scholarly journals Bojanov-Naidenov problem for positive (negative) parts of differentiable functions on the real domain

2018 ◽  
Vol 26 (1) ◽  
pp. 25 ◽  
Author(s):  
V.V. Kameneva ◽  
V.A. Kofanov
Keyword(s):  
Extremal Problem ◽  
Local Norm ◽  
The Real ◽  
Differentiable Functions ◽  
Real Domain

We solve the extremal problem $$$\| x^{(k)}_{\pm} \|_{L_p[a,b]} \rightarrow \sup$$$, $$$k = 0, 1, ..., r-1$$$, over the set of pairs $$$(x, I)$$$ of functions $$$x\in W^r_{\infty} (\mathbb{R})$$$ and intervals $$$I = [a,b]$$$ with restrictions on the local norm of function $$$x$$$ and the measure of support $$$\mu \{ \mathrm{supp}_{[a,b]} x^{(k)}_{\pm} \}$$$.

scholarly journals Inequalities of various metrics for the norms $$$\|x\|_{p,\delta} = \sup \bigl\{ \| x \|_{L_p[a,b]} \colon a,b\in \mathbb{R}, b-a\leqslant \delta \bigr\}$$$ of differentiable functions on the real domain

2018 ◽  
Vol 26 (1) ◽  
pp. 48
Author(s):  
V.A. Kofanov
Keyword(s):  
Real Line ◽  
Trigonometric Polynomials ◽  
The Real ◽  
Differentiable Functions ◽  
Real Domain

We prove sharp inequalities of various metrics for the norms $$$\| x \|_{p, \delta}$$$ of differentiable functions defined on the real line, trigonometric polynomials and periodic splines.

Linear Differential Equations in the Real Domain

1967 ◽  
Vol 51 (378) ◽  
pp. 364
Author(s):  
R. P. Gillespie ◽  
Kenneth S. Miller
Keyword(s):  
Differential Equations ◽  
Linear Differential Equations ◽  
The Real ◽  
Real Domain ◽  
Linear Differential

The Two-Sided Factorization of Ordinary Differential Operators

1980 ◽  
Vol 32 (5) ◽  
pp. 1045-1057 ◽  
Author(s):  
Patrick J. Browne ◽  
Rodney Nillsen
Keyword(s):  
Differential Operator ◽  
Linear Function ◽  
Bilinear Form ◽  
Differential Operators ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
Ordinary Differential Operators ◽  
Image Position

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

Most Infinitely Differentiable Functions are Nowhere Analytic

1973 ◽  
Vol 16 (4) ◽  
pp. 597-598 ◽  
Author(s):  
R. B. Darst
Keyword(s):  
Open Subset ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
First Category ◽  
Infinitely Differentiable Functions

We define a natural metric, d, on the space, C∞,, of infinitely differentiable real valued functions defined on an open subset U of the real numbers, R, and show that C∞, is complete with respect to this metric. Then we show that the elements of C∞, which are analytic near at least one point of U comprise a first category subset of C∞,.

Sign in / Sign up

Export Citation Format

Share Document