The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that $Ce{s}_p\left(I\right)$ is an interpolation space between $Ce{s}_{p₀}\left(I\right)$ and $Ce{s}_{p₁}\left(I\right)$ for 1 < p₀ < p₁ ≤ ∞ and 1/p = (1 - θ)/p₀ + θ/p₁ with 0 < θ < 1, where I = [0,∞) or [0,1]. The same result is true for Cesàro sequence spaces. On the other hand, $Ce{s}_p[0,1]$ is not an interpolation space between Ces₁[0,1] and $Ce{s}_{\infty }[0,1]$.

If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K(t,f;L^p\left(C\right),L^{\infty }\left(C\right))$ to quasicontinuous functions f ∈ QC is equivalent to $K(t,f;L^p\left(C\right)\cap QC,L^{\infty }\left(C\right)\cap QC)$. We apply this result to identify the interpolation space ${(L^{p₀,q₀}\left(C\right)\cap QC,L^{p₁,q₁}\left(C\right)\cap QC)}_{\theta ,q}$.

Let $\parallel ·\parallel$ be the uniform norm in the unit disk. We study the quantities $M_n\left(\alpha \right):=inf(\parallel zP\left(z\right)+\alpha \parallel -\alpha )$ where the infimum is taken over all polynomials $P$ of degree $n-1$ with $\parallel P\left(z\right)\parallel =1$ and $\alpha >0$. In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that ${inf}_{\alpha >0}{M}_n\left(\alpha \right)=1/n$. We find the exact values of $M_n\left(\alpha \right)$ and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

Let K be any subset of ${ℂ}^N$. We define a pluricomplex Green’s function $V_{K,\theta }$ for θ-incomplete polynomials. We establish properties of $V_{K,\theta }$ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of ${ℝ}^N$, we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on $K\setminus supp{\left(d{d}^c{V}_{K,\theta }\right)}^N$. We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $supp{\left(d{d}^c{V}_{K,\theta }\right)}^N$ when K is a compact...

A complete description of the real interpolation space $L={(L_{p₀}\left(\omega ₀\right),...,L_{pₙ}\left(\omega ₙ\right))}_{\theta ⃗,q}$ is given. An interesting feature of the result is that the whole measure space (Ω,μ) can be divided into disjoint pieces ${\Omega }_i$ (i ∈ I) such that L is an $l_q$ sum of the restrictions of L to ${\Omega }_i$, and L on each ${\Omega }_i$ is a result of interpolation of just two weighted $L_p$ spaces. The proof is based on a generalization of some recent results of the first two authors concerning real interpolation of vector-valued spaces.

Starting from Lagrange interpolation of the exponential function ${\mathrm{e}}^z$ in the complex plane, and using an integral representation formula for holomorphic functions on Banach spaces, we obtain Lagrange interpolating polynomials for representable functions defined on a Banach space $E$. Given such a representable entire funtion $f:E\to ℂ$, in order to study the approximation problem and the uniform convergence of these polynomials to $f$ on bounded sets of $E$, we present a sufficient growth condition on...

We present the complex interpolation of Besov and Triebel–Lizorkin spaces with generalized smoothness. In some particular cases these function spaces are just weighted Besov and Triebel–Lizorkin spaces. As a corollary of our results, we obtain the complex interpolation between the weighted Triebel–Lizorkin spaces ${\dot{F}}_{p_0,q_0}^{s_0}\left({\omega }_0\right)$ and ${\dot{F}}_{\infty ,q_1}^{s_1}\left({\omega }_1\right)$ with suitable assumptions on the parameters $s_0,s_1,p_0,q_0$ and $q_1$, and the pair of weights $({\omega }_0,{\omega }_1)$.

Let $P_k$ be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients $a_k$ in $f={\sum }_k{a}_k{P}_k$. A systematic use of the basic properties (including some nonstandard ones) of the polynomials $P_k$ results in obtaining a low order of the recurrence.

We find necessary and sufficient conditions under which the norms of the interpolation spaces ${(N₀,N₁)}_{\theta ,q}$ and ${(X₀,X₁)}_{\theta ,q}$ are equivalent on N, where N is the kernel of a nonzero functional ψ ∈ (X₀ ∩ X₁)* and $N_i$ is the normed space N with the norm inherited from $X_i$ (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator $T_{\theta }=S-2^{\theta }I$ (S...

We introduce the notion of the modulus of dentability defined for any point of the unit sphere S(X) of a Banach space X. We calculate effectively this modulus for denting points of the unit ball of the classical interpolation space $L¹+L^{\infty }.$ Moreover, a criterion for denting points of the unit ball in this space is given. We also show that none of denting points of the unit ball of $L¹+L^{\infty }$ is a LUR-point. Consequently, the set of LUR-points of the unit ball of $L¹+L^{\infty }$ is empty.

We consider the space ${}^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on ${}^{ℳ}$. A description of the space ${(}^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

We give a complete characterization of the positive trigonometric polynomials $Q(\theta ,\varphi )$ on the bi-circle, which can be factored as $Q(\theta ,\varphi )=|p(e^{i\theta },e^{i\varphi }){|}^2$ where $p(z,w)$ is a polynomial nonzero for $\left|z\right|=1$ and $\left|w\right|\le 1$. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight $\frac{1}{4{\pi }^{2}Q(\theta ,\varphi )}$ on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities...

We establish an interpolation theorem for a class of nonlinear operators in the Lebesgue spaces ${ℒ}^s\left(ℝⁿ\right)$ arising naturally in the study of elliptic PDEs. The prototype of those PDEs is the second order p-harmonic equation ${div\left|\nabla u\right|}^{p-2\nabla }u=div$. In this example the p-harmonic transform is essentially inverse to $div\left(\right|\nabla {|}^{p-2}\nabla )$. To every vector field $\in {ℒ}^q(ℝⁿ,ℝⁿ)$ our operator ${\mathscr{H}}_p$ assigns the gradient of the solution, ${\mathscr{H}}_p=\nabla u\in {ℒ}^p(ℝⁿ,ℝⁿ)$. The core of the matter is that we go beyond the natural domain of definition of this operator. Because of nonlinearity our...

The paper deals with the approximation by polynomials with integer coefficients in $L_p(0,1)$, 1 ≤ p ≤ ∞. Let $P_{n,r}$ be the space of polynomials of degree ≤ n which are divisible by the polynomial $x^r{(1-x)}^r$, r ≥ 0, and let $P_{n,r}^{\mathbb{Z}}\subset P_{n,r}$ be the set of polynomials with integer coefficients. Let $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ be the maximal distance of elements of $P_{n,r}$ from $P_{n,r}^{\mathbb{Z}}$ in $L_p(0,1)$. We give rather precise quantitative estimates of $\mu (P_{n,r}^{\mathbb{Z}};L₂)$ for n ≳ 6r. Then we obtain similar, somewhat less precise, estimates of $\mu (P_{n,r}^{\mathbb{Z}};L_p)$ for p ≠ 2. It follows that $\mu (P_{n,r}^{\mathbb{Z}};L_p)\asymp n^{-2r-2/p}$ as n → ∞. The results...

In this paper we investigate the factorization of the polynomials $f\left(x\right)x^n+g\left(x\right)\in \mathbb{Z}\left[x\right]$ in the special case where $f\left(x\right)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f\left(x\right)$ is monic and linear.

We apply pluripotential theory to establish results in ${ℝ}^k$ concerning uniform approximation by functions of the form wⁿPₙ where w denotes a continuous nonnegative function and Pₙ is a polynomial of degree at most n. Then we use our work to show that on the intersection of compact sections $\Sigma \subset {ℝ}^k$ a continuous function on Σ is uniformly approximable by θ-incomplete polynomials (for a fixed θ, 0 < θ < 1) iff f vanishes on θ²Σ. The class of sets Σ expressible as the intersection of compact...

For a real number $q>1$ and a positive integer $m$, let $Y_m\left(q\right):=\left\{{\sum }_{i=0}^n{ϵ}_i{q}^i:{ϵ}_i\in \left\{0,\pm 1,...,\pm m\right\},n=0,1,...\right\}$. In this paper, we show that $Y_m\left(q\right)$ is dense in $ℝ$ if and only if $q<m+1$ and $q$ is not a Pisot number. This completes several previous results and answers an open question raised by Erdös, Joó and Komornik [8].

We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in ${}_q\left[T\right]$ of degree d for which s consecutive coefficients $a=(a_{d-1},...,a_{d-s})$ are fixed. Our estimate asserts that $(d,s,a)={\mu }_{d}q+\left(q^{1/2}\right)$, where ${\mu }_d:={\sum }_{r=1}^d\left({(-1)}^{r-1}\right)/(r!)$. We also prove that $₂(d,s,a)=\mu {²}_{d}q²+\left(q^{3/2}\right)$, where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of ${}_q\left[T\right]$ of degree d with s consecutive coefficients fixed as above. Finally, we show that $₂(d,0)=\mu {²}_{d}q²+\left(q\right)$, where ₂(d,0) denotes the average second moment for...

Let D and ∂D denote the open unit disk and the unit circle of the complex plane, respectively. We denote by ₙ (resp. ${ₙ}^c$) the set of all polynomials of degree at most n with real (resp. complex) coefficients. We define the truncation operators Sₙ for polynomials $Pₙ\in {ₙ}^c$ of the form $Pₙ\left(z\right):={\sum }_{j=0}^n{a}_j{z}_j$, $a_j\in C$, by $Sₙ\left(Pₙ\right)\left(z\right):={\sum }_{j=0}^{n}a{̃}_j{z}_j$, $a{̃}_j:=a_j\left|a_j\right|min|a_j|,1$ (here 0/0 is interpreted as 1). We define the norms of the truncation operators by $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{real}:=su{p}_{Pₙ\in ₙ}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|)$, $\parallel Sₙ{\parallel }_{\infty ,\partial D}^{comp}:=su{p}_{Pₙ\in {ₙ}^c}(ma{x}_{z\in \partial D}\left|Sₙ\left(Pₙ\right)\left(z\right)\right|)/(ma{x}_{z\in \partial D}\left|Pₙ\left(z\right)\right|$. Our main theorem establishes the right order of magnitude of the above norms: there is an absolute constant c₁...

For n ∈ ℕ, L > 0, and p ≥ 1 let ${\kappa }_p(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0\left|\ge L\right({\sum }_{j=1}^n|a_j{|}^p$1/p$,$aj ∈ ℂ$,$such that ${(x-1)}^k$ divides P(x). For n ∈ ℕ and L > 0 let ${\kappa }_{\infty }(n,L)$ be the largest possible value of k for which there is a polynomial P ≠ 0 of the form $P\left(x\right)={\sum }_{j=0}^n{a}_j{x}^j$, $|a_0|\ge Lma{x}_{1\le j\le n}|a_j|$, $a_j\in ℂ$, such that ${(x-1)}^k$ divides P(x). We prove that there are absolute constants c₁ > 0 and c₂ > 0 such that $c_1\surd (n/L)-1\le {\kappa }_{\infty }(n,L)\le c_2\surd (n/L)$ for every L ≥ 1. This complements an earlier result of the authors valid for every n ∈ ℕ and L ∈...

Letting $T_n$ (resp. $U_n$) be the n-th Chebyshev polynomials of the first (resp. second) kind, we prove that the sequences ${\left(X^k{T}_{n-k}\right)}_k$ and ${\left(X^k{U}_{n-k}\right)}_k$ for n - 2⎣n/2⎦ ≤ k ≤ n - ⎣n/2⎦ are two basis of the ℚ-vectorial space ${}_n\left[X\right]$ formed by the polynomials of ℚ[X] having the same parity as n and of degree ≤ n. Also $T_n$ and $U_n$ admit remarkableness integer coordinates on each of the two basis.

Let A be a linear closed densely defined operator in a complex Banach space X. If A is of type ω (i.e. the spectrum of A is contained in a sector of angle 2ω, symmetric around the real positive axis, and $\parallel \lambda {(\lambda I-A)}^{-1}\parallel$ is bounded outside every larger sector) and has a bounded inverse, then A has a bounded $H^{\infty }$ functional calculus in the real interpolation spaces between X and the domain of the operator itself.