Showing papers on "Fourier series published in 1971"

Periodic orbits and classical quantization conditions

Abstract: The relation between the solutions of the time‐independent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half‐width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well‐defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropicgermanium.

Fourier analysis and approximation

Abstract: 0 Preliminaries.- 0 Preliminaries.- 0.1 Fundamentals on Lebesgue Integration.- 0.2 Convolutions on the Line Group.- 0.3 Further Sets of Functions and Sequences.- 0.4 Periodic Functions and Their Convolution.- 0.5 Functions of Bounded Variation on the Line Group.- 0.6 The Class BV2?.- 0.7 Normed Linear Spaces, Bounded Linear Operators.- 0.8 Bounded Linear Functional, Riesz Representation Theorems.- 0.9 References.- I Approximation by Singular Integrals.- 1 Singular Integrals of Periodic Functions.- 1.0 Introduction.- 1.1 Norm-Convergence and-Derivatives.- 1.1.1 Norm-Convergence.- 1.1.2 Derivatives.- 1.2 Summation of Fourier Series.- 1.2.1 Definitions.- 1.2.2 Dirichlet and Fejer Kernel.- 1.2.3 Weierstrass Approximation Theorem.- 1.2.4 Summability of Fourier Series.- 1.2.5 Row-Finite ?-Factors.- 1.2.6 Summability of Conjugate Series.- 1.2.7 Fourier-Stieltjes Series.- 1.3 Test Sets for Norm-Convergence.- 1.3.1 Norms of Some Convolution Operators.- 1.3.2 Some Applications of the Theorem of Banaeh-Steinhaus.- 1.3.3 Positive Kernels.- 1.4 Pointwise Convergence.- 1.5 Order of Approximation for Positive Singular Integrals.- 1.5.1 Modulus of Continuity and Lipschitz Classes.- 1.5.2 Direct Approximation Theorems.- 1.5.3 Method of Test Functions.- 1.5.4 Asymptotic Properties.- 1.6 Further Direct Approximation Theorems, Nikolski? Constants.- 1.6.1 Singular Integral of Fejer-Korovkin.- 1.6.2 Further Direct Approximation Theorems.- 1.6.3 Nikolski? Constants.- 1.7 Simple Inverse Approximation Theorems.- 1.8 Notes and Remarks.- 2 Theorems of Jackson and Bernstein for Polynomials of Best Approximation and for Singular Integrals.- 2.0 Introduction.- 2.1 Polynomials of Best Approximation.- 2.2 Theorems of Jackson.- 2.3 Theorems of Bernstein.- 2.4 Various Applications.- 2.5 1.- 4.2.1 The Case p = 2.- 4.2.2 The Case p ? 2.- 4.3 Finite Fourier-Stieltjes Transforms.- 4.3.1 Fundamental Properties.- 4.3.2 Inversion Theory.- 4.3.3 Fourier-Stieltjes Transforms of Derivatives.- 4.4 Notes and Remarks.- 5 Fourier Transforms Associated with the Line Group.- 5.0 Introduction.- 5.1 L1-Theory.- 5.1.1 Fundamental Properties.- 5.1.2 Inversion Theory.- 5.1.3 Fourier Transforms of Derivatives.- 5.1.4 Derivatives of Fourier Transforms, Moments of Positive Functions Peano and Riemann Derivatives.- 5.1.5 Poisson Summation Formula.- 5.2 Lp-Theory, 1 < p ? 2.- 5.2.1 The Case p = 2.- 5.2.2 The Case 1 2.- 5.2.3 Fundamental Properties.- 5.2.4 Summation of the Fourier Inversion Integral.- 5.2.5 Fourier Transforms of Derivatives.- 5.2.6 Theorem of Plancherel.- 5.3 Fourier-Stieltjes Transforms.- 5.3.1 Fundamental Properties.- 5.3.2 Inversion Theory.- 5.3.3 Fourier-Stieltjes Transforms of Derivatives.- 5.4 Notes and Remarks.- 6 Representation Theorems.- 6.0 Introduction.- 6.1 Necessary and Sufficient Conditions.- 6.1.1 Representation of Sequences as Finite Fourier or Fourier-Stieltjes Transforms.- 6.1.2 Representation of Functions as Fourier or Fourier-Stieltjes Transforms.- 6.2 Theorems of Bochner.- 6.3 Sufficient Conditions.- 6.3.1 Quasi-Convexity.- 6.3.2 Representation as L1/2? Transform.- 6.3.3 Representation as L1-Transform.- 6.3.4 A Reduction Theorem.- 6.4 Applications to Singular Integrals.- 6.4.1 General Singular Integral of Weierstrass.- 6.4.2 Typical Means.- 6.5 Multipliers.- 6.5.1 Multipliers of Classes of Periodic Functions.- 6.5.2 Multipliers on LP.- 6.6 Notes and Remarks.- 7 Fourier Transform Methods and Second-Order Partial Differential Equations.- 7.0 Introduction.- 7.1 Finite Fourier Transform Method.- 7.1.1 Solution of Heat Conduction Problems.- 7.1.2 Dirichlet's and Neumann's Problem for the Unit Disc.- 7.1.3 Vibrating String Problems.- 7.2 Fourier Transform Method in L1.- 7.2.1 Diffusion on an Infinite Rod.- 7.2.2 Dirichlet's Problem for the Half-Plane.- 7.2.3 Motion of an Infinite String.- 7.3 Notes and Remarks.- III Hilbert Transforms.- 8 Hilbert Transforms on the Real Line.- 8.0 Introduction.- 8.1 Existence of the Transform.- 8.1.1 Existence Almost Everywhere.- 8.1.2 Existence in L2-Norm.- 8.1.3 Existence in Lp-Norm, 1 ?.- 8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated Hilbert Transforms.- 8.2.1 Hilbert Formulae.- 8.2.2 Conjugates of Singular Integrals: 1 ?.- 8.2.3 Conjugates of Singular Integrals: p = 1.- 8.2.4 Iterated Hilbert Transforms.- 8.3 Fourier Transforms of Hilbert Transforms.- 8.3.1 Signum Rule.- 8.3.2 Summation of Allied Integrals.- 8.3.3 Fourier.- 8.3.4 Norm-Convergence of the Fourier Inversion Integral.- 8.4 Notes and Remarks.- 9 Hilbert Transforms of Periodic Functions.- 9.0 Introduction.- 9.1 Existence and Basic Properties.- 9.1.1 Existence.- 9.1.2 Hilbert Formulae.- 9.2 Conjugates of Singular Integrals.- 9.2.1 The Case 1 ?.- 9.2.2 Convergence in C2? and L1/2?.- 9.3 Fourier Transforms of Hilbert Transforms.- 9.3.1 Conjugate Fourier Series.- 9.3.2 Fourier Transforms of Derivatives of Conjugate Functions, the Classes (W~)xr2?'(V~)rx2?.- 9.3.3 Norm-Convergence of Fourier Series.- 9.4 Notes and Remarks.- IV Characterization of Certain Function Classes 355.- 10 Characterization in the Integral Case.- 10.0 Introduction.- 10.1 Generalized Derivatives, Characterization of the Classes Wrx2?.- 10.1.1 Riemann Derivatives in X2?-Norm.- 10.1.2 Strong Peano Derivatives.- 10.1.3 Strong and Weak Derivatives, Weak Generalized Derivatives.- 10.2 Characterization of the Classes Vr2?.- 10.3 Characterization of the Classes (V~)rx2?.- 10.4 Relative Completion.- 10.5 Generalized Derivatives in Lp-Norm and Characterizations for 1 ? p ?2.- 10.6 Generalized Derivatives in X(R)-Norm and Characterizations of the Classes Wrx(R) and Vrx(R).- 10.7 Notes and Remarks.- 11 Characterization in the Fractional Case.- 11.0 Introduction.- 11.1 Integrals of Fractional Order.- 11.1.1 Integral of Riemann-Liouville.- 11.1.2 Integral of M. Riesz.- 11.2 Characterizations of the Classes W[LP |?|?], V[LP |?|?], 1 ? p ? 2.- 11.2.1 Derivatives of Fractional Order.- 11.2.2 Strong Riesz Derivatives of Higher Order, the Classes V[LP |?|? ].- 11.3 The Operators R?{?} on Lp 1 ? p ? 2.- 11.3.1 Characterizations.- 11.3.2 Theorems of Bernstein-Titchmarsh and H. Weyl.- 11.4 The Operators R?(?} on 2?.- 11.5 Integral Representations, Fractional Derivatives of Periodic Functions.- 11.6 Notes and Remarks.- V Saturation Theory.- 12 Saturation for Singular Integrals on X2? and Lp, 1 ? p ? 2 433.- 12.0 Introduction.- 12.1 Saturation for Periodic Singular Integrals, Inverse Theorems.- 12.2 Favard Classes.- 12.2.1 Positive Kernels.- 12.2.2 Uniformly Bounded Multipliers.- 12.2.3 Functional Equations.- 12.3 Saturation in Lp, 1 ? p ? 2.- 12.3.1 Saturation Property.- 12.3.2 Characterizations of Favard Classes: p = 1.- 12.3.3 Characterizations of Favard Classes: 1 < p? 2.- 12.4 Applications to Various Singular Integrals.- 12.4.1 Singular Integral of Fejer.- 12.4.2 Generalized Singular Integral of Picard.- 12.4.3 General Singular Integral of Weierstrass.- 12.4.4 Singular Integral of Bochner-Riesz.- 12.4.5 Riesz Means.- 12.5 Saturation of Higher Order.- 12.5.1 Singular Integrals on the Real Line.- 12.5.2 Periodic Singular Integrals.- 12.6 Notes and Remarks.- 13 Saturation on X(R).- 13.0 Introduction.- 13.1 Saturation of D?(f x t) in X(R), Dual Methods.- 13.2 Applications to Approximation in Lp, 2 ?.- 13.2.1 Differences.- 13.2.2 Singular Integrals Satisfying (12.3.5).- 13.2.3 Strong Riesz Derivatives.- 13.2.4 The Operators R?{?}.- 13.2.5 Riesz and Fejer Means.- 13.3 Comparison Theorems.- 13.3.1 Global Divisibility.- 13.3.2 Local Divisibility.- 13.3.3 Special Comparison Theorems with no Divisibility Hypothesis.- 13.3.4 Applications to Periodic Continuous Functions.- 13.4 Saturation on Banach Spaces.- 13.4.1 Strong Approximation Processes.- 13.4.2 Semi-Groups of Operators.- 13.5 Notes and Remarks.- List of Symbols.- Tables of Fourier and Hilbert Transforms.

Convergence almost everywhere of certain singular integrals and multiple Fourier series

Abstract: T s Ts = o ( l o g l o g I~]), t~]-+ 0% for almost eve ry x in T,. I n Sections 1 to 3 in this pape r we prove among o ther things the following theorem, which generalizes the L p es t imate of the opera tor M* in [7]. TEEORE~. Assume that k is a Calderdn -Zygmund kernel defined in R', s ~_ 2, which has continuous derivatives of order ~ s 41 outside the origin. Let the operator M be defined by

On the convergence of multiple Fourier series

Abstract: Inequality (1) follows from the special case in which P is a triangle with a vertex at the origin; for any polygon breaks up into triangles, and the characteristic function of any triangle is a linear combination of characteristic functions of triangles with vertices at zero. Consequently, we can assume P has the form P= {(x, y)C£S\\ (x> y)-t

Whittaker's Cardinal Function in Retrospect*

Abstract: This paper exposes properties of the Whittaker cardinal function and illustrates the use of this function as a mathematical tool. The cardinal function is derived using the Paley-Wiener theorem. The cardinal function and the central-difference expansions arelinked through their similarities. A bound is obtained on the difference between the cardinal func- tion and the function which it interpolates. Several cardinal functions of a number of special functions are examined. It is shown how the cardinal function provides a link between Fourier series and Fourier transforms, and how the cardinal function may be used to solve integral equations.

An investigation of the array of circular-loop antennas

Abstract: A circular-loop antenna composed of N elements with arbitrary circumference arranged parallel in a row is analyzed by Fourier series expansion with emphasis on the existence of finite gaps at the driving points. The integral equations at the beginning are reduced to a series of linear simultaneous equations which contain only the Fourier coefficients of the electric currents of the same order. By appraising the current coefficients, the self-admittance is obtained as a sum of two parts: the finite series truncated at the number determined by the ratio of the circular-loop radius to the wire radius and the gap capacitance, and the mutual admittance is obtained as a finite series of fewer terms. Several computed curves concerning a parasitic endfire array of two elements are given, in which the computed values of the input admittance are substantiated by experiments, and radiation characteristics are examined to obtain a desirable configuration.

Piecewise Fourier Transformation for Picture Bandwidth Compression

Abstract: Recent advances in digital computer and optical technology have made image spectra determination practical. Pratt and Andrews [1] studied bandwidth compression using the Fourier transform of complete pictures. By treating pictures adaptively on a piecewise basis, picture detail is better represented. Also, subjective preferences of human vision can be used, which result in further improvements in picture quality. The original picture is sampled and then divided into small subsections. Each subsection is expanded in a two-dimensional Fourier series. The L -Fourier coefficients of largest absolute value are determined for each subsection, where L is proportional to the standard deviation of the picture samples in the subsection. The frequencies and complex amplitudes of those L -Fourier coefficients are transmitted. The number of quantization levels used for the Fourier coefficients in each subsection is made dependent on the standard deviation of the picture samples in the subsection, and the size of the quantum steps is made dependent on the magnitude of the largest Fourier coefficient of the subsection, aside from the average value. The frequencies of the coefficients correspond to positions in a twodimensional spatial frequency plane. These positions, or twodimensional frequencies, are transmitted by run-length coding. The process is adaptive in the sense that, its parameters vary from subsection to subsection of the picture in an effort to match the properties of the individual subsections. Subsection size and other important system constants are chosen with knowledge of the properties of human vision. We are able to obtain high-quality reconstructed pictures, using on the average 1.25 bits per picture point.

A Fast Fourier Transform for High-Speed Signal Processing

Abstract: An arbitrary-radix fast Fourier transform algorithm and a design of its implementing signal processing machine are introduced. The algorithm yields an implementation with a level of parallelism proportional to the radix r of factorization of the discrete Fourier transform, allows 100 percent utilization of the arithmetic unit, and yields properly ordered Fourier coefficients without the need for pre- or postordering of data.

Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature

Abstract: The conventional Fourier coefficient asymptotic expansion is derived by means of a specific contour integration. An adjusted expansion is obtained by deforming this contour. A corresponding adjustment to the Euler-Maclaurin expansion exists. The effect of this adjustment in the error functional for a general quadrature rule is investigated. It is the same as the effect of subtracting out a pair of complex poles from the integrand, using an unconventional subtraction function. In certain applications, the use of this subtraction function is of practical value. An incidental result is a direct proof of Erdelyi's formula for the Fourier coefficient asymptotic expansion, valid when f(x) has algebraic or logarithmic singularities, but is otherwise analytic. 1. The Fourier Coefficient Asymptotic Expansion. The Fourier coefficient asymptotic expansion (F.C.A.E.) (1.3) below is a classical formula which is elementary to derive using a standard application of the formula for integration by parts, namely, rl~~~~~~~~(v 1 (-)/2)

Method and apparatus for reordering data

Abstract: A reordering method and associated apparatus for storing and accessing samples of a physical process to facilitate the generation of Fourier series coefficients is disclosed. In particular, methods and apparatus for generating storage and retrieval patterns for a sequence of sampled data for facilitating the computation of fast Fourier transformation (FFT) of a plurality of overlapped records is disclosed.

A fourier approach to diffusion-nucleation calculations for thin foils

Abstract: The theory of diffusion and nucleation of Brown, Kelly and Mayer(2) is extended to the case of thin foils by expressing all the quantities involved as Fourier series. The method is then applied to the situation encountered during in situ electron irradiation experiments in high voltage electron microscopes. The results show how measurements of loop densities may be corrected for the effects of the surfaces and suggest areas for further experimental observations.

Complex helicity and the Sommerfeld-Watson transformation of group-theoretic expansions

Abstract: The connection of group-theoretic expansions (in both continuous and discrete bases) of a function defined onSO3 andSO2,1 and analytic on their complexification (except for certain singularities), is discussed. This is achieved by making various transformations of the Sommerfeld-Watson type. The analyticity assumptions needed for these transformations are considered and found to depend on the basis chosen for the representation functions. The applications of this work to multi-Reggeon expansions and, in particular, to the behaviour of Reggeon vertex functions are discussed. The connections between the various Sommerfeld-Watson and group-theoretic signatures are also given. As a preliminary, transforms are used to relate the Fourier series to the Fourier integral.

Almost Everywhere Convergence of Fourier Series on the Ring of Integers of a Local Field

Abstract: It is shown that the partial sums of the Fourier series of $L^p (\mathfrak{D})$-functions $(p > 1)$ converge almost everywhere (a.e.), where $\mathfrak{D}$ is the ring of integers in a local field K. This includes the case where K is a p-adic number field as well as the case where $\mathfrak{D}$ is the Walsh–Paley or dyadic group $2^\omega $. The techniques are essentially those used by Carleson [2] in establishing the a.e. convergence of trigonometric Fourier series for $L^2 ( - \pi ,\pi )$-functions as modified by Hunt [4] to obtain this same result for $L^p ( - \pi ,\pi )$-functions, $p > 1$. The necessary modifications for the local field setting are made in the context of the Sally’Taibleson [7] development of harmonic analysis on local fields and by use of Taibleson’s multiplier theorem [11]. These same results for $2^\omega $ have already been obtained by Billiard $(L^2 (2^\omega ))$ [1] and by Sjolin $(L^p (2^\omega ))$, $p > 1$) [8]. Many advantages (in particular the non-Archimidean nature of th...

One-dimensional theory

Abstract: 0 Preliminaries.- 0 Preliminaries.- 0.1 Fundamentals on Lebesgue Integration.- 0.2 Convolutions on the Line Group.- 0.3 Further Sets of Functions and Sequences.- 0.4 Periodic Functions and Their Convolution.- 0.5 Functions of Bounded Variation on the Line Group.- 0.6 The Class BV2?.- 0.7 Normed Linear Spaces, Bounded Linear Operators.- 0.8 Bounded Linear Functional, Riesz Representation Theorems.- 0.9 References.- I Approximation by Singular Integrals.- 1 Singular Integrals of Periodic Functions.- 1.0 Introduction.- 1.1 Norm-Convergence and-Derivatives.- 1.1.1 Norm-Convergence.- 1.1.2 Derivatives.- 1.2 Summation of Fourier Series.- 1.2.1 Definitions.- 1.2.2 Dirichlet and Fejer Kernel.- 1.2.3 Weierstrass Approximation Theorem.- 1.2.4 Summability of Fourier Series.- 1.2.5 Row-Finite ?-Factors.- 1.2.6 Summability of Conjugate Series.- 1.2.7 Fourier-Stieltjes Series.- 1.3 Test Sets for Norm-Convergence.- 1.3.1 Norms of Some Convolution Operators.- 1.3.2 Some Applications of the Theorem of Banaeh-Steinhaus.- 1.3.3 Positive Kernels.- 1.4 Pointwise Convergence.- 1.5 Order of Approximation for Positive Singular Integrals.- 1.5.1 Modulus of Continuity and Lipschitz Classes.- 1.5.2 Direct Approximation Theorems.- 1.5.3 Method of Test Functions.- 1.5.4 Asymptotic Properties.- 1.6 Further Direct Approximation Theorems, Nikolski? Constants.- 1.6.1 Singular Integral of Fejer-Korovkin.- 1.6.2 Further Direct Approximation Theorems.- 1.6.3 Nikolski? Constants.- 1.7 Simple Inverse Approximation Theorems.- 1.8 Notes and Remarks.- 2 Theorems of Jackson and Bernstein for Polynomials of Best Approximation and for Singular Integrals.- 2.0 Introduction.- 2.1 Polynomials of Best Approximation.- 2.2 Theorems of Jackson.- 2.3 Theorems of Bernstein.- 2.4 Various Applications.- 2.5 1.- 4.2.1 The Case p = 2.- 4.2.2 The Case p ? 2.- 4.3 Finite Fourier-Stieltjes Transforms.- 4.3.1 Fundamental Properties.- 4.3.2 Inversion Theory.- 4.3.3 Fourier-Stieltjes Transforms of Derivatives.- 4.4 Notes and Remarks.- 5 Fourier Transforms Associated with the Line Group.- 5.0 Introduction.- 5.1 L1-Theory.- 5.1.1 Fundamental Properties.- 5.1.2 Inversion Theory.- 5.1.3 Fourier Transforms of Derivatives.- 5.1.4 Derivatives of Fourier Transforms, Moments of Positive Functions Peano and Riemann Derivatives.- 5.1.5 Poisson Summation Formula.- 5.2 Lp-Theory, 1 < p ? 2.- 5.2.1 The Case p = 2.- 5.2.2 The Case 1 2.- 5.2.3 Fundamental Properties.- 5.2.4 Summation of the Fourier Inversion Integral.- 5.2.5 Fourier Transforms of Derivatives.- 5.2.6 Theorem of Plancherel.- 5.3 Fourier-Stieltjes Transforms.- 5.3.1 Fundamental Properties.- 5.3.2 Inversion Theory.- 5.3.3 Fourier-Stieltjes Transforms of Derivatives.- 5.4 Notes and Remarks.- 6 Representation Theorems.- 6.0 Introduction.- 6.1 Necessary and Sufficient Conditions.- 6.1.1 Representation of Sequences as Finite Fourier or Fourier-Stieltjes Transforms.- 6.1.2 Representation of Functions as Fourier or Fourier-Stieltjes Transforms.- 6.2 Theorems of Bochner.- 6.3 Sufficient Conditions.- 6.3.1 Quasi-Convexity.- 6.3.2 Representation as L1/2? Transform.- 6.3.3 Representation as L1-Transform.- 6.3.4 A Reduction Theorem.- 6.4 Applications to Singular Integrals.- 6.4.1 General Singular Integral of Weierstrass.- 6.4.2 Typical Means.- 6.5 Multipliers.- 6.5.1 Multipliers of Classes of Periodic Functions.- 6.5.2 Multipliers on LP.- 6.6 Notes and Remarks.- 7 Fourier Transform Methods and Second-Order Partial Differential Equations.- 7.0 Introduction.- 7.1 Finite Fourier Transform Method.- 7.1.1 Solution of Heat Conduction Problems.- 7.1.2 Dirichlet's and Neumann's Problem for the Unit Disc.- 7.1.3 Vibrating String Problems.- 7.2 Fourier Transform Method in L1.- 7.2.1 Diffusion on an Infinite Rod.- 7.2.2 Dirichlet's Problem for the Half-Plane.- 7.2.3 Motion of an Infinite String.- 7.3 Notes and Remarks.- III Hilbert Transforms.- 8 Hilbert Transforms on the Real Line.- 8.0 Introduction.- 8.1 Existence of the Transform.- 8.1.1 Existence Almost Everywhere.- 8.1.2 Existence in L2-Norm.- 8.1.3 Existence in Lp-Norm, 1 ?.- 8.2 Hilbert Formulae, Conjugates of Singular Integrals, Iterated Hilbert Transforms.- 8.2.1 Hilbert Formulae.- 8.2.2 Conjugates of Singular Integrals: 1 ?.- 8.2.3 Conjugates of Singular Integrals: p = 1.- 8.2.4 Iterated Hilbert Transforms.- 8.3 Fourier Transforms of Hilbert Transforms.- 8.3.1 Signum Rule.- 8.3.2 Summation of Allied Integrals.- 8.3.3 Fourier.- 8.3.4 Norm-Convergence of the Fourier Inversion Integral.- 8.4 Notes and Remarks.- 9 Hilbert Transforms of Periodic Functions.- 9.0 Introduction.- 9.1 Existence and Basic Properties.- 9.1.1 Existence.- 9.1.2 Hilbert Formulae.- 9.2 Conjugates of Singular Integrals.- 9.2.1 The Case 1 ?.- 9.2.2 Convergence in C2? and L1/2?.- 9.3 Fourier Transforms of Hilbert Transforms.- 9.3.1 Conjugate Fourier Series.- 9.3.2 Fourier Transforms of Derivatives of Conjugate Functions, the Classes (W~)xr2?'(V~)rx2?.- 9.3.3 Norm-Convergence of Fourier Series.- 9.4 Notes and Remarks.- IV Characterization of Certain Function Classes 355.- 10 Characterization in the Integral Case.- 10.0 Introduction.- 10.1 Generalized Derivatives, Characterization of the Classes Wrx2?.- 10.1.1 Riemann Derivatives in X2?-Norm.- 10.1.2 Strong Peano Derivatives.- 10.1.3 Strong and Weak Derivatives, Weak Generalized Derivatives.- 10.2 Characterization of the Classes Vr2?.- 10.3 Characterization of the Classes (V~)rx2?.- 10.4 Relative Completion.- 10.5 Generalized Derivatives in Lp-Norm and Characterizations for 1 ? p ?2.- 10.6 Generalized Derivatives in X(R)-Norm and Characterizations of the Classes Wrx(R) and Vrx(R).- 10.7 Notes and Remarks.- 11 Characterization in the Fractional Case.- 11.0 Introduction.- 11.1 Integrals of Fractional Order.- 11.1.1 Integral of Riemann-Liouville.- 11.1.2 Integral of M. Riesz.- 11.2 Characterizations of the Classes W[LP |?|?], V[LP |?|?], 1 ? p ? 2.- 11.2.1 Derivatives of Fractional Order.- 11.2.2 Strong Riesz Derivatives of Higher Order, the Classes V[LP |?|? ].- 11.3 The Operators R?{?} on Lp 1 ? p ? 2.- 11.3.1 Characterizations.- 11.3.2 Theorems of Bernstein-Titchmarsh and H. Weyl.- 11.4 The Operators R?(?} on 2?.- 11.5 Integral Representations, Fractional Derivatives of Periodic Functions.- 11.6 Notes and Remarks.- V Saturation Theory.- 12 Saturation for Singular Integrals on X2? and Lp, 1 ? p ? 2 433.- 12.0 Introduction.- 12.1 Saturation for Periodic Singular Integrals, Inverse Theorems.- 12.2 Favard Classes.- 12.2.1 Positive Kernels.- 12.2.2 Uniformly Bounded Multipliers.- 12.2.3 Functional Equations.- 12.3 Saturation in Lp, 1 ? p ? 2.- 12.3.1 Saturation Property.- 12.3.2 Characterizations of Favard Classes: p = 1.- 12.3.3 Characterizations of Favard Classes: 1 < p? 2.- 12.4 Applications to Various Singular Integrals.- 12.4.1 Singular Integral of Fejer.- 12.4.2 Generalized Singular Integral of Picard.- 12.4.3 General Singular Integral of Weierstrass.- 12.4.4 Singular Integral of Bochner-Riesz.- 12.4.5 Riesz Means.- 12.5 Saturation of Higher Order.- 12.5.1 Singular Integrals on the Real Line.- 12.5.2 Periodic Singular Integrals.- 12.6 Notes and Remarks.- 13 Saturation on X(R).- 13.0 Introduction.- 13.1 Saturation of D?(f x t) in X(R), Dual Methods.- 13.2 Applications to Approximation in Lp, 2 ?.- 13.2.1 Differences.- 13.2.2 Singular Integrals Satisfying (12.3.5).- 13.2.3 Strong Riesz Derivatives.- 13.2.4 The Operators R?{?}.- 13.2.5 Riesz and Fejer Means.- 13.3 Comparison Theorems.- 13.3.1 Global Divisibility.- 13.3.2 Local Divisibility.- 13.3.3 Special Comparison Theorems with no Divisibility Hypothesis.- 13.3.4 Applications to Periodic Continuous Functions.- 13.4 Saturation on Banach Spaces.- 13.4.1 Strong Approximation Processes.- 13.4.2 Semi-Groups of Operators.- 13.5 Notes and Remarks.- List of Symbols.- Tables of Fourier and Hilbert Transforms.

On the convergence rates of variational methods. I. Asymptotically diagonal systems

Abstract: We consider the problem of estimating the convergence rate of a variational solution to an inhomogeneous equation. This problem is not soluble in general without imposing conditions on both the class of expansion functions and the class of problems considered; we introduce the concept of "asymptotically diagonal systems," which is particularly appropriate for classical variational expansions as applied to elliptic partial differential equations. For such systems, we obtain a number of a priori estimates of the asymptotic convergence rate which are easy to compute, and which are likely to be realistic in practice. In the simplest cases these estimates reduce the problem of variational convergence to the simpler problem of Fourier series convergence, which is considered in a companion paper. We also produce estimates for the convergence rate of the individual expansion coefficients a. . thus categorising the convergence completely.

Stochastic Analysis of Periodic Hydrologic Process

Abstract: The harmonization of a stochastic process allows the Fourier-Stieltje's integral to represent a nonstationary stochastic process of the periodically correlated type. Therefore, the periodic runoff and precipitation processes can be represented by the Fourier series with random coefficients. The first order periodicity is explained by the periodicity of the first moment of the hydrologic variable. The periodicity in the covariance is explained by the harmonization of the stochastic component. While generating hydrologic records for the design of water resources systems, it is often assumed that the standardized hydrologic series or the residual series, after subtraction of the periodic and trend components, are stationary. Based on the Fourier series representation of the periodic process, a test for stationarity is developed. Observed monthly runoff and precipitation records are tested for stationarity as raw and transformed series.

The Propagation Matrix Method for the Band Problem with a Plane Boundary

Abstract: Solution of electronic problems involving plane surfaces on crystals requires solution of the band problem for real energy E in complex k space, and superposition of the generalized Bloch functions at the surface. A compact and general formulation of the problem of finding these Bloch functions and matching them across a plane makes use of a numerical matrix, the propagation matrix P, obtained from the Schrodinger equation. The eigenvectors of P are just the desired Bloch functions, and the eigenvalues give all k⊥ values at given E, k// (component parallel to the surface). Thus once P is found, the band problem is reduced to an ordinary eigenvalue problem; the bands can be followed along any line in k space parallel to k⊥ by varying k//; the potential may be complex (to describe inelastic scattering). A procedure for generating P by integration of a matrix equation has the advantage that a general anistropic potential can be used, but the disadvantage of a Fourier expansion parallel to the surface plane which does not hold well near the nucleus; hence it applies best for potentials that are weak or have a small number of Fourier coefficients. By generating P for a single layer by a two-dimensional version of KKR, this limitation is avoided for muffin-tin potentials.

Direct evaluation of Kα1 Fourier coefficients in X‐ray profile analysis

Abstract: Using a least-squares method of analyzing X-ray diffraction profiles, it is shown that one can calculate directly the Fourier coefficients of the Kα1 component from the total Kα doublet intensities. These Fourier coefficients can be calculated around any preferred points of the profile, e.g. the center of gravity or peak of the profile of Kα1.

A Moment Method Approach to the Viscosities of Simple Liquids

Abstract: It is shown that the expression for the Fourier components of the density-density correlation function in a fluid obtained from the linearized hydrodynamic equations can also be obtained by adopting a particularly simple form for the associated memory function The result is used to calculate the longitudinal viscosity of a fluid in terms of the moments of the space and time Fourier transform of the density-density correlation function S(q, ω)

Computation of LCAO Wave Functions for Ground States of Polymers and Solids

Abstract: The LCAO form of the Hartree–Fock method is discussed in its application to crystals. General formulae are given for obtaining Fourier coefficient of electronic density (in direct space) as well as of the band structure (in momentum space). Finally, it is shown that in its LCAO form, Slater–Hartree–Fock equations are very simple and that this method is of interest for numerical applications. Special integrals occurring in this formalism are evaluated for a Gaussian basis in the last part of this paper.