Convergence rate of fourier series and best approximations in the space Lp

1988 ◽  
Vol 39 (4) ◽  
pp. 389-398 ◽  
Author(s):  
A. I. Stepanets ◽  
A. K. Kushpel'
Keyword(s):  
Fourier Series ◽  
Convergence Rate ◽  
Best Approximations ◽  
The Space Lp

scholarly journals On a time fractional diffusion with nonlocal in time conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nguyen Hoang Tuan ◽  
Nguyen Anh Triet ◽  
Nguyen Hoang Luc ◽  
Nguyen Duc Phuong
Keyword(s):  
Fourier Series ◽  
Diffusion Equation ◽  
Convergence Rate ◽  
Exact Solutions ◽  
Mild Solution ◽  
Integral Condition ◽  
Fractional Diffusion ◽  
Well Posedness ◽  
Nonlocal Integral Condition ◽  
Regularized Solution

AbstractIn this work, we consider a fractional diffusion equation with nonlocal integral condition. We give a form of the mild solution under the expression of Fourier series which contains some Mittag-Leffler functions. We present two new results. Firstly, we show the well-posedness and regularity for our problem. Secondly, we show the ill-posedness of our problem in the sense of Hadamard. Using the Fourier truncation method, we construct a regularized solution and present the convergence rate between the regularized and exact solutions.

scholarly journals On the convergence rate for the estimation of impulse response function in the space Lp(T)

2018 ◽  
pp. 36-41
Author(s):  
I. Rozora
Keyword(s):  
Linear System ◽  
Convergence Rate ◽  
Impulse Response ◽  
Response Function ◽  
Estimation Error ◽  
Impulse Response Function ◽  
The Space Lp ◽  
Stochastic Linear System ◽  
Impulse Function ◽  
Active Research

The problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a ‘black box’ with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input–output cross-correlogram is taken as an estimator of the response function. An upper bound for the tail of the distribution of the estimation error is found that gives a convergence rate of estimator to impulse response function in the space Lp(T).

Estimates of Functions with Transformed Double Fourier Series

2021 ◽  
Vol 58 (1) ◽  
pp. 32-83
Author(s):  
Boris V. Simonov
Keyword(s):  
Fourier Series ◽  
Double Fourier Series ◽  
Moduli Of Smoothness ◽  
Best Approximations

The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.

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