**§ ****2 ****Generalized Fourier series and Fourier** - **Bessel series**

1. Generalized Fourier Series

If the continuous function is

_{} ( 1 )

Orthogonal on a certain interval , and if the function is absolutely integrable on , then the_{}_{}_{}

_{}

is the series of coefficients

_{}

is called the generalized Fourier series of functions with respect to the orthogonal function system ( 1 ) , denoted as_{}

*f* ( *x* ) ~_{}

_{}are called the Fourier coefficients of the orthogonal function system ( 1 ) ._{}

Let ( 1 ) be a standard orthogonal function system, that is, it satisfies

_{}

then

_{}

At this time, with respect to, there is Bessel's inequality_{}

_{} ( _{}is a square integrable function _{}

If for any square-integrable function, the closedness equation

_{}

It is said that the orthogonal function system at this time is closed ._{}

2. Fourier - Bessel series

[ Fourier - Bessel series ]

1 ^{} Let ^{o} be the positive root of the Bessel function (see Chapter 12), then the function system_{}_{}

_{}

Orthogonal by weight *x* on [0, 1] , i.e.

_{}

2 ^{o} For all functions that are absolutely integrable on [0, 1] , its Fourier - Bessel series can be made_{}

*f* ( *x* ) ~_{}

in the formula _{}

_{}are called the Fourier - Bessel coefficients of a function ._{}

3 ^{o} If it is continuous everywhere on [0, 1] except for a finite number of discontinuous points of the first kind and is segment-wise differentiable, then its Fourier - Bessel series converges at that time, and at the continuous points, the series The sum equals , at the discontinuity, the series sum equals ;_{}_{} _{}_{}

If it is absolutely integrable on [0 , 1] , continuous in the interval and has absolutely integrable derivatives, then its Fourier-Bessel series converges uniformly in every interval ;_{}_{}_{}_{}_{}

If it is absolutely integrable over [0 , 1] , continuous over the interval and has absolutely integrable derivatives, at the same time , then its Fourier - Bessel series converges uniformly over every interval ._{}_{}_{}_{}_{}_{}

[ Fourier - Bessel series of the second kind ]

1 ^{o} Let it be_{}

_{} ( *H* is a constant)

The positive root of , then, at that time , the function system_{}

_{}

Orthogonal by weight *x* on [0, 1] .

If it is absolutely integrable on [0 , 1] , then its generalized Fourier series with respect to the above orthogonal system is called a Fourier - Bessel series of the second kind , namely_{}_{}

_{}~_{}

in the formula _{}

_{}

2 ^{o} If the function is piecewise differentiable on [0, 1] (with at most a finite number of discontinuities of the first kind), then its Fourier - Bessel series of the second kind converges on 0 < *x* < 1 , and Equal at continuous points and equal at discontinuous points ;_{}_{}_{}_{}

If the function is continuous on [0 , 1] , twice differentiable (except for a finite number of points), and = 0, , bounded, then its second kind of Fourier - Bessel series is then , at each The interval [ ,1] (0< <1) is absolutely and uniformly convergent; at the same time , it is absolutely and uniformly converged on the entire interval [0 , 1] ._{}_{}_{}_{}_{}_{}_{}_{}

[ Fourier - Bessel series over the interval [0 , *l* ] ]

Suppose it is absolutely integrable on [0 , *l* ] , then its Fourier - Bessel series is_{}

^{
}*f* ( *x* ) ~_{}

in the formula
_{} _{}

For the Fourier - Bessel series of the second kind,

_{} _{}

Regarding the convergence of the series, we can only discuss the convergence of the corresponding Fourier - Bessel series on [0 , 1] by transforming , ._{}_{}_{}^{}

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