL;DR: Theorem 1 of (B2) was shown to be not correct in this paper, and a necessary and suf- ficient topological condition under which the smooth maps are strongly dense in the Sobolev spaces was shown.
Abstract: This paper addresses some topological and analytical issues concern- ing Sobolev mappings between compact Riemannian manifolds. Among the results we obtained are unified proofs of various generalizations of results obtained in a recent work of Brezis and Li. In particular we solved two conjectures in (BL). We also give a topological obstruction for the weak sequential density of smooth maps in a given Sobolev mapping space. Finally we show a necessary and suf- ficient topological condition under which the smooth maps are strongly dense in the Sobolev spaces. The earlier result, Theorem 1 of (B2), was shown to be not correct.

Journal ArticleDOI
TL;DR: In this article, the shortest-time problem on a Riemannian space with an external force was studied, and it was shown that the problem can be converted to a shortest path problem on the Randers space.
Abstract: In the paper, we study the shortest time problem on a Riemannian space with an external force. We show that such problem can be converted to a shortest path problem on a Randers space. By choosing an appropriate external force on the Euclidean space, we obtain a non-trivial Randers metric of zero flag curvature. We also show that any positively complete Randers metric with zero flag curvature must be locally Minkowskian.

Journal ArticleDOI
TL;DR: For the group O(p,q) of conformal transformations on R p−1,q−1, the authors of as mentioned in this paper gave a new construction of its minimal unitary representation via Euclidean Fourier analysis.

Journal ArticleDOI
TL;DR: In this article, the authors consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the thresholding greedy algorithm and show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation.
Abstract: We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space X has a basis and contains a complemented subspace with a symmetric basis and finite cotype then X has an almost greedy basis. We show that c0 is the only L∞ space to have a quasi-greedy basis. The Banach spaces which contain almost greedy basic sequences are characterized.

Journal ArticleDOI
TL;DR: In this article, almost global existence for solutions of quadratically quasi-linear systems of wave equations in 3D space dimensions was shown for the non-obstacle case using only the classical invariance of the wave operator under translations, spatial rotations and scaling.
Abstract: This article studies almost global existence for solutions of quadratically quasi linear systems of wave equations in three space dimensions. The approach here uses only the classical invariance of the wave operator under translations, spatial rotations, and scaling. Using these techniques we can handle wave equations in Minkowski space or Dirichlet-wave equations in the exterior of a smooth, star shaped obstacle. We can also apply our methods to systems of quasilinear wave equations having different wave speeds. This extends our work [11] for the semilinear case. Previous almost global ex istence theorems for quasilinear equations in three space dimensions were for the non-obstacle case. In [9], John and Klainerman proved almost global existence on Minkowski space for quadratic, quasilinear equations using the Lorentz invariance of the wave operator in addition to the symmetries listed above. Subsequently, in [14], Klainerman and Sideris obtained the same result for a class of quadratic, divergence-form nonlinearities without relying on Lorentz invariance. This line of thought was refined and applied to prove global-in-time results for null-form equa tions related to the theory of elasticity in Sideris [22], [23], and for multiple-speed systems of null-form quasilinear equations in Sideris and Tu [24], and Yokoyama [29]. The main difference between our approach and the earlier ones is that we ex ploit the 0(|x|-1) decay of solutions of wave equations with sufficiently decaying initial data as much as we involve the stronger 0(t~l) decay. Here, of course, x = (x\,X2,x$) is the spatial component, and t the time component, of a space time vector (t, x) G M+ x E3. Establishing 0(|x|_1) decay is considerably easier and can be achieved using only the invariance with respect to translations and spatial rotation. A weighted L2 space-time estimate for inhomogeneous wave equations (Proposition 3.1 below, from [11]) is important in making the spatial decay useful for the long-time existence argument. For semilinear systems, one can show almost global existence from small data using only this spatial decay [11]. For quasilinear systems, however, we also have to show that both first and second derivatives of u decay like 1/t. Fortunately, we can do this using a variant of some L1 ?> L?? estimates of John, H?rmander,

Journal ArticleDOI
TL;DR: In this article, Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves, which is a consequence of an equivalence between the perturbative expansion of super Yang-Mills theory and the $D$-instanton expansion of a certain string theory.
Abstract: Perturbative scattering amplitudes in Yang-Mills theory have many unexpected properties, such as holomorphy of the maximally helicity violating amplitudes. To interpret these results, we Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves. This in turn is apparently a consequence of an equivalence between the perturbative expansion of ${\cal N}=4$ super Yang-Mills theory and the $D$-instanton expansion of a certain string theory, namely the topological $B$ model whose target space is the Calabi-Yau supermanifold $\Bbb{CP}^{3|4}$.

Posted Content
Peter Gacs1
TL;DR: In this article, a general framework for the analysis of randomness in non-compact spaces is introduced, where the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution.
Abstract: The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend the theory to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary distributions (by Levin). We recall the main ideas and problems of Levin's theory, and report further progress in the same framework. - We allow non-compact spaces (like the space of continuous functions, underlying the Brownian motion). - The uniform test (deficiency of randomness) d_P(x) (depending both on the outcome x and the measure P should be defined in a general and natural way. - We see which of the old results survive: existence of universal tests, conservation of randomness, expression of tests in terms of description complexity, existence of a universal measure, expression of mutual information as "deficiency of independence. - The negative of the new randomness test is shown to be a generalization of complexity in continuous spaces; we show that the addition theorem survives. The paper's main contribution is introducing an appropriate framework for studying these questions and related ones (like statistics for a general family of distributions).

Journal ArticleDOI
TL;DR: In this paper, the long time behavior of solutions of the system of thermoelasticity of type III in a bounded domain of ℝn (n = 1,2,3) and in the whole space of n is analyzed.
Abstract: This paper is devoted to analyzing the long time behavior of solutions of the system of thermoelasticity of type III in a bounded domain of ℝn (n = 1,2,3) and in the whole space ℝn. For the first case, we introduce a decoupled system that allows to reduce the problem of the asymptotic behavior for the original system to a suitable observability inequality for the Lame system. In this way most of the existing results for the classical system of thermoelasticity are shown to hold for this system too. In particular, we show that: (1) For most domains the energy of the system does not decay uniformly; (2) Under suitable conditions on the domain that may be described in terms of Geometric Optics, the energy of the system decays exponentially; and (3) For most domains in two space dimensions, the energy of smooth solutions decays polynomially. For the problem in the whole space ℝn, first, based on Fourier analysis and Lyapunov's second method, we show that the energy of longitudinal and thermal waves of the system decays as that of the classical heat equation (while that of the transversal wave component is conservative). Then, by means of a careful spectral analysis, we give a sharp description on the decay rate of the high frequency longitudinal and thermal waves of the system.

Journal ArticleDOI
TL;DR: In this article, the authors study the conformality problems associated with quasiregular mappings in space and propose an approach based on the concept of the infinitesimal space and some new Grotzsch-Teichmuller type modulus estimates that are expressed in terms of the mean value of the dilatation coefficients.
Abstract: We study the conformality problems associated with quasiregular mappings in space. Our approach is based on the concept of the infinitesimal space and some new Grotzsch-Teichmuller type modulus estimates that are expressed in terms of the mean value of the dilatation coefficients.

Journal ArticleDOI
TL;DR: In this paper, the authors have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis, where ''uniform'' here means independence from parameters in compact spaces.
Abstract: In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. `Uniform' here means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, normed linear spaces, uniformly convex spaces as well as inner product spaces.

Journal ArticleDOI
TL;DR: In this article, the Zamolodchikov c-theorem was used to demonstrate that the target space ADM energy of the UV fixed point is greater than that of the IR fixed point.
Abstract: We study renormalization group flows in unitary two dimensional sigma models with asymptotically flat target spaces. Applying an infrared cutoff to the target space, we use the Zamolodchikov c-theorem to demonstrate that the target space ADM energy of the UV fixed point is greater than that of the IR fixed point: spacetime energy decreases under world-sheet RG flow. This result mirrors the well understood decrease of spacetime Bondi energy in the time evolution process of tachyon condensation.

Book ChapterDOI
01 Jan 2003
TL;DR: In this article, the authors reveal the power of descriptive set theory in penetrating the structure of Banach spaces and give a strong flavor of this aspect of the Banach space theory.
Abstract: The goal of this chapter is to reveal the power of descriptive set theory in penetrating the structure of Banach spaces. The chapter is divided into three subchapters, each with its own introduction. Subchapters one, two and three were mostly written by the second, third, and first authors, respectively. Space limitations forced us to leave out many fundamental results and on-going research in this exciting interface. We do hope, however, that our article gives a strong flavor of this aspect of Banach space theory.

Journal ArticleDOI
TL;DR: In this article, the existence of an extended Chebyshev basis has been shown to be guaranteed for the Hermite interpolation problem in the space Cn generated by 1,t, …, tn-2, cos t, sin t.
Abstract: We analyze the connection between two ideas of apparently different nature. On one hand, the existence of an extended Chebyshev basis, which means that the Hermite interpolation problem has always a unique solution. On the other hand, the existence of a normalized totally positive basis, which means that the space is suitable for design purposes. We prove that the intervals where the existence of a normalized totally positive basis is guaranteed are those intervals where the existence of an extended Chebyshev basis of the space of derivatives can be ensured. We apply our results to the spaces Cn generated by 1,t, …, tn-2, cos t, sin t. In particular, C5 is a space suitable for design which permits the exact reproduction of remarkable parametric curves, including lines and circles with a single control polygon. We prove that this space has the minimal dimension for this purpose.