Is this function in the L^2(0,1) space for certain values of C_1 and C_2?

In summary, the conversation discusses the concept of a function being an element of space ##L^2(0,1)## and the condition that the integral of its absolute value squared must be finite. The conversation then moves on to the difficulty of calculating this integral for certain functions, such as ##f(x)=x^{-1}(C_1+C_2 \ln x)##, and the question of whether this function is in ##L^2(0,1)## for some constants ##C_1## and ##C_2##. The conversation then briefly mentions the concept of Lebesgue differentiation, which is related to Lebesgue integration. Ultimately, the conversation does not provide a conclusive answer to the question of the
  • #1
LagrangeEuler
717
20
If some function is element of space ##L^2(0,1)## then
[tex]\int^1_0|f(x)|^2dx< \infty[/tex]. What in the case when it is not so simple to calculate this integral. For example ##f(x)=x^{-1}(C_1+C_2 \ln x)##. How to find is it this function in ##L^2(0,1)## for some ##C_1,C_2##?
 
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  • #2
It is never (other than both constants = 0). Integral diverges at x=0.
 
  • #3
Please help me to define Relation between Lebesgue Differentiation and Lebesgue integration?
 
  • #4
Amal Chacko said:
Please help me to define Relation between Lebesgue Differentiation and Lebesgue integration?
I have never seen the term Lebesgue differentiation. Lebesgue integration is a theory of integration based on measure theory.
 
  • #5
  • #7
Ah, yes, Hawkeye's link is clearly better, more direct than mine.
 

What is a "L^2(0,1) space function"?

A "L^2(0,1) space function" refers to a function that belongs to the L^2 space on the interval [0,1]. This space consists of all square-integrable functions on this interval, meaning that the integral of the squared absolute value of the function is finite.

What is the significance of the L^2 space?

The L^2 space is a fundamental concept in functional analysis and is widely used in many areas of mathematics, physics, and engineering. It is a complete vector space, which allows for the use of powerful tools from linear algebra and functional analysis to study functions in this space.

How is the norm defined in the L^2 space?

The norm in the L^2 space is defined as the square root of the integral of the squared absolute value of the function. This norm measures the size of the function in the L^2 space and is used to define important concepts such as convergence and orthogonality.

What is the difference between the L^2 space and other function spaces?

The L^2 space differs from other function spaces in that it focuses on the square-integrable functions, rather than just the integrable or continuous functions. This allows for a wider range of functions to be studied, including those that may not be continuous but still have finite integral values.

How is the L^2 space used in practical applications?

The L^2 space has many practical applications, such as in signal processing, image and audio compression, and quantum mechanics. It is also used in the study of differential equations, where it provides a framework for understanding the behavior of solutions.

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