# Questions tagged [real-analysis]

Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

3,579 questions
Filter by
Sorted by
Tagged with
118 views

### Condition on kernel convolution operator

I am studying O'Neil's convolution inequality. Let $\Phi_1$ and $\Phi_2$ be $N$-functions, with $$\Phi_i(2t)\approx \Phi_i(t), \quad i=1,2$$ with $t\gg 1$ and let $k \in M_+(\mathbf R^n)$ is the ...
111 views

800 views

### Was Cantor aware of Lebesgue theory of integration?

Georg Cantor died in 1919, more than ten years after appearance of the Lebesgue theory of measure and integration at the beginning of the twentieth century. Lebesgue theory has a deep connection with ...
133 views

218 views

### Uniformly approximating a function and its derivative using polynomials

I'm struggling either proving or disproving the following statement: Let $K\subset \mathbb{R}$ be compact, and $S = \mathrm{span}\{p_k, k = 0, 1, \ldots\}$, where $p_k$'s are polynomials over $K$. If ...
278 views

166 views

### Is there a function $f$ that is a finite sum of functions with finite products of the inputs of $f$ as inputs with this property?

Note: This question aims to be a generalization of Is it possible to create a polynomial $p(x)$ with this relation between $p(0)$ and $p(c)$? and Is it possible to create a polynomial $p(x)$ with this ...
486 views

### Interpolation inequality $\int_{\mathbb R} u^3 dx \le \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$ [closed]
Let $u \in C^\infty(\mathbb R)$. Is it true that the following interpolation inequality holds? $$\int_{\mathbb R} u^3 dx \lesssim \int_{\mathbb R} (u')^2 dx + \int_{\mathbb{R}} u^2 dx$$