Showing papers on "Fourier series published in 2004"

Functions of a Complex Variable

Abstract: In earlier chapters, complex-valued functions appeared in connection with Fourier series expansions. In this context, while the function assumes complex values, the argument of the function is real-valued. There is a highly developed theory of (complex-valued) functions of a complex-valued argument. This theory contains some remarkably powerful results which are applicable to a variety of problems.

Measurement and Data Analysis for Engineering and Science

Abstract: Fundamentals of Experimentation Introduction Experiments Chapter Overview Experimental Approach Role of Experiments The Experiment Classification of Experiments Plan for Successful Experimentation Hypothesis Testing* Design of Experiments* Factorial Design* Problems Bibliography Fundamental Electronics Chapter Overview Concepts and Definitions Circuit Elements RLC Combinations Elementary DC Circuit Analysis Elementary AC Circuit Analysis Equivalent Circuits* Meters* Impedance Matching and Loading Error* Electrical Noise* Problems Bibliography Measurement Systems: Sensors and Transducers Chapter Overview Measurement System Overview Sensor Domains Sensor Characteristics Physical Principles of Sensors Electric Piezoelectric Fluid Mechanic Optic Photoelastic Thermoelectric Electrochemical Sensor Scaling* Problems Bibliography Measurement Systems: Other Components Chapter Overview Signal Conditioning, Processing, and Recording Amplifiers Filters Analog-to-Digital Converters Smart Measurement Systems Other Example Measurement Systems Problems Bibliography Measurement Systems: Calibration and Response Chapter Overview Static Response Characterization by Calibration Dynamic Response Characterization Zero-Order System Dynamic Response First-Order System Dynamic Response Second-Order System Dynamic Response Measurement System Dynamic Response Problems Bibliography Measurement Systems: Design-Stage Uncertainty Chapter Overview Design-Stage Uncertainty Analysis Design-Stage Uncertainty Estimate of a Measurand Design-Stage Uncertainty Estimate of a Result Problems Bibliography Signal Characteristics Chapter Overview Signal Classification Signal Variables Signal Statistical Parameters Problems Bibliography The Fourier Transform Chapter Overview Fourier Series of a Periodic Signal Complex Numbers and Waves Exponential Fourier Series Spectral Representations Continuous Fourier Transform Continuous Fourier Transform Properties* Discrete Fourier Transform Fast Fourier Transform Problems Bibliography Digital Signal Analysis Chapter Overview Digital Sampling Digital Sampling Errors Windowing* Determining a Sample Period Problems Bibliography Probability Chapter Overview Relation to Measurements Basic Probability Concepts Sample versus Population Plotting Statistical Information Probability Density Function Various Probability Density Functions Central Moments Probability Distribution Function Problems Bibliography Statistics Chapter Overview Normal Distribution Normalized Variables Student's t Distribution Rejection of Data Standard Deviation of the Means Chi-Square Distribution Pooling Samples* Problems Bibliography Uncertainty Analysis Chapter Overview Modeling and Experimental Uncertainties Probabilistic Basis of Uncertainty Identifying Sources of Error Systematic and Random Errors Quantifying Systematic and Random Errors Measurement Uncertainty Analysis Uncertainty Analysis of a Multiple-Measurement Result Uncertainty Analyses for Other Measurement Situations Uncertainty Analysis Summary Finite-Difference Uncertainties* Uncertainty Based upon Interval Statistics* Problems Bibliography Regression and Correlation Chapter Overview Least-Squares Approach Least-Squares Regression Analysis Linear Analysis Higher-Order Analysis* Multi-Variable Linear Analysis* Determining the Appropriate Fit Regression Confidence Intervals Regression Parameters Linear Correlation Analysis Signal Correlations in Time* Problems Bibliography Units and Significant Figures Chapter Overview English and Metric Systems Systems of Units SI Standards Technical English and SI Conversion Factors Prefixes Significant Figures Problems Bibliography Technical Communication Chapter Overview Guidelines for Writing Technical Memo Technical Report Oral Technical Presentation Problems Bibliography A Glossary B Symbols C Review Problem Answers Index

Fourier Analysis and Approximation of Functions

Abstract: 1. Representation Theorems.- 1.1 Theorems on representation at a point.- 1.2 Integral operators. Convergence in Lp-norm and almost everywhere.- 1.3 Multidimensional case.- 1.4 Further problems and theorems.- 1.5 Comments to Chapter 1.- 2. Fourier Series.- 2.1 Convergence and divergence.- 2.2 Two classical summability methods.- 2.3 Harmonic functions and functions analytic in the disk.- 2.4 Multidimensional case.- 2.5 Further problems and theorems.- 2.6 Comments to Chapter 2.- 3. Fourier Integral.- 3.1 L-Theory.- 3.2 L2-Theory.- 3.3 Multidimensional case.- 3.4 Entire functions of exponential type. The Paley-Wiener theorem.- 3.5 Further problems and theorems.- 3.6 Comments to Chapter 3.- 4. Discretization. Direct and Inverse Theorems.- 4.1 Summation formulas of Poisson and Euler-Maclaurin.- 4.2 Entire functions of exponential type and polynomials.- 4.3 Network norms. Inequalities of different metrics.- 4.4 Direct theorems of Approximation Theory.- 4.5 Inverse theorems. Constructive characteristics. Embedding theorems.- 4.6 Moduli of smoothness.- 4.7 Approximation on an interval.- 4.8 Further problems and theorems.- 4.9 Comments to Chapter 4.- 5. Extremal Problems of Approximation Theory.- 5.1 Best approximation.- 5.2 The space Lp. Best approximation.- 5.3 Space C. The Chebyshev alternation.- 5.4 Extremal properties for algebraic polynomials and splines.- 5.5 Best approximation of a set by another set.- 5.6 Further problems and theorems.- 5.7 Comments to Chapter 5.- 6. A Function as the Fourier Transform of A Measure.- 6.1 Algebras A and B. The Wiener Tauberian theorem.- 6.2 Positive definite and completely monotone functions.- 6.3 Positive definite functions depending only on a norm.- 6.4 Sufficient conditions for belonging to Ap and A*.- 6.5 Further problems and theorems.- 6.6 Comments to Chapter 6.- 7. Fourier Multipliers.- 7.1 General properties.- 7.2 Sufficient conditions.- 7.3 Multipliers of power series in the Hardy spaces.- 7.4 Multipliers and comparison of summability methods of orthogonal series.- 7.5 Further problems and theorems.- 7.6 Comments to Chapter 7.- 8. Summability Methods. Moduli of Smoothness.- 8.1 Regularity.- 8.2 Applications of comparison. Two-sided estimates.- 8.3 Moduli of smoothness and K-functionals.- 8.4 Moduli of smoothness and strong summability in Hp(D), 0erences.- Author Index.- Topic Index.

Lifting-Line Analysis for Twisted Wings and Washout-Optimized Wings

Abstract: A more practical form of the analytical solution for the effects of geometric and aerodynamic twist (washout) on the low-Mach-number performance of a finite wing of arbitrary planform is presented. This infinite series solution is based on Prandtl's classical lifting-line theory and the Fourier coefficients are presented in a form that only depends on wing geometry

Local Fourier Transforms and Rigidity for D-Modules

Abstract: Local Fourier transforms, analogous to the l-adic local Fourier transforms [14], are constructed for connections over k((t)). Following a program of Katz [12], a meromorphic connection on a curve is shown to ber igid, i.e. determined by local data at the singularities, if and only if a certain infinitesimal rigidity conditionis satisfied. As in [12],the argument uses local Fourier transforms to prove an invariance result for the rigidity index under global Fourier transform. A key technical tool is the notion of good lattice pairs for a connection[5].

Θ-summability of Fourier series

Abstract: A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejer operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H p to L p (1/2

Bohr's inequality for uniform algebras

Abstract: We prove a uniform algebra analogue of a classical inequality of Bohr's concerning Fourier coefficients of bounded holomorphic functions. The classical inequality follows trivially.

Real Analysis and Foundations

Abstract: Number Systems The Real Numbers The Complex Numbers Sequences Convergence of Sequences Subsequences Limsup and Liminf Some Special Sequences Series of Numbers Convergence of Series Elementary Convergence Tests Advanced Convergence Tests Some Special Series Operations on Series Basic Topology Open and Closed Sets Further Properties of Open and Closed Sets Compact Sets The Cantor Set Connected and Disconnected Sets Perfect Sets Limits and Continuity of Functions Basic Properties of the Limit of a Function Continuous Functions Topological Properties and Continuity Classifying Discontinuities and Monotonicity Differentiation of Functions The Concept of Derivative The Mean Value Theorem and Applications More on the Theory of Differentiation The Integral Partitions and the Concept of Integral Properties of the Riemann Integral Another Look at the Integral Advanced Results on Integration Theory Sequences and Series of Functions Partial Sums and Pointwise Convergence More on Uniform Convergence Series of Functions The Weierstrass Approximation Theorem Elementary Transcendental Functions Power Series More on Power Series: Convergence Issues The Exponential and Trigonometric Functions Logarithms and Powers of Real Numbers Differential Equations Picard's Existence and Uniqueness Theorem Power Series Methods Introduction to Harmonic Analysis The Idea of Harmonic Analysis The Elements of Fourier Series An Introduction to the Fourier Transform Fourier Methods and Differential Equations Functions of Several Variables A New Look at the Basic Concepts of Analysis Properties of the Derivative The Inverse and Implicit Function Theorems Advanced Topics Metric Spaces Topology in a Metric Space The Baire Category Theorem The Ascoli-Arzela Theorem Normed Linear Spaces What Is This Subject About? What Is a Normed Linear Space? Finite-Dimensional Spaces Linear Operators The Three Big Results Applications of the Big Three Appendix I: Elementary Number Systems Appendix II: Logic and Set Theory Appendix III: Review of Linear Algebra Table of Notation Glossary Bibliography Index Exercises are included at the end of each section.

Integrated volatility measuring from unevenly sampled observations

Abstract: This paper derives the linear interpolation bias of realized volatility. To avoid the bias, the Fourier series estimator has been proposed by Malliavin and Mancino (2002). We examine the theoretical relationship between the Fourier estimator and realized volatility and show that the latter is the most efficient estimator in the class of the former.

A comparative study of Fourier descriptors and Hu's seven moment invariants for image recognition

Abstract: The paper evaluates and compares the performance of Fourier descriptors and Hu's seven moment invariants for recognizing images with different spatial resolutions. Both Fourier descriptors and Hu's seven moment invariants have the preferred invariance property against image transformations, including scale change, translation and rotation. However, spatial resolution thresholds exist for both of them. In our experiment with the image recognition engine, for Fourier descriptors, with feature vectors composed by the first 10 elements of the series, the spatial resolution should not be less than 64/spl times/64 to achieve 100% recognition. For Hu's seven moment invariants, the minimum spatial resolution is 128/spl times/128.

A successive order of scattering model for solving vector radiative transfer in the atmosphere

Abstract: A full vector radiative transfer model for vertically inhomogeneous plane–parallel media has been developed by using the successive order of scattering approach. In this model, a fast analytical expansion of Fourier decomposition is implemented and an exponent-linear assumption is used for vertical integration. An analytic angular interpolation method of post-processing source function is also implemented to accurately interpolate the Stokes vector at arbitrary angles for a given solution. It has been tested against the benchmarks for the case of randomly orientated oblate spheroids, illustrating a good agreement for each stokes vector (within 0.01%). Sensitivity tests have been conducted to illustrate the accuracy of vertical integration and angle interpolation approaches. The contribution of each scattering order for different optical depths and single scattering albedos are also analyzed.

Approximation Theory: From Taylor Polynomials to Wavelets

Abstract: The book is a concisely written introduction to approximation theory. The presentation of results demonstrates the dynamic nature of mathematics and is enriched by illustrative examples, which help to develop intuition. The book consists of five chapters, each followed by exercises. The first is devoted to function approximation with polynomials. It begins with a general introduction to the idea of approximation, then deals with the Weierstrass and Taylor theorems. In the second chapter the authors focus on the approxima- tion properties related to infinite series of functions. The basic idea of signal transmission is also presented. The next chapter indicates how Fourier series and Fourier transform can be used as tools to represent functions and how pro- perties of a function are reflected in the expansion coefficients. The purpose of the last two chapters is to introduce wavelets as natural continuation of the previous material. They give the reader an understanding of the fundamental concepts, sketch the history of wavelets and present their application in signal processing. The explanation for how wavelet expansions reflect the local beha- viour of a function and how wavelets can be used to detect jumps in a signal is also provided. The book ends with an outline for the frames and Gabor systems, which are frequently used as alternatives to wavelet systems. The authors focus on ideas rather than on technical details, so that proofs are generally omitted, and just a few are included in the appendices. Minimal prerequisites (elementary calculus) make the book comprehensible to undergra- duate students of mathematics, mathematical physics and engineering. Readers are, at the same time, led towards the advanced literature in approximation theory.

Wave propagation in inhomogeneous layered media: solution of forward and inverse problems

Abstract: Wave propagation in anisotropic inhomogeneous layered media due to high frequency impact loading is studied using a new Spectral Layer Element (SLE). The element can model functionally graded materials (FGM), where the material property variation is assumed to follow an exponential function. The element is exact for a single parameter model which describes both moduli and density variation. This novel element is formulated using the method of partial wave technique (PWT) in conjunction with linear algebraic methodology. The matrix structure of finite element (FE) formulation is retained, which substantially simplifies the modeling of a multi-layered structure. The developed SLE has an exact dynamic stiffness matrix as it uses the exact solution of the governing elastodynamic equation in the frequency domain as its interpolation function. The mass distribution is modeled exactly, and, as a result, the element gives the exact frequency response of each layer. Hence, one element may be as large as one complete layer which results in a system size being very small compared to conventional FE systems. The Fast-Fourier Transform (FFT) and Fourier series are used for the inversion to the time/space domain. The formulated element is further used to study the stress distribution in multi-layered media. As a natural application, Lamb wave propagation in an inhomogeneous plate is studied and the time domain description is obtained. Further, the advantage of the spectral formulation in the solution of inverse problems, namely the force identification and system identification is investigated. Constrained nonlinear optimization technique is used for the material property identification, whereas the transfer function approach is taken for the impact force identification.

Fourier embeddings and Mihlin‐type multiplier theorems

Abstract: Recent theorems on singular convolution operators are combined with new Fourier embedding results to prove strong multiplier theorems on various function spaces (including Besov, Lebesgue–Bochner, and Hardy). All the results apply to operator-valued multipliers acting on vector-valued functions, but some of them are new even in the scalar case. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Azimuthal harmonic coefficients of the microwave backscattering from a non-Gaussian ocean surface with the first-order SSA model

Abstract: In this paper, the first-order small slope approximation is applied to a rough sea surface with non-Gaussian statistics, for which the third- and the fourth-order statistics are taken into account in the calculation of the radar cross section. From the Cox and Munk slope distribution, the higher order statistic moments are derived, and behaviors of the corresponding correlation functions are assumed. We show that the fourth order (related to the peakedness or kurtosis) is isotropic, whereas the third order (related to the skewness) has a behavior as cos(/spl psi/), where /spl psi/ is the wave direction along the wind direction. Thus, using the Elfouhaily et al. sea height spectrum, related to the second-order statistics, we show that the normalized radar backscattering cross section (NRBCS) can be expanded as an even Fourier series in cos(n/spl phi/) (where n is a positive integer), for which the harmonic coefficients require only a single integration over the radial distance. This result is consistent with experimental data done for microwave frequencies. In addition, we show for microwave frequencies (like C- and Ku-bands) that the Fourier series can be truncated up to the second order, since the higher order harmonic coefficients vanish. The NRBCS is also compared with empirical backscattering models CMOD2-I3 and SASS-II, valid in C- and Ku-bands, according to the scattering angle and the wind direction. The first-order harmonic coefficient predicts the surface asymmetry along the upwind and downwind directions, whereas the second-order harmonic coefficient describes the surface asymmetry along the upwind and crosswind directions.

Block Thresholding and Sharp Adaptive Estimation in Severely Ill-Posed Inverse Problems

Abstract: We consider the problem of solving linear operator equations from noisy data under the assumptions that the singular values of the operator decrease exponentially fast and that the underlying solution is also exponentially smooth in the Fourier domain. We suggest an estimator of the solution based on a running version of block thresholding in the space of Fourier coefficients. This estimator is shown to be sharp adaptive to the unknown smoothness of the solution.

Closed-form solutions to assess multilayered-plate theories for various thermal stress problems

Abstract: This paper represents a further development of the first author’s works on twodimensional modeling for thermal stress analysis of multilayered composite plates. The governing equations are written by referring to the unified compact formulation. These equations have been obtained in a form that is not affected by the order of the expansion in the thickness plate direction z or by variable descriptions (layer-wise models and equivalent single layers models). Classical theories based on the principle of virtual displacements and advanced mixed theories based on the Reissner mixed variational theorem are both considered. As a result, a large variety of theories are derived and compared. The temperature profile TP in the direction z is calculated by solving the heat conduction problem and it is compared to the case in which TP is assumed linear in z. Exact closed-form solutions have been derived for the case of the in-plane harmonic distribution of displacements, transverse stress variables, and temperature fields. The Fourier expansion was then used to solve problems related to uniform, triangular, bitriangular (tentlike), and localized in-plane distribution of temperature. In some cases more than 25 theories were compared. The effect of transverse shear deformation, the zig-zag form of displacement fields, and interlaminar continuity of transverse stresses (both shear and normal components) have been evaluated in the framework of both classical and mixed theories.

Robust Uncertainty Principles: Exact Signal Reconstruction from Highly Incomplete Frequency Information

Abstract: This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal $f \in \C^N$ and a randomly chosen set of frequencies $\Omega$ of mean size $\tau N$. Is it possible to reconstruct $f$ from the partial knowledge of its Fourier coefficients on the set $\Omega$? A typical result of this paper is as follows: for each $M > 0$, suppose that $f$ obeys $$ # \{t, f(t) eq 0 \} \le \alpha(M) \cdot (\log N)^{-1} \cdot # \Omega, $$ then with probability at least $1-O(N^{-M})$, $f$ can be reconstructed exactly as the solution to the $\ell_1$ minimization problem $$ \min_g \sum_{t = 0}^{N-1} |g(t)|, \quad \text{s.t.} \hat g(\omega) = \hat f(\omega) \text{for all} \omega \in \Omega. $$ In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for $\alpha$ which depends on the desired probability of success; except for the logarithmic factor, the condition on the size of the support is sharp. The methodology extends to a variety of other setups and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one or two-dimensional) object from incomplete frequency samples--provided that the number of jumps (discontinuities) obeys the condition above--by minimizing other convex functionals such as the total-variation of $f$.

Aquifer Response to Sinusoidal or Arbitrary Stage of Semipervious Stream

Abstract: Analytical expressions for the aquifer responses, viz., groundwater head, rate of flow and cumulative volume of flow, to a generalized sinusoidal stage of semipervious streams considering the stream boundary resistance, are derived. The analytical aquifer responses to a linear stream stage and to a typical analytical flood wave that was used by Cooper and Rorabaugh, are also derived. For a zero-stream resistance, the aquifer responses converge to those for a fully penetrating stream. Also, two analytical methods, a ramp kernel method and a Fourier series method, for obtaining the aquifer responses to an arbitrary temporal stage of sempervious stream, are developed. The analytical expressions of the ramp kernels for different aquifer responses are developed. The ramp kernel method is found superior to the conventional convolution that uses numerical integration or pulse kernels for obtaining the convolution integral. In the Fourier series method, the aquifer responses to sinusoidal stage are used along with Fourier series. The results obtained using both methods are in close agreement. The new methods are also applicable to fully penetrating streams by assigning a zero value to the stream resistance.

Séries trigonométriques à coefficients arithmétiques

Abstract: By exploiting the recent analytic tool of friable summation, this work describes a new approach to a class of problems in multiplicative number theory and Fourier series theory originated by H. Davenport. A definite answer to the last original question of Davenport of this type, which was still open, as well as a number of other applications, is given.

Exact artificial boundary conditions for the Schrödinger equation in $R ^2$

Abstract: In this paper, we propose a class of exact artificial boundary conditions for the numerical solution of the Schrodinger equation on unbounded domains in two-dimensional cases. After we introduce a circular artificial boundary, we get an initial-boundary problem on a disc enclosed by the artificial boundary which is equivalent to the original problem. Based on the Fourier series expansion and the special functions techniques, we get the exact artificial boundary condition and a series of approximating artificial boundary conditions. When the potential function is independent of the radiant angle θ, the problem can be reduced to a series of one-dimensional problems. That can reduce the computational complexity greatly. Our numerical examples show that our method gives quite good numerical solutions with no numerical reflections.

Carleson's Theorem: Proof, Complements, Variations

Abstract: Carleson's Theorem from 1965 states that the partial Fourier sums of a square integrable function converge pointwise. We prove an equivalent statement on the real line, following the method developed by the author and C. Thiele. This theorem, and the proof presented, is at the center of an emerging theory which complements the statement and proof of Carleson's theorem. An outline of these variations is also given.