Fourier Analysis of the C. I. E. Colorimetric Reference System

The Fourier coefficients are investigated for such a periodic time function whose magnitude keeps constant during the time of every half-period. In this case the relationship between the mean value and the Fourier coefficients is achieved using appropriate mathematical theory. It was applied successfully to the study of torsional harmonic excitations due to gas pressure in a four-cycle IC engine. A formula relating the truncated Fourier coefficient series to MIP has been established with very little error. The series simulation of Fourier coefficients is utilized in the determination and analysis of excitations.

Computations of Fourier coefficients and related integrals of the associated Legendre functions with a new method along with their application to spherical harmonics analysis and synthesis are presented. The method incorporates a stable three-step recursion equation that can be processed separately for each colatitudinal Fourier wavenumber. Recursion equations for the zonal and sectorial modes are derived in explicit single-term formulas to provide accurate initial condition. Stable computations of the Fourier coefficients as well as the integrals needed for the projection of Legendre functions are demonstrated for the ultra-high degree of 10,800 corresponding to the resolution of one arcmin. Fourier coefficients, computed in double precision, are found to be accurate to 15 significant digits, indicating that the normalized error is close to the machine round-off error. The orthonormality, evaluated with Fourier coefficients and related integrals, is shown to be accurate to O(10−15) for degrees and orders up to 10,800. The Legendre function of degree 10,800 and order 5,000, synthesized from Fourier coefficients, is accurate to the machine round-off error. Further extension of the method to even higher degrees seems to be realizable without significant deterioration of accuracy. The Fourier series is applied to the projection of Legendre functions to the high-resolution global relief data of the National Geophysical Data Center of the National Oceanic and Atmospheric Administration, and the spherical harmonic degree variance (power spectrum) of global relief data is discussed.

Non-parametric frequency identification and estimation in mean function for almost periodically correlated time series

A system of transparent layers on top of an arbitrary underlying subsystem is considered. It is shown that the layer refractive index and the subsystem generalized Fresnel coefficients can be expressed in a simple and explicit way by the Fourier coefficients of the ellipsometric function ρ with the layer thickness as an expansion parameter. Thus the inverse ellipsometric problem is reduced to a much simpler and well-defined problem of finding these Fourier coefficients. Analysis shows that the ellipsometric inverse task must be considered separately, depending on whether the modulus of the s-polarization Fresnel coefficient for the ambient–layer interface is smaller than, equal to, or greater than that for the layer–substrate system.

In two preceding papers by Jagodzinski (Phys. Stat. Sol.(a)146(1994) 477; Acta Crystallogr. A52(1996) 675), a new method of phase determination has been described. Using latent lattices of point scatterers for a single kind of atoms, the structure is generated by shift vectors, displacing the atoms from the origins of the latent lattice into their correct positions. The mathematical treatment of this procedure resulted in very complex formulae for structure factorsE′(h), normalised such thatE′(0) = 1. The three components of the displacement vectors define a new real and reciprocal shift space. The method is briefly described in the first three sections of this paper. There is an important difference between one and higher dimensions: the number of latent lattices with different shift functions is infinite, but only very few of them are optimal. Phases and amplitudes of the Fourier coefficients of the shift function are strongly correlated. This property results from the transformation of diffraction data and the prohibition of trespassing of points, caused by large Fourier coefficients of the shift function. In addition to the well known influence of strong reflections, the approximate [unk]-symmetries of the diffraction data and the areas of minimum intensity play an outstanding role for the selection of the optimal latent lattice. Its corresponding shift function has small displacements and small amplitudes of the Fourier coefficients. In the case of complicated structures, these properties demand a careful computer evaluation of the intensity distribution in reciprocal space, in order to determine the optimal latent lattice of a given structure. – The unit cell of the structure containsM(= number of atoms) cells of the latent lattice. As long asMis a product of three integersMxMyMz, there is no difficulty in finding the optimal latent lattice in diffraction. Each set of directions has to be tested in reciprocal space. In all other cases difficulties arise which may only be overcome with the aid of latent sublattices (smaller cell, more atoms in the structure), or latent superlattices (larger cell, less atoms in the structure). Both cases are discussed, and it is shown that the existence of the various types of latent lattices may be found in the normal diffraction pattern, applying the criteria derived in this paper. The meaning of latent sub- and superlattices is discussed from the structural point of view, and phase relations, caused by the prohibition of trespassing of points, are derived. The great power of the new method will be discussed in a later paper, where differences of shift functions are introduced. Their important consequences for phase determination will be demonstrated with the aid of certain systems of linear equations.

In his letter to the Editor of "Mathematical Notes," the referee pointed out the incompleteness of the proof of Lemma 3. Indeed, the proof should have been provided with some reservations, which was not done.First, it should be noted the functions f taking negative values on [0, π/2] can be excluded.For functions f ∈ H[ω] that are nonnegative on [0, π/2], we denote

We compare rates of decay of the cosine Fourier coefficients of a 2φ-periodic function f with that of Riemann sum of f over [0, 2φ] minus .

In this paper we study the Fourier coefficients of theta functions attached to Dirichlet characters at cusps other than infinity. The method is based on expressing them in terms of explicit elements of the adelic Schwartz space and studying the action of the adelic metaplectic group on these elements. We derive explicit formulae for the Fourier coefficients at all cusps. For the sake of simplicity, some restrictions are placed on the Dirichlet characters considered.

The implication is a transformation of a Harker synthesis that has the property of indicating the possible positions of atoms in the crystal structure. The location of the atoms is subject to a certain ambiguity which can be defined by an ambiguity factor, m = M/q. M is a basic ambiguity characteristic of the symmetry axis and arising from group products, while q is a degeneracy factor expressing a particular kind of equivalence of these products. The ambiguity factor can also be determined more easily with the aid of missing spectra. It is shown that M is the modulus of the transformation from the implication cell to the crystal cell, and 1/q is the fraction of spectra present. The ambiguity factor of any implication synthesis can, therefore, be easily determined from the symmetry and characteristic missing spectra of the space group. The ambiguity of an implication can be resolved with the aid of m Patterson line syntheses. The implication synthesis, therefore, provides a theoretical device for untangling the Patterson synthesis. This suggests that there should exist a simple relation between the Fourier coefficients in the implication function and the corresponding coefficients in the Fourier expression of the projected electron density. The details of this relation are dealt with in another paper.

We present a package that allows the computation of the trigonometric Fourier coefficients of a smooth function. The function can be provided as a subprogram or as a data list of function values at equally spaced points. The computational cost of the algorithm does not depend on the required number of Fourier coefficients. Numerical results of comparative tests with a standard integrator for oscillatory functions are also reported.

Boolean function is one of the most fundamental computation models in theoretical computer science. The two most common representations of Boolean functions are Fourier transform and real polynomial form. Applying analytic tools under these representations to the study Boolean functions has led to fruitful research in many areas such as complexity theory, learning theory, inapproximability, pseudorandomness, metric embedding, property testing, threshold phenomena, social choice, etc. In this thesis, we focus on \emph{sparse representations} of Boolean function in both Fourier transform and polynomial form, and obtain the following new results. A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function $f$ over $\mathbb{F}_2^{n}$ has the same absolute value, namely $|\hat{f}(\alpha)|=1/2^k$ for every $\alpha$ in the Fourier support of $f$, then $f$ must be the indicator function of some affine subspace of dimension $n-k$. Here we slightly generalize their result, and show that Boolean functions whose Fourier coefficients take values in the set $\{-2/2^k, -1/2^k, 0, 1/2^k, 2/2^k\}$ are indicator functions of two disjoint affine subspaces of dimension $n-k$ or four disjoint affine subspaces of dimension $n-k-1$. Our main technical tools are results from additive combinatorics which offer tight bounds on the affine span size of a subset of $\mathbb{F}_2^{n}$ when the doubling constant of the subset is small. For polynomial representation of Boolean function, we study the distribution of the number of non-zero coefficients of \emph{random} Boolean functions. For a random Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$, i.e., a function whose value at each point on the Boolean cube is chosen independently and uniformly at random from $\{0,1\}$, in real polynomial representation, we give several bounds and concentration results about the distribution of the sparsity of $f$.

Riemann and Lebesgue-type integrations can be employed to represent operators on normed function spaces whose norms are not stronger than sup-norm by T(f) = f f dp where y is determined by the action of T on the simple functions. The real-valued absolutely continuous functions on [0, 1] are not in the closure of the simple functions in the BV norm, and hence such an integral representation of an operator is not obtainable. In this paper the authors develop a v-integral whose structure depends on fundamental functions different than simple functions. This integral is as computable as the Riemann integral. By using these fundamental functions, the authors are able to obtain a direct, analytic representation of the linear functionals on AC which are continuous in the B V norm in terms of the v-integral. Further, the v-integral gives a characterization of the dual of AC in terms of the space of fundamentally bounded set functions which are convex with respect to length. This space is isometrically isomorphically identified with the space of Lipschitz functions anchored at zero with the norm given by the Lipschitz constant, which in turn is isometrically isomorphic to L-. Hence a natural identification exists between the classical representation and the one given in this paper. The results are extended to the vector setting.

An explicit representation of Verblunsky coefficients

Using Fourier series to analyse mass imperfections in vibratory gyroscopes

We consider Fourier coefficients for functions of bounded variation with respect to general orthonormal systems (GONS). We show that in general case the sequence of coefficients may have an arbitrarily prescribed order of vanishing. In this connection we seek for a class of GONS such that Fourier coefficients of a function from V(0, 1) satisfy the same inequality as in the case of classical systems (trigonometric, Walsh, and Haar ones). In the present paper we study this problem and related issues.