Related papers: Analog for the Wiener Lemma for Wolff-Denjoy Serie…
An abstract theory of Fourier series in locally convex topological vector spaces is developed. An analog of Fej\'{e}r's theorem is proved for these series. The theory is applied to distributional solutions of Cauchy-Riemann equations to…
We show a Wolff-Denjoy type theorem in complete geodesic spaces in the spirit of Beardon's framework that unifies several results in this area. In particular, it applies to strictly convex bounded domains in $\mathbb{R}^{n}$ or…
The product of any number of Legendre functions, under a restricted domain, can be expanded by the corresponding Legendre polynomials, with the coefficient being the sinc function. While an analogous expansion can be made for any number of…
We give a short proof of Wolff-Denjoy theorem for (not necessarily smooth) strictly convex domains. With similar techniques we are also able to prove a Wolff-Denjoy theorem for weakly convex domains, again without any smoothness assumption…
We establish a Wiman-Valiron theory for a polynomial series based on the Wilson operator $\mathcal{D}_\mathrm{W}$. For an entire function $f$ of order smaller than $\frac13$, this theory includes (i) an estimate which shows that $f$ behaves…
In this paper we present generalisations of Paley-Wiener type theorems to Mellin and (Laplace-)Fourier transforms of rapidly decreasing smooth functions with positive support and log-polyhomogeneous asymptotic expansion at zero. This…
We establish a Wiman-Valiron theory of a polynomial series based on the Askey-Wilson operator $\mathcal{D}_q$, where $q\in(0,1)$. For an entire function $f$ of log-order smaller than $2$, this theory includes (i) an estimate which shows…
In this article, we consider the family of functions $f$ meromorphic in the unit disk $\ID=\{z :\,|z| < 1\}$ with a pole at the point $z=p$, a Taylor expansion \[f(z)= z+\sum_{k=2}^{\infty} a_kz^k, \quad |z|<p, \] and satisfying the…
Let $W_0(\mathbb R)$ be the Wiener Banach algebra of functions representable by the Fourier integrals of Lebesgue integrable functions. It is proven in the paper that, in particular, a trigonometric series $\sum\limits_{k=-\infty}^\infty…
We define an almost periodic extension of the Wiener algebras in the quaternionic setting and prove a Wiener-Levy type theorem for it, as well as extending the theorem to the matrix-valued case. We prove a Wiener-Hopf factorization theorem…
A Lemma of Riemann--Lebesgue type for Fourier--Jacobi coefficients is derived. Via integral representations of Dirichlet--Mehler type for Jacobi polynomials its proof directly reduces to the classical Riemann--Lebesgue Lemma for Fourier…
Let $J$ denote the interval either $(0,1]$ or $ [1, \infty)$. A positive function $f$ on $J$ with $f(1) =1$ is reffered to as a Weierstrass function if it fulfils the double inequality for $x,y \in J$: $$f(x) + f(y) -1 \leq f(xy) \leq…
We define and study the counterpart of the Wiener algebra in the quaternionic setting, both for the discrete and continuous case. We prove a Wiener-L\'evy type theorem and a factorization theorem. We give applications to Toeplitz and…
We consider the systems of rational functions $\{\Phi_n(z)\}, ~n \in \mathbb{Z}$, defined by fixed set points ${\bf a}:=\{a_k\}_{k=0}^{\infty}, ~ (\mathop{\rm Im} a_k>0)$, ${\bf b}:=\{b_k\}_{k=1}^{\infty}, ~ (\mathop{\rm Im} b_k<0)$ and is…
Wiener-Hopf plus Hankel operators acting between Lebesgue spaces on the real line are studied in view of their invertibility, one sided-invertibility, Fredholm, and semi-Fredholm properties. This is done in two different cases: (i) when the…
The results presented in this paper are refinements of some results presented in a previous paper. Three such refined results are presented. The first one relaxes one of the basic hypotheses assumed in the previous paper, and thus extends…
A simple version for the extension of the Taylor theorem to the operator functions was found. The expansion was done with respect to a value given by a diagonal matrix for the non-commutative case, and the coefficients are given both by…
We study multiplicative functions $f$ satisfying $|f(n)|\le 1$ for all $n$, the associated Dirichlet series $F(s):=\sum_{n=1}^{\infty} f(n) n^{-s}$, and the summatory function $S_f(x):=\sum_{n\le x} f(n)$. Up to a possible trivial…
We formulate and derive a generalization of an orthogonal rational-function basis for spectral expansions over the infinite or semi-infinite interval. The original functions, first presented by Wiener are a mapping and weighting of the…
In this paper, we give the general solution of the functional equation $$\big\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\big\}=\big\{\|x+y\|,\|x-y\|\big\}\qquad(x,y\in X)$$ where $f:X\to Y$ and $X,Y$ are inner product spaces. Related equations are also…