TY - JOUR AB - The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz–Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the q-dimensional sphere Sq, and investigate how well continuous Lp-norms of polynomials f of maximum degree n on the sphere Sq can be discretized by positively weighted Lp-sum of finitely many samples, and discuss the distortion between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points ξ1,…,ξN on Sq, the dimension q, and the degree n of the polynomials. AU - Filbir, F. AU - Hielscher, R.* AU - Jahn, T.* AU - Ullrich, T.* C1 - 70282 C2 - 55482 CY - 525 B St, Ste 1900, San Diego, Ca 92101-4495 Usa TI - Marcinkiewicz–Zygmund inequalities for scattered and random data on the q-sphere. JO - Appl. Comput. Harmon. Anal. VL - 71 PB - Academic Press Inc Elsevier Science PY - 2024 SN - 1063-5203 ER - TY - JOUR AB - We propose a signal analysis tool based on the sign (or the phase) of complex wavelet coefficients, which we call a signature. The signature is defined as the fine-scale limit of the signs of a signal's complex wavelet coefficients. We show that the signature equals zero at sufficiently regular points of a signal whereas at salient features, such as jumps or cusps, it is non-zero. At such feature points, the orientation of the signature in the complex plane can be interpreted as an indicator of local symmetry and antisymmetry. We establish that the signature rotates in the complex plane under fractional Hilbert transforms. We show that certain random signals, such as white Gaussian noise and Brownian motions, have a vanishing signature. We derive an appropriate discretization and show the applicability to signal analysis. AU - Storath, M.* AU - Demaret, L. AU - Massopust, P.* C1 - 46757 C2 - 37794 CY - San Diego SP - 199-223 TI - Signal analysis based on complex wavelet signs. JO - Appl. Comput. Harmon. Anal. VL - 42 IS - 2 PB - Academic Press Inc Elsevier Science PY - 2015 SN - 1063-5203 ER - TY - JOUR AB - Many current problems dealing with big data can be cast efficiently as function approximation on graphs. The information in the graph structure can often be reorganized in the form of a tree; for example, using clustering techniques. The objective of this paper is to develop a new system of orthogonal functions on weighted trees. The system is local, easily implementable, and allows for scalable approximations without saturation. A novelty of our orthogonal system is that the Fourier projections are uniformly bounded in the supremum norm. We describe in detail a construction of wavelet-like representations and estimate the degree of approximation of functions on the trees. AU - Chui, C.K.* AU - Filbir, F. AU - Mhaskar, H.N.* C1 - 31844 C2 - 34834 CY - San Diego SP - 489-509 TI - Representation of functions on big data: Graphs and trees. JO - Appl. Comput. Harmon. Anal. VL - 38 IS - 3 PB - Academic Press Inc Elsevier Science PY - 2014 SN - 1063-5203 ER - TY - JOUR AB - Fusion frames enable signal decompositions into weighted linear subspace components. For positive integers p, we introduce p-fusion frames, a sharpening of the notion of fusion frames. Tight p-fusion frames are closely related to the classical notions of designs and cubature formulas in Grassmann spaces and are analyzed with methods from harmonic analysis in the Grassmannians. We define the p-fusion frame potential, derive bounds for its value, and discuss the connections to tight p-fusion frames. AU - Bachoc, C.* AU - Ehler, M. C1 - 11801 C2 - 30817 SP - 1-15 TI - Tight p-fusion frames. JO - Appl. Comput. Harmon. Anal. VL - 35 IS - 1 PB - Academic Press Elsevier Science PY - 2013 SN - 1063-5203 ER - TY - JOUR AB - We investigate the reconstruction problem of limited angle tomography. Such problems arise naturally in applications like digital breast tomosynthesis, dental tomography, electron microscopy etc. Since the acquired tomographic data is highly incomplete, the reconstruction problem is severely ill-posed and the traditional reconstruction methods, such as filtered backprojection (FBP), do not perform well in such situations. To stabilize the reconstruction procedure additional prior knowledge about the unknown object has to be integrated into the reconstruction process. In this work, we propose the use of the sparse regularization technique in combination with curvelets. We argue that this technique gives rise to an edge-preserving reconstruction. Moreover, we show that the dimension of the problem can be significantly reduced in the curvelet domain. To this end, we give a characterization of the kernel of limited angle Radon transform in terms of curvelets and derive a characterization of solutions obtained through curvelet sparse regularization. In numerical experiments, we will present the practical relevance of these results. AU - Frikel, J. C1 - 7386 C2 - 29693 SP - 117-141 TI - Sparse regularization in limited angle tomography. JO - Appl. Comput. Harmon. Anal. VL - 34 IS - 1 PB - Elsevier PY - 2013 SN - 1063-5203 ER - TY - JOUR AB - The purpose of this article is to introduce a new class of kernels on SO(3) for approximation and interpolation, and to estimate the approximation power of the associated spaces. The kernels we consider arise as linear combinations of Green's functions of certain differential operators on the rotation group. They are conditionally positive definite and have a simple, closed-form expression, lending themselves to direct implementation via, e.g., interpolation or least-squares approximation. To gauge the approximation power of the underlying spaces, we introduce an approximation scheme providing precise Lp error estimates for linear schemes, namely with Lp approximation order conforming to the Lp smoothness of the target function. AU - Hangelbroeck, T.* AU - Schmid, D. C1 - 5559 C2 - 28410 SP - 169-184 TI - Surface spline approximation on SO(3). JO - Appl. Comput. Harmon. Anal. VL - 31 IS - 2 PB - Elsevier PY - 2011 SN - 1063-5203 ER - TY - JOUR AB - Based on a result of Rosier and Voit for ultraspherical polynomials, we derive an uncertainty principle for compact Riemannian manifolds M. The frequency variance of a function in L-2(M) is therein defined by means of the radial part of the Laplace-Beltrami operator. The proof of the uncertainty rests upon Dunkl theory. In particular, a special differential-difference operator is constructed which plays the role of a generalized root of the radial Laplacian. Subsequently, we prove with a family of Gaussian-like functions that the deduced uncertainty is asymptotically sharp. Finally, we specify in more detail the uncertainty principles for well-known manifolds like the d-dimensional unit sphere and the real projective space. AU - Erb, W. C1 - 5982 C2 - 27894 CY - San Diego SP - 182-197 TI - Uncertainty principles on compact Riemannian manifolds. JO - Appl. Comput. Harmon. Anal. VL - 29 IS - 2 PB - Acad. Press Inc. Elsevier Sci. PY - 2010 SN - 1063-5203 ER - TY - JOUR AB - Circular and spherical mean data arise in various models of thermoacoustic and photoacoustic tomography which are rapidly developing modalities for in vivo imaging. We describe variants of a summability type reconstruction method adapted to this type of data. Among the highlights of the resulting algorithms, suggested by the results of numerical experiments, is the feature that the detectors need not lie on a regular curve or surface, such as a circle or a sphere. Several such numerical examples are included here. AU - Filbir, F. AU - Hielscher, R. AU - Madych, W.R.* C1 - 4273 C2 - 28163 CY - San Diego SP - 111-120 TI - Reconstruction from circular and spherical mean data. JO - Appl. Comput. Harmon. Anal. VL - 29 IS - 1 PB - Acad. Press Inc. Elsevier Sci. PY - 2010 SN - 1063-5203 ER - TY - JOUR AB - We consider shift-invariant multiresolution spaces generated by rotation-covariant functions {rho} in R2. To construct corresponding scaling and wavelet functions, {rho} has to be localized with an appropriate multiplier, such that the localized version is an element of L2(R2).We consider several classes of multipliers and show a new method to improve regularity and decay properties of the corresponding wavelets. The wavelets are complex-valued functions, which are approximately rotation-covariant and therefore behave as Wirtinger differential operators. Moreover, our class of multipliers gives a novel approach for the construction of polyharmonic B-splines with better polynomial reconstruction properties. AU - Forster, B. AU - Blu, T.* AU - van de Ville, D.* AU - Unser, M.* C1 - 3748 C2 - 28270 SP - 240-265 TI - Shift-invariant spaces from rotation-covariant functions. JO - Appl. Comput. Harmon. Anal. VL - 25 IS - 2 PB - Elsevier PY - 2008 SN - 1063-5203 ER - TY - JOUR AU - Führ, H. AU - Wild, M. C1 - 3726 C2 - 23960 SP - 184-201 TI - Characterizing wavelet coefficient decay of discrete-time signals. JO - Appl. Comput. Harmon. Anal. VL - 20 PY - 2006 SN - 1063-5203 ER - TY - JOUR AB - We prove the boundedness of a general class of Fourier multipliers, in particular of the Hilbert transform, on modulation spaces. In general, however, the Fourier multipliers in this class fail to be bounded on Lp spaces. The main tools are Gabor frames and methods from time–frequency analysis. AU - Bényi, A.* AU - Grafakos, L.* AU - Gröchenig, K. AU - Okoudjou, K.* C1 - 645 C2 - 22813 SP - 131-139 TI - A class of Fourier multipliers for modulation spaces. JO - Appl. Comput. Harmon. Anal. VL - 19 IS - 1 PY - 2005 SN - 1063-5203 ER - TY - JOUR AB - The theory of localized frames is refined to include quasi-Banach spaces and spaces with multiple generators. Applications are given to nonlinear approximation with frames and to the convergence of the iterative frame algorithm in finer norms, and to the characterization of Besov spaces with wavelet frames. AU - Cordero, E.* AU - Gröchenig, K. C1 - 2108 C2 - 21897 SP - 29-47 TI - Localization of frames II. JO - Appl. Comput. Harmon. Anal. VL - 17 IS - 1 PY - 2004 SN - 1063-5203 ER - TY - JOUR AB - A general approach for biorthogonal local trigonometric bases in the two-overlapping setting was given by Chui and Shi. In this paper, we give error estimates for the approximation with such basis functions. In particular, it is shown that for a partition of the real axis into small intervals one obtains better approximation order if polynomials are reproduced locally. Furthermore, smooth trigonometric bases are constructed, which reproduce constants resp. linear functions by only one resp. a small number of basis functions for each interval. AU - Bittner, K. C1 - 20812 C2 - 18879 SP - 75-102 TI - Error estimates and reproduction of polynomials for biorthogonal local trigonometric bases. JO - Appl. Comput. Harmon. Anal. VL - 6 IS - 1 PY - 1999 SN - 1063-5203 ER - TY - JOUR AB - In this paper, the reproduction of trigonometric polynomials with two-overlapping local cosine bases is investigated. This study is motivated by the need to represent most effectively a Fourier series in the form of a localized cosine series for the purpose of local analysis, thus providing a vehicle for the transition from classical harmonic analysis to analysis by Wilson-type wavelets. It is shown that there is one and only one class, which is a one-parameter family, of window functions that allows pointwise reproduction of all global harmonics, where the parameter is the order of smoothness of the window functions. It turns out that this class of window functions is also optimal in the sense that all global harmonics are reproduced by using a minimal number of the local trigonometric basis functions. AU - Bittner, K. AU - Chui, C.K.* C1 - 20881 C2 - 18938 SP - 382-399 TI - From local cosine bases to global harmonics. JO - Appl. Comput. Harmon. Anal. VL - 6 IS - 3 PY - 1999 SN - 1063-5203 ER -