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In [[mathematical analysis]], many generalizations of [[Fourier series]] have proved to be useful.
They are all special cases of decompositions over an [[orthonormal basis]] of an [[inner product space]].
Here we consider that of [[square-integrable]] functions defined on an [[Interval (mathematics)|interval]] of the [[real line]], which is important, among others, for [[interpolation]] theory.
 
==Definition==
 
Consider a set of [[square-integrable]] functions with values in <math> \mathbb{F}=\mathbb{C}\mbox{ or }\mathbb{R}</math>,
 
:<math>\Phi = \{\varphi_n:[a,b]\rightarrow \mathbb{F}\}_{n=0}^\infty,</math>
 
which are pairwise [[orthogonal]] for the [[inner product]]
 
:<math>\langle f, g\rangle_w = \int_a^b f(x)\,\overline{g}(x)\,w(x)\,dx</math>
 
where ''w''(''x'') is a [[weight function]], and <math>\overline\cdot</math> represents [[complex conjugation]], i.e. <math>\overline{g}(x)=g(x)</math> for <math> \mathbb{F}=\mathbb{R}</math>.
 
The '''generalized Fourier series''' of a [[square-integrable]] function ''f'': [''a'', ''b''] → <math> \mathbb{F}</math>,
with respect to Φ, is then
 
:<math>f(x) \sim \sum_{n=0}^\infty c_n\varphi_n(x),</math>
 
where the coefficients are given by
 
:<math>c_n = {\langle f, \varphi_n \rangle_w\over \|\varphi_n\|_w^2}.</math>
 
If Φ is a complete set, i.e., an [[orthonormal basis]] of the space of all square-integrable functions on [''a'', ''b''], as opposed to a smaller orthonormal set,
the relation <math>\sim \,</math> becomes equality in the ''[[L2 space|L²]]'' sense, more precisely modulo |·|<sub>''w''</sub> (not necessarily pointwise, nor [[almost everywhere]]).
 
== Example (Fourier–Legendre series) ==
The [[Legendre polynomials]] are solutions to the [[Sturm–Liouville theory|Sturm–Liouville problem]]
 
: <math> \left((1-x^2)P_n'(x)\right)'+n(n+1)P_n(x)=0</math>
 
and because of Sturm-Liouville theory, these polynomials are eigenfunctions of the problem and are solutions orthogonal with respect to the inner product above with unit weight. So we can form a generalized Fourier series (known as a Fourier–Legendre series) involving the Legendre polynomials, and
 
:<math>f(x) \sim \sum_{n=0}^\infty c_n P_n(x),</math>
 
:<math>c_n = {\langle f, P_n \rangle_w\over \|P_n\|_w^2}</math>
 
As an example, let us calculate the Fourier–Legendre series for ''&fnof;''(''x'')&nbsp;=&nbsp;cos&nbsp;''x'' over [&minus;1,&nbsp;1]. Now,
 
:<math>
\begin{align}
c_0 & = \sin{1} = {\int_{-1}^1 \cos{x}\,dx \over \int_{-1}^1 (1)^2 \,dx} \\
c_1 & = 0 = {\int_{-1}^1 x \cos{x}\,dx \over \int_{-1}^1 x^2 \, dx} = {0 \over 2/3 } \\
c_2 & = {5 \over 2} (6 \cos{1} - 4\sin{1}) = {\int_{-1}^1 {3x^2 - 1 \over 2} \cos{x} \, dx \over \int_{-1}^1 {9x^4-6x^2+1 \over 4} \, dx} = {6 \cos{1} - 4\sin{1} \over 2/5 }
\end{align}
</math>
 
and a series involving these terms
 
:<math>c_2P_2(x)+c_1P_1(x)+c_0P_0(x)= {5 \over 2} (6 \cos{1} - 4\sin{1})\left({3x^2 - 1 \over 2}\right) + \sin{1}(1)</math>
:<math>= \left({45 \over 2} \cos{1} - 15 \sin{1}\right)x^2+6 \sin{1} - {15 \over 2}\cos{1}</math>
 
which differs from cos ''x'' by approximately 0.003, about&nbsp;0. It may be advantageous to use such Fourier–Legendre series since the eigenfunctions are all polynomials and hence the integrals and thus the coefficients are easier to calculate.
 
== Coefficient theorems ==
Some theorems on the coefficients ''c''<sub>''n''</sub> include:
 
===Bessel's inequality===
 
:<math>\sum_{n=0}^\infty |c_n|^2\leq\int_a^b|f(x)|^2\,dx.</math>
 
===Parseval's theorem===
 
If Φ is a complete set,
 
:<math>\sum_{n=0}^\infty |c_n|^2 = \int_a^b|f(x)|^2\, dx.</math>
 
==See also==
*[[Orthogonality]]
*[[Orthogonal function]]
*[[Eigenfunctions]]
*[[Vector space]]
*[[Function space]]
*[[Topological vector space]]
*[[Hilbert space]]
*[[Banach space]]
 
[[Category:Fourier analysis]]

Latest revision as of 01:15, 4 December 2014

When looking for the appropriate hemorrhoid treatment you will want to consider a few important factors such as, which one you think you want, if there is a remarkable amount of recovery required, plus how extended it takes to receive results. In this article you'll learn the answers to all of these issues, providing we the answer you should find the right hemorrhoid treatment.

If you have been seeing blood on your toilet paper, you are probably inside the late stages of the ailment and need a pretty efficient thrombosed hemorrhoids treatment.

Take a "sitz" bath with warm water. Do this 3 or 4 time a day for 15 minutes every time. This treatment is very efficient at reducing pain and swelling and is absolutely simple to do.

Lunch plus Dinner. Gradually add real fruits plus veggies to the food, plus substitute entire grains for white flour and pasta for an additional fiber punch.

There are two kinds of hemorrhoids- internal plus exterior. Both are the outcome of swollen veins in the anal region. Internal hemorrhoids could be hard to discover since they are not noticeable. You'll just find out later whenever it begins to bleed. On the alternative hand, exterior hemorrhoids is felt as a difficult lump inside the anal opening. These are generally very noticeable due to the fact which they are swollen, red, itchy, and really painful.

Sitz Bath: This system is regarded as the many well-known methods selected to relieve sufferers of the pain caused by hemorrhoids. A sitz bath is a tub filled with warm water, when you want we can add several important oils to your bathtub water. You should soak a rectum to the warm water for at least 15 minutes. Do this 3 instances a day and it may greatly reduce the swelling plus the pain of the the hemorrhoids.

Using these techniques we can tame your hemorrhoids. If you have stubborn hemorrhoids we could like to look into several advanced natural hemorrhoid treatments, this involves the effective Chinese way.

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