Approximation of analytic functions by discrete shifts of the Hurwitz zeta-function in short intervals

In the paper, we prove theorems in short intervals on approximation of analytic functions by shifts ζ ( s + ikh , α ), h > 0, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>k</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:math> k∈ N₀ , of the Hurwitz zeta-function. Two cases are discussed. In the first case, it is assumed that the set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>log</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>+</m:mo> <m:mi>α</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mi>m</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mi>π</m:mi> <m:mo>/</m:mo> <m:mi>h</m:mi> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \(log (m+α )\,:\,m∈ N₀),2π /h\ is linearly independent over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="double-struck">Q</m:mi> </m:math> Q , and it is obtained that the set of the above shifts approximating every analytic function defined on the strip <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">D</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">C</m:mi> <m:mspace width="0.17em"/> <m:mo>:</m:mo> <m:mspace width="0.17em"/> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> <m:mo><</m:mo> <m:mi>σ</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> D=∈ C\,:\,1/2< σ < 1\ has a positive lower density (or density) in the interval of length H , h −1 ( Nh ) 27/82 ≤ H ≤ ( Nh ) 1/2 h −1 . The second case is devoted to arbitrary 0 < α < 1, α ≠ 1/2. It is obtained that there exists a closed non-empty set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="script">F</m:mi> </m:mrow> <m:mrow> <m:mi>h</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:math> Fh,α of analytic functions defined on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">D</m:mi> </m:math> D such that, for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mrow> <m:mi mathvariant="script">F</m:mi> </m:mrow> <m:mrow> <m:mi>h</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:math>