grandes-ecoles

Q1a Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Show that $\Gamma^{s}(x_{0})$ is a vector subspace of $\mathcal{C}$, then that, for all real numbers $s_{1}$ and $s_{2}$ satisfying $0 \leq s_{1} \leq s_{2} < 1$, we have $\Gamma^{s_{2}}(x_{0}) \subset \Gamma^{s_{1}}(x_{0})$. Finally, determine $\Gamma^{0}(x_{0})$.
Recall: Let $x_{0} \in [0,1]$. For all $s \in [0,1[$, $\Gamma^{s}(x_{0})$ is the subset of $\mathcal{C}$ formed by functions $f$ which satisfy: $$\sup_{x \in [0,1] \backslash \{x_{0}\}} \frac{|f(x) - f(x_{0})|}{|x - x_{0}|^{s}} < +\infty .$$

Q1b Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Let $f \in \mathcal{C}$. If $f$ is differentiable at $x_{0}$, show that $f \in \Gamma^{s}(x_{0})$ for all $s \in [0,1[$.

Q1c Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

Show that for all $x_{0} \in ]0,1[$, there exists $f \in \mathcal{C}$ non-differentiable at $x_{0}$ such that for all $s \in [0,1[$, $f \in \Gamma^{s}(x_{0})$.

Q2 Sequences and Series Limit Evaluation Involving Sequences View

Let $p : [0,1] \rightarrow \mathbf{R}$, $x \mapsto \sqrt{|1 - 4x^{2}|}$. Determine the pointwise Hölder exponent of $p$ at $\frac{1}{2}$.
Recall: For all $f \in \mathcal{C}$ and all $x_{0} \in [0,1]$, $$\alpha_{f}(x_{0}) = \sup \{s \in [0,1[ \mid f \in \Gamma^{s}(x_{0})\} .$$

Q3a Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by $$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$ Show that $\omega_{f}$ is increasing, and continuous at 0.

Q3b Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by $$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$ Show that for all $h, h' \in [0,1]$ such that $h \leq h'$, $\omega_{f}$ satisfies $$\omega_{f}(h') \leq \omega_{f}(h) + \omega_{f}(h' - h) .$$

Q3c Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View

For all $f \in \mathcal{C}$, the function $\omega_{f} : [0,1] \rightarrow \mathbf{R}_{+}$ is defined by $$\omega_{f}(h) = \sup \{|f(x) - f(y)| \mid x, y \in [0,1] \text{ and } |x - y| \leq h\} .$$ Using the result of question 3b, deduce that $\omega_{f}$ is continuous on $[0,1]$.

Q4a Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $s \in [0,1[$. Suppose that the function $h \mapsto \frac{\omega_{f}(h)}{h^{s}}$ is bounded on $]0,1]$. For all $x_{0} \in [0,1]$, show that $f \in \Gamma^{s}(x_{0})$.

Q4b Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View

Let $q : [0,1] \rightarrow \mathbf{R}$ defined by $$\left\{ \begin{array}{l} q(x) = x \cos\left(\frac{\pi}{x}\right) \text{ for } x > 0 \\ q(0) = 0 \end{array} \right.$$ Show that for all $x_{0} \in [0,1]$, $\alpha_{q}(x_{0}) = 1$, but that $\frac{\omega_{q}(h)}{\sqrt{h}}$ does not tend to 0 when $h$ tends to 0.

Q5a Sequences and Series Functional Equations and Identities via Series View

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Show that for all $j \in \mathbf{N}$ and all $k \in \mathcal{T}_{j+1}$, there exists a unique integer $k' \in \mathcal{T}_{j}$ such that $$[k 2^{-j-1}, (k+1) 2^{-j-1}] \subset [k' 2^{-j}, (k'+1) 2^{-j}]$$ Specify the relationship between $k$ and $k'$.

Q5b Sequences and Series Evaluation of a Finite or Infinite Sum View

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Calculate $\theta_{j,k}(\ell 2^{-j-1})$ for all $j \in \mathbf{N}$, $k \in \mathcal{T}_{j}$, $\ell \in \mathcal{T}_{j+1}$.

Q5c Proof Proof That a Map Has a Specific Property View

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Show that for all $(j, k) \in \mathcal{I}$, the function $\theta_{j,k}$ is continuous, affine on each interval of the form $[\ell 2^{-n}, (\ell+1) 2^{-n}]$ where $n > j$ and $\ell \in \mathcal{T}_{n}$.

Q5d Proof Direct Proof of an Inequality View

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Prove that for all $(j, k) \in \mathcal{I}$ and $(x, y) \in [0,1]^{2}$, we have $$|\theta_{j,k}(x) - \theta_{j,k}(y)| \leq 2^{j+1} |x - y|$$

Q6 Proof Computation of a Limit, Value, or Explicit Formula View

In the rest of the second part, $f$ is an element of $\mathcal{C}_{0}$. For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Show that $\lim_{j \rightarrow +\infty} \max_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| = 0$.

Q7a Proof Computation of a Limit, Value, or Explicit Formula View

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ For all $(j, k) \in \mathcal{I}$, $(i, \ell) \in \mathcal{I}$, calculate $c_{j,k}(\theta_{i,\ell})$.

Q7b Proof Direct Proof of a Stated Identity or Equality View

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. Let $a_{j,k}$ be a family of real numbers indexed by $(j, k) \in \mathcal{I}$. We denote $b_{j} = \max_{k \in \mathcal{T}_{j}} |a_{j,k}|$, and we suppose that the series $\sum b_{j}$ is convergent.
For all $j \in \mathbf{N}$, let $f_{j}^{a}$ be the function defined by $$f_{j}^{a}(x) = \sum_{k \in \mathcal{T}_{j}} a_{j,k} \theta_{j,k}(x)$$ Show that the series $\sum f_{j}^{a}$ is uniformly convergent on $[0,1]$ towards a function denoted $f^{a}$, which belongs to $\mathcal{C}_{0}$ and which satisfies, for all $(j, k) \in \mathcal{I}$, $c_{j,k}(f^{a}) = a_{j,k}$.

Q8a Proof Bounding or Estimation Proof View

For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Suppose $f$ is of class $\mathcal{C}^{1}$. Show that there exists a constant $M \geq 0$ such that for all $(j, k) \in \mathcal{I}$, $|c_{j,k}(f)| \leq M 2^{-j}$.
Deduce that the sequence of functions $S_{n} f$ is uniformly convergent on $[0,1]$ when $n$ tends to $\infty$.

Q8b Proof Bounding or Estimation Proof View

For all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Suppose $f$ is of class $\mathcal{C}^{2}$. Show that there exists a constant $M' \geq 0$ such that for all $(j, k) \in \mathcal{I}$, $|c_{j,k}(f)| \leq M' 4^{-j}$.

Q9a Proof Direct Proof of a Stated Identity or Equality View

For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ Show that for all $n \in \mathbf{N}$ and all $\ell \in \mathcal{T}_{n+1}$, the function $S_{n} f$ is affine on the interval $[\ell 2^{-n-1}, (\ell+1) 2^{-n-1}]$.

Q9b Proof Direct Proof of a Stated Identity or Equality View

For all $n \in \mathbf{N}$, let $S_{n} f$ be the function of $\mathcal{C}_{0}$ defined by $$S_{n} f = \sum_{j=0}^{n} \sum_{k \in \mathcal{T}_{j}} c_{j,k}(f) \theta_{j,k}$$ where, for all $(j, k) \in \mathcal{I}$, $$c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2} .$$ Let $n \in \mathbf{N}$. Suppose that for all $\ell \in \mathcal{T}_{n}$, $(S_{n-1} f)(\ell 2^{-n}) = f(\ell 2^{-n})$. Show that we also have that for all $\ell \in \mathcal{T}_{n+1}$, $(S_{n} f)(\ell 2^{-n-1}) = f(\ell 2^{-n-1})$.
One may distinguish cases according to the parity of $\ell$.

Q9c Proof Deduction or Consequence from Prior Results View

Deduce that for all $n \in \mathbf{N}$ and all $\ell \in \mathcal{T}_{n+1}$, $(S_{n} f)(\ell 2^{-n-1}) = f(\ell 2^{-n-1})$.

Q10a Proof Deduction or Consequence from Prior Results View

Deduce from question 9 that for all $f$ in $\mathcal{C}_{0}$, $\lim_{n \rightarrow +\infty} \|f - S_{n} f\|_{\infty} = 0$.

Q10b Proof Proof That a Map Has a Specific Property View

Let $n \in \mathbf{N}$. Show that $S_{n}$ is a projector on $\mathcal{C}_{0}$, whose subordinate norm (to $\|\cdot\|_{\infty}$) equals 1.

Q11a Proof Direct Proof of an Inequality View

Let $s \in ]0,1[$. Show that if $a, b \geq 0$, then $a^{s} + b^{s} \leq 2^{1-s}(a+b)^{s}$.

Q11b Proof Bounding or Estimation Proof View

Let $s \in ]0,1[$. Show that if $f \in \Gamma^{s}(x_{0}) \cap \mathcal{C}_{0}$, then there exists a real number $c_{1} > 0$, such that for all $(j, k) \in \mathcal{I}$, we have $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s} .$$ Recall that $c_{j,k}(f) = f\left(\left(k + \frac{1}{2}\right) 2^{-j}\right) - \frac{f(k 2^{-j}) + f((k+1) 2^{-j})}{2}$.

Q12 Sequences and Series Limit Evaluation Involving Sequences View

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$.
Show that there exists a unique $n_{0} \in \mathbf{N}$ such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$.

Q13 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We recall that $\widetilde{k}_{j}(x)$ is the integer part of $2^{j} x$. We set $$W_{j} = \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x) - \theta_{j,k}(x_{0})|$$ Show that $$W_{j} \leq (|c_{j,\widetilde{k}_{j}(x)}(f)| + |c_{j,\widetilde{k}_{j}(x_{0})}(f)|) 2^{j+1} |x - x_{0}|$$

Q14a Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$.
Show that for $j \leq n_{0}$, we have $$W_{j} \leq 4 c_{1} 2^{(1-s)j} 3^{s} |x - x_{0}|$$

Q14b Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$.
Deduce that, by setting $c_{2} = 8(2^{1-s} - 1)^{-1} (3/2)^{s} c_{1}$, $$\sum_{j=0}^{n_{0}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x) - \theta_{j,k}(x_{0})| \leq c_{2} |x - x_{0}|^{s}$$

Q15 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We recall that $\widetilde{k}_{j}(x_{0})$ is the integer part of $2^{j} x_{0}$.
Show that for all $j \in \mathbf{N}$, $|c_{j,\widetilde{k}_{j}(x_{0})}(f)| \leq 2^{s(1-j)} c_{1}$. Deduce, by setting $c_{3} = (1 - 2^{-s})^{-1} 2^{s} c_{1}$, $$\sum_{j=n_{0}+1}^{+\infty} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x_{0})| \leq c_{3} |x - x_{0}|^{s}$$

Q16 Sequences and Series Limit Evaluation Involving Sequences View

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$. We suppose that $\|f\|_{\infty} = 1$.
Show that there exists a unique $n_{1} \in \mathbf{N}$ such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.

Q17 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$, and $n_{1}$ the unique natural integer such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.
Show that for all $n \geq n_{1}$, we have $$\|f - S_{n} f\|_{\infty} \leq 2^{s+1} |x - x_{0}|^{s}$$ One may use the results of questions 9a and 9c.

Q18a Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. Let $n_{0}$ be the unique natural integer such that $2^{-n_{0}-1} < |x - x_{0}| \leq 2^{-n_{0}}$, and $n_{1}$ the unique natural integer such that $\omega_{f}(2^{-n_{1}-1}) < 2^{-n_{0} s} \leq \omega_{f}(2^{-n_{1}})$.
Show that when $n_{0} < n_{1}$, we have $$\sum_{j=n_{0}+1}^{n_{1}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x)| \leq c_{1} 3^{s} (n_{1} - n_{0}) |x - x_{0}|^{s} .$$

Q18b Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We fix $x_{0}$, $s$, $c_{1}$ and $x \in [0,1] \backslash \{x_{0}\}$. We suppose furthermore that the function $\omega_{f}$ satisfies property $(\mathcal{P}_{2})$: for all integer $N \geq 1$, there exists a real number $c_{4}(N) > 0$, such that for all $h \in ]0,1]$, $$\omega_{f}(h) \leq c_{4}(N) (1 + |\log_{2} h|)^{-N}$$ For all integer $N \geq 1$, we set $c_{5}(N) = 3^{s} c_{1} (c_{4}(N))^{1/N}$. Show that $$n_{1} - n_{0} \leq n_{1} + 1 \leq \left(\frac{c_{4}(N)}{\omega_{f}(2^{-n_{1}})}\right)^{\frac{1}{N}}$$ and deduce $$\sum_{j=n_{0}+1}^{n_{1}} \sum_{k \in \mathcal{T}_{j}} |c_{j,k}(f)| |\theta_{j,k}(x)| \leq c_{5}(N) |x - x_{0}|^{(1 - \frac{1}{N})s}$$

Q19 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View

Throughout the third part, $f \in \mathcal{C}_{0}$ satisfies property $(\mathcal{P}_{1})$: there exist $x_{0} \in [0,1]$, $s \in ]0,1[$ and $c_{1} \in ]0, +\infty[$, such that for all $(j, k) \in \mathcal{I}$, $$|c_{j,k}(f)| \leq c_{1} (2^{-j} + |k 2^{-j} - x_{0}|)^{s}$$ We suppose furthermore that $\omega_{f}$ satisfies property $(\mathcal{P}_{2})$: for all integer $N \geq 1$, there exists $c_{4}(N) > 0$ such that for all $h \in ]0,1]$, $\omega_{f}(h) \leq c_{4}(N)(1 + |\log_{2} h|)^{-N}$.
Deduce from the above that $\alpha_{f}(x_{0}) \geq s$.
One may distinguish the cases $n_{0} \geq n_{1}$ and $n_{0} < n_{1}$.

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Papers (191)

2013 x-ens-maths2__mp