Is the Integral of f(x)/x Sin(nx) Zero as n Approaches Infinity?

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In summary, square integrable functions have a finite integral when squared over their entire domain, and this concept is used to determine the convergence of certain functions in analysis and calculus. Square integrability is stricter than regular integrability, and examples of square integrable functions include polynomials, trigonometric functions, and exponential functions. It is important in mathematics because it allows for various operations and is a key concept in the study of Fourier series. To determine if a function is square integrable, one can calculate its integral over its entire domain and take the square root, or use mathematical theorems and techniques to prove its square integrability.
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Logarythmic
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If

[tex]\frac{f(x)}{x}[/tex]

is square integrable, why does

[tex]\int_a^b \frac{f(x)}{x} \sin{nx} dx \longrightarrow 0[/tex]

when

[tex]n \rightarrow \infty[/tex]

?
 
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This is not an answer, but maybe a hint is contained in here. I think there is some theorem that will let you prove a square integrable function on a finite interval is absolutely integrable on that interval, and that demands a certain smoothness of the function. The integral of any smooth function multiplied by an infinite frequency sinusoid must be zero.
 

Related to Is the Integral of f(x)/x Sin(nx) Zero as n Approaches Infinity?

1. What does it mean for a function to be square integrable?

Square integrable means that the function has a finite integral when squared over its entire domain. This is a mathematical concept used to determine the convergence of certain functions and is often used in analysis and calculus.

2. How is square integrability different from regular integrability?

Square integrability is a stricter requirement than regular integrability. While a function can be integrable without being square integrable, a square integrable function must also be integrable. This means that the integral of the function over its entire domain must be finite in both cases, but for square integrability, the integral of the function squared must also be finite.

3. What are some examples of square integrable functions?

Some common examples of square integrable functions include polynomials, trigonometric functions, and exponential functions. In general, any function that decays or oscillates sufficiently fast as the independent variable approaches infinity will be square integrable.

4. Why is square integrability important in mathematics?

Square integrability is important because it allows us to determine the convergence of certain functions and to perform various mathematical operations on them. It is also a key concept in the study of Fourier series and other areas of mathematics.

5. How do you determine if a function is square integrable?

To determine if a function is square integrable, you can calculate its integral over its entire domain and then take the square root of that value. If the result is a finite number, then the function is square integrable. Alternatively, you can use certain mathematical theorems and techniques to prove the square integrability of a function.

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