Displaying similar documents to “Growth orders of Cesàro and Abel means of uniformly continuous operator semi-groups and cosine functions”

On solutions of differential equations with ``common zero'' at infinity

Árpád Elbert, Jaromír Vosmanský (1997)

Archivum Mathematicum

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The zeros c k ( ν ) of the solution z ( t , ν ) of the differential equation z ' ' + q ( t , ν ) z = 0 are investigated when lim t q ( t , ν ) = 1 , | q ( t , ν ) - 1 | d t < and q ( t , ν ) has some monotonicity properties as t . The notion c κ ( ν ) is introduced also for κ real, too. We are particularly interested in solutions z ( t , ν ) which are “close" to the functions sin t , cos t when t is large. We derive a formula for d c κ ( ν ) / d ν and apply the result to Bessel differential equation, where we introduce new pair of linearly independent solutions replacing the usual pair J ν ( t ) , Y ν ( t ) . We show the concavity of c κ ( ν ) for...

Less than one implies zero

Felix L. Schwenninger, Hans Zwart (2015)

Studia Mathematica

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In this paper we show that from an estimate of the form s u p t 0 | | C ( t ) - c o s ( a t ) I | | < 1 , we can conclude that C(t) equals cos(at)I. Here ( C ( t ) ) t 0 is a strongly continuous cosine family on a Banach space.

Generalized trigonometric functions in complex domain

Petr Girg, Lukáš Kotrla (2015)

Mathematica Bohemica

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We study extension of p -trigonometric functions sin p and cos p to complex domain. For p = 4 , 6 , 8 , , the function sin p satisfies the initial value problem which is equivalent to (*) - ( u ' ) p - 2 u ' ' - u p - 1 = 0 , u ( 0 ) = 0 , u ' ( 0 ) = 1 in . In our recent paper, Girg, Kotrla (2014), we showed that sin p ( x ) is a real analytic function for p = 4 , 6 , 8 , on ( - π p / 2 , π p / 2 ) , where π p / 2 = 0 1 ( 1 - s p ) - 1 / p . This allows us to extend sin p to complex domain by its Maclaurin series convergent on the disc { z : | z | < π p / 2 } . The question is whether this extensions sin p ( z ) satisfies (*) in the sense of differential equations in complex domain. This...

On the range of a closed operator in an L 1 -space of vector-valued functions

Ryotaro Sato (2005)

Commentationes Mathematicae Universitatis Carolinae

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Let X be a reflexive Banach space and A be a closed operator in an L 1 -space of X -valued functions. Then we characterize the range R ( A ) of A as follows. Let 0 λ n ρ ( A ) for all 1 n < , where ρ ( A ) denotes the resolvent set of A , and assume that lim n λ n = 0 and sup n 1 λ n ( λ n - A ) - 1 < . Furthermore, assume that there exists λ ρ ( A ) such that λ ( λ - A ) - 1 1 . Then f R ( A ) is equivalent to sup n 1 ( λ n - A ) - 1 f 1 < . This generalizes Shaw’s result for scalar-valued functions.

Weighted integrability of double cosine series with nonnegative coefficients

Chang-Pao Chen, Ming-Chuan Chen (2003)

Studia Mathematica

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Let f c ( x , y ) j = 1 k = 1 a j k ( 1 - c o s j x ) ( 1 - c o s k y ) with a j k 0 for all j,k ≥ 1. We estimate the integral 0 π 0 π x α - 1 y β - 1 ϕ ( f c ( x , y ) ) d x d y in terms of the coefficients a j k , where α, β ∈ ℝ and ϕ: [0,∞] → [0,∞]. Our results can be regarded as the trigonometric analogues of those of Mazhar and Móricz [MM]. They generalize and extend Boas [B, Theorem 6.7].