Showing papers on "Fourier series published in 1975"

Digital signal analysis

Abstract: Review of least squares, orthogonality and the Fourier series review of continuous transforms transfer functions and convolution sampling and measurement of signals the discrete Fourier transform the fast Fourier transform the z-transform non-recursive digital systems digital and continuous systems simulation of continuous systems analogue and digital filter design review of random functions correlation and power spectra least-squares system design random sequences and spectral estimation.

Modular Forms Associated to Real Quadratic Fields.

Abstract: The purpose of this paper is to construct modular forms, both for SL27Z (and certain of its congruence subgroups) and for the Hilbert modular group of a real quadratic field. In w 1 we fix a real quadratic field K and even integer k > 2 and construct a series of functions co,,(Zl, z2) (m=0, 1, 2, . . .) which are modular forms of weight k for the Hilbert modular group SL2(9 ((9=ring of integers in K). The form co o is a multiple of the Hecke-Eisenstein series for K, while all of the other co,. are cusp forms. The Fourier expansion of co,. (z 1, z2) is calculated in w 2; each Fourier coefficient is expressed as an infinite sum whose typical term is the product of a finite exponential sum (analogous to a Kloosterman sum) and a Bessel function of order k 1 . The main result is that, for any points z 1 and z 2 in the upper half-plane .~, the numbers m k-1 co,,(z 1, z2) (m= 1, 2, ...) are the Fourier coefficients of a modular form (in one variable) of weight k. More precisely, let D be the discriminant of K, e.=(D/ ) the character of K, and S(D, k, ~) the space of cusp forms of weight k for Fo(D ) with character e; then for fixed z 1, zzc .~, the function

Fourier Series with Respect to General Orthogonal Systems

Abstract: Terminology. Preliminary Information.- I. Convergence of Fourier Series in the Classical Sense. Lebesgue Functions of Bounded Systems.- 1. The Fundamental Inequality.- 2. The Logarithmic Growth of the Lebesgue Functions. Divergence of Fourier Series.- 3. Series with Decreasing Coefficients.- 4. Generalizations, Counterexamples, Problems.- 5. The Stability of the Orthogonalization Operator.- II. Convergence Almost Everywhere Conditions on the Coefficients.- 1. The Class S?.- 2. Garsia's Theorem.- 3. The Coefficients of Convergent Series in Complete Systems.- 4. Extension of a System of Functions to an ONS.- III. Properties of Complete Systems the Role of the Haar System.- 1. The Basic Construction.- 2. Divergent Fourier Series.- 3. Bases in Function Spaces and Majorants of Fourier Series.- 4. Fourier Coefficients of Continuous Functions.- 5. Some More Results about the Haar System.- IV. Series from L2 and Peculiarities of Fourier Series from the Spaces Lp.- 1. The Matrices Ak.- 2. Lebesgue Functions and Convergence Almost Everywhere.- 3. Convergence of Fourier Series of Functions from Various Classes.- 4. Sums of Fourier Series.- 5. Conditional Bases in Hubert Space.

A Fourier series method for numerical Kramers-Kronig analysis

Abstract: Using a time domain method the Kramers-Kronig integrals are derived without recourse to complex analysis (except in evaluating the Fourier transform of sgn (t)). From the time domain result, a Fourier series method for numerical evaluation of causality relations is derived. This method eliminates the need to use numerical integration, the use of logarithms in evaluating the function and the consideration of Cauchy principal parts. Through the use of the fast Fourier transform algorithm the calculation can be vary rapid. The accuracy of the technique is considered.

Fourier-series techniques applied to distance protection

Abstract: For distance-relay applications, the fundamental system impedance as seen from the relaying point is used as the basis for the decision to give the trip signal. This system impedance can be calculated from knowledge of the fundamental frequency components of the voltage and current waveforms, these, in turn, being determined by processing appropriate waveforms sampled at the relay point. Fourier-series analysis is one method suitable for processing the sampled waveforms and for determining their fundamental components. The analysis is designed for continuous, periodic waveforms and, in these circumstances, separates the various harmonic components of the waveform. Its behaviour under the nonperiodic conditions of fault incidence and transient interference will be demonstrated.

Fourier series on spheres

Abstract: A spectral representation consisting of a two‐dimensional Fourier series for use on a sphere is described. The method is applied to the advection of a passive scalar field over the poles and is compared to the pseudo‐spectral and grid‐point representations. The results show that the double Fourier series method compares favourably with both the pseudo‐spectral and grid‐point schemes.

A Fourier method for the numerical solution of Poisson’s equation

Abstract: A method for the solution of Poisson's equation in a rectangle, based on the relation between the Fourier coefficients for the solution and those for the right-hand side, is developed. The Fast Fourier Transform is used for the computation and its influence on the accuracy is studied. Error estimates are given and the method is shown to be second order accurate under certain general conditions on the smoothness of the solution. The accuracy is found to be limited by the lack of smoothness of the periodic extension of the inhomogeneous term. Higher order methods are then derived with the aid of special solutions. This reduces the problem to a case with sufficiently smooth data. A comparison of accuracy and efficiency is made between our Fourier method and the Buneman algorithm for the solution of the standard finite difference formulae.

A simple proof of Wiener’s 1/ theorem

Abstract: We give a simple proof of Wiener's theorem on the reciprocals of absolutely convergent Fourier series. One of Wiener's famous theorems states that if /(x) has an absolutely convergent Fourier series and never vanishes, then 1//(x) has an absolutely convergent Fourier series. The original proof of this utilized the so-called localization principle which depended in turn on the special nature of the "triangle" and "'trapezoid" functions. A beautiful modern proof emerges from the work of Gelfond which involves function algebras and his fundamental notion of maximal ideal spaces. Thus neither of these proofs is particularly simple. We propose, here, to give a rather simple elementary proof. Though many readers will have no difficulty in recognizing such items as the 'tspectral norm", the "openness of invertible elements", and other overlaps with older proofs, we do point out that this proof is self-contained and quite direct. We write, as usual, 00 00 ||zan eix| E la nl, and we recall the triangle inequalities II/ + gll II 1111 + llgill I1/ gl l II I 111 lgill We also will need the inequalities Max1/| < 11/1 < Max|/| + 2 Max1/'1, the first of which is trivial, while the second follows at once from Schwarz' inequality and Parseval's theorem. We have, namely, writing /(x) = ane that I ao0 I < Max I / (x) I while Received by the editors February 25, 1974. AMS (MOS) subject classifications (1970). Primary 42A16, 42A28; Secondary 43A20.

Lebesgue's inequality in a uniform metric and on a set of full measure

Abstract: Letf be a continuous periodic function with Fourier sums Sn(f), and let En(f)=En be the best approximation tof by trigonometric polynomials of order n. The following estimate is proved: $$||f - S_n (f)|| \leqslant c\sum olimits_{v = n}^{2u} {\frac{{E_v }}{{v - n + 1}}} .$$ (Here c is an absolute constant.) This estimate sharpens Lebesgue's classical inequality for “fast” decreasing E v . The sharpness of this estimate is proved for an arbitrary class of functions having a given majorant of best approximations. Also investigated is the sharpness of the corresponding estimate for the rate of convergence of a Fourier series almost everywhere.

Vibrations of constrained cylindrical shells

Abstract: A study is made of the vibration characteristics of a cylindrical shell with arbitrary boundary conditions and with several intermediate constraint positions between the ends. The solution is obtained using a Rayleigh-Ritz procedure in which the axial modal displacements are constructed from simple Fourier series expressions. Geometric boundary conditions that are not identically satisfied are enforced with Lagrange multipliers. Unwanted geometric boundary conditions, forced to be zero due to the nature of the assumed series, are released through the mechanism of Stokes' transformation. For the problem without intermediate constraint, comparison with other investigators yields excellent agreement. For the problem with intermediate constraint, results are presented for a wide variety of constraint positions and types, boundary conditions and circumferential mode numbers.

Estimation for rotational processes with one degree of freedom--Part II: Discrete-time processes

Abstract: General error criteria and probability distributions on the circle are studied in connection with estimation by using their Fourier series representations. Conditional probability densities for certain discrete-time folded normal processes, which are analogous to the continuous-time processes associated with the bilinear problems considered in Part I of this series, are computed. An intrinsic physical difference between the discrete-time and continuous-time problems is discussed, and the complexity of the estimation equations in the discrete-time case is analyzed in this setting. Suboptimal sequential filtering schemes are briefly discussed. In addition, Fourier analysis of conditional probability distributions exposes the inherent rich structure in quite general classes of estimation problems on the circle.

On the eigenvalues of the Orr-Sommerfeld equation

Abstract: The characteristic function defining the eigenvalues of the Orr-Sommerfeld equation is discussed and it is shown how the expected analytic properties of this function can be exploited to generate series expansions defining eigenvalues within the circle of convergence This technique is applied to the modes arising in the Blasius flat-plate boundary layer (treated as a parallel flow), for which the complex wavenumber α can be expanded as a convergent power series in the complex frequency parameter β in various regions of the β plane Such power series are effectively equivalent to Fourier expansions and the properties of the latter are used to find the coefficientsA square-root singularity in the relationship between α and β is found and it is shown how α can, nevertheless, be expressed in terms of β as the sum of one regular series and the square root of a second regular series The loci of the real and imaginary parts of α have been computed from these series and show the behaviour in the neighbourhood of the branch pointThe series description provides a particularly simple and rapid method of evalauting eigenvalues and their derivatives within any given region

Scattering of spherical pressure pulses by a hard cylinder

Abstract: The transient scattering of spherical pressure pulses by an infinitely long acoustically hard circular cylinder is analyzed. The pressure in the neighborhood of the reflected wavefront is calculated by the method of series expansion in conjunction with the transport equations. From the wave equation, a simple integral formula is derived by means of which the solution to the present three‐dimensional problem can be effectively constructed from the two‐dimensional line source solution. Thus, the pressure in the neighborhood of the diffracted wavefront is computed based upon Friedlander’s diffraction formulas pertaining to the scatttering of a cylindrical pulse by a hard cylinder. Outside the neighborhood of the scattered wavefronts, the solution is obtained by the Fourier series and integral transform techniques and is accurately computed by this formula. By combining the various solutions, the true time histories of the scattered pressure are inferred for various locations on the cylinder. Some features of...

A fourier series in an arbitrary bounded orthonormal system that diverges on a set of positive measure

Abstract: It is shown that every system of collectively bounded orthonormal functions admits an integrable function whose Fourier series diverges on a set of positive measure.Bibliography: 6 titles.

Equivalent circuit of a ceramic ring transducer operated in the dipole mode

Abstract: An equivalent circuit for a piezoelectric ring transducer excited in its dipole mode and its remaining odd‐order curcumferential extensional modes is derived. The analysis is accomplished by expressing the applied electric field excitation by a Fourier series and then combining solutions appropriately. The equivalent network presented here has a voltage–current terminal pair and a force–velocity terminal pair to each Fourier component which yield the transducer performance for the dipole and higher‐order modes.Subject Classification: 85.52; 40.85.

Absolute convergence of Fourier series

Abstract: In this paper we generalize the Bernshtein-Szasz and Zygmund-Szasz theorems on the absolute convergence of Fourier series in the classes Lip α.

Walsh-to-Fourier Spectral Conversion for Periodic Waves

Abstract: Relations are developed for the determination of the Fourier spectra of frequency-limited periodic waves from truncated Walsh spectra. The matrix conversion process is simplest if the highest-order Walsh coefficient in the spectru to be converted is 2n, where n is an integer. For such cases, compensation for truncation consists of a diagonal matrix that premultiplies the Walsh to Fourier conversion matrix and the elements of which are [(sinx)/x]-2 terms. Element values range between unity and less than ?2/4. The same compensation matrix is used for determning the Walsh spectra. of sequency-limited waves from 2n Fourier expansion terms. Examples are included which demonstrate the spectral conversion processes, Walsh to Fourier and Fourier to Walsh.

Derivation of a Mathematical Equation for the EEG and the General Solution Within the Brain and in Space

Abstract: A mathematical model for the EEG is presented here based on physical and anatomical characteristics of the brain. The important finding is that the system is characterised by a diffusion equation whose general solution represents a Fourier series expansion for the EEG at any point of the zone under consideration.

Vibrations of circular cylindrical shells

Abstract: An exact solution method for the free vibration problem of thin circular cylindrical shells is presented. The differential equations of motion are solved directly with the use of simple Fourier series as the modal displacement functions. Stokes' transformation is exploited to obtain correct series expressions for the derivatives of the Fourier series. From this method an explicit expression of the exact frequency equation can be obtained for any kind of boundary conditions. The accuracy of the present method is checked against available data. The proposed method is then used to find the modal characteristics of the thermal liner of the Fast Test Reactor (FTR). The numerical results obtained are compared with finite element method solutions. (auth)