19 results on '"Fourier series"'
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2. An analytical filter design method for guided wave phased arrays.
- Author
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Kwon, Hyu-Sang and Kim, Jin-Yeon
- Subjects
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ELECTRIC filter design & construction , *PHASED array antennas , *TRANSDUCERS , *INVERSE problems , *APPROXIMATION theory , *LEAST squares , *FOURIER series - Abstract
This paper presents an analytical method for designing a spatial filter that processes the data from an array of two-dimensional guided wave transducers. An inverse problem is defined where the spatial filter coefficients are determined in such a way that a prescribed beam shape, i.e., a desired array output is best approximated in the least-squares sense. Taking advantage of the 2 π -periodicity of the generated wave field, Fourier-series representation is used to derive closed-form expressions for the constituting matrix elements. Special cases in which the desired array output is an ideal delta function and a gate function are considered in a more explicit way. Numerical simulations are performed to examine the performance of the filters designed by the proposed method. It is shown that the proposed filters can significantly improve the beam quality in general. Most notable is that the proposed method does not compromise between the main lobe width and the sidelobe levels; i.e. a narrow main lobe and low sidelobes are simultaneously achieved. It is also shown that the proposed filter can compensate the effects of nonuniform directivity and sensitivity of array elements by explicitly taking these into account in the formulation. From an example of detecting two separate targets, how much the angular resolution can be improved as compared to the conventional delay-and-sum filter is quantitatively illustrated. Lamb wave based imaging of localized defects in an elastic plate using a circular array is also presented as an example of practical applications. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
3. Approximation and existence of Schauder bases in Müntz spaces of L1 functions.
- Author
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Ludkowski, Sergey V.
- Subjects
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APPROXIMATION theory , *EXISTENCE theorems , *SCHAUDER bases , *MATHEMATICAL functions , *ISOMORPHISM (Mathematics) - Abstract
An approximation of functions in Müntz spaces M Λ , L 1 of L 1 functions is studied with the help of Fourier series. Müntz spaces are considered with the Müntz condition and the gap condition imposed. It is proved that up to an isomorphism such spaces are contained in Weil–Nagy's class. Moreover, existence of Schauder bases in Müntz spaces M Λ , L 1 is investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
4. Approximation of rough functions.
- Author
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Barnsley, M.F., Harding, B., Vince, A., and Viswanathan, P.
- Subjects
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APPROXIMATION theory , *EXISTENCE theorems , *UNIQUENESS (Mathematics) , *DIFFERENTIABLE functions , *INTERPOLATION , *FOURIER analysis - Abstract
For given p ∈ [ 1 , ∞ ] and g ∈ L p ( R ) , we establish the existence and uniqueness of solutions f ∈ L p ( R ) , to the equation f ( x ) − a f ( b x ) = g ( x ) , where a ∈ R , b ∈ R ∖ { 0 } , and | a | ≠ | b | 1 / p . Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
5. On multivariate de la Vallée Poussin-type projection operators.
- Author
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Németh, Zsolt
- Subjects
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FOURIER series , *PARTIAL sums (Series) , *ESTIMATION theory , *MULTIVARIATE analysis , *APPROXIMATION theory - Abstract
This paper deals with the de la Vallée Poussin means of the triangular partial sums of multivariate Fourier series. We determine the exact order of the corresponding operator norms. The lower estimation of these norms will be extended to a class of projection operators having similar projection properties as the de la Vallée Poussin mean. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
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6. On optimal wavelet reconstructions from Fourier samples: Linearity and universality of the stable sampling rate.
- Author
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Adcock, B., Hansen, A.C., and Poon, C.
- Subjects
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WAVELETS (Mathematics) , *OPTIMAL control theory , *STATISTICAL sampling , *APPROXIMATION theory , *MATHEMATICAL constants - Abstract
Abstract: In this paper we study the problem of computing wavelet coefficients of compactly supported functions from their Fourier samples. For this, we use the recently introduced framework of generalized sampling. Our first result demonstrates that using generalized sampling one obtains a stable and accurate reconstruction, provided the number of Fourier samples grows linearly in the number of wavelet coefficients recovered. For the class of Daubechies wavelets we derive the exact constant of proportionality. Our second result concerns the optimality of generalized sampling for this problem. Under some mild assumptions we show that generalized sampling cannot be outperformed in terms of approximation quality by more than a constant factor. Moreover, for the class of so-called perfect methods, any attempt to lower the sampling ratio below a certain critical threshold necessarily results in exponential ill-conditioning. Thus generalized sampling provides a nearly-optimal solution to this problem. [Copyright &y& Elsevier]
- Published
- 2014
- Full Text
- View/download PDF
7. Improved approximation guarantees for sublinear-time Fourier algorithms
- Author
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Iwen, Mark A.
- Subjects
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APPROXIMATION theory , *ALGORITHMS , *FOURIER transforms , *FOURIER series , *LOGARITHMIC functions , *FUNCTIONAL analysis - Abstract
Abstract: In this paper modified variants of the sparse Fourier transform algorithms from Iwen (2010) [32] are presented which improve on the approximation error bounds of the original algorithms. In addition, simple methods for extending the improved sparse Fourier transforms to higher dimensional settings are developed. As a consequence, approximate Fourier transforms are obtained which will identify a near-optimal k-term Fourier series for any given input function, , in time (neglecting logarithmic factors). Faster randomized Fourier algorithm variants with runtime complexities that scale linearly in the sparsity parameter k are also presented. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
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8. A theoretical basis for the Harmonic Balance Method
- Author
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García-Saldaña, Johanna D. and Gasull, Armengol
- Subjects
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HARMONIC analysis (Mathematics) , *HEURISTIC algorithms , *FOURIER series , *APPROXIMATION theory , *NUMERICAL analysis , *PERIODIC functions , *DIFFERENTIAL equations - Abstract
Abstract: The Harmonic Balance Method provides a heuristic approach for finding truncated Fourier series as an approximation to the periodic solutions of ordinary differential equations. Another natural way for obtaining these types of approximations consists in applying numerical methods. In this paper we recover the pioneering results of Stokes and Urabe that provide a theoretical basis for proving that near these truncated series, whatever is the way they have been obtained, there are actual periodic solutions of the equation. We will restrict our attention to one-dimensional non-autonomous ordinary differential equations, and we apply the obtained results to a concrete example coming from a rigid cubic system. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
9. The residue harmonic balance for fractional order van der Pol like oscillators
- Author
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Leung, A.Y.T., Yang, H.X., and Guo, Z.J.
- Subjects
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VAN der Pol oscillators (Physics) , *HARMONIC analysis (Mathematics) , *ITERATIVE methods (Mathematics) , *APPROXIMATION theory , *NONLINEAR differential equations , *FOURIER series , *DAMPING (Mechanics) , *LIMIT cycles - Abstract
Abstract: When seeking a solution in series form, the number of terms needed to satisfy some preset requirements is unknown in the beginning. An iterative formulation is proposed so that when an approximation is available, the number of effective terms can be doubled in one iteration by solving a set of linear equations. This is a new extension of the Newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series. When Fourier series is employed, the method is called the residue harmonic balance. In this paper, the fractional order van der Pol oscillator with fractional restoring and damping forces is considered. The residue harmonic balance method is used for generating the higher-order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator. The highly accurate solutions to angular frequency and limit cycle of the fractional order van der Pol equations are obtained analytically. The results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives. The influence of the fractional order on the geometry of the limit cycle is investigated for the first time. [Copyright &y& Elsevier]
- Published
- 2012
- Full Text
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10. Fourier sums with exponential weights on : and cases
- Author
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Mastroianni, G. and Notarangelo, I.
- Subjects
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FOURIER series , *EXPONENTIAL sums , *ORTHOGONAL polynomials , *APPROXIMATION theory , *MATHEMATICAL analysis , *NUMERICAL analysis - Abstract
Abstract: We study the behavior of the Fourier sums in orthonormal polynomial systems, related to exponential weights on , in weighted and uniform metrics. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
11. Method for identifying models of nonlinear systems using linear time periodic approximations
- Author
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Sracic, Michael W. and Allen, Matthew S.
- Subjects
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NONLINEAR systems , *APPROXIMATION theory , *FOURIER series , *MODAL models , *COMBINATORIAL dynamics , *PERTURBATION theory , *MATRICES (Mathematics) , *FLOQUET theory - Abstract
Abstract: This work presents a new method for identifying models of nonlinear systems from experimental measurements. The system is first forced to oscillate in stable periodic orbit, and then a small impulsive disturbance force is used to perturb the system slightly from that orbit. One then measures the response until the system returns to the periodic orbit. If the nonlinearities in the system are sufficiently smooth and the perturbation from the periodic orbit is sufficiently small, then one can linearize the perturbed response about the periodic orbit and approximate the system as linear time periodic. One of a variety of methods can then be used to extract the time varying modal model of the system from the response. The extracted modes can be used to construct a time periodic state transition matrix and state coefficient matrix, which describe the system''s nonlinear dynamics over a range of the states. The resulting model for the nonlinear system encompasses that portion of the state space is traversed by the system during its periodic orbit. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
12. A Fourier-series-based kernel-independent fast multipole method
- Author
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Zhang, Bo, Huang, Jingfang, Pitsianis, Nikos P., and Sun, Xiaobai
- Subjects
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FOURIER series , *KERNEL functions , *PARTICLES , *NUMERICAL analysis , *BIOLOGICAL systems , *APPROXIMATION theory , *GREEN'S functions - Abstract
Abstract: We present in this paper a new kernel-independent fast multipole method (FMM), named as FKI-FMM, for pairwise particle interactions with translation-invariant kernel functions. FKI-FMM creates, using numerical techniques, sufficiently accurate and compressive representations of a given kernel function over multi-scale interaction regions in the form of a truncated Fourier series. It provides also economic operators for the multipole-to-multipole, multipole-to-local, and local-to-local translations that are typical and essential in the FMM algorithms. The multipole-to-local translation operator, in particular, is readily diagonal and does not dominate in arithmetic operations. FKI-FMM provides an alternative and competitive option, among other kernel-independent FMM algorithms, for an efficient application of the FMM, especially for applications where the kernel function consists of multi-physics and multi-scale components as those arising in recent studies of biological systems. We present the complexity analysis and demonstrate with experimental results the FKI-FMM performance in accuracy and efficiency. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
13. Model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds
- Author
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Gabale, Amit P. and Sinha, S.C.
- Subjects
- *
NONLINEAR systems , *INVARIANT manifolds , *PARTIAL differential equations , *APPROXIMATION theory , *FOURIER series , *FLOQUET theory , *MECHANICAL loads , *DAMPING (Mechanics) - Abstract
Abstract: A methodology for determining reduced order models of periodically excited nonlinear systems with constant as well as periodic coefficients is presented. The approach is based on the construction of an invariant manifold such that the projected dynamics is governed by a fewer number of ordinary differential equations. Due to the existence of external and parametric periodic excitations, however, the geometry of the manifold varies with time. As a result, the manifold is constructed in terms of temporal and dominant state variables. The governing partial differential equation (PDE) for the manifold is nonlinear and contains time-varying coefficients. An approximate technique to find solution of this PDE using a multivariable Taylor–Fourier series is suggested. It is shown that, in certain cases, it is possible to obtain various reducibility conditions in a closed form. The case of time-periodic systems is handled through the use of Lyapunov–Floquet (L–F) transformation. Application of the L–F transformation produces a dynamically equivalent system in which the linear part of the system is time-invariant; however, the nonlinear terms get multiplied by a truncated Fourier series containing multiple parametric excitation frequencies. This warrants some structural changes in the proposed manifold, but the solution procedure remains the same. Two examples; namely, a 2-dof mass–spring–damper system and an inverted pendulum with periodic loads, are used to illustrate applications of the technique for systems with constant and periodic coefficients, respectively. Results show that the dynamics of these 2-dof systems can be accurately approximated by equivalent 1-dof systems using the proposed methodology. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
14. -convergence of Fourier sums with exponential weights on
- Author
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Mastroianni, G. and Notarangelo, I.
- Subjects
- *
STOCHASTIC convergence , *FOURIER analysis , *ORTHOGONAL polynomials , *APPROXIMATION theory , *OPERATOR theory , *MATHEMATICAL singularities , *EXPONENTS - Abstract
Abstract: In order to approximate functions defined on and having exponential singularities at the endpoints of the interval, we study the behavior of some modified Fourier Sums in an orthonormal system related to exponential weights. We give necessary and sufficient conditions for the boundedness of the related operators in suitable weighted -spaces, with . Then, in these spaces, these processes converge with the order of the best polynomial approximation. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
15. Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier Series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions
- Author
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Boyd, John P. and Yu, Fu
- Subjects
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SPECTRAL theory , *INTERPOLATION , *NUMERICAL solutions to Poisson's equation , *POLYNOMIALS , *FOURIER series , *BESSEL functions , *APPROXIMATION theory , *STOCHASTIC convergence , *DEGREES of freedom - Abstract
Abstract: We compare seven different strategies for computing spectrally-accurate approximations or differential equation solutions in a disk. Separation of variables for the Laplace operator yields an analytic solution as a Fourier–Bessel series, but this usually converges at an algebraic (sub-spectral) rate. The cylindrical Robert functions converge geometrically but are horribly ill-conditioned. The Zernike and Logan–Shepp polynomials span the same space, that of Cartesian polynomials of a given total degree, but the former allows partial factorization whereas the latter basis facilitates an efficient algorithm for solving the Poisson equation. The Zernike polynomials were independently rediscovered several times as the product of one-sided Jacobi polynomials in radius with a Fourier series in θ. Generically, the Zernike basis requires only half as many degrees of freedom to represent a complicated function on the disk as does a Chebyshev–Fourier basis, but the latter has the great advantage of being summed and interpolated entirely by the Fast Fourier Transform instead of the slower matrix multiplication transforms needed in radius by the Zernike basis. Conformally mapping a square to the disk and employing a bivariate Chebyshev expansion on the square is spectrally accurate, but clustering of grid points near the four singularities of the mapping makes this method less efficient than the rest, meritorious only as a quick-and-dirty way to adapt a solver-for-the-square to the disk. Radial basis functions can match the best other spectral methods in accuracy, but require slow non-tensor interpolation and summation methods. There is no single “best” basis for the disk, but we have laid out the merits and flaws of each spectral option. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
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16. The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials
- Author
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Navas, Luis M., Ruiz, Francisco J., and Varona, Juan L.
- Subjects
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MOBIUS function , *FOURIER series , *BERNOULLI polynomials , *EULER polynomials , *FOURIER analysis , *NUMBER theory , *APPROXIMATION theory - Abstract
Abstract: Hurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
17. Exact estimates of the rate of approximation of convolution operators
- Author
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Draganov, Borislav R.
- Subjects
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ESTIMATES , *APPROXIMATION theory , *MATHEMATICAL convolutions , *OPERATOR theory , *SINGULAR integrals , *FOURIER series , *MULTIPLIERS (Mathematical analysis) - Abstract
Abstract: The paper presents a method for establishing direct and strong converse inequalities in terms of -functionals for convolution operators acting in homogeneous Banach spaces of multivariate functions. The method is based on the behaviour of the Fourier transform of the kernel of the convolution operator. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
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18. High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations
- Author
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Lyon, Mark and Bruno, Oscar P.
- Subjects
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CONTINUATION methods , *GIBBS phenomenon , *APPROXIMATION theory , *NUMERICAL analysis , *PARTIAL differential equations , *FOURIER series , *FINITE element method - Abstract
Abstract: A new PDE solver was introduced recently, in Part I of this two-paper sequence, on the basis of two main concepts: the well-known Alternating Direction Implicit (ADI) approach, on one hand, and a certain “Fourier Continuation” (FC) method for the resolution of the Gibbs phenomenon, on the other. Unlike previous alternating direction methods of order higher than one, which only deliver unconditional stability for rectangular domains, the new high-order FC-AD (Fourier-Continuation Alternating-Direction) algorithm yields unconditional stability for general domains—at an cost per time-step for an N point spatial discretization grid. In the present contribution we provide an overall theoretical discussion of the FC-AD approach and we extend the FC-AD methodology to linear hyperbolic PDEs. In particular, we study the convergence properties of the newly introduced FC(Gram) Fourier Continuation method for both approximation of general functions and solution of the alternating-direction ODEs. We also present (for parabolic PDEs on general domains, and, thus, for our associated elliptic solvers) a stability criterion which, when satisfied, ensures unconditional stability of the FC-AD algorithm. Use of this criterion in conjunction with numerical evaluation of a series of singular values (of the alternating-direction discrete one-dimensional operators) suggests clearly that the fifth-order accurate class of parabolic and elliptic FC-AD solvers we propose is indeed unconditionally stable for all smooth spatial domains and for arbitrarily fine discretizations. To illustrate the FC-AD methodology in the hyperbolic PDE context, finally, we present an example concerning the Wave Equation—demonstrating sixth-order spatial and fourth-order temporal accuracy, as well as a complete absence of the debilitating “dispersion error”, also known as “pollution error”, that arises as finite-difference and finite-element solvers are applied to solution of wave propagation problems. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
19. Almost everywhere convergence of Fejér and logarithmic means of subsequences of partial sums of the Walsh–Fourier series of integrable functions
- Author
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Gát, György
- Subjects
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STOCHASTIC convergence , *LOGARITHMIC functions , *MATHEMATICAL sequences , *PARTIAL sums (Series) , *FOURIER series , *INTEGRAL functions , *APPROXIMATION theory - Abstract
Abstract: The aim of this paper is to prove some a.e. convergence results of Fejér and logarithmic means of subsequences of partial sums of Walsh–Fourier series of integrable functions. We prove for lacunary sequences that the means of the partial sums converges to a.e. Besides, for every convex tending to and every integrable function the logarithmic means of the partial sums converges to a.e. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
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