## Archived Dissertations

12-1970

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

#### First Advisor

Frank M. Choleweiuski

John Muelly

A. E. Schwertz

#### Abstract

Let ν ≥ 0 be a real number and set, J(x) = cνx^1/2-νJ ν-1/2^(x) Where cν = 2^ν-1/2Γ(ν+1/2) and Jν-1/2(x) is Bessel's function of the first kind of order ν - 1/2. Let, dμ(x) = cν^-1x2νdx. Then L p/ν (1≤p≤∞) is defined as all those real measureable functioned defined on [0, 1] such that ||f|| p,ν = [ 1/0|f(x)| pdμ(x)] ^1/p 2ν+1/ν+1 then the Fourier-Dini series of f is Riesz summable to f(x) for almost all points x in [0,1]. It is also shown that if ε L2/ν then lim/k->∞ SNK f(x) =f(x) for almost all points x in [0,1] where SNK f(x) denotes the Nk^th partial sum of the Fourier-Dini series of f and the sequence {Nk}^∞ k=1 satisfies inf Nk+1/Nk > 1. Finally, it is show that if f ε Lp/ν for p >2ν+1/ν+1 then each series, Σ(n=0)^∞ f(n)/log(n+2) J(λnx)j(n) and Σ(n=0)^∞ f(n)/(1+n)^α J(λnx)j(n) α for >0 is convergent at almost all points x in [0,1].

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