Summability of the Heine and Neumann Series of Legendre Polynomials

1966 ◽  
Vol 18 ◽  
pp. 1261-1263 ◽  
Author(s):  
Amnon Jakimovski
Keyword(s):  
Holomorphic Function ◽  
Jordan Curve ◽  
Legendre Polynomials ◽  
Closed Interval ◽  
Neumann Series ◽  
Image Position

With a holomorphic function f(z) defined in a domain H which includes the closed interval [—1, 1] we associate the Neumann series1where Pn(z), Qn(t) are, respectively, the nth Legendre polynomials of the first and second kind and γ is a closed and rectifiable Jordan curve which includes [— 1, 1] in its interior and is included, together with its interior, in H.

p-Radial Exceptional Sets and Conformal Mappings

2007 ◽  
Vol 50 (4) ◽  
pp. 579-587
Author(s):  
Piotr Kot
Keyword(s):  
Holomorphic Function ◽  
Unit Disc ◽  
Conformal Mappings ◽  
Exceptional Sets ◽  
Image Position

AbstractFor p > 0 and for a given set E of type Gδ in the boundary of the unit disc ∂ we construct a holomorphic function f ∈ such thatIn particular if a set E has a measure equal to zero, then a function f is constructed as integrable with power p on the unit disc .

scholarly journals The Product of Two Legendre Polynomials

1953 ◽  
Vol 1 (3) ◽  
pp. 121-125 ◽  
Author(s):  
John Dougall
Keyword(s):  
Linear Function ◽  
Spherical Harmonics ◽  
Legendre Polynomials ◽  
Image Position ◽  
Two Sides

1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we haveThe earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams [2]. The proof now to be given follows different lines from his.

Asymptotics for Minimal Discrete Riesz Energy on Curves in ℝd

2004 ◽  
Vol 56 (3) ◽  
pp. 529-552 ◽  
Author(s):  
A. Martínez-Finkelshtein ◽  
V. Maymeskul ◽  
E. A. Rakhmanov ◽  
E. B. Saff
Keyword(s):  
Jordan Curve ◽  
Hausdorff Measure ◽  
Compact Sets ◽  
Minimizing Sequences ◽  
Point Sets ◽  
Dimensional Hausdorff Measure ◽  
One Dimensional ◽  
Riesz Kernel ◽  
Riesz Energy ◽  
Image Position

AbstractWe consider the s-energy for point sets 𝒵 = {𝒵k,n: k = 0, …, n} on certain compact sets Γ in ℝd having finite one-dimensional Hausdorff measure,is the Riesz kernel. Asymptotics for the minimum s-energy and the distribution of minimizing sequences of points is studied. In particular, we prove that, for s ≥ 1, the minimizing nodes for a rectifiable Jordan curve Γ distribute asymptotically uniformly with respect to arclength as n → ∞.

scholarly journals On the Fredholm integral equation associated with pairs of dual integral equations

1969 ◽  
Vol 16 (3) ◽  
pp. 185-194 ◽  
Author(s):  
V. Hutson
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Bessel Function ◽  
Integral Equations ◽  
Fredholm Integral Equation ◽  
Unknown Function ◽  
Sufficient Conditions ◽  
Neumann Series ◽  
Dual Integral Equations ◽  
Image Position

Consider the Fredholm equation of the second kindwhereand Jv is the Bessel function of the first kind. Here ka(t) and h(x) are given, the unknown function is f(x), and the solution is required for large values of the real parameter a. Under reasonable conditions the solution of (1.1) is given by its Neumann series (a set of sufficient conditions on ka(t) for the convergence of this series is given in Section 4, Lemma 2). However, in many applications the convergence of the series becomes too slow as a→∞ for any useful results to be obtained from it, and it may even happen that f(x)→∞ as a→∞. It is the aim of the present investigation to consider this case, and to show how under fairly general conditions on ka(t) an approximate solution may be obtained for large a, the approximation being valid in the norm of L2(0, 1). The exact conditions on ka(t) and the main result are given in Section 4. Roughly, it is required that 1 -ka(at) should behave like tp(p>0) as t→0. For example, ka(at) might be exp ⌈-(t/ap)⌉.

A Relation Between Ultraspherical and Jacobi Polynomial Sets

1953 ◽  
Vol 5 ◽  
pp. 301-305 ◽  
Author(s):  
Fred Brafman
Keyword(s):  
Jacobi Polynomial ◽  
Legendre Polynomials ◽  
Jacobi Polynomials ◽  
Ultraspherical Polynomials ◽  
Image Position ◽  
Special Case

The Jacobi polynomials may be defined bywhere (a)n = a (a + 1) … (a + n — 1). Putting β = α gives the ultraspherical polynomials which have as a special case the Legendre polynomials .

A Regular Singular Functional

1956 ◽  
Vol 8 ◽  
pp. 53-68
Author(s):  
A. D. Martin
Keyword(s):  
Singular Point ◽  
Closed Interval ◽  
Joint Paper ◽  
Quadratic Functionals ◽  
Image Position ◽  
Regular Functional

1. Introduction. In a joint paper with Leighton (2), the author considered quadratic functionals of the type1.1 (0 < a < b)in which x = 0 is a singular point of the functional which is otherwise regular on [0, b]. The hypothesis on a regular functional includes the assumption that r is continuous and positive on a closed interval [0, b].

The extremal values of Legendre polynomials and of certain related functions

1950 ◽  
Vol 46 (4) ◽  
pp. 549-554 ◽  
Author(s):  
R. Cooper
Keyword(s):  
Legendre Polynomials ◽  
Extreme Value ◽  
Positive Zero ◽  
Right Hand ◽  
Extremal Values ◽  
Image Position ◽  
The Right

1. The tabulated values of the Legendre polynomials suggest that the right-hand minimum of Pn(x) changes monotonically as n increases. Let xr, n be the value of x which gives the rth extreme value to the left of 1 of Pn(x). Then we can show thatwhere jr is the rth pösitive zero of J1(z), and that after some term the sequence Pn(xr, n) is monotonic with the moduli of the terms decreasing. We cannot, however, show that the sequence is monotonic from the place at which its terms become significant.

A theorem on positive harmonic functions

1949 ◽  
Vol 45 (2) ◽  
pp. 207-212 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Unit Circle ◽  
Harmonic Functions ◽  
Closed Interval ◽  
Positive Harmonic Function ◽  
Image Position ◽  
Function Conjugate ◽  
Positive Harmonic Functions

1. Let z = reiθ, and let h(z) denote a (regular) positive harmonic function in the unit circle r < 1. Then h(r) (1−r) and h(r)/(1 − r) tend to limits as r → 1. The first limit is finite; the second may be infinite. Such properties of h can be obtained in a straightforward way by using the fact that we can writewhere α(phgr) is non-decreasing in the closed interval (− π, π). Another method is to writewhere h* is a harmonic function conjugate to h. Then the functionhas the property | f | < 1 in the unit circle. Such functions have been studied by Julia, Wolff, Carathéodory and others.

On the right lower circular density of two-dimensional Jordan curves in the plane

1968 ◽  
Vol 64 (1) ◽  
pp. 67-70
Author(s):  
D. G. Larman
Keyword(s):  
Jordan Curve ◽  
Two Dimensional ◽  
End Points ◽  
Jordan Curves ◽  
End Point ◽  
Dimensional Measure ◽  
Image Position ◽  
The Right

We use J(a, b) to denote a Jordan curve of positive two-dimensional measure in the plane, with end-points a and b. If υ is a point of J(a, b), we define the right lower arc density at υ bywhere J( υ, υ′) is the largest arc, whose left-end point is υ, which is contained in the disc c(υ, r).

An integral representation for a generalised variation of a function

1974 ◽  
Vol 11 (2) ◽  
pp. 225-229 ◽  
Author(s):  
A.M. Russell
Keyword(s):  
Integral Representation ◽  
Classical Result ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Variation Of A Function ◽  
Image Position

In this note we present sufficient conditions for the continuity of the total kth variation of a function defined on a closed interval [a, b]. We also give an integral representation for total feth variation, thus obtaining an extension of the classical result

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