scholarly journals Lebesgue constants for double Hausdorff means

1985 ◽  
Vol 31 (2) ◽  
pp. 199-214
Author(s):  
F. Ustina
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Lebesgue Constant ◽  
Mean Value ◽  
Lebesgue Constants ◽  
Jump Discontinuities ◽  
The Mean ◽  
Image Position ◽  
Partial Extension ◽  
Hausdorff Means

As is well known, the divergence of the set of constants known as the Lebesgue constants corresponding to a particular method of summability implies the existence of a continuous, periodic function whose Fourier series, summed by the method, diverges at a point, and of another such function the sums of whose Fourier series converge everywhere but not uniformly in the neighborhood of some point.In 1961, Lorch and Newman established that if L(n; g) is the nth Lebesgue constant for the Hausdorff summability method corresponding to the weight function g(u), thenwherewhere the summation is taken over the jump discontinuities {εk} of g(u) and M{f(u)} denotes the mean value of the almost periodic function f(u).In this paper, a partial extension of this result to the two dimensional analogue is obtained. This extension is summarized in Theorem 1.3.

scholarly journals IV. On the annual and semi-annual seismic periods

1894 ◽  
Vol 54 (326-330) ◽  
pp. 82-85
Keyword(s):  
Periodic Function ◽  
Mean Value ◽  
Earthquake Frequency ◽  
The Mean

Method of Investigation .—The method adopted is similar to that employed by Dr. C. G. Knott in his paper on “Earthquake Frequency.” If f ( θ ) be a periodic function of θ, then f ( θ ) = a 0 + a 1 cos ( θ + α 1 ) + a 2 cos (2 θ + α 2 ) + .... + a n cos ( nθ + α n + ...., from which it follows that 1/π∫ θ + π/2 θ - π/2 f ( θ ) dθ = a 0 + 2 a 1 /π cos (θ + α 1 ) - 2 a 3 /3π cos (3θ + α 3 ) + ... + 2 a n sin nπ /2/ nπ cos ( nθ + α n + ..... This latter expression gives the mean value of f ( θ ) through an interval π /2 on either side of θ . From it, all terms involving even multiples of θ are eliminated, and the coefficients of all terms after the second are diminished to a greater extent than that of the second.

scholarly journals Microstructure of the Galactic Magnetic Field

1958 ◽  
Vol 8 ◽  
pp. 952-954
Author(s):  
K. Serkowski
Keyword(s):  
Magnetic Field ◽  
Position Angle ◽  
Mean Value ◽  
Open Clusters ◽  
Electric Vector ◽  
Stokes Parameters ◽  
Galactic Magnetic Field ◽  
The Mean ◽  
Image Position

The polarization of the stars in open clusters, explained on the basis of the Davis-Greenstein theory, gives some information on the microstructure of the galactic magnetic field.The polarization is most conveniently described by the parameters Q, U, proportional to the Stokes parameters and defined by where p is the amount of polarization, θ is the position angle of the electric vector, and θ̄ is the mean value of θ for the region under consideration.

scholarly journals On approximation in generalized Zygmund class

2019 ◽  
Vol 52 (1) ◽  
pp. 370-387
Author(s):  
Hare Krishna Nigam
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Conjugate Function ◽  
Degree Of Approximation ◽  
Zygmund Class ◽  
Conjugate Fourier Series ◽  
Product Method ◽  
Hausdorff Means

AbstractHere, we estimate the degree of approximation of a conjugate function {\tilde g} and a derived conjugate function {\tilde g'} , of a 2π-periodic function g \in Z_r^\lambda , r ≥ 1, using Hausdorff means of CFS (conjugate Fourier series) and CDFS (conjugate derived Fourier series) respectively. Our main theorems generalize four previously known results. Some important corollaries are also deduced from our main theorems. We also partially review the earlier work of the authors in respect of order of the Euler-Hausdorff product method.

scholarly journals On the mean value of the enumeration function for multiplicative partitions

1990 ◽  
Vol 41 (3) ◽  
pp. 407-410 ◽  
Author(s):  
Cao Hui-Zong ◽  
Ku Tung-Hsin
Keyword(s):  
Natural Number ◽  
Mean Value ◽  
The Mean ◽  
Image Position

Let g(n) denote the number of multiplicative partitions of the natural number n. We prove that

The mean value theorem in second order arithmetic

2001 ◽  
Vol 66 (3) ◽  
pp. 1353-1358 ◽  
Author(s):  
Christopher S. Hardin ◽  
Daniel J. Velleman
Keyword(s):  
Mean Value ◽  
Second Order ◽  
Mean Value Theorem ◽  
Natural Numbers ◽  
Second Order Arithmetic ◽  
Elementary Analysis ◽  
Logical Connectives ◽  
The Mean ◽  
Order Arithmetic ◽  
Image Position

This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2].We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is .The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the formwhere φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]).Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.

scholarly journals Chemical Evolution of the Galactic Disk and the Radial Metallicity Gradient

1976 ◽  
Vol 72 ◽  
pp. 207-208
Author(s):  
M. Mayor
Keyword(s):  
Chemical Evolution ◽  
Galactic Disk ◽  
Sodium Concentration ◽  
Mean Value ◽  
Heavy Elements ◽  
Solar Neighbourhood ◽  
Chemical Gradient ◽  
The Galaxy ◽  
The Mean ◽  
Image Position

An analysis of the kinematical and photometric properties of about 600dF stars and 600 gG-gK stars permits the estimation of the radial chemical gradient in the Galaxy. The mean value in the solar neighbourhood obtained for all of these stars is: The values of [Fe/H] used for this estimation are deduced for the dF stars using uvby β photometric measurements and for the gG-gK stars from a list published by Hansen and Kjaergaard. An estimate of the chemical gradient using UBV photometry of dG stars in the solar neighbourhood gives a similar value. For all the samples studied (dF, dG or giants) the order of magnitude for the gradient is the same. However, for the youngest stars in these samples the metallicity gradient could be larger: Such a value may be affected by dynamical perturbations of the galactic disk.The values published by Hansen and Kjaergaard for the sodium concentration in giant star atmospheres also indicate a radial galactic gradient of the same order.If only the dF stars which are sufficiently evolved to allow an age estimate are considered, then a very distinct correlation is found between age and metallicity: An important fraction of the heavy elements actually present in the solar neighbourhood seems to have synthetized during the life of the galactic disk.The two derivatives and are not independent, but are connected by the chemical evolution of the galactic disk. Some elementary deductions show the coherency of these two estimates.The intrinsic dispersion of metallicities, at a given age and birthplace, is somewhat lower than the admitted values. It has not been possible to find any significant variation with age of this quantity from the present observational material. The simultaneous variation of σ2w and [Fe/H] as function of age is evidence for a z stratification in the mean abundance of the heavy elements. The ratio between the mean metallicity in the plane and at z = 500 pc is estimated to be about a factor of two.Finally it is shown that the interpretation of the kinematical diagrams for different groups of given metallicity is ambiguous. A relation as e vs [Fe/H] depends not only on the chemical and kinematical history of the Galaxy but is also strongly dependent on the observational errors of [Fe/H] and on criteria used to define the sample.A paper containing the above results has been submitted for publication in Astronomy and Astrophysics.

scholarly journals New Determination of the Polar Motion from 1890 to 1969

1972 ◽  
Vol 48 ◽  
pp. 12-13 ◽  
Author(s):  
E. P. Fedorov ◽  
A. A. Rorsun ◽  
S. P. Major ◽  
N. T. Panchenko ◽  
V. K. Tarady ◽  
...  
Keyword(s):  
Polar Motion ◽  
Normal Values ◽  
Mean Value ◽  
Polar Coordinates ◽  
Periodic Part ◽  
Equal Weight ◽  
The Mean ◽  
Image Position ◽  
Almost All ◽  
Academy Of Sciences

To obtain the coordinates of the Earth's pole almost all series of systematic latitude observations that continued for more than two years have been utilized. They are listed in Table I which comprises 92 series of observation at 72 observatories.Computation was made by the following stages. As initial data we used normal values of latitude φ1, φ2, ……. φn, i.e. the means of instantaneous latitudes over successive intervals of time. These values were smoothed using Whittaker's numerical method which is capable of giving the most probable curve of latitude variation. The smoothed values φ′ satisfy the following condition where hi is a measure of precision, λ2 an arbitrary number by means of which the degree of smoothing is set, and Δ3 designates the third difference of φ′. Whittaker's method was applied in different modifications according to whether or not the normal values of φ′i had an equal weight and were given at equidistant moments of time.For the origin of the system of coordinates we adopted the mean pole of the epoch of observation. Because of this the data given in Table II represent only the periodic part of the polar motion in the region of frequency from 0.77 to 2 cycles per year. In this connection the sequence of φ′ was subjected to filtration in order to eliminate variation of the mean latitude.Coordinates of the pole were computed in two approximations. First, it was assumed that all the series are of the same accuracy and so they were taken with an equal weight.The polar coordinates obtained on this assumption are denoted by x1, y1 and shown in the second and third columns of Table II. The divergences of the smoothed values φ′i from the latitudes computed with x1, y1 were denoted by zκi where the index κ designates the number of a series. Then for the second approximation each series of observation was taken with the weight inversely proportional to the mean value of for this series. The polar coordinates obtained in the second approximation are denoted by x2, y2 and given in the last two columns of Table II.The full paper with the tables will be published by the Ukrainian Academy of Sciences as a separate book.

On the absolute summability of some series related to a Fourier series

1970 ◽  
Vol 67 (1) ◽  
pp. 29-45 ◽  
Author(s):  
B. K. Ray
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Absolute Summability ◽  
The Absolute ◽  
Image Position

1.Introduction. 1.1. Let f(t) be a periodic function with period 2π and integrable in the Lebesgue sense over ( -π,π). We assume as we may without loss of generality, that the Fourier series of f(t) is .

scholarly journals Curves with zero derivative in F-spaces

1981 ◽  
Vol 22 (1) ◽  
pp. 19-29 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Characteristic Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Map ◽  
Left And Right ◽  
The Mean ◽  
Image Position ◽  
Locally Convex

Let X be an F-space (complete metric linear space) and suppose g:[0, 1] → X is a continuous map. Suppose that g has zero derivative on [0, 1], i.e.for 0≤t≤1 (we take the left and right derivatives at the end points). Then, if X is locally convex or even if it merely possesses a separating family of continuous linear functionals, we can conclude that g is constant by using the Mean Value Theorem. If however X* = {0} then it may happen that g is not constant; for example, let X = Lp(0, 1) (0≤p≤1) and g(t) = l[0,t] (0≤t≤1) (the characteristic function of [0, t]). This example is due to Rolewicz [6], [7; p. 116].

The Lebesgue Constants for (γ, r) Summation of Fourier Series

1963 ◽  
Vol 6 (2) ◽  
pp. 179-182 ◽  
Author(s):  
Lee Lorch ◽  
Donald J. Newman
Keyword(s):  
Fourier Series ◽  
Imaginary Part ◽  
Complex Number ◽  
Circle Method ◽  
Summation Method ◽  
Lebesgue Constants ◽  
Taylor Method ◽  
Image Position

The (γ, r) summation method, 0 < r < 1, is the "circle method" employed by G. H. Hardy and J. E. Littlewood. It is also known as the Taylor method. Its Lebesgue constants, say L(Tr, n), n = 1, 2, …, were studied by K. Ishiguro [1] in the notation L*(n;1-r). He noted that1where Im{z} denotes the imaginary part of the complex number z, and proved that2Here3

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