2021/09/05 by Ramazan Akgün, Akgün, Ramazan
Mathematics · #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2109.02083
openalex publication_date 2021/09/05 · openalex created_date 2022/08/31 · openalex updated_date 2026/07/28
Present work contains a method to obtain Jackson and Stechkin type inequalities of approximation by integral functions of finite degree (IFFD) in some variable exponent Lebesgue space of real functions defined on \boldsymbolR:=( -∞ ,+∞ ) . To do this we employ a transference theorem which produce norm inequalities starting from norm inequalities in C(\boldsymbolR), the class of bounded uniformly continuous functions defined on \boldsymbolR. Let B⊆ \boldsymbolR be a measurable set, p( x) :B→ \lbrack 1,∞ ) be a measurable function. For the class of functions f belonging to variable exponent Lebesgue spaces Lp( x) (B) we consider difference operator ( I-Tδ) rf( ⋅ ) under the condition that p(x) satisfies the Log Hölder continuity condition and 1≤ \mathop\rm ess inf\nolimitsx∈ Bp(x), \mathop\rm ess sup\nolimitsx∈ Bp(x)<∞ where I is the identity operator, r∈ N:=\ 1,2,3,⋯ \ , δ≥ 0 and Tδf( x) =(1)/(δ)∫\nolimits0δf( x+t) dt ) is the forward Steklov operator. We obtain main properties of difference operator \Vert ( I-Tδ) rf\Vert p( ⋅ ) in Lp( x) ( B) . We give proof of direct and inverse theorems of approximation by IFFD in Lp( x) ( \boldsymbolR).