Showing papers on "Fourier series published in 1972"

Fourier Descriptors for Plane Closed Curves

Abstract: A method for the analysis and synthesis of closed curves in the plane is developed using the Fourier descriptors FD's of Cosgriff [1]. A curve is represented parametrically as a function of arc length by the accumulated change in direction of the curve since the starting point. This function is expanded in a Fourier series and the coefficients are arranged in the amplitude/phase-angle form. It is shown that the amplitudes are pure form invariants as well as are certain simple functions of phase angles. Rotational and axial symmetry are related directly to simple properties of the Fourier descriptors. An analysis of shape similarity or symmetry can be based on these relationships; also closed symmetric curves can be synthesized from almost arbitrary Fourier descriptors. It is established that the Fourier series expansion is optimal and unique with respect to obtaining coefficients insensitive to starting point. Several examples are provided to indicate the usefulness of Fourier descriptors as features for shape discrimination and a number of interesting symmetric curves are generated by computer and plotted out.

Fourier Preprocessing for Hand Print Character Recognition

Abstract: A pattern-recognition method, making use of Fourier transformations to extract features which are significant for a pattern, is described. The ordinary Fourier coefficients are difficult to use as input to categorizers because they contain factors dependent upon size and rotation as well as an arbitrary phase angle. From these Fourier coefficients, however, other more useful features can easily be derived. By using these derived property constants, a distinction can be made between genuine shape constants and constants representing size, location, and orientation. The usefulness of the method has been tested with a computer program that was used to classify 175 samples of handprinted letters, e.g., 7 sets of the 25 letters A to Z. In this test, 98 percent were correctly recognized when a simple nonoptimized decision method was used. The last section contains some considerations of the technical realizability of a fast preprocessing system for reading printed text.

Fourier series and integrals

Abstract: Historical Introduction. Fourier Series. Fourier Integrals. Fourier Integrals and Complex Function Theory. Fourier Series and Integrals on Groups. Additional Reading. Bibliography.

A new numerical approach to the calculation of electromagnetic scattering properties of two-dimensional bodies of arbitrary cross section

Abstract: A new method is introduced for formulating the scattering problem in which the scattered fields (and the interior fields in the case of a dielectric scatterer) are represented in an expansion in terms of free-space modal wave functions in cylindrical coordinates, the coefficients of which are the unknowns. The boundary conditions are satisfied using either an analytic continuation procedure, in which the far-field pattern (in Fourier series form) is continued into the near field and the boundary conditions are applied at the surface of the scatterer; or the completeness of the modal wave functions, to approximately represent the fields in the interior and exterior regions of the scatterer directly. The methods were applied to the scattering of two-dimensional cylindrical scatterers of arbitrary cross section and only the TM polarization of the excitation is considered. The solution for the coefficients of the modal wave functions are obtained by inversion of a matrix which depends only on the shape and material of the scatterer. The methods are illustrated using perfectly conducting square and elliptic cylinders and elliptic dielectric cylinders. A solution to the problem of multiple scattering by two conducting scatterers is also obtained using only the matrices characterizing each of the single scatterers. As an example, the method is illustrated by application to a two-body configuration.

Convergence, uniqueness, and summability of multiple trigonometric series

Abstract: In this paper our primary interest is in developing further insight into convergence properties of multiple trigonometric series, with emphasis on the problem of uniqueness of trigonometric series Let E be a subset of positive (Lebesgue) measure of the k dimensional torus The principal result is that the convergence of a trigonometric series on E forces the boundedness of the partial sums almost everywhere on E where the system of partial sums is the one associated with the system of all rectangles situated symmetrically about the origin in the lattice plane with sides parallel to the axes If E has a countable complement, then the partial sums are bounded at every point of E This result implies a uniqueness theorem for double trigonometric series, namely, that if a double trigonometric series converges unrestrictedly rectangularly to zero everywhere, then all the coefficients are zero Although uniqueness is still conjectural for dimensions greater than two, we obtain partial results and indicate possible lines of attack for this problem We carry out an extensive comparison of various modes of convergence (eg, square, triangular, spherical, etc) A number of examples of pathological double trigonometric series are displayed, both to accomplish this comparison and to indicate the "best possible" nature of some of the results on the growth of partial sums We obtain some compatibility relationships for summability methods and finally we present a result involving the (C, a, 0) summability of multiple Fourier series Introduction The main interest of this paper will be the theory of multiple trigonometric series Multiple Fourier series (the most important type of multiple trigonometric series) will be discussed only in connection with the theory of uniqueness and again in the last chapter For the definitions of any unfamiliar terms used in the introduction the reader is referred to ?1 One of the main difficulties in multiple series arises in connection with the usual consistency theorems for summation methods In order to maintain the validity of the typical theorem "convergence implies summability," even in the case of Poisson summation one has to have the added condition that all partial sums be bounded If one attempts to restrict himself to regular methods of forming the partial sums, it is easy to construct examples where this condition fails However, by introducing unrestricted rectangular partial sums, convergence of a multiple trigonometric Received by the editors January 22, 1971 AMS 1970 subject classifications Primary 42A92, 42A48, 42A20, 42A24, 40B05; Secondary 40G10, 40A05, 40D15

A Recurrence Technique for Expanding a Function in Spherical Harmonics

Abstract: A recurrence technique is described that enables the use of proven efficient Fourier transform techniques to be applied to the expansion of a given function in terms of spherical harmonics.

Determination of the jump of a function of bounded p-variation by its Fourier series

Abstract: A formula is obtained for the jump of a function of bounded p-variation at a given point in terms of derivatives of partial sums of its Fourier series.

Low order models representing realizations of turbulence

Abstract: The equations governing two-dimensional turbulence are written as an infinite system of ordinary differential equations, in which the dependent variables are the coefficients in the expansion of the vorticity field in a double Fourier series. The variables are sorted into sets which correspond to consecutive bands in the wavenumber spectrum; within each set it is supposed that the separate variables will exhibit statistically similar behaviour. A low order model is then constructed by retaining only a few variables within each set. Multiplicative factors are introduced into the equations to compensate for the reduced number of terms in the summations. Like the original equations, the low order equations conserve kinetic energy and enstrophy, apart from the effects of external forcing and viscous dissipation. A special case is presented in which the bands are half octaves and there is effectively only one dependent variable per set. Solutions of these equations are compared with conventional numerical simulations of turbulence, and agree reasonably well, although the nonlinear effects are somewhat underestimated.

The convergence almost everywhere of Legendre series

Abstract: It is proved that the Legendre series of an LP function converges almost everywhere, provided 4/3 1, then its Fourier series converges p.p. By combining his theorem with standard equiconvergence theorems we can prove the first part of the following result. THEOREM. Iff E LPfor some p in the range 4/3

A new expansion in the theory of polar liquids

Abstract: The Boltzmann factor for a pair of molecules interacting as coplanar nonpolarizable point dipoles is expanded as a double Fourier series in the orientation angles enabling the angle integrations in the partition function to be performed. This leads to an expansion for the thermodynamic properties of such a system as a perturbation on those of a spherically symmetric liquid.

The ^{} behavior of eigenfunction expansions

Abstract: We investigate the extent to which the eigenf unction expansions arising from a large class of two-point boundary value problems behave like Fourier series expansions in the norm of LP(O, .1), 1 < p < oo. We obtain our results by relating Green's function to the Hilbert transform.

Continuously updating fourier coefficients every sampling interval

Abstract: A method and apparatus for continually producing updated Fourier coefficient values of an input signal during each sample time. The Fourier coefficient F(k Omega ) is first calculated for any sequence of N samples of the input signal f(zT). To calculate the Fourier coefficients for the next ensuing sample time, the previously calculated Fourier coefficient value is updated by the addition of the product of the reference value and the difference between the new sample and the sample which occurred N samples earlier in time. This process is continued for each new sample.

Integration and Harmonic Analysis on Compact Groups

Abstract: General Introduction Acknowledgements Part I. Integration and the Riesz representation theorem: 1. Preliminaries regarding measures and integrals 2. Statement and discussion of Riesz's theorem 3. Method of proof of RRT: preliminaries 4. First stage of extension of I 5. Second stage of extension of I 6. The space of integrable functions 7. The a- measure associated with I: proof of the RRT 8. Lebesgue's convergence theorem 9. Concerning the necessity of the hypotheses in the RRT 10. Historical remarks 11. Complex-valued functions Part II. Harmonic analysis on compact groups 12. Invariant integration 13. Group representations 14. The Fourier transform 15. The completeness and uniqueness theorems 16. Schur's lemma and its consequences 17. The orthogonality relations 18. Fourier series in L2(G) 19. Positive definite functions 20. Summability and convergence of Fourier series 21. Closed spans of translates 22. Structural building bricks and spectra 23. Closed ideals and closed invariant subspaces 24. Spectral synthesis problems 25. The Hausdorff-Young theorem 26. Lacunarity.

An integral analogue of Taylor’s series and its use in computing Fourier transforms

Abstract: In this paper, an integral analogue of Taylor's series f(z) = f ', Dof(z o)(z -z o) lo/ P( + 1) dc is discussed. D 'f(z) is a fractional derivative of order w. Extensions of this integral are also given, one of which is an integral analogue of Lagrange's expansion. These integrals are shown to be generalizations of the Fourier integral theorem. Several special cases of these integrals are computed, and a table of Fourier transforms emerges. 1. Introduction. The fractional derivative of order a of the function f(z) with respect to g(z), Dc(2)f(z), is a generalization of the familiar derivative daf(z)/(dg(z))a to nonintegral values of a. In the author's previous papers on the fractional calculus, three distinct features evolved: (1) Certain formulas familiar from the elementary calculus were shown to be special cases of more general expressions involving fractional derivatives. These included Taylor's series (5), Leibniz rule (3), (6), (7), the chain rule (4), and Lagrange's expansion (5). (2) Through the fractional calculus, we were able to relate formulas familiar from the study of Fourier analysis to the above-mentioned calculus relations. Thus, it was shown that the generalized Taylor's series could be viewed as an extension of the Fourier series (5), and that the generalized Leibniz rule was an extension of Parseval's relation (7). (3) Most of the important special functions can be represented by fractional derivatives of elementary functions, such as J,(z) = ir-K"2(2z)- Dz2 _-/2(cos z)/z. We found that our extensions of calculus formulas, when combined with fractional derivative representations for the higher functions, produced interesting series re- lations involving the special functions. This paper continues our study of the fractional calculus by exposing the three features outlined above for certain integrals which are related to Taylor's series and Lagrange's expansion. We find it useful to distinguish three special cases: Case 1. The expression

The finite prism in analysis of thick simply supported bridge boxes

Abstract: A COMBINATION OF FOURIER SERIES AND FINITE ELEMENT EXPANSION IS USED TO STUDY THREE-DIMENSIONAL STRUCTURES OF CONSTANT CROSS-SECTION SWEEPING A SOLID VOLUME ALONG A STRAIGHT OR CURVED GENERATOR. THE THREE-DIMENSIONAL PROBLEM IS REDUCED TO A SERIES OF TWO-DIMENSIONAL ONES BECAUSE OF DECOUPLING OF TERMS, THUS OFTEN RESULTING IN CONSIDERABLE ECONOMY. THE ACCURACY OF THE METHOD IS SHOWN IN STRAIGHT AND CURVED BEAM EXAMPLES AND A PRACTICAL APPLICATION TO A TYPICAL CONCRETE BRIDGE BOX IS GIVEN. /AUTHOR/

Design of spatial filters and their application to high-resolution aeromagnetic data

Abstract: Methods for the design of spatial filters are discussed in this paper For a given response of a one‐dimensional filter, the weighting coefficients are calculated by solving a set of simultaneous equations with a simple matrix inversion procedure In the case of a two‐dimensional filter, the method for obtaining the coefficients of a double Fourier series representing a set of given values is used to design the spatial operator The problems connected with the length of the operator and the choice of a suitable decay in the high‐frequency response are discussed in detail In order to show the usefulness of these methods, the paper presents several examples of operators designed for computing the vertical gradient, the second vertical derivative, and downward continuation of potential field data A two‐dimensional vertical gradient filter is applied to the total field data obtained during a high‐resolution aeromagnetic survey over an area in the Precambrian Shield of Northeastern Ontario The calculated gr

Analysis of Closed-Loop Conveyor Systems

Abstract: A closed-loop conveyor system having a single loading station, a single unloading station, and operating with time-varying input and output flow rates, is analyzed. The balance of flow on the conveyor is represented by a difference equation. Solutions of that difference equation appear naturally in terms of a Fourier series expansion. An important description of the system is its frequency response. Singularities in the frequency response represent cases of incompatibility. Incompatibility is shown to depend on the ratio T/P of conveyor period to work-cycle period, and on the presence of harmonics in input and output flow rates. Solutions for several specific cases are presented graphically.

From characteristic function to distribution function via fourier analysis

Abstract: In this paper characteristic functions of probability distributions are considered. The numerical calculation of the distribution function when the characteristic function is known is discussed and two different methods are presented.

Bounds on the truncation error of periodic signals

Abstract: Bounds on the truncation error as a function of the number of terms used in a Fourier series are obtained. Two different error criteria, based on the Hilbert norm and the Chebyshev norm, are used.

Absolute convergence of fourier series with respect to complete orthonormal systems

Abstract: This article gives a survey of results on absolute convergence of Fourier series with respect to the trigonometric system, the Haar system and arbitrary complete orthonormal systems. It clarifies which of the propositions on absolute convergence of trigonometric Fourier series and Fourier-Haar series remain valid for arbitrary complete orthonormal systems.

Superharmonic Resonance in Piecewise-Linear System : Effect of Damping and Stability Problem

Abstract: This paper deals with perfect Fourier series solutions for superharmonic vibrations in various piecewise-linear systems with viscous damping, and gives a stability criterion for these solutions utilizing Hill's infinite determinant. Resonance curves for superharmonic vibrations of the 3rd order and stability charts for vibrations of Type I are constructed. The results using analog computer are shown to confirm these curves and charts.

Fourier transforms of homogeneous distribution

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