Polynomial interpolation and asymptotic representations for zeta functions

Michael I. Ganzburg

Displaying similar documents to “Polynomial interpolation and asymptotic representations for zeta functions”

Polynomial interpolation and approximation in $ℂ^{d}$

T. Bloom, L. P. Bos, J.-P. Calvi, N. Levenberg (2012)

Annales Polonici Mathematici

Similarity:

We update the state of the subject approximately 20 years after the publication of T. Bloom, L. Bos, C. Christensen, and N. Levenberg, Polynomial interpolation of holomorphic functions in ℂ and ℂⁿ, Rocky Mountain J. Math. 22 (1992), 441-470. This report is mostly a survey, with a sprinkling of assorted new results throughout.

Some infinite sums identities

Meher Jaban, Sinha Sneh Bala (2015)

Czechoslovak Mathematical Journal

Similarity:

We find the sum of series of the form $\sum_{i = 1}^{\infty} \frac{f (i)}{i^{r}}$ for some special functions $f$ . The above series is a generalization of the Riemann zeta function. In particular, we take $f$ as some values of Hurwitz zeta functions, harmonic numbers, and combination of both. These generalize some of the results given in Mező’s paper (2013). We use multiple zeta theory to prove all results. The series sums we have obtained are in terms of Bernoulli numbers and powers of $π$ .

A counterexample to the Γ-interpolation conjecture

Adama S. Kamara (2015)

Annales Polonici Mathematici

Similarity:

Agler, Lykova and Young introduced a sequence $C_{ν}$ , where ν ≥ 0, of necessary conditions for the solvability of the finite interpolation problem for analytic functions from the open unit disc into the symmetrized bidisc Γ. They conjectured that condition $C_{n - 2}$ is necessary and sufficient for the solvability of an n-point interpolation problem. The aim of this article is to give a counterexample to that conjecture.

Ergodic Universality Theorems for the Riemann Zeta-Function and other $L$ -Functions

Jörn Steuding (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We prove a new type of universality theorem for the Riemann zeta-function and other $L$ -functions (which are universal in the sense of Voronin’s theorem). In contrast to previous universality theorems for the zeta-function or its various generalizations, here the approximating shifts are taken from the orbit of an ergodic transformation on the real line.

Hausdorff dimension of a fractal interpolation function

Guantie Deng (2004)

Colloquium Mathematicae

Similarity:

We obtain a lower bound for the Hausdorff dimension of the graph of a fractal interpolation function with interpolation points $(i / N, y_{i}) : i = 0, 1, . . ., N$ .

Measure of weak noncompactness under complex interpolation

Andrzej Kryczka, Stanisław Prus (2001)

Studia Mathematica

Similarity:

Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón’s complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if T: A₀ → B₀ or T: A₁ → B₁ is weakly compact, then so is $T : A_{[θ]} \to B_{[θ]}$ for all 0 < θ < 1, where $A_{[θ]}$ and $B_{[θ]}$ are interpolation spaces with respect to the pairs (A₀,A₁) and (B₀,B₁). Some formulae for this measure and relations to other quantities measuring weak noncompactness are...

Interpolation of Cesàro sequence and function spaces

Sergey V. Astashkin, Lech Maligranda (2013)

Studia Mathematica

Similarity:

The interpolation properties of Cesàro sequence and function spaces are investigated. It is shown that $C e s_{p} (I)$ is an interpolation space between $C e s_{p ₀} (I)$ and $C e s_{p ₁} (I)$ 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, $C e s_{p} [0, 1]$ is not an interpolation space between Ces₁[0,1] and $C e s_{\infty} [0, 1]$ .

Lions-Peetre reiteration formulas for triples and their applications

Irina Asekritova, Natan Krugljak, Lech Maligranda, Lyudmila Nikolova, Lars-Erik Persson (2001)

Studia Mathematica

Similarity:

We present, discuss and apply two reiteration theorems for triples of quasi-Banach function lattices. Some interpolation results for block-Lorentz spaces and triples of weighted $L_{p}$ -spaces are proved. By using these results and a wavelet theory approach we calculate (θ,q)-spaces for triples of smooth function spaces (such as Besov spaces, Sobolev spaces, etc.). In contrast to the case of couples, for which even the scale of Besov spaces is not stable under interpolation, for triples we...

Interpolation by bivariate polynomials based on Radon projections

B. Bojanov, I. K. Georgieva (2004)

Studia Mathematica

Similarity:

For any given set of angles θ₀ < ... < θₙ in [0,π), we show that a set of $(\binom{n + 2}{2})$ Radon projections, consisting of k parallel X-ray beams in each direction $θ_{k}$ , k = 0, ..., n, determines uniquely algebraic polynomials of degree n in two variables.

$Ω_{\pm}$ -results of the error term in the mean square formula of the Riemann zeta-function in the critical strip

Yuk-Kam Lau, Kai-Man Tsang (2001)

Acta Arithmetica

Similarity:

Sur l'élimination des "infinis" en théorie quantique des champs : la régularisation zeta à l'épreuve de l'interprétation de Colombeau ou vice versa

Charpentier Éric

Similarity:

Zeta regularization is one of the heuristic techniques used, specifically, in quantum field theory (QFT) in order to extract from divergent expressions some finite and well-defined values. Colombeau’s theory of generalized functions (containing the distributions and allowing their multiplication) provides a mathematically rigorous setting for QFT, where only convergent expressions appear. The aim of this paper is to compare these two points of view, both in their underlying logic and...

On interpolation error on degenerating prismatic elements

Ali Khademi, Sergey Korotov, Jon Eivind Vatne (2018)

Applications of Mathematics

Similarity:

We propose an analogue of the maximum angle condition (commonly used in finite element analysis for triangular and tetrahedral meshes) for the case of prismatic elements. Under this condition, prisms in the meshes may degenerate in certain ways, violating the so-called inscribed ball condition presented by P. G. Ciarlet (1978), but the interpolation error remains of the order $O (h)$ in the $H^{1}$ -norm for sufficiently smooth functions.

Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

Thomas Christ, Justas Kalpokas (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments $I_{k, l} (T) = \int_{0}^{T} {| ζ^{(l)} (\frac{1}{2} + i t) |}^{2 k} d t$ , where $l$ is a non-negative integer and $k \geq 1$ a rational number. In particular, these lower bounds are of the expected order of magnitude for $I_{k, l} (T)$ .

Exact Kronecker constants of Hadamard sets

Kathryn E. Hare, L. Thomas Ramsey (2013)

Colloquium Mathematicae

Similarity:

A set S of integers is called ε-Kronecker if every function on S of modulus one can be approximated uniformly to within ε by a character. The least such ε is called the ε-Kronecker constant, κ(S). The angular Kronecker constant is the unique real number α(S) ∈ [0,1/2] such that κ(S) = |exp(2πiα(S)) - 1|. We show that for integers m > 1 and d ≥ 1, $α 1, m, . . ., m^{d - 1} = (m^{d - 1} - 1) / (2 (m^{d} - 1))$ and α1,m,m²,... = 1/(2m).

The size of the Lerch zeta-function at places symmetric with respect to the line $ℜ (s) = 1 / 2$

Ramūnas Garunkštis, Andrius Grigutis (2019)

Czechoslovak Mathematical Journal

Similarity:

Let $ζ (s)$ be the Riemann zeta-function. If $t \geq 6.8$ and $σ > 1 / 2$ , then it is known that the inequality $| ζ (1 - s) | > | ζ (s) |$ is valid except at the zeros of $ζ (s)$ . Here we investigate the Lerch zeta-function $L (λ, α, s)$ which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters $λ = α$ it is still possible to obtain a certain version of the inequality $| L (λ, λ, 1 - \bar{s}) | > | L (λ, λ, s) |$ .