Titlebook: Numerical Fourier Analysis; Gerlind Plonka,Daniel Potts,Manfred Tasche Book 20181st edition Springer Nature Switzerland AG 2018 fourier an

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ated applications in various fields of science and engineering. It is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order. The fractional derivative has been used in various physical problems, such as frequency-dependent damping behavior of structures, biolo

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Fourier Series,agation. Section 1.2 introduces some basic notions such as Fourier coefficients and Fourier series of a 2.-periodic function. The convolution of 2.-periodic functions is handled in Sect. 1.3. Section 1.4 presents main results on the pointwise and uniform convergence of Fourier series. For a 2.-perio

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Fourier Transforms,ned on the Banach space .. The main properties of the Fourier transform are handled, such as the Fourier inversion formula and the convolution property. Then, in Sect. 2.2, the Fourier transform is introduced as a bijective mapping of the Hilbert space . onto itself by the theorem of Plancherel. The

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Discrete Fourier Transforms,ation of Fourier coefficients, Fourier transforms or trigonometric interpolation, lead to the DFT. We also present barycentric formulas for interpolating trigonometric polynomials. In Sect. 3.2, we study the basic properties of the Fourier matrix and of the DFT. In particular, we consider the eigenv

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Numerical Applications of DFT,nal interpolation by translates .(⋅ −.) with ., where . is a compactly supported, continuous function. In this approximation method, the cardinal Lagrange function is of main interest. Applying this technique, we compute the multidimensional Fourier transform by the method of attenuation factors. Th

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Prony Method for Reconstruction of Structured Functions,der the parameter estimation problem, where the classical Prony method and its relatives are described. In Sect. 10.2, we study frequently used methods for solving the parameter estimation problem, namely MUSIC (MUltiple Signal Classification), APM (Approximate Prony Method), and ESPRIT (Estimation