For every element x** in the double dual of a separable Banach space X there exists the sequence $\left(x^{\left(2n\right)}\right)$ of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class $B_1\left(X\right)╲B_{1/2}\left(X\right)$ (resp. to the class $B_{1/4}\left(X\right)$) as the elements with the sequence $\left(x^{\left(2n\right)}\right)$ equivalent to the usual basis of ${\ell }^1$ (resp. as the elements with the sequence $(x^{(4n-2)}-x^{\left(4n\right)})$ equivalent to the...

Bessaga and Pełczyński showed that if $c_0$ embeds in the dual $X^{*}$ of a Banach space X, then ${\ell }^1$ embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of ${\ell }^1$ contains a copy of ${\ell }^1$ that is complemented in ${\ell }^1$. Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of $L^1[0,1]$ contains a copy of ${\ell }^1$ that is complemented in $L^1[0,1]$. In this note a traditional sliding hump argument is used to establish a simple mapping property of...

Višek [3] and Culpin [1] investigated infinite binary sequence $X=(X_1,X_2,\cdots )$ with $X_i$ taking values $0$ or $1$ at random. They investigated also real mappings $H\left(X\right)$ which have the uniform distribution on $[0;1]$ (notation $𝒰(0;1)$). The problem for $n$-ary sequences is dealt with in this paper.

It is proved that if ${\phi }_n$ is a complete orthonormal system of bounded functions and ɛ>0, then there exists a measurable set E ⊂ [0,1] with measure |E|>1-ɛ, a measurable function μ(x), 0 < μ(x) ≤ 1, μ(x) ≡ 1 on E, and a series of the form ${\sum }_{k=1}^{\infty }{c}_k{\phi }_k\left(x\right)$, where $c_k\in l_q$ for all q>2, with the following properties: 1. For any p ∈ [1,2) and $f\in L_{\mu }^p[0,1]=f:{ʃ}_0^1{\left|f\left(x\right)\right|}^p\mu \left(x\right)dx<\infty$ there are numbers ${\varepsilon }_k$, k=1,2,…, ${\varepsilon }_k$ = 1 or 0, such that $li{m}_{n\to \infty }{ʃ}_0^1{|{\sum }_{k=1}^n{\varepsilon }_k{c}_k{\phi }_k\left(x\right)-f\left(x\right)|}^p\mu \left(x\right)dx=0.$ 2. For every p ∈ [1,2) and $f\in L_{\mu }^p[0,1]$ there are a function $g\in L^1[0,1]$ with g(x) = f(x) on E and numbers ${\delta }_k$, k=1,2,…, ${\delta }_k=1$ or 0,...

Let $U$ be an infinite-dimensional separable Fréchet space with a topology defined by a family of norms. Let $F$ be an infinite-dimensional Banach space. Then $F$ is the inductive limit of a family of spaces equal to $E$. The choice of suitable classes of Fréchet spaces allows to give characterizations of ultrabornological spaces using the result above.. Let $\Omega$ be a non-empty open set in the euclidean $n$-dimensional space $R^n$. Then $F$ is the inductive limit of a family of spaces equal to $\mathbf{D}\left(\Omega \right)$. ...

We investigate the Fourier transforms of functions in the Sobolev spaces $W_1^{r_1,...,r_n}$. It is proved that for any function $f\in W_1^{r_1,...,r_n}$ the Fourier transform f̂ belongs to the Lorentz space $L^{n/r,1}$, where $r=n{({\sum }_{j=1}^{n}1/r_j)}^{-1}\le n$. Furthermore, we derive from this result that for any mixed derivative $D^{s}f(f\in C_0^{\infty },s=(s_1,...,s_n))$ the weighted norm $\parallel {\left(D^{s}f\right)}^{\wedge }{\parallel }_{L^1\left(w\right)}(w\left(\xi \right)={\left|\xi \right|}^{-n})$ can be estimated by the sum of $L^1$-norms of all pure derivatives of the same order. This gives an answer to a question posed by A. Pełczyński and M. Wojciechowski.

Let $\Phi \left(t\right)={ʃ}_0^{t}a\left(s\right)ds$ and $\Psi \left(t\right)={ʃ}_0^{t}b\left(s\right)ds$, where a(s) is a positive continuous function such that ${ʃ}_1^{\infty }a\left(s\right)/sds=\infty$ and b(s) is quasi-increasing and $li{m}_{s\to \infty }b\left(s\right)=\infty$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants $c_1$ and $s_0$ such that ${ʃ}_1^{s}a\left(t\right)/tdt\ge c_{1}b\left(c_{1}s\right)$ for all $s\ge s_0$; (jj) there exist positive constants $c_2$ and $c_3$ such that ${ʃ}_0^{2\pi }\Psi (\left(c_2\right)/{\left(\right|⨍|}_{}\left)\right|⨍\left(x\right)\left|\right)dx\le c_3+c_3{ʃ}_0^{2\pi }{\Phi (1/(\left|⨍\right|}_{}\left)\right)Mf\left(x\right)dx$ for all $⨍\in L^1\left(\right)$.

Let $Q={\left(Q_k\right)}_{k\ge 0},Q_0=1,Q_{k+1}=q_k{Q}_k,q_k\ge 2$, be a Cantor scale, ${𝐙}_Q$ the compact projective limit group of the groups $𝐙/Q_k𝐙$, identified to ${\prod }_{0\le j\le k-1}𝐙/q_j𝐙$, and let $\mu$ be its normalized Haar measure. To an element $x=\{a_0,a_1,a_2,\cdots \},0\le a_k\le q_{k+1}-1$, of ${𝐙}_Q$ we associate the sequence of integral valued random variables $x_k={\sum }_{0\le j\le k}{a}_j{Q}_j$. The main result of this article is that, given a complex $𝐐$-multiplicative function $g$ of modulus $1$, we have $$\underset{x_k\to x}{lim}(\frac{1}{x_k}\sum _{n\le x_k-1}g\left(n\right)-\prod _{0\le j\le k}\frac{1}{q_j}\sum _{0\le a\le q_j}g\left(a{Q}_j\right))=0\mu \text{-a.e}.$$

Let $P(\lambda ,D)={\sum }_{\left|\alpha \right|\le m}{a}_{\alpha }\left(\lambda \right)D^{\alpha }$ be a differential operator with constant coefficients $a_{\alpha }$ depending analytically on a parameter $\lambda$. Assume that the family $\{$ P($\lambda$,D)$\}$ is of constant strength. We investigate the equation $P(\lambda ,D){𝔣}_{\lambda }\equiv g_{\lambda }$ where ${𝔤}_{\lambda }$ is a given analytic function of $\lambda$ with values in some space of distributions and the solution ${𝔣}_{\lambda }$ is required to depend analytically on $\lambda$, too. As a special case we obtain a regular fundamental solution of P($\lambda$,D) which depends analytically on $\lambda$. This result answers a question of L. Hörmander. ...