Difference between revisions of "Kernel of a summation method"
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| − | + | A function $ K _ {n} ( t) $( | |
| + | depending on a parameter) the values of which are the averages of the given method of summation applied to the series | ||
| − | + | $$ \tag{1 } | |
| + | { | ||
| + | \frac{1}{2} | ||
| + | } + \sum _ {\nu = 1 } ^ \infty \cos \nu t. | ||
| + | $$ | ||
| − | + | The kernel of a summation method gives an integral representation of the averages of the method in the summation of [[Fourier series|Fourier series]]. If the summation method is defined by a transformation of sequences into sequences using a matrix $ \| a _ {nk} \| _ {n,k= 0 } ^ \infty $, | |
| + | then the kernel of this method is the function | ||
| − | + | $$ | |
| + | K _ {n} ( t) = \sum _ {k = 0 } ^ \infty a _ {nk} D _ {k} ( t), | ||
| + | $$ | ||
| − | + | where $ D _ {k} ( t) $ | |
| + | are the partial sums of the series (1): | ||
| − | + | $$ \tag{2 } | |
| + | D _ {k} ( t) = \ | ||
| + | { | ||
| + | \frac{1}{2} | ||
| + | } + \sum _ {\nu = 1 } ^ { k } \cos \nu t = \ | ||
| + | |||
| + | \frac{\sin \{ ( k + {1 / 2 } ) t \} }{2 \sin ( t / 2) } | ||
| + | . | ||
| + | $$ | ||
| + | |||
| + | In this case the averages of the Fourier series for a $ 2 \pi $- | ||
| + | periodic function $ f $ | ||
| + | can be expressed in terms of $ f $ | ||
| + | and the kernel by the formula | ||
| + | |||
| + | $$ | ||
| + | \sigma _ {n} ( f, x) = \ | ||
| + | { | ||
| + | \frac{1} \pi | ||
| + | } | ||
| + | \int\limits _ {- \pi } ^ \pi f ( t) K _ {n} ( t - x) dt. | ||
| + | $$ | ||
In particular, the kernel of the method of arithmetical averages (cf. [[Arithmetical averages, summation method of|Arithmetical averages, summation method of]]) has the form | In particular, the kernel of the method of arithmetical averages (cf. [[Arithmetical averages, summation method of|Arithmetical averages, summation method of]]) has the form | ||
| − | + | $$ | |
| + | K _ {n} ( t) = \ | ||
| + | |||
| + | \frac{2}{n + 1 } | ||
| + | |||
| + | \left [ | ||
| + | |||
| + | \frac{\sin \{ ( n + 1) t /2 \} }{2 \sin ( t/2) } | ||
| + | |||
| + | \right ] ^ {2} , | ||
| + | $$ | ||
and is called the Fejér kernel. The kernel of the [[Abel summation method|Abel summation method]] is given by | and is called the Fejér kernel. The kernel of the [[Abel summation method|Abel summation method]] is given by | ||
| − | + | $$ | |
| + | K ( r, t) = \ | ||
| + | { | ||
| + | \frac{1}{2} | ||
| + | } | ||
| + | |||
| + | \frac{1 - r ^ {2} }{1 - 2r \cos t + r ^ {2} } | ||
| + | ,\ \ | ||
| + | 0 \leq r < 1, | ||
| + | $$ | ||
| − | and is called the Poisson kernel. The function | + | and is called the Poisson kernel. The function $ D _ {k} ( t) $ |
| + | in (2) is called the Dirichlet kernel. | ||
| − | The function | + | The function $ \overline{K}\; _ {n} ( t) $ |
| + | whose values are the averages of a summation method applied to the series | ||
| − | + | $$ | |
| + | \sum _ {\nu = 1 } ^ \infty \sin \nu t | ||
| + | $$ | ||
is called the conjugate kernel of the summation method. | is called the conjugate kernel of the summation method. | ||
Latest revision as of 22:14, 5 June 2020
A function $ K _ {n} ( t) $(
depending on a parameter) the values of which are the averages of the given method of summation applied to the series
$$ \tag{1 } { \frac{1}{2} } + \sum _ {\nu = 1 } ^ \infty \cos \nu t. $$
The kernel of a summation method gives an integral representation of the averages of the method in the summation of Fourier series. If the summation method is defined by a transformation of sequences into sequences using a matrix $ \| a _ {nk} \| _ {n,k= 0 } ^ \infty $, then the kernel of this method is the function
$$ K _ {n} ( t) = \sum _ {k = 0 } ^ \infty a _ {nk} D _ {k} ( t), $$
where $ D _ {k} ( t) $ are the partial sums of the series (1):
$$ \tag{2 } D _ {k} ( t) = \ { \frac{1}{2} } + \sum _ {\nu = 1 } ^ { k } \cos \nu t = \ \frac{\sin \{ ( k + {1 / 2 } ) t \} }{2 \sin ( t / 2) } . $$
In this case the averages of the Fourier series for a $ 2 \pi $- periodic function $ f $ can be expressed in terms of $ f $ and the kernel by the formula
$$ \sigma _ {n} ( f, x) = \ { \frac{1} \pi } \int\limits _ {- \pi } ^ \pi f ( t) K _ {n} ( t - x) dt. $$
In particular, the kernel of the method of arithmetical averages (cf. Arithmetical averages, summation method of) has the form
$$ K _ {n} ( t) = \ \frac{2}{n + 1 } \left [ \frac{\sin \{ ( n + 1) t /2 \} }{2 \sin ( t/2) } \right ] ^ {2} , $$
and is called the Fejér kernel. The kernel of the Abel summation method is given by
$$ K ( r, t) = \ { \frac{1}{2} } \frac{1 - r ^ {2} }{1 - 2r \cos t + r ^ {2} } ,\ \ 0 \leq r < 1, $$
and is called the Poisson kernel. The function $ D _ {k} ( t) $ in (2) is called the Dirichlet kernel.
The function $ \overline{K}\; _ {n} ( t) $ whose values are the averages of a summation method applied to the series
$$ \sum _ {\nu = 1 } ^ \infty \sin \nu t $$
is called the conjugate kernel of the summation method.
References
| [1] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |
| [2] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) |
Kernel of a summation method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_a_summation_method&oldid=47490