MINIMUM NORM PROPERTIES OF EVEN DEGREE POLYNOMIAL SPLINES WITH RESPECT TO FRACTIONAL DIFFERENTIATION OPERATORS

2006 ◽  
Vol 39 (11) ◽  
pp. 112-117
Author(s):  
Wolfgang A. Halang
Keyword(s):  
Minimum Norm ◽  
Fractional Differentiation ◽  
Polynomial Splines ◽  
Degree Polynomial

Spline Functions on the Circle: Cardinal L-Splines Revisited

1980 ◽  
Vol 32 (6) ◽  
pp. 1459-1473 ◽  
Author(s):  
Charles A. Micchelli ◽  
A. Sharma
Keyword(s):  
Point Of View ◽  
Spline Functions ◽  
Roots Of Unity ◽  
Minimum Norm ◽  
Piecewise Polynomial ◽  
Polynomial Splines ◽  
Exponential Polynomials ◽  
The Subject ◽  
Piecewise Exponential ◽  
The Difference

Although the literature on splines has grown vastly during the last decade [11], the study of polynomial splines on the circle seems to have suffered neglect. The first to study the subject in depth seem to be Ahlberg, Nilson and Walsh [1]. Almost at the same time I. J. Schoenberg [8] studied the problem of interpolation at the roots of unity by splines and its relation to quadrature on the circle. For discrete polynomial splines on the circle we refer to [5]. M. Golomb [3] also considers interpolation by a class of “spline” functions in the complex plane but his point of view is based on minimum norm properties of spline functions. Perhaps the reason for this neglect may be attributed to the fact that one can pass from the circle to the line by means of the transformation z → exp 2-πix. This changes the problem on the circle into periodic interpolation on the line with the difference that instead of interpolation by piecewise polynomial, we now consider piecewise exponential polynomials with complex exponents.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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