The Lebesgue Integral: Riesz Method

1995 ◽  
pp. 50-86
Author(s):  
Soo Bong Chae
Keyword(s):  
Lebesgue Integral ◽  
Riesz Method

Characterization of decomposition integrals extending Lebesgue integral

2021 ◽  
Author(s):  
Jun Li ◽  
Radko Mesiar ◽  
Yao Ouyang ◽  
Adam Šeliga
Keyword(s):  
Lebesgue Integral

Lebesgue Integral

2021 ◽  
Author(s):  
Liviu C. Florescu
Keyword(s):  
Lebesgue Integral

scholarly journals The double Fourier series of a discontinuous function

1924 ◽  
Vol 106 (737) ◽  
pp. 299-314 ◽  
Keyword(s):  
Fourier Series ◽  
Finite Number ◽  
Double Fourier Series ◽  
General Term ◽  
Discontinuous Function ◽  
Lebesgue Integral ◽  
Function Of Two Variables

A function of two variables may be expanded in a double Fourier series, as a function of one variable is expanded in an ordinary Fourier series. Purpose that the function f ( x, y ) possesses a double Lebesgue integral over the square (– π < π ; – π < y < π ). Then the general term of the double Fourier series of this function is given by cos = є mn { a mn cos mx cos ny + b mn sin mx sin ny + c mn cos mx sin ny + d mn sin mx cos ny } There є 00 = ¼, є m0 = ½ ( m > 0), є 0n = ½ ( n > 0), є ms = 1 ( m > 0, n >0). the coefficients are given by the formulæ a mn = 1/ π 2 ∫ π -π ∫ π -π f ( x, y ) cos mx cos ny dx dy , obtained by term-by-term integration, as in an ordinary Fourier series. Ti sum of a finite number of terms of the series may also be found as in the ordinary theory. Thus ∫ ms = Σ m μ = 0 Σ n v = 0 A μ v = 1/π 2 ∫ π -π ∫ π -π f (s, t) sin( m +½) ( s - x ) sin ( n + ½) ( t - y )/2 sin ½ ( s - x ) 2 sin ½ ( t - y ) if f ( s , t ) is defined outside the original square by double periodicity, we have sub S ms = 1/π 2 ∫ π 0 ∫ π 0 f ( x + s , y + t ) + f ( x + s , y - t ) + f ( x - s , y + t ) + f ( x - s , y - t ) sin ( m + ½) s / 2 sin ½ s sin ( n + ½) t / 2 sin ½ t ds dt .

Novel Lebesgue-integral-based approach to improved results for neural networks with additive time-varying delay components

2017 ◽  
Vol 354 (16) ◽  
pp. 7543-7565 ◽  
Author(s):  
Deqiang Zeng ◽  
Ruimei Zhang ◽  
Shouming Zhong ◽  
Guowu Yang ◽  
Yongbin Yu ◽  
...  
Keyword(s):  
Neural Networks ◽  
Time Varying ◽  
Lebesgue Integral ◽  
Time Varying Delay ◽  
Varying Delay

More on Lebesgue Integral and Measure

2014 ◽  
pp. 527-573
Author(s):  
Houshang H. Sohrab
Keyword(s):  
Lebesgue Integral

The Lebesgue Integral

Real Analysis ◽  
2002 ◽  
pp. 123-170
Author(s):  
Emmanuele DiBenedetto
Keyword(s):  
Lebesgue Integral

scholarly journals A Bibliometric Analysis for Lebesgue Measure Integration in Optimization

2021 ◽  
Vol 2 (2) ◽  
pp. 54-63
Author(s):  
Endang Rusyaman ◽  
Devi Munandar ◽  
Diah Chaerani ◽  
Dwindi Agryanti Johar ◽  
Rizky Ashgi
Keyword(s):  
Lebesgue Measure ◽  
Integral Theory ◽  
Lebesgue Integral ◽  
Set Domain ◽  
Real Numbers ◽  
Search Results ◽  
Pure Mathematics ◽  
Bibliographic Data ◽  
Mathematical Problems ◽  
Integration Concept

In solving mathematical problems so far, Riemann's integral theory is quite adequate for solving pure mathematics and applications problems. But not all problems can be solved using this integration, such as a discontinuous function that is not Riemann's integration. Lebesgue integral is an integration concept based on measure and can solve finite and unlimited function problems and be solved in a more general set domain. One of the bases of this integration is the Lebesgues measure includes the set of real numbers, where the length of the interval is the endpoints. The alternative use of this integral is widely used in various studies such as partial differential equations, quantum mechanics, and probabilistic analysis, requiring the integration of arbitrary set functions. This paper will show a comprehensive bibliometric survey of peer-reviewed articles referring to Lebesgue measure in integration. Search results are obtained 832 papers in the google scholar database and 997 papers using Lebesgue measure integration in optimization. It can also be seen that the research have 4 clusters and 3 clusters respectively with scattered keywords for each cluster. Finally, using bibliographic data can be obtained Lebesgues measure in integration and optimization supports many of the research and provides productive citations to citing the study.

scholarly journals Measure and Integral

1947 ◽  
Vol 36 ◽  
pp. 1-12
Author(s):  
W. W. Rogosinski
Keyword(s):  
University Education ◽  
Research Worker ◽  
Lebesgue Integral ◽  
Modern Analysis ◽  
The Common ◽  
Classical Conception

1. It is now nearly half a century since H. Lebesgue, whose obituary the reader may have seen in Nature not so long ago, created his theory of the integral which since then has superseded in modern analysis the classical conception due to B. Riemann. It is, I think, regrettable that knowledge of the Lebesgue integral seems to be still largely confined to the research worker. There is nothing unduly abstract or unnatural in this theory, nor anything in the proofs which would be too difficult for a good honours student to grasp. If the aim of university education be the teaching of general ideas and methods rather than that of technicalities, then the modern notion of the integral should not be omitted from the mathematical syllabus. It is the purpose of this purely expository note to sketch the build up of both the Riemann and the Lebesgue integral on the common geometrical basis of “measure” and thus to make evident to the uninitiated reader the striking advantages of the new integral.

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