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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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...

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

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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

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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).

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Hafedh Herichi, Michel L. Lapidus (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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We survey some of the universality properties of the Riemann zeta function $ζ (s)$ and then explain how to obtain a natural quantization of Voronin’s universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in [60]. It also makes an essential use of the functional analytic framework developed by the authors in [25] for rigorously studying the spectral operator...

Weak-type operators and the strong fundamental lemma of real interpolation theory

N. Krugljak, Y. Sagher, P. Shvartsman (2005)

Studia Mathematica

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We prove an interpolation theorem for weak-type operators. This is closely related to interpolation between weak-type classes. Weak-type classes at the ends of interpolation scales play a similar role to that played by BMO with respect to the $L^{p}$ interpolation scale. We also clarify the roles of some of the parameters appearing in the definition of the weak-type classes. The interpolation theorem follows from a K-functional inequality for the operators, involving the Calderón operator....

Interpolation of quasicontinuous functions

Joan Cerdà, Joaquim Martín, Pilar Silvestre (2011)

Banach Center Publications

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If C is a capacity on a measurable space, we prove that the restriction of the K-functional $K (t, f; L^{p} (C), L^{\infty} (C))$ to quasicontinuous functions f ∈ QC is equivalent to $K (t, f; L^{p} (C) \cap Q C, L^{\infty} (C) \cap Q C)$ . We apply this result to identify the interpolation space ${(L^{p ₀, q ₀} (C) \cap Q C, L^{p ₁, q ₁} (C) \cap Q C)}_{θ, q}$ .