Inequality of Bernstein

In mathematics, the inequality of Bernstein is a result of analysis. It makes it possible to compare the higher Borne of a function having a particular form and that of its derivative.

In its general form, the inequality applies to a function of the following form

$f\; (T)\; =\; \backslash \; sum\_\; \{k=1\}\; ^p\; \backslash \; alpha\_k\; e^\; \{I\; \backslash \; lambda\_k\; T\}$

with complex coefficients $\backslash \; alpha\_k$ and real and distinct coefficients $\backslash \; lambda\_k$. The inequality states thus

$\backslash |f\text{'}\; \backslash |\_\; \backslash \; infty\; \backslash \; Leq\; \backslash \; max\; \backslash \; limits\_\; \{1\; \backslash \; Leq\; K\; \backslash \; Leq\; p\}|\backslash \; lambda\_k|\backslash \; cdot\; \backslash |F\; \backslash |\_\backslash infty$

Demonstration

One will note

$\backslash \; Lambda\; =\; \backslash \; max\; \backslash \; limits\_\; \{1\; \backslash \; Leq\; K\; \backslash \; Leq\; p\}|\backslash \; lambda\_k|$

One can bring back if this constant has a value chosen, for example $\backslash \; Lambda=\; \backslash \; frac\; \backslash \; pi2$, by carrying out the change of variables $u=\; \backslash \; frac\; \{\backslash \; pi\; T\}\; \{2\; \backslash \; Lambda\}$. It will be supposed that $\backslash \; Lambda$ has this value in the continuation.

One uses the following formula

$\backslash \; forall\; X\; \backslash \; in\; \backslash \; frac\; \backslash \; pi2,\; \backslash \; qquad\; x=\; \backslash \; sum\_\; \{n=-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}$

\ gamma_n e^{inx} with

$\backslash \; gamma\; \_\; \{2n\}\; =0,\; \backslash \; qquad\; \backslash \; gamma\_\; \{2n+1\}\; =\; \backslash \; frac\; \{2\; (-\; 1)\; ^\; \{n+1\}\; I\}\; \{\backslash \; pi\; (2n+1)\; ^2\},$

formulate resulting from the theory of the Fourier series. It is indeed about the development in Fourier series of a function triangle.

If one breaks up the factors $\backslash \; lambda\_k$ appearing in the derivative of F using this formula,

$f\text{'}\; (T)\; =\; \backslash \; sum\_\; \{k=1\}\; ^p\; \backslash \; left\; (\backslash \; sum\_\; \{n=-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; gamma\_n$

e^ {in \ lambda_k} \ right) I \ alpha_k e^ {I \ lambda_k T} = I \ sum_ {n=- \ infty} ^ {+ \ infty} \ gamma_n \ sum_ {k=1} ^p \ alpha_k e^ {I \ lambda_k (t+n)}

Finally the derivative is expressed like

$f\text{'}\; (T)\; =\; I\; \backslash \; sum\_\; \{n=-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; \backslash \; gamma\_n$

F (t+n)

What can be raised by

$|f\text{'}\; (T)|\backslash \; Leq\; \backslash \; left\; (\backslash \; sum\_\; \{n=-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; |\; \backslash \; gamma\_n$

|\ right) \ cdot \|F \|_\infty

However for $t=\; \backslash \; frac\; \backslash \; pi\; 2$, all the terms $\backslash \; gamma\_n\; e^\; \{int\}$ are real positive, therefore

$\backslash \; sum\_\; \{n=-\; \backslash \; infty\}\; ^\; \{+\; \backslash \; infty\}\; |\; \backslash \; gamma\_n$

|= \ frac \ pi 2

What is well the desired property:

$\backslash |f\text{'}\; \backslash |\_\; \backslash \; infty\; \backslash \; Leq\; \backslash \; Lambda\; \backslash \; cdot\; \backslash |F\; \backslash |\_\backslash infty$

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