Some analytic function algebras

AbstractIn this article, we define a convolution operator and study its boundedness on mixed-norm spaces. In particular, we obtain a well-known result on the boundedness of composition operators given by Avetisyan and Stević in [K. Avetisyan and S. Stević, The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc, Appl. Math. Comput. 213 2009, 2, 304–311]. Also we consider the adjoint {\mathcal{A}^{b,c}} for {b>0} of two parameter families of Cesáro averaging operators and prove the boundedness on Besov mixed-norm spaces {B_{\alpha+(c-1)}^{p,q}} for {c>1}.

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On finite sums of periodic functions

Georgian Mathematical Journal ◽

10.1515/gmj-2019-2076 ◽

2020 ◽

Vol 27 (2) ◽

pp. 265-269

Author(s):

Alexander Kharazishvili

Keyword(s):

Analytic Function ◽

Real Line ◽

Real Analytic Function ◽

Periodic Functions ◽

Uniform Limit ◽

The Real ◽

Pointwise Limit ◽

Finite Sums ◽

Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

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Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator

Open Mathematics ◽

10.1515/math-2021-0028 ◽

2021 ◽

Vol 19 (1) ◽

pp. 329-337

Author(s):

Huo Tang ◽

Kaliappan Vijaya ◽

Gangadharan Murugusundaramoorthy ◽

Srikandan Sivasubramanian

Keyword(s):

Analytic Function ◽

Differential Operator ◽

Lower Bounds ◽

Analytic Functions ◽

Function Class ◽

Partial Sums ◽

Generalized Function ◽

Inclusion Relations ◽

Alpha Beta

Abstract Let f k ( z ) = z + ∑ n = 2 k a n z n {f}_{k}\left(z)=z+{\sum }_{n=2}^{k}{a}_{n}{z}^{n} be the sequence of partial sums of the analytic function f ( z ) = z + ∑ n = 2 ∞ a n z n f\left(z)=z+{\sum }_{n=2}^{\infty }{a}_{n}{z}^{n} . In this paper, we determine sharp lower bounds for Re { f ( z ) / f k ( z ) } {\rm{Re}}\{f\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}\left(z)\} , Re { f k ( z ) / f ( z ) } {\rm{Re}}\{{f}_{k}\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}f\left(z)\} , Re { f ′ ( z ) / f k ′ ( z ) } {\rm{Re}}\{{f}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}_{k}^{^{\prime} }\left(z)\} and Re { f k ′ ( z ) / f ′ ( z ) } {\rm{Re}}\{{f}_{k}^{^{\prime} }\left(z)\hspace{-0.08em}\text{/}\hspace{-0.08em}{f}^{^{\prime} }\left(z)\} , where f ( z ) f\left(z) belongs to the subclass J p , q m ( μ , α , β ) {{\mathcal{J}}}_{p,q}^{m}\left(\mu ,\alpha ,\beta ) of analytic functions, defined by Sălăgean ( p , q ) \left(p,q) -differential operator. In addition, the inclusion relations involving N δ ( e ) {N}_{\delta }\left(e) of this generalized function class are considered.

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Rational Krylov methods for fractional diffusion problems on graphs

BIT Numerical Mathematics ◽

10.1007/s10543-021-00881-0 ◽

2021 ◽

Author(s):

Michele Benzi ◽

Igor Simunec

Keyword(s):

Analytic Function ◽

Diffusion Equation ◽

Graph Laplacian ◽

Fractional Diffusion ◽

Singular Matrix ◽

Krylov Methods ◽

Fractional Powers ◽

Directed Networks ◽

Rank One ◽

Diffusion Problems

AbstractIn this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian L as a product $$f(L^T) \varvec{b}$$ f ( L T ) b , where f is a non-analytic function involving fractional powers and $$\varvec{b}$$ b is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $$f(L^T) \varvec{b}$$ f ( L T ) b to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

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