Finding function, simplifying the summation

In summary: Assuming a is an integer, I would start by seeing what a constant solution would look like. Since f is not given as continuous, you could then see what changing f at only a countable set of points looks like (e.g. at a2-m, m=1, 2...). That still satisfies (c). Admittedly that's backwards; just because it satisfies (c) does not mean it should satisfy the given condition, but it might yield some insight.
  • #1
Saitama
4,243
93

Homework Statement


Let ## n \geq 2## be a fixed integer. ##f(x)## be a bounded function defined in ##f:(0,a) \rightarrow R## satisfying
[tex]f(x)=\frac{1}{n^2}\sum_{r=0}^{(n-1)a} f\left(\frac{x+r}{n}\right)[/tex]
then ##f(x)## =
a)-f(x)
b)2f(x)
c)f(2x)
d)nf(x)


Homework Equations





The Attempt at a Solution


I see no way of simplifying the summation. I need a few hints to start with.

Any help is appreciated. Thanks!
 
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  • #2
I don't see any way to simplify it at the first glance. If I spot anything, I will edit this post.

Edit: Don't try to simplify the sum, because it won't turn out so nicely anyway. Instead, inspect the functional equations given to you in the options and try substituting them back into the original sum formula.
 
Last edited:
  • #3
Millennial said:
Don't try to simplify the sum, because it won't turn out so nicely anyway. Instead, inspect the functional equations given to you in the options and try substituting them back into the original sum formula.

That gives a), b) and d), what about the c) option?
 
  • #4
(a), (b) and (d) are equivalent to f(x)=0, no?
What does the range limit mean when (n-1)a is not an integer?
 
  • #5
I figured out a), b) and d). :)

haruspex said:
What does the range limit mean when (n-1)a is not an integer?

I can't understand what you ask me here.
 
  • #6
Pranav-Arora said:
I can't understand what you ask me here.
The OP doesn't say whether a is an integer. If it isn't, (n-1)a need not be either. In that case I'm not sure how to interpret the sum. Does it mean the sum up to the greatest integer <= (n-1)a?
Anyway, assuming a is an integer, I would start by seeing what a constant solution would look like. Since f is not given as continuous, you could then see what changing f at only a countable set of points looks like (e.g. at a2-m, m=1, 2...). That still satisfies (c). Admittedly that's backwards; just because it satisfies (c) does not mean it should satisfy the given condition, but it might yield some insight.
 

Related to Finding function, simplifying the summation

1. What is the purpose of finding function and simplifying the summation?

The purpose of finding function and simplifying the summation is to make complex mathematical equations or expressions more manageable and understandable. It allows for easier analysis and interpretation of data or patterns.

2. How do you find the function of a given set of data?

To find the function of a given set of data, you can use various methods such as curve fitting, regression analysis, or interpolation. These methods involve finding the mathematical relationship between the independent and dependent variables in the data set.

3. What is the difference between summation and simplification?

Summation involves adding together a series of numbers or terms, while simplification involves reducing a complex expression or equation into a simpler form. Summation is a step in the process of simplification.

4. Can you simplify a summation without knowing the function?

Yes, it is possible to simplify a summation without knowing the underlying function. This can be done by using mathematical properties and techniques such as the distributive property, factoring, or using known summation formulas.

5. How can finding function and simplifying the summation be applied in real-life situations?

Finding function and simplifying the summation can be applied in various fields such as finance, engineering, and data analysis. It can help in making predictions, optimizing processes, and identifying patterns in large sets of data.

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