On Mean Convergence of Fourier‐Bessel Series of Negative Order

<jats:p>The expansion of <jats:italic>f</jats:italic> ∈ <jats:italic>L<jats:sup>p</jats:sup></jats:italic>(0, 1) Fourier series of Bessel functions of order <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/sapm1971503281-math-0003.gif" xlink:title="equation image" /> converges to <jats:italic>f</jats:italic> in <jats:italic>L<jats:sup>p</jats:sup></jats:italic> whenever <jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" position="anchor" xlink:href="graphic/sapm1971503281-math-0001.gif"><jats:alt-text>urn:x-wiley:00222526:media:sapm1971503281:sapm1971503281-math-0001</jats:alt-text></jats:graphic></jats:disp-formula> </jats:p><jats:p>Let <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/sapm1971503281-math-0004.gif" xlink:title="equation image" /> be the space of <jats:italic>p</jats:italic>‐integrable functions with respect to the measure <jats:italic>t dt</jats:italic> and <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/sapm1971503281-math-0005.gif" xlink:title="equation image" /> where {<jats:italic>s</jats:italic><jats:sub><jats:italic>n</jats:italic></jats:sub>}, <jats:italic>n</jats:italic> = 1, 2, …, is the set of positive zeros of <jats:italic>J</jats:italic><jats:sub><jats:italic>v</jats:italic></jats:sub>. Then, the expansion of <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/sapm1971503281-math-0006.gif" xlink:title="equation image" /> in a Fourier series of functions <jats:italic>ψ</jats:italic><jats:sub><jats:italic>n</jats:italic></jats:sub>, −1 < ν < −½, converges to <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/sapm1971503281-math-0007.gif" xlink:title="equation image" /> in <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/sapm1971503281-math-0008.gif" xlink:title="equation image" /> whenever <jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" position="anchor" xlink:href="graphic/sapm1971503281-math-0002.gif"><jats:alt-text>urn:x-wiley:00222526:media:sapm1971503281:sapm1971503281-math-0002</jats:alt-text></jats:graphic></jats:disp-formula> </jats:p>