Here is a persuasive advertisement for ghostwriting services for the mathematical discipline:
**Unlock the Secrets of Mathematical Inequalities with Our Expert Ghostwriting Services**
Are you struggling to derive mathematical inequalities involving S2, M, δ, n, and x? Do you need help with summation and lemma 10? Look no further! Our team of expert ghostwriters specializes in crafting high-quality mathematical content that will help you achieve academic success.
**Our Services:**
* Customized mathematical derivations and proofs
* Expertise in summation and lemma 10
* Well-structured and easy-to-follow mathematical arguments
* Accurate and error-free calculations
* Fast turnaround times without compromising quality
**Why Choose Us:**
* Our ghostwriters hold advanced degrees in mathematics and have extensive experience in academic writing
* We understand the intricacies of mathematical notation and formatting
* We guarantee original and plagiarism-free content
* Our services are confidential and discreet, ensuring your complete anonymity
* We offer competitive pricing and flexible payment options
**Get Ahead of the Curve with Our Expert Ghostwriting Services**
Don’t let mathematical inequalities hold you back from achieving academic success. Let our team of expert ghostwriters help you unlock the secrets of mathematical derivations and proofs. Order now and receive a high-quality, customized mathematical solution that will impress your professors and peers alike!
\[\Rightarrow S_{2} \leq 2M\sum_{\left|k-nx\right|\geq\delta n}r_{k}\left(x\right)=2M \sum_{\left|k-nx\right|\geq\delta n}r_{k}\left(x\right)\frac{\left(k-nx \right)^{2}}{\left(k-nx\right)^{2}}\] \[\Rightarrow S_{2} \leq \frac{2M}{\delta^{2}n^{2}}\sum_{k=0}^{n}r_{k}\left(x\right)\left( k-nx\right)^{2}\]and by the above Lemma 10
\[S_{2}\leq\frac{2M}{\delta^{2}n^{2}}nx\left(1-x\right).\]
Using the fact that maximum of the function \(x\left(1-x\right)\) on \(\left[0,1\right]\) is \(\frac{1}{4}\) we obtain
\[S_{2}\leq\frac{2M}{\delta^{2}n}\frac{1}{4}<\frac{2M}{\delta^{2}n}<\epsilon\]
for all \(n\geq N>\frac{2M}{\delta^{2}\epsilon}.\) Note that \(N\) depends only on \(\epsilon,\) so we have shown
\[\left|f\left(x\right)-\left(B_{n}\left(f\right)\right)\left(x\right)\right| \leq 2\epsilon\]
for all \(x\in\left[0,1\right]\) whenever \(n\geq N.\) This shows
\[B_{n}\left(f\right)\rightrightarrows f\text{ on }\left[0,1\right]\]
which completes the proof of Bernstein’s theorem.
Proof of Lemma 10.: By the binomial theorem we already observed that
\[1=\left(x+\left(1-x\right)\right)^{n}=\sum_{k=0}^{n}\binom{n}{k}x^{k}\left(1- x\right)^{n-k}=\sum_{k=0}^{n}r_{k}\left(x\right).\]
Consider the binomial theorem again. Differentiating both sides with respect to \(x\) yields:
\[\left(x+y\right)^{n}=\sum_{k=0}^{n}\binom{n}{k}x^{k}y^{n-k}\]
\[n\left(x+y\right)^{n-1}=\sum_{k=0}^{n}k\binom{n}{k}x^{k-1}y^{n-k}.\]
Then multiplying by \(x\) gives:
\[nx\left(x+y\right)^{n-1}=\sum_{k=0}^{n}k\binom{n}{k}x^{k}y^{n-k}.\]
Setting \(y=1-x\) will give
\[nx=\sum_{k=0}^{n}kr_{k}\left(x\right).\]
Next we differentiate both sides of the binomial expansion twice and multiply by \(x^{2}.\) Thus we obtain
\[n\left(n-1\right)\left(x+y\right)^{n-2}x^{2}=x^{2}\sum_{k=0}^{n}k\left(k-1 \right)\binom{n}{k}x^{k-2}y^{n-k}.\]Substituting \(y=1-x\) yields
\[n\left(n-1\right)x^{2} = \sum_{k=0}^{n}k\left(k-1\right)r_{k}\left(x\right)\] \[= \sum_{k=0}^{n}k^{2}r_{k}\left(x\right)-\sum_{k=0}^{n}kr_{k}\left(x\right)\] \[\Rightarrow n\left(n-1\right)x^{2}+nx=\sum_{k=0}^{n}k^{2}r_{k} \left(x\right).\]
Now
\[\sum_{k=0}^{n}\left(k-nx\right)^{2}r_{k}\left(x\right) = \sum_{k=0}^{n}\left(k^{2}-2knx+n^{2}x^{2}\right)r_{k}\left(x\right)\] \[= \sum_{k=0}^{n}k^{2}r_{k}\left(x\right)-2nx\sum_{k=0}^{n}kr_{k} \left(x\right)+n^{2}x^{2}\sum_{k=0}^{n}r_{k}\left(x\right)\]
and we have expression for all of the above sums appearing on the right-hand side. Thus
\[\sum_{k=0}^{n}\left(k-nx\right)^{2}r_{k}\left(x\right) = nx+n\left(n-1\right)x^{2}-2nx\left(nx\right)+n^{2}x^{2}\] \[= nx-nx^{2}=nx\left(1-x\right)\ \ \text{as desired.}\]
**Remark 53**.: Others proofs of the Weierstrass theorem are known. Landau’s proof is indirect but closely follows the original ideas of Weierstrass; Lebesgue’s proof is a very clever, simple proof. In the above we presented Bernstein’s proof which was motivated by the theory of probability. For details of these proofs we refer the reader to [20], pp. 152-154. A nice introduction to approximation theory can be found in [15]; for a detailed discussion of the Weierstrass and approximation theory we refer the reader to [40] and a generalization of the Weierstrass approximation theorem can be found in M. H. Stone’s paper [51].
### Exercises
1. Show that \(n\)th Bernstein polynomial for \(f(x)=e^{x}\) is \[B_{n}(x)=[1+(e^{\frac{1}{n}}-1)x]^{n}.\]
2. Prove that \(B_{n}(e^{x})\) converges uniformly to \(e^{x}\) on \([0,1]\).
2. Show that the map \(B_{n}\) is linear and monotone.
3. Prove that if \(f\) is a continuous function on the closed and bounded interval \([a,b]\), then for any \(\epsilon>0\), there is a piecewise linear function \(F\) which approximates \(f\) uniformly within \(\epsilon\) on the interval.
Hint: Divide the interval \([a,b]\) into \(N\) equal subintervals at points
\[a=x_{0} Let \(P_{k}\) be a point \((x_{k},y_{k})\) where \(y_{k}=f(x_{k})\) and define the function \(F\) on each subinterval by \[F(x)=\frac{(x_{k+1}-x)y_{k}+(x-x_{k})y_{k+1}}{x_{k+1}-x_{k}}.\] 4. 1. Show that if \(f\in C[a,b]\), and if \(\int_{a}^{b}x^{n}f(x)dx=0\) for each \(n=1,2,…\), then \(f=0\). 5. Apply the Weierstrass approximation theorem to show that if a function \(f\) has a continuous derivative of order \(n\) on an interval \([a,b]\), then for each \(\epsilon>0\) there is a polynomial \(P\) such that \[|f(x)-P(x)|<\epsilon,\ \ |f^{\prime}(x)-P^{\prime}|<\epsilon,\ldots,|f^{n}(x)-P ^{n}(x)|<\epsilon\] for all \(x\in[a,b]\). 6. Let \(f\in C[0,1]\) have _Lipschitz constant 1_, then show that \(||B_{n}f-f||\leq\frac{1}{\sqrt{n}}\). Recall that \(f(x)\) defined on an interval \(I\) is said to be _Lipschitz with constant \(L>0\)_ provided that \(|f(x)-f(y)|\leq L|x-y|\) for all \(x,y\in I\). 7. Show that there exists a sequence of polynomials with integral coefficients converging to \(f\in C[0,1]\) uniformly if \(f(0)=f(1)=0\). 8. Deduce the approximation theorem by trigonometric polynomials from Weierstrass’ approximation theorem by polynomials. 9. Suppose \(f\) is a continuously differentiable function on \([0,1]\) and \(B_{n}f(x)\) is the Bernstein polynomial. Show that \(B_{n}^{\prime}f(x)\) converges uniformly to \(f^{\prime}(x)\) on the interval \([0,1]\). 10. Given any function \(f\in C[0,1]\) and any number \(\epsilon>0\), show that there exists a polynomial \(p\), all of whose coefficients are rational numbers, such that \(||p-f||<\epsilon\). ### 2.5 Functional Equations Functional equations are equations that define their objects (functions) implicitly by their properties (equations) rather than by explicit formulas. Thus they are very much like algebraic equations except that the unknown quantities are functions rather than real numbers. There are some functional equations where one can guess functions that satisfy them. For example \(f(x)-f(-x)=0\) is satisfied by \(f(x)=x^{2}\), \(f(x)=\cos x\) or \(f(x)=|-x|\), while the function \(f(x)=2x\) satisfies the functional equation \(f(x+y)=f(x)+f(y)\), but the function \(f(x)=x^{2}\) does not. This can be seen algebraically or by noting that the area \(x^{2}+y^{2}\) is a subset of the total area \((x+y)^{2}\) (Figure 2.16). Similarly, the functional equation \(f(xy)=f(x)+f(y)\) is satisfied by all logarithmic functions while the functional equation \(f(x+y)=f(x)f(y)\) is satisfied by all exponential functions. Given a functional equation, it is natural to ask the following questions: what function(s) are a solution to that equation? Or what values of the domain of the function satisfy the relationship given by the functional equation? Sometimes more than one functional equation exists for a single function. For example there are three different functional equations for the gamma function \(\Gamma(x)\), where the gamma function is defined as \[\Gamma(x)=\int_{0}^{\infty}e^{-t}t^{x-1}dt.\]
2. For a \(f\in C[a,b]\), the _moments_ of \(f\) are the numbers \[\mu_{n}=\int_{a}^{b}x^{n}f(x)dx,\] where \(n=0,1,2,…\). Prove that two continuous functions defined on \([a,b]\) are identical if they have the same sequence of moments.
发表回复