Cesaro Approximations

Brown University, Applied Mathematics

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Cesaro Approximations

Ernesto Cesàro (1859 – 1906) was an Italian mathematician who introduced another definition of convergence of infinite series. According to him, an infinite series

\[ \sum_{n\ge 0} a_n \]

converges to the value of S if

\[ S= \lim_{n\to \infty} \,\frac{S_0 + S_1 + \cdots + S_n}{n+1} , \]

where

\[ S_n = \sum_{k=0}^n a_k \]

are finite partial sums.

Essentially, The Hungarian mathematician Lipot Fejer (who was born in 1880 as Leopold Weiss in Jewish family and changed his name around 1900) discovered in 1900 what is now known as Fejer's theorem:

If \( F: \mathbb{C} \mapsto R \) is a continuous function with period \( 2\pi ,\) so \( f(-\pi ) = f(\pi ) ,\) then the sequence \( \sigma_n \) of Cesaro means of the sequence \( \{ s_n \} \) of partial sums of Fourier series

\[ \sigma_n = \frac{1}{n} \,\sum_{k=0}^{n-1} s_k , \qquad s_n = \sum_{k=-n}^n c_k \,e^{{\bf j} kx} , \]

where

\[ c_k = \frac{1}{2\pi} \, \int_{-\pi}^{\pi} f(t) \, e^{-{\bf j} kx} \,{\text d} x , \]

converges uniformly to f(x) on the interval \( [-\pi , \pi ] . \) A more general form of the theorem applies to functions which are not necessarily continuous. In particular, suppose f(x) is absolutely integrable on the interval \( [-\pi , \pi ] . \) If the left and right limits \( f(x_0 \pm 0) \mbox{ of } f(x) \) exist at \( x_0 , \) or if both limits are infinite of the same sign, then

\[ \sigma_n (x_0 ) \,\mapsto \, \frac{1}{2} \left[ f(x_0 +0) + f(x_0 -0) \right] . \]

Fejer's result says that for f(x) continuous and periodic, the convergence of the Cesaro sum of the trigonometric polynomials comprising the Fourier series is uniform. You may compare the result to the Weierstrass theorem, which gives uniform convergence of an approximating sequence of algebraic polynomials for a function continuous and defined over a compact domain. Both provide the density of such polynomials in the space of continous functions over an interval.

Both theorems can be proved similarly by expressing the sequence of approximating polynomials as a sequence of convolutions of the function with positive approximations of the Dirac delta function, which is convolution's idealized identity function. It is a fact that, over an interval, the convolutions of continuous function with a sequence of positive approximations of the identity is a uniform approximation of the function.

For the simple partial sum of the Fourier series, the inverse of convolution is known as the Dirichlet kernel. While it approximates the identity, is unfortunately not positive; indeed, the Dirichlet kernel is

\[ D_n (x) = \frac{\sin (nx+x/2)}{\sin (x/2)} \]

The Cesaro sum of the Fourier series, that is, the average of partial sums, yields after inversion of convolution the average of the Dirichlet kernels -- it turns out that this average of Dirichlet kernels, known as the Fejer kernel, is in fact positive!

\[ K_n (x) = \frac{1}{n+1} \, \frac{\sin^2 (nx/2)}{\sin^2 (x/2)} . \]

Therefore for a continuous function the Cesaro sums of the terms of the Fourier series converge uniformly to the continous function so that, like algebraic polynomials under the Weirstrass theorem, trigonometric polynomials are in fact dense in the space of continous functions over a compact interval.

Application of Fejer's Theorem

Let \( f(x) \) be a continuous periodic function on the interval \( [-\pi , \pi ] \) so \( f(-\pi ) = f(\pi ). \) Let \( C(f ; n; x) \) denote the n-th Cesaro sum of its Fourier series at x in the sense that if

\[ a_n = \frac{1}{2\pi} \, \int_{-\pi}^{\pi} f(t) \, e^{-{\bf j} nx} \, {\text d} t \]

is the n-th Fourier coefficient of f, then

\begin{align*} C(f ; 0; x) &= a_0 , \\ C(f ; 1; x) &= \frac{1}{2} \left( a_{-1} e^{-{\bf j} x} + 2\,a_0 + a_1 \,e^{{\bf j}x} \right) , \\ C(f ; 2; x) &= \frac{1}{3} \left( a_{-2} \,e^{-2{\bf j} x} + 2\,a_{-1} e^{-{\bf j} x} + 3\,a_0 + 2\,a_1 \,e^{{\bf j}x} + a_2 e^{2{\bf j} x}\right) , \end{align*}

and in general

\[ C(f ; n; x) = \sum_{k=-n}^n \frac{n+1 -|k|}{n+1} \,a_k \,e^{k{\bf j} x} . \]

Here j is the unit vector on the complex plane so that \( {\bf j}^2 =-1 . \) In general, we have

\[ C(f ; n; x) = \frac{1}{2\pi} \,\int_{-\pi}^{\pi} f(t)\, K_n (x-t)\,{\text d}t , \]

where \( K_n (x) \) is the Fejer's kernel:

\[ (n+1)\,K_n (x) = \left( \sum_{k=0}^n e^{{\bf j} (k-n/2)x} \right)^2 . \]

Examples

Example 1:

Example 2:

Example 3:

Example 4:

Working In MuPAD

The syntax is exactly the same, only this time, we will be ussing "ilaplace" instead of "laplace".

ilaplace::addpattern(pat, s, t, res)

ilaplace(1/`λ`^5,`λ`,t)

ilaplace(1/(`λ`^2+1),`λ`,t)

Cesaro Approximations

Brown University, Applied Mathematics